FLUID MECHANICS AND PIPE FLOW: TURBULENCE, SIMULATION AND DYNAMICS

FLUID MECHANICS AND PIPE FLOW: TURBULENCE, SIMULATION AND DYNAMICS No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services.

FLUID MECHANICS AND PIPE FLOW: TURBULENCE, SIMULATION AND DYNAMICS

DONALD MATOS AND

CRISTIAN VALERIO EDITORS

Nova Science Publishers, Inc. New York

Copyright © 2009 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Fluid mechanics and pipe flow : turbulence, simulation, and dynamics / editors, Donald Matos and Cristian Valerio. p. cm. Includes bibliographical references and index. ISBN 978-1-61668-990-2 (E-Book) 1. Fluid mechanics. 2. Pipe--Fluid dynamics. I. Matos, Donald. II. Valerio, Cristian. TA357.F5787 2009 620.1'06--dc22 2009017666

Published by Nova Science Publishers, Inc. Ô New York

CONTENTS Preface

vii

Chapter 1

Solute Transport, Dispersion, and Separation in Nanofluidic Channels Xiangchun Xuan

1

Chapter 2

H2O in the Mantle: From Fluid to High-Pressure Hydrous Silicates N.R. Khisina, R. Wirth and S. Matsyuk

27

Chapter 3

On the Numerical Simulation of Turbulence Modulation in TwoPhase Flows K. Mohanarangam and J.Y. Tu

41

Chapter 4

A Review of Population Balance Modelling for Multiphase Flows: Approaches, Applications and Future Aspects Sherman C.P. Cheung, G.H. Yeoh and J.Y. Tu

117

Chapter 5

Numerical Analysis of Heat Transfer and Fluid Flow for ThreeDimensional Horizontal Annuli with Open Ends Chun-Lang Yeh

171

Chapter 6

Convective Heat Transfer in the Thermal Entrance Region of Parallel Flow Double-Pipe Heat Exchangers for Non-Newtonian Fluids Ryoichi Chiba, Masaaki Izumi and Yoshihiro Sugano

205

Chapter 7

Numerical Simulation of Turbulent Pipe Flow M. Ould-Rouis and A.A. Feiz

231

Chapter 8

Pipe Flow Analysis of Uranium Nuclear Heating with Conjugate Heat Transfer G.H. Yeoh and M.K.M. Ho

269

Chapter 9

First and Second Law Thermodynamics Analysis of Pipe Flow Ahmet Z. Sahin

317

vi

Contents

Chapter 10

Single-Phase Incompressible Fluid Flow in Mini- and Micro-channels Lixin Cheng

343

Chapter 11

Experimental Study of Pulsating Turbulent Flow through a Divergent Tube Masaru Sumida

365

Chapter 12

Solution of an Airfoil Design Inverse Problem for a Viscous Flow Using a Contractive Operator Jan Šimák and Jaroslav Pelant

379

Chapter 13

Some Free Boundary Problems in Potential Flow Regime Using the Level Set Method M. Garzon, N. Bobillo-Ares and J.A. Sethian

399

Chapter 14

A New Approach for Polydispersed Turbulent Two-Phase Flows: The Case of Deposition in Pipe-Flows S. Chibbaro

441

Index

455

PREFACE Fluid mechanics is the study of how fluids move and the forces that develop as a result. Fluids include liquids and gases and fluid flow can be either laminar or turbulent. This book presents a level set based methodology that will avoid problems in potential flow models with moving boundaries. A review of the state-of-the-art population balance modelling techniques that have been adopted to describe the nature of dispersed phase in multiphase problems is presented as well. Recent works that are aimed at putting forward the main ideas behind a new theoretical approach to turbulent wall-bounded flows are examined, including a state-ofthe-art review on single-phase incompressible fluid flow. Recent breakthrough in nanofabrication has stimulated the interest of solute separation in nanofluidic channels. Since the hydraulic radius of nanochannels is comparable to the thickness of electric double layers, the enormous electric fields inherent to the latter generate transverse electromigrations causing charge-dependent solute distributions over the channel cross-section. As a consequence, the non-uniform fluid flow through nanochannels yields charge-dependent solute speeds enabling the separation of solutes by charge alone. In Chapter 1 we develop a theoretical model of solute transport, dispersion and separation in electroosmotic and pressure-driven flows through nanofluidic channels. This model provides a basis for the optimization of solute separation in nanochannels in terms of selectivity and resolution as traditionally defined. As presented in Chapter 2, infrared spectroscopic data show that nominally anhydrous olivine (Mg,Fe)2SiO4 contains traces of H2O, up to several hundred wt. ppm of H2O (Miller et al., 1987; Bell et al., 2004; Koch-Muller et al., 2006; Matsyuk & Langer, 2004) and therefore olivine is suggested to be a water carrier in the mantle (Thompson, 1992). Protonation of olivine during its crystallization from a hydrous melt resulted in the appearance of intrinsic OH-defects (Libowitsky & Beran, 1995). Mantle olivine nodules from kimberlites were investigated with FTIR and TEM methods (Khisina et al., 2001, 2002, 2008). The results are the following: (1) Water content in xenoliths is lower than water content in xenocrysts. From these data we concluded that kimberlite magma had been saturated by H2O, whereas adjacent mantle rocks had been crystallized from water-depleted melts. (2) Extrinsic water in olivine is represented by high-pressure phases, 10Å-Phase Mg3Si4O10(OH)2.nH2O and hydrous olivine n(Mg,Fe)2SiO4.(H2MgSiO4), both of which belong to the group of Dense Hydrous Magnesium Silicates (DHMS), which were synthesized in laboratory high-pressure experiments (Prewitt & Downs, 1999). The DHMS were regarded as possible mineral carriers for H2O in the mantle; however, they were not found in natural material until quite recently.

viii

Donald Matos and Cristian Valerio

Our observations demonstrate the first finding of the 10Å-Phase and hydrous olivine as a mantle substance. (3) 10Å-Phase, which occurred as either nanoinclusions or narrow veins in olivine, is a ubiquitous nano-mineral of kimberlite and closely related to olivine. (4) There are two different mechanisms of the 10Å-Phase formation: (a) purification of olivine from OHbearing defects resulting in transformation of olivine to the 10 Å-Phase with the liberation of water fluid; and (b) replacement of olivine for the 10Å-Phase due to hydrous metasomatism in the mantle in the presence of H2O fluid. With the increase of computational power, computational modelling of two-phase flow problems using computational fluid dynamics (CFD) techniques is gradually becoming attractive in the engineering field. The major aim of Chapter 3 is to investigate the Turbulence Modulation (TM) of dilute two phase flows. Various density regimes of the two-phase flows have been investigated in this paper, namely the dilute Gas-Particle (GP) flow, LiquidParticle (LP) flow and also the Liquid-Air (LA) flows. While the density is quite high for the dispersed phase flow for the gas-particle flow, the density ratio is almost the same for the liquid particle flow, while for the liquid-air flow the density is quite high for the carrier phase flow. The study of all these density regimes gives a clear picture of how the carrier phase behaves in the presence of the dispersed phases, which ultimately leads to better design and safety of many two-phase flow equipments and processes. In order to carry out this approach, an Eulerian-Eulerian Two-Fluid model, with additional source terms to account for the presence of the dispersed phase in the turbulence equations has been employed for particulate flows, whereas Population Balance (PB) have been employed to study the bubbly flows. For the dilute gas-particle flows, particle-turbulence interaction over a backward-facing step geometry was numerically investigated. Two different particle classes with same Stokes number and varied particle Reynolds number are considered in this study. A detailed study into the turbulent behaviour of dilute particulate flow under the influence of two carrier phases namely gas and liquid was also been carried out behind a sudden expansion geometry. The major endeavour of the study is to ascertain the response of the particles within the carrier (gas or liquid) phase. The main aim prompting the current study is the density difference between the carrier and the dispersed phase. While the ratio is quite high in terms of the dispersed phase for the gas-particle flows, the ratio is far more less in terms of the liquid-particle flows. Numerical simulations were carried out for both these classes of flows and their results were validated against their respective sets of experimental data. For the Liquid-Air flows the phenomenon of drag reduction by the injection of micro-bubbles into turbulent boundary layer has been investigated using an Eulerian-Eulerian two-fluid model. Two variants namely the Inhomogeneous and MUSIG (MUltiple-SIze-Group) based on Population balance models are investigated. The simulated results were benchmarked against the experimental findings and also against other numerical studies explaining the various aspects of drag reduction. For the two Reynolds number cases considered, the buoyancy with the plate on the bottom configuration is investigated, as from the experiments it is seen that buoyancy seem to play a role in the drag reduction. The under predictions of the MUSIG model at low flow rates was investigated and reported, their predictions seem to fair better with the decrease of the break-up tendency among the micro-bubbles. Population balance modelling is of significant importance in many scientific and industrial instances such as: fluidizations, precipitation, particles formation in aerosols, bubbly and droplet flows and so on. In population balance modelling, the solution of the population balance equation (PBE) records the number of entities in dispersed phase that

Preface

ix

always governs the overall behaviour of the practical system under consideration. For the majority of cases, the solution evolves dynamically according to the “birth” and “death” processes of which it is tightly coupled with the system operation condition. The implementation of PBE in conjunction with the Computational Fluid Dynamics (CFD) is thereby becoming ever a crucial consideration in multiphase flow simulations. Nevertheless, the inherent integrodifferential form of the PBE poses tremendous difficulties on its solution procedures where analytical solutions are rare and impossible to be achieved. In Chapter 4, we present a review of the state-of-the-art population balance modelling techniques that have been adopted to describe the phenomenological nature of dispersed phase in multiphase problems. The main focus of the review can be broadly classified into three categories: (i) Numerical approaches or solution algorithms of the PBE; (ii) Applications of the PBE in practical gas-liquid multiphase problems and (iii) Possible aspects of the future development in population balance modelling. For the first category, details of solution algorithms based on both method of moment (MOM) and discrete class method (CM) that have been proposed in the literature are provided. Advantages and drawbacks of both approaches are also discussed from the theoretical and practical viewpoints. For the second category, applications of existing population balance models in practical multiphase problems that have been proposed in the literature are summarized. Selected existing mathematical closures for modelling the “birth” and “death” rate of bubbles in gas-liquid flows are introduced. Particular attention is devoted to assess the capability of some selected models in predicting bubbly flow conditions through detail validation studies against experimental data. These studies demonstrate that good agreement can be achieved by the present model by comparing the predicted results against measured data with regards to the radial distribution of void fraction, Sauter mean bubble diameter, interfacial area concentration and liquid axial velocity. Finally, weaknesses and limitations of the existing models are revealed are suggestions for further development are discussed. Emerging topics for future population balance studies are provided as to complete the aspect of population balance modelling. Study of the heat transfer and fluid flow inside concentric or eccentric annuli can be applied in many engineering fields, e.g. solar energy collection, fire protection, underground conduit, heat dissipation for electrical equipment, etc. In the past few decades, these studies were concentrated in two-dimensional research and were mostly devoted to the investigation of the effects of convective heat transfer. However, in practical situation, this problem should be three-dimensional, except for the vertical concentric annuli which could be modeled as two-dimensional (axisymmetric). In addition, the effects of heat conduction and radiation should not be neglected unless the outer cylinder is adiabatic and the temperature of the flow field is sufficiently low. As the author knows, none of the open literature is devoted to the investigation of the conjugated heat transfer of convection, conduction and radiation for this problem. The author has worked in industrial piping design area and is experienced in this field. The author has also employed three-dimensional body-fitted coordinate system associated with zonal grid method to analyze the natural convective heat transfer and fluid flow inside three-dimensional horizontal concentric or eccentric annuli with open ends. Owing to its broad application in practical engineering problems, Chapter 5 is devoted to a detailed discussion of the simulation method for the heat transfer and fluid flow inside threedimensional horizontal concentric or eccentric annuli with open ends. Two illustrative problems are exhibited to demonstrate its practical applications.

x


In Chapter 6, the conjugated Graetz problem in parallel flow double-pipe heat exchangers is analytically solved by an integral transform method—Vodicka’s method—and an analytical solution to the fluid temperatures varying along the radial and axial directions is obtained in a completely explicit form. Since the present study focuses on the range of a sufficiently large Péclet number, heat conduction along the axial direction is considered to be negligible. An important feature of the analytical method presented is that it permits arbitrary velocity distributions of the fluids as long as they are hydrodynamically fully developed. Numerical calculations are performed for the case in which a Newtonian fluid flows in the annulus of the double pipe, whereas a non-Newtonian fluid obeying a simple power law flows through the inner pipe. The numerical results demonstrate the effects of the thermal conductivity ratio of the fluids, Péclet number ratio and power-law index on the temperature distributions in the fluids and the amount of exchanged heat between the two fluids. Many experimental and numerical studies have been devoted to turbulent pipe flows due to the number of applications in which theses flows govern heat or mass transfer processes: heat exchangers, agricultural spraying machines, gasoline engines, and gas turbines for examples. The simplest case of non-rotating pipe has been extensively studied experimentally and numerically. Most of pipe flow numerical simulations have studied stability and transition. Some Direct Numerical Simulations (DNS) have been performed, with a 3-D spectral code, or using mixed finite difference and spectral methods. There is few DNS of the turbulent rotating pipe flow in the literature. Investigations devoted to Large Eddy Simulations (LES) of turbulence pipe flow are very limited. With DNS and LES, one can derive more turbulence statistics and determine a well-resolved flow field which is a prerequisite for correct predictions of heat transfer. However, the turbulent pipe flows have not been so deeply studied through DNS and LES as the plane-channel flows, due to the peculiar numerical difficulties associated with the cylindrical coordinate system used for the numerical simulation of the pipe flows. Chapter 7 presents Direct Numerical Simulations and Large Eddy Simulations of fully developed turbulent pipe flow in non-rotating and rotating cases. The governing equations are discretized on a staggered mesh in cylindrical coordinates. The numerical integration is performed by a finite difference scheme, second-order accurate in space and time. The time advancement employs a fractional step method. The aim of this study is to investigate the effects of the Reynolds number and of the rotation number on the turbulent flow characteristics. The mean velocity profiles and many turbulence statistics are compared to numerical and experimental data available in the literature, and reasonably good agreement is obtained. In particular, the results show that the axial velocity profile gradually approaches a laminar shape when increasing the rotation rate, due to the stability effect caused by the centrifugal force. Consequently, the friction factor decreases. The rotation of the wall has large effects on the root mean square (rms), these effects being more pronounced for the streamwise rms velocity. The influence of rotation is to reduce the Reynolds stress component 〈Vr'Vz'〉 and to increase the two other stresses 〈Vr'Vθ'〉 and 〈Vθ'Vz'〉. The effect of the Reynolds number on the rms of the axial velocity (〈Vz'2〉1/2) and the distributions of 〈Vr'Vz'〉 is evident, and it increases with an increase in the Reynolds number. On the other hand, the 〈Vr'Vθ'〉profiles appear to be nearly independent of the Reynolds number. The present DNS and LES predictions will be helpful for developing more accurate turbulence models for heat transfer and fluid flow in pipe flows.

Preface

xi

The field of computational fluid dynamics (CFD) has evolved from an academic curiosity to a tool of practical importance. Applications of CFD have become increasingly important in nuclear engineering and science, where exacting standards of safety and reliability are paramount. The newly-commissioned Open Pool Australian Light-water (OPAL) research reactor at the Australian Nuclear Science and Technology Organisation (ANSTO) has been designed to irradiate uranium targets to produce molybdenum medical isotopes for diagnosis and radiotherapy. During the irradiation process, a vast amount of power is generated which requires efficient heat removal. The preferred method is by light-water forced convection cooling—essentially a study of complex pipe flows with coupled conjugate heat transfer. Feasibility investigation on the use of computational fluid dynamics methodologies into various pipe flow configurations for a variety of molybdenum targets and pipe geometries are detailed in Chapter 8. Such an undertaking has been met with a number of significant modeling challenges: firstly, the complexity of the geometry that needed to be modeled. Herein, challenges in grid generation are addressed by the creation of purpose-built bodyfitted and/or unstructured meshes to map the intricacies within the geometry in order to ensure numerical accuracy as well as computational efficiency in the solution of the predicted result. Secondly, various parts of the irradiation rig that are required to be specified as composite solid materials are defined to attain the correct heat transfer characteristics. Thirdly, the use of an appropriate turbulence model is deemed to be necessary for the correct description of the fluid and heat flow through the irradiation targets, since the heat removal is forced convection and the flow regime is fully turbulent, which further adds to the complexity of the solution. As complicated as the computational fluid dynamics modeling is, numerical modeling has significantly reduced the cost and lead time in the molybdenum-target design process, and such an approach would not have been possible without the continual improvement of computational power and hardware. This chapter also addresses the importance of experimental modeling to evaluate the design and numerical results of the velocity and flow paths generated by the numerical models. Predicted results have been found to agree well with experimental observations of pipe flows through transparent models and experimental measurements via the Laser Doppler Velocimetry instrument. In Chapter 9, the entropy generation for during fluid flow in a pipe is investigated. The temperature dependence of the viscosity is taken into consideration in the analysis. Laminar and turbulent flow cases are treated separately. Two types of thermal boundary conditions are considered; uniform heat flux and constant wall temperature. In addition, various crosssectional pipe geometries were compared from the point of view of entropy generation and pumping power requirement in order to determine the possible optimum pipe geometry which minimizes the exergy losses. Chapter 10 aims to present a state-of-the-art review on single-phase incompressible fluid flow in mini- and micro-channels. First, classification of mini- and micro-channels is discussed. Then, conventional theories on laminar, laminar to turbulent transition and turbulent fluid flow in macro-channels (conventional channels) are summarized. Next, a brief review of the available studies on single-phase incompressible fluid flow in mini- and microchannels is presented. Some experimental results on single phase laminar, laminar to turbulent transition and turbulent flows are presented. The deviations from the conventional friction factor correlations for single-phase incompressible fluid flow in mini and microchannels are discussed. The effect factors on mini- and micro-channel single-phase fluid flow are analyzed. Especially, the surface roughness effect is focused on. According to this review,

xii


the future research needs have been identified. So far, no systematic agreed knowledge of single-phase fluid flow in mini- and micro-channels has yet been achieved. Therefore, efforts should be made to contribute to systematic theories for microscale fluid flow through very careful experiments. In Chapter 11, an experimental investigation was conducted of pulsating turbulent flow in a conically divergent tube with a total divergence angle of 12°. The experiments were carried out under the conditions of Womersley numbers of α =10∼40, mean Reynolds number of Reta =20000 and oscillatory Reynolds number of Reos =10000 (the flow rate ratio of η = 0.5). Time-dependent wall static pressure and axial velocity were measured at several longitudinal stations and the distributions were illustrated for representative phases within one cycle. The rise between the pressures at the inlet and the exit of the divergent tube does not become too large when the flow rate increases, it being moderately high in the decelerative phase. The profiles of the phase-averaged velocity and the turbulence intensity in the cross section are very different from those for steady flow. Also, they show complex changes along the tube axis in both the accelerative and decelerative phases. Chapter 12 deals with a numerical method for a solution of an airfoil design inverse problem. The presented method is intended for a design of an airfoil based on a prescribed pressure distribution along a mean camber line, especially for modifying existing airfoils. The main idea of this method is a coupling of a direct and approximate inverse operator. The goal is to find a pseudo-distribution corresponding to the desired airfoil with respect to the approximate inversion. This is done in an iterative way. The direct operator represents a solution of a flow around an airfoil, described by a system of the Navier-Stokes equations in the case of a laminar flow and by the k−ω model in the case of a turbulent flow. There is a relative freedom of choosing the model describing the flow. The system of PDEs is solved by an implicit finite volume method. The approximate inverse operator is based on a thin airfoil theory for a potential flow, equipped with some corrections according to the model used. The airfoil is constructed using a mean camber line and a thickness function. The so far developed method has several restrictions. It is applicable to a subsonic pressure distribution satisfying a certain condition for the position of a stagnation point. Numerical results are presented. Recent advances in the field of fluid mechanics with moving fronts are linked to the use of Level SetMethods, a versatile mathematical technique to follow free boundaries which undergo topological changes. A challenging class of problems in this context are those related to the solution of a partial differential equation posed on a moving domain, in which the boundary condition for the PDE solver has to be obtained from a partial differential equation defined on the front. This is the case of potential flow models with moving boundaries. Moreover, the fluid front may carry some material substance which diffuses in the front and is advected by the front velocity, as for example the use of surfactants to lower surface tension. We present a Level Set based methodology to embed this partial differential equations defined on the front in a complete Eulerian framework, fully avoiding the tracking of fluid particles and its known limitations. To show the advantages of this approach in the field of Fluid Mechanics we present in Chapter 13 one particular application: the numerical approximation of a potential flow model to simulate the evolution and breaking of a solitary wave propagating over a slopping bottom and compare the level set based algorithm with previous front tracking models.

Preface

xiii

Chapter 14 is basically a review of recent works that is aimed at putting forward the main ideas behind a new theoretical approach to turbulent wall-bounded flows, notably pipe-flows, in which complex physics is involved, such as combustion or particle transport. Pipe flows are ubiquitous in industrial applications and have been studied intensively in the last century, both from a theoretical and experimental point of view. The result of such a strong effort is a good comprehension of the physics underlying the dynamics of these flows and the proposition of reliable models for simple turbulent pipe-flows at large Reynolds number Nevertheless, the advancing of engineering frontiers casts a growing demand for models suitable for the study of more complex flows. For instance, the motion and the interaction with walls of aerosol particles, the presence of roughness on walls and the possibility of drag reduction through the introduction of few complex molecules in the flow constitute some interesting examples of pipe-flows with some new complex physics involved. A good modeling approach to these flows is yet to come and, in the commentary, we support the idea that a new angle of attack is needed with respect to present methods. In this article, we analyze which are the fundamental features of complex two-phase flows and we point out that there are two key elements to be taken into account by a suitable theoretical model: 1) These flows exhibit chaotic patterns; 2) The presence of instantaneous coherent structures radically change the flow properties. From a methodological point of view, two main theoretical approaches have been considered so far: the solution of equations based on first principles (for example, the Navier-Stokes equations for a single phase fluid) or Eulerian models based on constitutive relations. In analogy with the language of statistical physics, we consider the former as a microscopic approach and the later as a macroscopic one. We discuss why we consider both approaches unsatisfying with regard to the description of general complex turbulent flows, like two-phase flows. Hence, we argue that a significant breakthrough can be obtained by choosing a new approach based upon two main ideas: 1) The approach has to be mesoscopic (in the middle between the microscopic and the macroscopic) and statistical; 2) Some geometrical features of turbulence have to be introduced in the statistical model. We present the main characteristics of a stochastic model which respects the conditions expressed by the point 1) and a method to fulfill the point 2). These arguments are backed up with some recent numerical results of deposition onto walls in turbulent pipe-flows. Finally, some perspectives are also given.

In: Fluid Mechanics and Pipe Flow Editors: D. Matos and C. Valerio, pp. 1-26

ISBN: 978-1-60741-037-9 © 2009 Nova Science Publishers, Inc.

Chapter 1

SOLUTE TRANSPORT, DISPERSION, AND SEPARATION IN NANOFLUIDIC CHANNELS Xiangchun Xuan* Department of Mechanical Engineering, Clemson University, Clemson, SC 29634-0921, USA

Abstract Recent breakthrough in nanofabrication has stimulated the interest of solute separation in nanofluidic channels. Since the hydraulic radius of nanochannels is comparable to the thickness of electric double layers, the enormous electric fields inherent to the latter generate transverse electromigrations causing charge-dependent solute distributions over the channel cross-section. As a consequence, the non-uniform fluid flow through nanochannels yields charge-dependent solute speeds enabling the separation of solutes by charge alone. In this chapter we develop a theoretical model of solute transport, dispersion and separation in electroosmotic and pressure-driven flows through nanofluidic channels. This model provides a basis for the optimization of solute separation in nanochannels in terms of selectivity and resolution as traditionally defined.

1. Introduction Solute transport and separation in micro-columns (e.g., micro capillaries and chip-based microchannels) have been a focus of research and development in electrophoresis and chromatography communities for many years. Recent breakthrough in nanofabrication has initiated the study of these topics among others in nanofluidic channels [1-3]. Since the hydraulic radius of nanochannels is comparable to the thickness of electric double layers (EDL), the enormous electric fields inherent to the latter generate transverse electromigrations causing charge-dependent solute distributions over the channel cross-section [4-7]. As a consequence, the non-uniform fluid flow in nanochannels yields charge-dependent solute speeds enabling the separation of solutes by charge alone [8,9]. Such charge-based solute *

E-mail address: [email protected]. Tel: (864) 656-5630. Fax: (864) 656-7299

2

Xiangchun Xuan

separation was first proposed and implemented by Pennathur and Santiago [10] and Garcia et al. [11] in electroosmotic flow through nanoscale channels, termed nanochannel electrophoresis. As a matter of fact, this separation may also happen in pressure-driven flow along nanoscale channels, termed here as nanochannel chromatography for comparison, which was first demonstrated theoretically by Griffiths and Nilson [12] and Xuan and Li [13], and later experimentally verified by Liu’s group [14]. So far, a number of theoretical studies have been conducted on the transport [4-6,913,15,19], dispersion [7,9,12,15-19] and separation [4,9-13,15,19] of solutes in free solutions through nanofluidic channels. This chapter combines and unites the works from ourselves in this area [6,13,15,17-19], and is aimed to develop a general analytical model of solute transport, dispersion and separation in nanochannels. It is important to note that this model applies only to point-like solutes. For those with a finite size, one must consider the hydrodynamic and electrostatic interactions among solutes, electric field, and flow fluid, and as well the Steric interactions between solutes and channel walls etc [20].

2. Nomenclature a Bi cb ci Ci ci,0 Ci,0 Di

channel half-height defined function, = exp(−ziΨ) bulk concentration of the background electrolyte concentration of solute species i bulk concentration of solute species i concentration of solute species i at the channel centerline initial concentration of solute species i at the channel centerline solute diffusion coefficient effective solute diffusion coefficient

E Est F hi j Ki L P Pe rji R Rji t T ui

ui

axial electric field streaming potential field Faraday’s constant reduced theoretical plate height electric current density hydrodynamic dispersion channel length hydrodynamic pressure drop per unit channel length Peclet number solute selectivity Universal gas constant resolution time coordinate absolute temperature axial solute speed mean solute speed

vi Wi

solute mobility half width of the initially injected solute zone

Di′

Solute Transport, Dispersion, and Separation in Nanofluidic Channels x Xi y zi Z

3

streamwise or longitudinal coordinate the central location of the injected solute zone transverse coordinate valence of ions electrokinetic “figure of merit”

Greek Symbols β ε γ κ χi λb μ ψ Ψ Ψ0

ρe σb σi σt ζ ζ*

non-dimensional product of fluid properties, = λbμ/εRT permittivity apparent viscosity ratio reciprocal of Debye length dispersion coefficient molar conductivity of the background electrolyte fluid viscosity electrical double layer potential non-dimensional EDL potential EDL potential at the channel center net charge density bulk electric conductivity of background electrolyte standard deviation of solute peak distribution standard deviation of solute peak distribution in the time domain zeta potential non-dimensional zeta potential

Subscripts e p i

electroosmosis related pressure-driven related solute species i

3. Fluid Flow in Nanochannels Given the fact that the width (in micrometers) of state-of-the-art nanofluidic channels is usually much larger than their depth (in nanometers) [1-3], we consider the solute transport in fluid flow through a long straight nanoslit, see Figure 1 for the schematic. The flow may be electric field-driven, i.e., electroosmotic, or pressure-driven. For simplicity, the electrolyte solution is assumed symmetric with unit-charge, e.g., KCl. As the time scale for fluid flow (in the order of nanoseconds) is far less than that of solute transport (typically of tens of seconds), we assume a steady-state, fully-developed incompressible fluid motion, which in a slit channel is governed by

4

Xiangchun Xuan

y

Solute zone u + viziFE

u

a

x

Figure 1. Schematic of solute transport in a slit nanochannel (only the top half is illustrated due to the symmetry).

μ

d 2u + P + ρe E = 0 dy 2

(3-1)

where μ is the fluid viscosity, u the axial fluid velocity, y the transverse coordinate originating from the channel axis, P the pressure drop per unit channel length, and E the axial electric field either externally applied in electroosmotic flow or internally induced in pressure-driven flow (i.e., the so-called streaming potential field) [21-24]. The net charge density, ρe, is solved from the Poisson equation [25]

d 2ψ ρ e = −ε 2 dy

(3-2)

where ε is the fluid permittivity and ψ is the EDL potential. Invoking the no-slip condition for Eq. (3-1) and the zeta potential condition for Eq. (3-2) on the channel wall (i.e., y = a), one can easily obtain

u = u p + ue up =

(3-3)

a2 ⎛ y2 ⎞ ⎜1 − ⎟ P 2μ ⎝ a 2 ⎠

(3-4)

εζ μ

(3-5)

ue = −

⎛ Ψ⎞ ⎜1 − ∗ ⎟ E ⎝ ζ ⎠

where up is the pressure-driven fluid velocity, ue the electroosmotic fluid velocity, a the halfheight of the channel, and Ψ = Fψ/RT and ζ* = Fζ/RT the dimensionless forms of the EDL and wall zeta potentials with F the Faraday’s constant, R the universal gas constant and T the absolute fluid temperature. It is noted that the contribution of charged solutes to the net charge density ρe has been neglected. This is reasonable as long as the solute concentration is much lower than the ionic concentration of the background electrolyte, which is fulfilled in typical solute separations. Under such a condition, it is also safe to assume a uniform zeta potential on the channel wall.

Solute Transport, Dispersion, and Separation in Nanofluidic Channels

5

The non-dimensional EDL potential in Eq. (3-5), Ψ, may be solved from the PoissonBoltzmann equation [25]

d 2Ψ = κ 2 sinh ( Ψ ) 2 dy where

(3-6)

κ = 2 F 2 cb ε RT is the inverse of the so-called Debye screening length with cb the

bulk concentration of the background electrolyte. We recognize that the assumed Boltzmann distribution of electrolyte ions in Poisson-Boltzmann equation might be questionable in nanoscale channels, especially in those with strong EDL overlapping [26,27]. However, this equation has been successfully used to explain the experimentally measured electric conductance and streaming current in variable nanofluidic channels [28-32], and is thus still employed here. For the case of a small magnitude of ζ (e.g., |ζ| < 25 mV or |ζ*| < 1) which is actually desirable for sensitive solute separations in nanochannels as demonstrated by Griffiths and Nilson [12], one may use the Debye-Huckel approximation to simplify Eq. (3-6) as [21,25]

d 2Ψ = κ 2Ψ dy 2

(3-7)

It is then straightforward to obtain

Ψ =ζ*

cosh (κ y ) cosh (κ a )

(3-8)

where κa may be viewed as the normalized channel half-height. It is important to note that for a given fluid and channel combination, the wall zeta potential will in general vary with κa [28,31,32]. One option to address this is to use a surface-charge based potential parameter for scaling instead of zeta potential [16]. In this work and other studies [6-14], the zeta potential is used directly, because it may be readily determined through experiment and provides a direct measure of the electroosmotic mobility. The area-averaged fluid velocity

u may be written in terms of the Poiseuille and

electroosmotic components

u = u p + ue up =

a2 P 3μ

(3-9)

(3-10)

6

Xiangchun Xuan

ue = −

where " =

εζ μ

⎡ tanh (κ a ) ⎤ ⎢1 − ⎥E κ a ⎣ ⎦

(3-11)

∫ (") d ( y a ) signifies an area-averaged quantity. a

0

3.1. Electroosmotic Flow For electroosmotic flow, no pressure gradient is present, and so the fluid motion in Eq. (3-3) is described by

u p = 0 and ue = −

εζ ⎡ cosh (κ y ) ⎤ ⎢1 − ⎥E μ ⎣ cosh (κ a ) ⎦

Electroosmotic velocity profile

1 10

(3-12)

20

5

0.8 2

0.6 0.4

κa = 1 0.2 0 0

0.2

0.4

0.6

0.8

1

y/a Figure 2. Radial profile of the normalized electroosmotic fluid velocity, ue/UHS = 1 − cosh(κy)/cosh(κa), at different κa values. All symbols are referred to the nomenclature.

Figure 2 shows the profile of electroosmotic velocity normalized by the so-called HelmholtzSmoluchowski velocity UHS = −εζE/μ [25], i.e., ue/UHS = 1 − cosh(κy)/cosh(κa), in a slit channel with different κa values. When κa > 10, the curves are almost plug-like except near the channel wall and the bulk velocity is equal to UHS. These are the typical features of electroosmotic flow when there is little or zero EDL overlapping. The profiles at κa < 5 become essentially parabolic resembling the traditional pressure-driven flow. Moreover, the maximum velocity along the channel centerline is significantly lower than UHS and decreases with κa, indicating a vanishingly small electroosmotic mobility when κa approaches 0.


7

3.2. Pressure-Driven Flow For pressure-driven flow, the downstream accumulation of counter-ions results in the development of a streaming potential field [21-24]. This induced electric field, Est, can be determined from the condition of zero electric current though the channel. If an equal mobility for the positive and negative ions of the electrolyte is assumed, the electric current density, j, in pressure-driven flow is given as [21-24]

j = ρeu + σ b cosh ( Ψ ) Est

(3-13)

where σb = cbλb is the bulk conductivity of the electrolyte with λb being the molar conductivity. Referring to Eqs. (3-2) and (3-3), one may rewrite the last equation as

j = −ε

d 2 Ψ RT ( u p + ue ) + cbλb cosh ( Ψ ) Est dy 2 F

(3-14)

Note that the surface conductance of the outer diffusion layer in the EDL has been considered through the cosine hyperbolic function in Eq. (3-14) (which reduces to 1 at Ψ = 0). The contribution of the inner Stern layer conductance [33] to the electric current is, however, ignored. Readers may be referred to Davidson and Xuan [34] for a discussion of this issue in electrokinetic streaming effects. Integrating j in Eq. (3-14) over the channel cross-section and using the zero electric current condition in a steady-state pressure-driven flow yield

g1 ζ * Est = P cb F ( g 2 + β g3 ζ *2 )

(3-15)

tanh (κa ) tanh (κa ) 1 ⎛ y⎞ , g2 = and g 3 = ∫ cosh (Ψ )d ⎜ ⎟ (3-16) − 2 κa cosh (κa ) κa ⎝a⎠ 0 a

g1 = 1 −

Therefore, the fluid motion in pressure-driven flow is characterized as

up =

εζ a2 ⎛ y2 ⎞ ⎜1 − 2 ⎟ P and ue = − μ 2μ ⎝ a ⎠

⎡ cosh (κ y ) ⎤ ⎢1 − ⎥ Est cosh κ a ( ) ⎣ ⎦

(3-17)

Area-averaging the two velocity components in the last equation and combining them with Eq. (3-15) lead to

ue up

= −Z

(3-18)

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Xiangchun Xuan

where Z is previously termed electrokinetic “figure of merit” as it gauges the efficiency of electrokinetic energy conversion [35,36], and defined as

Z=

(κ a )

2

(g

β=

3g12

(3-19)

∗2 ) 2 + β g3 ζ

λb μ εRT

(3-20)

where κ2 = 2F2cb/εRT has been invoked during the derivation, and β is a non-dimensional product of fluid properties whose reciprocal was termed Levine number by Griffiths and Nilson [37]. Apparently, Z depends on three non-dimensional parameters, β, κa and ζ*, among which β spans in the range of 2 ≤ β ≤ 10 and ζ* spans in the rage of −8 ≤ ζ* ≤ 0 [33] for typical aqueous solutions. Moreover, Z is unconditionally positive and less than unity due to the entropy generation in non-equilibrium electrokinetic flow [38]. The curves of Z at ζ* = −1 (or ζ ≈ −25 mV) and β = 2 and 10, respectively, are displayed in Figure 3 as a function of κa. One can see that Z achieves the maximum at around κa = 2, indicating that nanochannels with a strong EDL overlapping are the necessary conditions for efficient electrokinetic energy conversion. For more information about Z and its function in electrokinetic energy conversion, the reader is referred to Xuan and Li [36].

0.15

β = 10 0.12

Z

0.09 0.06

β=2

0.03 0.00 0.1

1

κa

10

100

Figure 3. The electrokinetic “figure of merit” as a function of κa at ζ* = −1 and β = 2 and 10, respectively.


9

Based on Eq. (3-18), the effects of streaming potential on the average fluid speed, i.e.,

u in Eq. (3-9), in an otherwise pure pressure-driven flow, are characterized by

u up

= 1− Z

(3-21)

This equation also provides a measure of the so-called electro-viscous effects in micro/nanochannels [39]. If the concept of apparent viscosity is employed to characterize such retardation effects, the apparent viscosity ratio γ is readily derived as

γ = 1 (1 − Z )

(3-22)

For more information on this topic, the reader is referred to Li [39] and Xuan [40].

4. Solute Transport in Nanochannels Solute transport in nanochannels is governed by the Nernst-Planck equation in the absence of chemical reactions, which under the assumption of fully-developed fluid flow is written as

∂ci ∂c ∂ 2c ∂ 2c ∂ ⎛ ∂ψ ⎞ + ui i = Di 2i + Di 2i + vi zi F ⎜ ci ⎟ ∂t ∂x ∂x ∂y ∂y ⎝ ∂y ⎠

(4-1)

ui = ue + u p + vi zi FE

(4-2)

where ci is the concentration of solute species i, t the time coordinate, ui the local solute speed (a combination of electroosmosis, pressure-driven motion, and electrophoresis), x the axial coordinate originating from the channel inlet (see Figure 1), Di the molecular diffusion coefficient, vi the solute mobility, and zi the solute charge number. Note that the product viziF represents the solute electrophoretic mobility. Integrating Eq. (4.1) over the channel crosssection eliminates the last two terms on the right hand side due to the impermeable wall conditions

∂ ci ∂ ui ci ∂ 2 ci + = Di ∂t ∂x ∂x 2

(4-3)

where again ... indicates the area-average over the channel cross-section as defined earlier. Since the time scale for transverse solute diffusion in nanochannels (characterized by a2/Di, which is about 100 μs when a = 100 nm and Di = 1×10−10 m2/s) is much shorter than

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Xiangchun Xuan

that for longitudinal solute transport (typically of tens of seconds), it is reasonable to assume that solute species are at a quasi-steady equilibrium in the y direction, i.e.,

0 = Di

∂ 2 ci ∂ ⎛ ∂ψ ⎞ + vi zi F ⎜ ci ⎟ 2 ∂y ∂y ⎝ ∂y ⎠

(4-4)

Integrating Eq. (4.4) twice and using the Nernst-Einstein relation [33], vi = Di/RT, one obtains

ci ( x, y, t ) = ci ,0 ( x, t ) exp ⎡⎣ − zi ( Ψ − Ψ 0 ) ⎤⎦

(4-5)

where ci,0 is the solute concentration at the channel centerline where the local EDL potential is defined as Ψ0 (non-zero in the presence of EDL overlapping). Substituting Eq. (4.5) into Eq. (4.3) and considering the hydrodynamic dispersion due to the velocity non-uniformity over the channel cross-section [41,42], one may obtain

∂ci ,0 ∂t

+ ui

∂ci ,0 ∂x

= Di′

∂ 2 ci ,0 ∂x 2

ui = uip + uie + vi zi FE uip =

u p Bi Bi

and uie =

ue Bi Bi

(4-6) (4-7)

(4-8)

where ui is the mean solute speed (i.e., zone velocity) with uip and uie being its components due to pressure-driven and electroosmotic flows, Di′ the effective diffusion coefficient which is a combination of molecule diffusion and hydrodynamic dispersion and will be addressed in the next section, and Bi = exp(−ziΨ) the like-Boltzmann distribution of solutes in the crossstream direction. It is the dependence of ui on the charge number zi that enables the chargedbased solute separation in nanochannels. For an initially uniform concentration Ci,0 of solute species i along the channel axis, a closed-form solution to Eq. (4-6) is given by [5]

ci ,0 =

⎛ x − X i − ui t ⎞ ⎤ Ci ,0 ⎡ ⎛ Wi − x + X i + ui t ⎞ ⎢erf ⎜ ⎟ + erf ⎜ ⎟⎥ ⎟ ⎜ 2 D′t ⎟ ⎥ 2 ⎢⎣ ⎜⎝ 2 Di′t i ⎠ ⎝ ⎠⎦

(4-9)

where erf denotes the error function, and Wi is the half width of the initial solute zone with its center being located at Xi. As such, the electrokinetic transport of solute species in nanochannels is described by


ci ( x, y, t ) =

Ci 2

⎡ ⎛W − x + X +u t ⎞ ⎛ x − X i − ui t ⎞ ⎤ i i ⎢erf ⎜ i ⎟ + erf ⎜ ⎟ ⎥ exp ( − zi Ψ ) ⎟ ⎜ 2 D′t ⎟ ⎥ 2 Di′t ⎢⎣ ⎜⎝ i ⎠ ⎝ ⎠⎦

11

(4-10)

where Ci = Ci ,0 exp ( zi Ψ 0 ) is the bulk solute concentration at zero potential outside the slit nanochannel, refer to Eq. (4-5). zv = +2 zv = +1 zv = 0 zv = −1 zv = −2 0.2 Cmax

0.4 Cmax

0.6 Cmax

0.8 Cmax

Cmax

Figure 4. Transport of solutes with zi = [+2, −2] through a 100 nm deep channel in nanochannel chromatography. Other parameters are referred to the text. Reprinted with permission from [13].

Figure 4 illustrates the transport of an initially Wi = 1 μm wide plug of solutes with zi = [+2, −2] (from top to bottom) through a 100 nm deep (i.e., a = 50 nm) channel 5 s after a pressure gradient P = 1×108 Pam-1 was imposed. Note that only the top half of the channel is shown due to symmetry. The ionic concentration of the background electrolyte is cb = 1 mM, corresponding to κa ≈ 5. The other two non-dimensional parameters are assumed to be β = 4 and ζ* = −2 (or ζ = −50 mV), both of which are typical to aqueous solutions as indicated above. As to the validity of the Debye-Huckel approximation at ζ* = −2, we have recently demonstrated using numerical simulation the fairly good accuracy of Eq. (3-7) in predicting the solute migration velocity [5]. As shown, positive solutes are concentrated to near the negatively charged wall due to the solute-wall electrostatic interactions [9,11], or in essence the transverse electromigration in response to the induced EDL field [5,10,12]. Moreover, the higher the charge number is, the closer the solutes are to the walls. As the fluid velocity near no-slip walls is slower than its average, positive solutes move slower than neutral solutes that are still uniformly distributed over the channel cross-section. Conversely, negative solutes are repelled by the negatively changed walls and concentrated to the region close to the channel center. Hence, they move faster than neutral solutes as seen in Figure 4. As the fluid velocity profile is available in Eq. (3-12) for electroosmotic flow and in Eq. (3-17) for pressure-driven flow, the mean speed of solutes, ui , is readily obtained from Eq. (4-7). Figure 5 compares the mean speed of solutes with zi = [−2, +2] in electroosmotic flow with an electric field of 4 kV/m. One can see that ui of all five solutes decreases when κa gets smaller. This reduction may be explained by the overall lower electroosmotic velocity at a smaller κa, as demonstrated in Figure 2. When κa > 100, negatively charged solutes move slower than positive ones due to their opposite electrophoresis to fluid electroosmosis

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Xiangchun Xuan

(identical to the curve with zi = 0). When κa gets smaller than 100, however, negatively solutes start moving faster than positively solutes as the latter ones are concentrated to the EDL region within which the fluid has a slower speed than the bulk as explained above. The relative magnitude of ui between the solutes of like charges is, however, a complex function of both the charge number zi, which determines the velocity component due to fluid flow, i.e., uip + uie in Eq. (4-7), and the solute mobility vi, which determines the velocity component due to solute electrophoresis, i.e., the most right term in Eq. (4-7). When κa further decreases to less than 1, the EDL potential becomes nearly flat due to the strong EDL overlapping (see Figure 2), and so the order of ui for the three solutes at large κa (i.e., microchannel electrophoresis) is recovered.

Figure 5. Comparison of the mean speeds of solutes with zi = [−2, +2] as a function of κa in nanochannel electrophoresis. The solute diffusivity is assumed to be constant, Di = 5×10−11 m2/s. Other parameters are referred to the text.

Combining Eqs. (4-7) and (4-8) provides a measure of the streaming potential effects on the solute mean speed in pressure-driven flow,

u B + vi zi F Bi Est ui = 1+ e i uip u p Bi

(4-11)

It is obvious that the last equation is reduced to Eq. (3-21) for neutral solutes, i.e., zi = 0 and thus Bi = 1. Figure 6 shows the ratio ui uip as a function of κa. As expected, streaming


13

potential effects reduce the solute speed due to the induced electroosmotic backflow. This reduction varies with the solute charge zi and attains the extreme at about κa = 3, where the figure of merit Z (refer to Figure 3) approaches its maximum indicating the largest streaming potential effects. In both the high and low limits of κa, streaming potential effects become negligible, i.e., Z → 0, see Figure 3. Accordingly, ui uip reduces to 1 for all solutes at large

κa while varying with zi at small κa because of the finite solute mobility [15]. 1.1 −2 −1

1 ui/uip

0 +1

0.9

zi = +2 0.8

0.7 0.1

1

10

100

κa Figure 6. Effects of streaming potential on the solute mean speed in nanochannel chromatography.

5. Solute Dispersion in Nanochannels As up and ue vary over the channel cross-section (refer to Eqs. (3-4) and (3-5), and Figure 2), they both contribute to the spreading of solutes along the flow direction, which is termed hydrodynamic dispersion or Taylor dispersion [41,42]. The general formula for calculating this dispersion is given by [43,44]

a2 Ki = Di

y Bi−1 ⎡ ∫ Bi ( ui − ui ) dy′⎤ ⎣⎢ 0 ⎦⎥

2

(5-1)

Bi

Referring to Eqs. (4-2) and (4-7), one may then rewrite the last equation as

Ki =

(

a2 Fip u p Di

2

+ Fipe u p ue + Fie ue

2

)

(5-2)

14

Xiangchun Xuan y Fip = Bi−1 ⎡ ∫ Bi ( u ∗p − uip∗ ) dy′⎤ ⎣⎢ 0 ⎦⎥

y Fie = Bi−1 ⎡ ∫ Bi ( ue∗ − uie∗ ) dy′⎤ ⎢⎣ 0 ⎥⎦

2

Bi

2

Bi

−1

−1

y y Fipe = 2 Bi−1 ⎡ ∫ ( u ∗p − uip∗ ) Bi dy′⎤ ⎡ ∫ ( ue∗ − uie∗ ) Bi dy′⎤ ⎢⎣ 0 ⎥⎦ ⎢⎣ 0 ⎥⎦ ∗

where um = um

(5-3a)

(5-3b)

Bi

−1

(5-3c)

um and uim∗ = uim um (m = p and e). Note that the three terms, Fip, Fipe

and Fie in Eq. (5-2) represent the contributions to dispersion due to the pressure-driven flow, the coupling between pressure-driven and electroosmotic flows, and the electroosmotic flow, respectively. Hydrodynamic dispersion is often expressed in terms of a non-dimensional dispersion coefficient χi [45],

K i = χ i Pei2 Di

(5-4)

where the Peclet number Pei in this case may be defined with respect to the mean solute speed, i.e., Pei = ui a Di [12,19], or to the area-averaged fluid velocity, i.e.,

(

)

Pei = u p + ue a Di [15-18,45]. Using the solute speed-based Peclet number, χi becomes dependent on the solute diffusivity Di which complicates the analysis. This is because the solute mobility vi in ui [see Eq. (4-7)] is coupled to Di via the Nernst-Einstein relation, vi = Di/RT. Such dependence doesn’t occur if χi is defined using the fluid velocitybased Peclet number. Here, we employ the latter definition in keeping with the dispersion studies of neutral solutes in the literature [45]. As such, the dispersion coefficient χi may be easily obtained from Eq. (5-2) as

χi =

Fip u p

2

+ Fipe u p ue + Fie ue

(

u p + ue

)

2

2

(5-5)

It is important to note that χi is independent of the solute speed or the driving force of the flow while Ki (in the unit of Di) not. Instead, χi is primarily determined by the flow type (pressure- or electric field-driven), channel structure (including shape and size) and solute charge number.


15

5.1. Electroosmotic Flow In electroosmotic flow, the hydrodynamic dispersion in Eq. (5-2) is reduced to

K i = Fie

a 2 ue

2

(5-6)

Di

Accordingly, the dispersion coefficient in Eq. (5.5) is simplified as

χ i = Fie

(5-7)

χi for nanochannel electrophoresis

0.1 +2

0.01 +1 zi = 0

0.001 −1 −2

0.0001 0.1

1

10

100

κa Figure 7. Illustration of dispersion coefficient χi of solutes with zi = [−2, +2] in nanochannel electrophoresis as a function of κa.

Figure 7 shows χi of solutes with zi = [−2, +2] in nanochannel electrophoresis as a function of κa. We see that in the entire range of κa, χi of positive solutes is larger than that of neutral ones while χi of negative solutes is smaller than the latter. This is because positive solutes are concentrated to near the channel walls where the velocity gradients are large while negative ones are concentrated to the channel centerline where the velocity gradients are small (refer to Figure 4). Moreover, the higher the charge number zi, the larger is χi for positive solutes and the smaller for negative ones. In the low limit of κa (i.e., the narrowest channel), the EDL potential is essentially uniform over the channel cross-section (refer to Figure 2), and so is the solute distribution regardless of the charge number. As a consequence, χi of all charged solutes approach that of neutral solutes, i.e., 2/105. Note that this value is equal to the dispersion coefficient of neutral solutes in a pure pressure-driven flow indicating the resemblance between pressure-driven and electroosmotic flows in very small

16

Xiangchun Xuan

nanochannels. This aspect will be revisited shortly. In the high limit of κa (i.e., the widest channel), the EDL thickness is so thin compared to the channel height that the solute distribution becomes once again uniform across the channel (the EDL potential is, here, uniformly zero while equal to the wall zeta potential in the low limit of κa). Therefore, the hydrodynamic dispersion in electroosmotic flow, or the so-called electrokinetic dispersion [46], decreases with the square of κa and ultimately converges to zero [47,48].

5.2. Pressure-Driven Flow In pressure-driven flow with consideration of streaming potential, the hydrodynamic dispersion is obtained from Eq. (5-2) as

K i = Fip

a2 u p

2

Di

(1 − δ

i2

Z + δi3Z 2 )

δ i 2 = Fipe Fip and δ i 3 = Fie Fip

(5-8)

(5-9)

during which Eqs. (3-17) and (3-18) have been invoked and Z is the electrokinetic “figure of merit” as defined in Eq. (3-19). It is apparent that the streaming potential induced electroosmotic backflow produces two additional dispersions in pressure-driven flow: one is the electrokinetic dispersion due to electroosmotic flow itself, the term with δi3 in Eq. (5-8), which tends to increase the total dispersion, and the other is due to the coupling of pressuredriven and electroosmotic flows, the term with δi2 in Eq. (5-8), which tends to decrease the total dispersion. The latter phenomenon has been employed previously to reduce the hydrodynamic dispersion in capillary electrophoresis where a pressure-driven backflow is intentionally introduced to partially compensate the non-uniformity in electroosmotic velocity profile [49,50]. If streaming potential effects are ignored, i.e., for a pure pressure-driven flow with Z = 0, Eq. (5-8) reduces to

K ip = Fip

2

a2 u p

(5-10)

Di

such that

Ki = 1 − δi 2 Z + δi3Z 2 K ip as

(5-11)

Similarly, the dispersion coefficient in Eq. (5-5) for real pressure-driven flow is obtained

χ i = Fip

1 − δi 2 Z + δi3Z 2

(1 − Z )

2

(5-12)


17

where Fip = χip is the dispersion coefficient for a pure pressure-driven flow by analogy to Eq. (5-7) in a pure electroosmotic flow. We thus have

χi 1 − δ i 2 Z + δ i 3 Z 2 K = =γ2 i 2 K ip χ ip (1 − Z )

(5-13)

Therefore, χi/χip differs from Ki/Kip by only the square of the apparent viscosity ratio γ, see the definition in Eq. (3-22). As γ is independent of the solute charge number zi, it is expected that the variation of χi/χip with respect to zi will be identical to that of Ki/Kip.

χi for nanochannel chromatography

1 +2

0.1

+1 zi = 0 −1

0.01

−2

0.001 0.1

1

κa

10

100

Figure 8. Dispersion coefficient χi of solutes with zi = [−2, +2] in nanochannel chromatography as a function of κa.

Figure 8 shows χi of solutes with zi = [−2, +2] as a function of κa in nanochannel chromatography. Due to the same reason as stated above for nanochannel electrophoresis, χi of positive solutes is larger than that of neutral ones while χi of negative solutes is the smallest. In both the high and low limits of κa, the flow-induced streaming potential is negligible, see Eq. (3-18) and Figure 3. Hence, the electroosmotic back flow and the induced solute electrophoresis vanish, yielding χi = 2/105 regardless of the solute charge [51]. It is important to note that χi of neutral solutes in pressure-driven flow is not uniformly 2/105 as accepted in the literature. Due to the effects of flow-induced streaming potential, χi is increased by the electroosmotic backflow [i.e., δi3Z2 term in Eq. (5-12)] even though the coupled dispersion term [i.e., −δi2Z term in Eq. (5-12)] drops for neutral solutes [17].

18

Xiangchun Xuan

1.15 1.1

χi/χip

Ki/Kip or χi/χip

1.05

zi

1 0.95 0.9 0.85 Ki/Kip

0.8

zi

0.75 0.1

1

10

κa

100 .

Figure 9. Effects of streaming potential on the ratio of solute dispersion, Ki/Kip, and the ratio of dispersion coefficient, χi/χip, in nanochannel chromatography as a function of κa. Adapted with permission from [18].

Figure 9 displays the effects of streaming potential on the ratio of solute dispersion, Ki/Kip, and the ratio of dispersion coefficient, χi/χip, in nanochannel chromatography as a function of κa. In all cases, Ki/Kip is less than 1 indicating that streaming potential effects result in a decrease in hydrodynamic dispersion. This reduction, as a consequence of the induced electroosmotic backflow, gets larger (i.e., Ki/Kip deviates further away from 1) when the solute charge zi increases. The optimum κa at which Ki/Kip achieves its extreme increases slightly with zi. In contrast to the decrease in solute dispersion, the dispersion coefficient is increased by the effects of streaming potential, i.e., χi/χip > 1. These dissimilar trends stem from the dependence of γ on κa, see Eqs. (3-19), (3-22) and (5-13). As streaming potential effects increase (or in other words, the electrokinetic “figure of merit” Z increases), the electroosmotic backflow increases causing a decrease in Ki/Kip (and Ki/Kip < 1) while an increase in γ (and γ > 1). The net result is the observed variation of χi/χip with respect to κa. The increase in χi/χip is more sensitive to zi than the decrease in Ki/Kip. However, the trend that χi/χip varies with respect to zi is consistent with Ki/Kip as pointed out earlier. In addition, χi/χip attains a maximum at a larger value of κa than that at which Ki/Kip is minimized.

5.3. Neutral Solutes For neutral solutes (zi = 0), closed-form formulae are available for the functions Fm (m = ip, ipe, ie) defined in Eq. (5-3) which in turn determine δi2 and δi3 through Eq. (5-9). Specifically, we find [17]


Fip = 2 105 Fipe =

Fie =

ω2 2 (1 − ω )

(5-14)

ω ⎡2 2 6 ⎛ 1 − ω ⎞⎤ + ⎢ − ⎥ (1 − ω ) ⎢⎣15 (κ a )2 (κ a )4 ⎜⎝ ω ⎟⎠ ⎥⎦

(5-15)

⎡1 ⎤ 2 3 1 − − 2 ⎢ + ⎥ 2 2 2 2 ⎢⎣ 3 (κ a ) 2ω (κ a ) 2ω (κ a ) cosh (κ a ) ⎥⎦

ω=

19

tanh (κ a ) κa

(5-16)

(5-17)

Figure 10 compares the magnitude of δi2 and δi3 [see their definitions in Eq. (5-9)] for neutral solutes as a function of κa. As shown, δi2 is always larger than δi3. In the low limit of κa, δi2 approaches 2 while δi3 approaches 1, and the square bracketed terms in Eq. (5-2) thus reduces to ( + )2 reflecting the similarity of pressure-driven and electroosmotic flow profiles in very narrow nanochannels. In the high limit of κa, both δi2 and δi3 approach zero because the streaming potential is negligible and the electroosmotic velocity profile becomes essentially plug-like. Note that Eq. (5-14) gives the well-known hydrodynamic dispersion coefficient of neutral solutes in a pure pressure-driven flow between two parallel plates [51]. Moreover, Eq. (5-16) is identical to that derived by Griffiths and Nilson [47,48] which gives the electrokinetic dispersion coefficient of neutral solutes in a pure electroosmotic flow between two parallel plates. 2

δi2

δ

1.5

1

δi3

0.5

0 0.1

1

10 κa

100

1000

Figure 10. Plot of δi2 and δ i3 for neutral solutes as a function of κa. Adapted with permission from [17].

20

Xiangchun Xuan

6. Solute Separation in Nanochannels Solute separation is typically characterized by retention, selectivity, plate height (or plate number), and resolution [43], of which retention and plate height are related to only one type of solutes. In contrast, selectivity and resolution are both dependent on two types of solute species, and thus provide a direct measure of the separation performance of solutes. As plate height is involved in the definition of resolution, see Eq. (6-8), it will still be considered below along with the selectivity and resolution for a comprehensive understanding of solute separation in nanofluidic channels. In order to emphasize the advantage of electrophoresis and chromatography in nanochannels over those taking place in micro-columns, we focus on the solutes with a similar electrophoretic mobility, or specifically, viziF = constant. This is equivalent to assuming a constant charge-to-size ratio or a constant product, Dz = Dizi, of solute charge and diffusivity because solute size is inversely proportional to its diffusivity via the NernstEinstein relation [33]. Such solutes are unable to be separated in free solutions through pressure-driven or electroosmotic microchannel flows. A typical value of the solute chargediffusivity product, Dz = 1×10−10 m2/s, was selected in the following demonstrations while the solute charge number zi may be varied from −4 to +4. The ratio of channel length to halfheight was fixed at L/a = 104 for convenience even though we recognize that fixing the channel length might be a wiser option when the channel height is varied.

6.1. Selectivity Selectivity, rji, is defined as the ratio of the mean speeds of solutes i and j

rji = ui u j

(6-1)

and should be larger than 1 as traditionally defined [43]. A larger rji indicates a better separation. Figure 11 compares the selectivity, rji, of (a) positive and (b) negative solutes in nanochannel chromatography (solid lines) and nanochannel electrophoresis (dashed lines). It is important to note that the indices of rji, which indicate the charge values of the two solutes to be separated, are switched between positive and negative solutes in order that rji > 1 as traditional defined [43]. Specifically, we use r21, r32, and r43 for positive solutes (or more generally, solutes with ziζ* < 0) as those with higher charges are concentrated in a region of smaller fluid speed (i.e., closer to the channel wall) and thus move slower. Note that the solute electrophoretic mobility has been assumed to remain unvaried. In contrast, negative solutes (or solutes with ziζ* > 0) with higher charges appear predominantly in the region of larger fluid speed (closer to the channel center) and thus move faster. Therefore, we need to use r12, r23, and r34 for negative solutes. This index switch also applies to the resolution, Rji, which will be illustrated in Figure 12.


4

21

(a)

Selectivity, rji

r43 3 r32

2

r43

r21 r32 r21

1 0.1

1

10

100

κa

Selectivity, rji

1.2

(b)

r12

1.1

r23 r12

r34

r23 r34

1.0 0.1

1

10

100

κa Figure 11: Selectivity, rji, of (a) positive and (b) negative solutes in nanochannel chromatography (solid lines) and nanochannel electrophoresis (dashed lines). Reprinted with permission from [19].

One can see in Figure 11a that the selectivity, rji, of positive solutes in nanochannel chromatography is always greater than that of the same pair of solutes in nanochannel electrophoresis. This discrepancy gets larger when the solute charge number zi increases. Meanwhile, the optimal κa value at which rji is maximized increases for both chromatography and electrophoresis though it is always smaller in the former case. The discrepancy between

22

Xiangchun Xuan

these two optimal κa also increases with the rise of zi. For negative solutes, Figure 11b shows a significantly lower rji than positive solutes in nanochannel chromatography. Moreover, rji decreases when the solute charge number increases. The optimal κa at which rji is maximized is also smaller than that for positive solutes, and decreases (but only slightly) with zi. All these results apply equally to rji of negative solutes in nanochannel electrophoresis except at around κa = 0.6 where rji varies rapidly with κa. Within this region of κa, the electrophoretic velocity of negative solutes is close to the fluid electroosmotic velocity [more accurately, uie in Eq. (4-7)] while in the opposite direction. Therefore, the real solute speed is essentially so small that even a trivial difference in the solute speed (essentially the difference in uie as the solute electrophoretic velocity is constant due to the fixed charge-to-size ratio) could yield a large rji. It is, however, important to note that the speed of negative solutes could be reversed in nanochannel electrophoresis when κa is less than a threshold value (e.g., κa = 0.6 in Figure 11b). In other words, solutes migrate to the anode side instead of the cathode side along with the electrolyte solution. In such circumstances, it is very likely that only one of the two solute species migrates toward the detector no matter the detector is placed in the cathode or the anode side of the channel. Another consequence is that the maximum rji in nanochannel electrophoresis might be achieved with a fairly long analysis time, which makes the separation practically meaningless. We therefore expect that solutes with a constant electrophoretic mobility can be better separated in nanochannel chromatography than in nanochannel electrophoresis. Moreover, solutes with ziζ* < 0 can be separated more easily than can those with ziζ* > 0.

6.2. Plate Height Plate height, Hi, is the spatial variance of the solute peak distribution,

σ i2 , divided by the

migration distance, L, within a time period of ti. It is often expressed in the following dimensionless form of a reduced plate height, hi [42,45]

H i σ i2 2 Di′ti 2 Di′ = = = a aL aL au i

(6-2)

Di′ = Di + K i = Di (1 + χ i Pei2 )

(6-3)

hi =

where Di′ is the effective diffusion coefficient due to a combination of hydrodynamic dispersion Ki [see Eq. (5-4)] and molecular diffusion Di. Note that other sources of dispersion such as injection and detection (refer to [46,52,53] for detail) have been neglected for simplicity. Following Griffiths and Nilson’s analysis [12], we may combine Eqs. (6-2) and (6-3) to rewrite the reduced plate height as


hi = 2(1 Pei + χ i Pei )

23 (6-4)

Therefore, hi attains its minimum

hi ,min = 4 χ i at Pei ,opt = 1

χi

(6-5)

In other words, there exists an optimal value for the mean solute speed, ui ,opt = Di a

χi ,

and thus an optimal electric field in nanochannel electrophoresis or an optimal pressure gradient in nanochannel chromatography, at which the separation efficiency is maximized. As hi is a function of solely the dispersion coefficient χi that has been demonstrated in Figures 7 and 8 for nanochannel electrophoresis and chromatography, respectively, its variations with respect to zi and κa are not repeated here for brevity.

6.3. Resolution Resolution, Rji, can be defined in two different ways: the one introduced by Giddings [54], i.e., Eq. (6-6), and the one adopted by Huber [55] and Kenndler et al. [56-58], i.e., Eq. (6-7),

R ji =

t j − ti

(6-6)

2(σ t ,i + σ t , j )

R ji =

t j − ti

(6-7)

σ t ,i

where t is the migration time as defined in Eq. (6-2) and σt is the standard deviation of solute peak distribution in the time domain. Consistent with the solute selectivity rji, a larger value of Rji indicates a better separation. Substituting ti = L ui , t j = L u j and σ t ,i = σ i ui into the last equation leads to

R ji =

L

σi

(r

ji

− 1) =

La hi

(r

ji

− 1)

(6-8)

Referring back to Eq. (6-5), it is straightforward to obtain

R ji ,max =

La hi ,min

(r

ji

− 1) at Pei ,opt = 1

χi

(6-9)

24

Xiangchun Xuan

because the selectivity rji is independent of the solute Peclet number. Therefore, when the plate height of one type of solute is minimized, the separation resolution of this solute from another type of solute may reach the maximum value. Figure 12 compares the maximum resolution, Rji,max, of positive and negative solutes (as labeled) in nanochannel chromatography (solid lines) and nanochannel electrophoresis (dashed lines). The indices of Rji,max are assigned following those of the selectivity, rji, in Figure 11, to ensure Rji,max > 0. One can see that Rji,max of positive solutes in chromatography is larger than that of negative ones throughout the range of κa. In electrophoresis, the former also yield a better resolution if κa > 1. When κa < 1, Rji,max of negative solutes increases and reaches the extremes at κa = 0.6 due to the sudden rise in the selectivity (refer to Figure 11b) as explained above. Within the same range of κa, Rji,max of positive solutes continues decreasing when κa decreases and thus becomes smaller than that of negative solutes. Interestingly, chromatography and electrophoresis offer a comparable resolution for both types of solutes in nanoscale channels if κa > 1.

Positive solutes

Maximum resolution, Rji,max

100

R32

R32

R43

R43 R21 R21

10

R12 Negative solutes

R23 R34

1 0.1

1

10

100

κa Figure 12. Maximum resolution, Rji,max, of positive and negative solutes in nanochannel chromatography (solid lines) and nanochannel electrophoresis (dashed lines). Reprinted with permission from [19].

It is also noted in Figure 12 that the optimum channel size for both separation methods appears to be 1 < κa < 10. In other words, the best channel half-height for solute separation in nanochannels will be 10 nm < a < 100 nm if 1 mM electrolyte solutions are used. In this context, the optimum Peclet number to achieve the maximum resolution in a channel of κa = 5 (or a ≈ 50 nm) will be Pei,opt = O(4) because hi,min = O(1). Although this Peclet number (corresponding to the mean solute speed of the order of 8 mm/s) seems a little too high in current nanofluidics, it indicates that large fluid flows are preferred in both nanochannel


25

chromatography and nanochannel electrophoresis for high throughputs and separation efficiencies.

7. Conclusion We have developed an analytical model to study the transport, dispersion and separation of solutes (both charged and non-charged) in electroosmotic and pressure-driven flows through nanoscale slit channels. This model explains why solutes can be separated by charge in nanochannels, and provides compact formulas for calculating the migration speed and hydrodynamic dispersion of solutes. It also presents a simple approach to optimizing the separation performance in nanochannels, which has been applied particularly to solutes with a similar electrophoretic mobility. In addition, we would like to point out that the model or the approach developed in this work can be readily extended to one-dimensional round nanotubes [12,15-17,19] and to even two-dimensional rectangular nanochannels [7,59,60].

References [1] Prakash, S.; Piruska, A.; Gatimu, E. N.; Bohn, P. W. et al. IEEE Sensor. J. 2008, 8, 441450. [2] Abgrall, P.; and Nguyen, N. T. Anal. Chem. 2008, 80, 2326-2341. [3] Schoh, R. B.; and Han, J. Y.; Renaud, P. Rev. Modern Phys. 2008, 80, 839-883. [4] Pennathur, S.; and Santiago, J. G. Anal. Chem. 2005, 77, 6772-6781. [5] Petsev, D. N. J. Chem. Phys. 2005, 123, 244907. [6] Xuan, X.; and Li, D. Electrophoresis 2006, 27, 5020-5031. [7] Dutta, D. J. Colloid. Interf. Sci. 2007, 315, 740-746. [8] Pennathur, S.; Baldessari, F.; Santiago, J. G.; Kattah, M. G. et al. Anal. Chem. 2007, 79, 8316-8322. [9] Das, S.; and Chakraborty, S. Electrophoresis, 2008, 29, 1115-1124. [10] Pennathur, S.; and Santiago, J. G. Anal. Chem. 2005, 77, 6782-6789. [11] Garcia, A. L.; Ista, L. K.; Petsev, D. N. et al. Lab Chip 2005, 5, 1271-1276. [12] Griffiths, S. K.; and Nilson, R. N. Anal. Chem. 2006, 78, 8134-8141. [13] Xuan, X.; and Li, D. Electrophoresis 2007, 28, 627-634. [14] Wang, X.; Kang , J.; Wang, S.; Lu, J.; and Liu, S. J Chromatography A 2008, 1200, 108-113. [15] Xuan, X. J. Chromatography A 2008, 1187, 289-292. [16] De Leebeeck, A.; and Sinton, D. Electrophoresis 2006, 27, 4999-5008. [17] Xuan, X.; and Sinton, D. Microfluid. Nanofluid. 2007, 3, 723-728. [18] Xuan, X. Anal. Chem. 2007, 79, 7928-7932. [19] Xuan, X. Electrophoresis 2008, 29, 3737-3743. [20] Israelachvili, J. Intermolecular & Surface Forces, Academic Press, 2nd edition, San Diego, CA, 1991. [21] Burgreen, D., and Nakache, F. R., J. Phys. Chem. 1964, 68, 1084-1091. [22] Rice, C. L. and Whitehead, R., J. Phys. Chem. 1965, 69, 4017-4024. [23] Hildreth, D., J. Phys. Chem. 1970, 74, 2006-2015.

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[24] Li, D. Colloid. Surf. A 2001, 191, 35-57. [25] Hunter, R. J. Zeta potential in colloid science, principles and applications, Academic Press, New York, 1981. [26] Qu, W.; and Li, D. J. Colloid and Interface Sci. 2000, 224, 397-407. [27] Taylor, J.; and Ren, C. L. Microfluid. Nanofluid. 2005, 1, 356-363. [28] Stein, D.; Kruithof, M.; and Dekker, C. Phys. Rev. Lett. 2004, 93, 035901. [29] Karnik, R.; Fan, R.; Yue, M. et al. Nano Lett. 2005, 5, 943-948. [30] Fan, R.; Yue, M.; Karnik, R. et al. Phys. Rev. Lett. 2005, 95, 086607. [31] Van der Heyden, F. H.J.; Stein, D.; and Dekker, C. Phys. Rev. Lett. 2005, 95, 116104. [32] Van Der Hayden, F. H. J.; Bonthius, D. J.; Stein, D.; Meyer, C.; and Dekker, C. Nano Lett. 2007, 7, 1022-1025. [33] Probstein, R. F. Physicochemical hydrodynamics, John Willey & Sons, New York, 1995. [34] Davidson, C., and Xuan, X., Electrophoresis 2008, 29, 1125-1130. [35] Morrison, F. A.; and Osterle, J. F. J. Chem. Phys. 1965, 43, 2111-2115. [36] X. Xuan, and D. Li, J. Power Source, 156 (2006) 677-684. [37] Griffiths, S. K.; and Nilson, R. H. Electrophoresis 2005, 26, 351-361. [38] Xuan, X.; and Li, D J Micromech. Microeng. 2004, 14, 290-298. [39] Li, D., Electrokinetics in Microfluidics, Elsevier Academic Press, Burlington, MA 2004. [40] Xuan, X. Microfluid. Nanofluid. 2008, 4, 457-462. [41] Taylor, G. I. Proc. Roy. Soc. London A 1953, 219, 186-203. [42] Aris, R. Proc. Roy. Soc. London A 1956, 235, 67-77. [43] Giddings, J. C. Unified separation science, John Wiley & Sons, Inc., New York, 1991. [44] Martin, M.; Giddings, J. C. J. Phys. Chem. 1981, 85, 727-733. [45] Dutta, D.; Ramachandran, A.; and Leighton, D. T. Microfluid. Nanofluid. 2006, 2, 275-290. [46] Ghosal, S., Annu. Rev. Fluid Mech. 2006, 38, 309-338. [47] Griffiths, S. K., Nilson, R. H. Anal. Chem. 1999, 71, 5522-5529. [48] Griffiths, S. K., and Nilson, R. N., Anal. Chem. 2000, 72, 4767-4777. [49] Datta, R. Biotechnol. Prog. 1990, 6, 485-493. [50] Datta, R.; and Kotamarthi, V. R AICHE J. 1990, 36. 916-926. [51] Wooding, R. A., J. Fluid. Mech. 1960, 7, 501-515. [52] Gas, B.; Stedry, M.; and Kenndler, E. Electrophoresis 1997, 18, 2123-2133. [53] Gas, B.; Kenndler, E. Electrophoresis 2000, 21, 3888-3897. [54] Giddings, J. C. Sep. Sci. 1969, 4, 181-189. [55] Huber, J. F. K. Fresenius' Z. Anal. Chem. 1975, 277, 341-347. [56] Kenndler, E. J Cap. Elec. 1996, 3, 191-198. [57] Schwer, C.; and Kenndler, E. Chromatographia 1992, 33, 331-335. [58] Kenndler, E.; and Fridel, W. J Chromatography 1992, 608, 161-170. [59] Dutta, D. Electrophoresis 2007, 28, 4552-4560. [60] Dutta, D. Anal. Chem. 2008, 80, 4723-4730.



Chapter 2

H2O IN THE MANTLE: FROM FLUID TO HIGH-PRESSURE HYDROUS SILICATES N.R. Khisina1,*, R. Wirth2 and S. Matsyuk3 1

Institute of Geochemistry and Analytical Chemistry of Russian Academy of Sciences, Kosygin st. 19, 119991 Moscow, Russia 2 GeoForschungZentrum Potsdam, Germany 3 Institute of Geochemistry, Mineralogy and Ore Formation, National Academy of Sciences of Ukraine, Paladin Ave., 34, 03680 Kiev-142, Ukraine

Abstract Infrared spectroscopic data show that nominally anhydrous olivine (Mg,Fe)2SiO4 contains traces of H2O, up to several hundred wt. ppm of H2O (Miller et al., 1987; Bell et al., 2004; Koch-Muller et al., 2006; Matsyuk & Langer, 2004) and therefore olivine is suggested to be a water carrier in the mantle (Thompson, 1992). Protonation of olivine during its crystallization from a hydrous melt resulted in the appearance of intrinsic OH-defects (Libowitsky & Beran, 1995). Mantle olivine nodules from kimberlites were investigated with FTIR and TEM methods (Khisina et al., 2001, 2002, 2008). The results are the following: (1) Water content in xenoliths is lower than water content in xenocrysts. From these data we concluded that kimberlite magma had been saturated by H2O, whereas adjacent mantle rocks had been crystallized from water-depleted melts. (2) Extrinsic water in olivine is represented by highand hydrous olivine pressure phases, 10Å-Phase Mg3Si4O10(OH)2.nH2O n(Mg,Fe)2SiO4.(H2MgSiO4), both of which belong to the group of Dense Hydrous Magnesium Silicates (DHMS), which were synthesized in laboratory high-pressure experiments (Prewitt & Downs, 1999). The DHMS were regarded as possible mineral carriers for H2O in the mantle; however, they were not found in natural material until quite recently. Our observations demonstrate the first finding of the 10Å-Phase and hydrous olivine as a mantle substance. (3) 10Å-Phase, which occurred as either nanoinclusions or narrow veins in olivine, is a ubiquitous nano-mineral of kimberlite and closely related to olivine. (4) There are two different mechanisms of the 10Å-Phase formation: (a) purification of olivine from OHbearing defects resulting in transformation of olivine to the 10 Å-Phase with the liberation of water fluid; and (b) replacement of olivine for the 10Å-Phase due to hydrous metasomatism in the mantle in the presence of H2O fluid. *

E-mail address: [email protected]

28

N.R. Khisina, R. Wirth and S. Matsyuk

Introduction The presence of water in the mantle, either as H2O molecules or OH- groups, has been the subject of long-term interest in geochemistry and geophysics because of the dramatic H2O influence on the melting and the physical properties of mantle rocks. Nowdays the concept of “wet” and heterogeneous mantle is universally accepted among petrologists (Kamenetsky et al., 2004; Sobolev & Chaussidon, 1996; Katayama et al., 2005). However, which minerals could serve as deep water storage in the mantle is yet to be a widely-discussed topic. The problem is that the mantle material available for a direct investigation is very limited and restricted by mantle nodules trapped by kimberlitic or basaltic magma from the depth. Among the presumed candidates considered for water storage in the mantle were the so-called DHMS phases (dense hydrous magnesium silicates) synthesized in laboratory experiments at P-T conditions of the mantle (Prewitt & Downs, 1999); however, DHMS phases have not yet been found as macroscopic minerals in mantle material. The main mineral of the mantle is olivine (Mg,Fe)2SiO4, which is stable at mantle pressures up to ~ 15 GPa; with increasing pressure the olivine ( α-phase) transforms to higher density structure of wadsleite (β-phase). The P-Tconditions of α-β transition in olivine correspond to the depth of ~ 400 km; according to geophysical data, the mantle has a discontinuity at this depth, specified as a boundary between the upper mantle and transition zone. Infrared spectroscopic data show that olivine contains traces of H2O up to several hundreds wt. ppm of H2O (Bell et al., 2003; Koch-Müller et al., 2006; Kurosawa et al., 1997; Matsyuk & Langer, 2004; Miller et al., 1987); therefore, nominally anhydrous olivine is considered as water storage in the mantle (Thompson, 1992). The highest water content, such as about 400 wt. ppm of H2O, was registered for olivine samples from mantle peridotite nodules in kimberlites. Water in olivine occurs as either OHor H2O. There are two modes of “water” occurrence in olivine: intrinsic and extrinsic. An intrinsic mode represents the OH- incorporated into the olivine structure and is considered a water-derived defect complex either associated with a metal vacancy {vMe, 2OH-} or by a vacancy at the Si site {vSi,4OH-} (Beran & Putnis, 1983; Beran & Libovitzky, 2006; Lemaire et al., 2004). Extrinsic water is possessed by inclusions, either solid or fluid, and occurs as OH- or H2O. Recent TEM investigations of olivine nodules from Udachnaya kimberlite (Yakuyia) revealed the nanoinclusions of hydrous magnesium silicates represented by highpressure phases, 10Å-Phase and hydrous olivine, partially replaced by low-pressure serpentine + talc assemblage (Wirth & Khisina, 1998; Khisina et al., 2001; Khisina & Wirth, 2002, 2008a, 2008b). 10Å-Phase and hydrous olivine belong to the group of DHMS phases. This first finding of DHMS phases in mantle material (Wirth & Khisina, 1998; Khisina et al., 2001; Khisina & Wirth, 2002, 2008a, 2008b) provides direct evidence of the DHMS occurrence in the mantle and specifies them as nanominerals of non-magmatic origin, closely related to olivine. The amount of water incorporated into the olivine structure depends on the P-T conditions as well as on the chemical environment and olivine composition (Fe/Mg ratio in olivine), and increases with increasing water activity, oxygen fugacity, pressure and temperature (Kohlstedt et al., 1996; Bai & Kohlstedt, 1993; Zhao et al., 2004). Diffusion of hydrogen in olivine is very fast (Kohlstedt & Mackwell, 1998); therefore, due to interaction of olivine and surrounding melt the initial water content in olivine can be changed under changes of either P, or T, or fO2 or fH2O during a post-crystallization stage. Olivine remained

H2O in the Mantle: From Fluid to High-Pressure Hydrous Silicates

29

in a host melt can loose water due to decompression; olivine trapped by a foreign melt can become either deprotonated or secondarily protonated, depending on whether the foreign melt is lower or higher by water concentration (by water activity) in comparison to the host melt, correspondingly. Experiments by Peslier and Luhr (2006) and Mosenfelder et al. (2006a, 2006b) demonstrate rapid processes of deprotonation and secondary protonation of olivine. FTIR and TEM data provide the information about H2O content and a mode of water occurrence in olivine. Here we suggest a way to reconstruct the P-history of the kimberlite process and elucidate the water behavior at different stages of the kimberlite process. The collected data on the H2O occurrence in mantle olivine nodules represented by xenoliths, xenocrysts and phenocrysts from Yakutian kimberlites (Udachnaya, Obnazennaya, Mir, Kievlyanka, Slyudyanka, Vtorogodnitza and Bazovayua pipes) are used here as a guide for tracing H2O behavior from fluid to high-pressure DHMS phases in the mantle. We show here that nominally anhydrous olivine is a carrier of water in the mantle and can be used as indicator of P-f(H2O) regime in the mantle.

Samples and Collected Data Sample Description Typical kimberlites сontain xenocrysts and xenoliths of mantle and crustal origin embedded into a fine- to coarse-grained groundmass of crystallized kimberlite melt (Sobolev V.S. et al, 1972; Sobolev N.V., 1974; Pokhilenko et al., 1993; Ukhanov et al., 1988; Matsyuk et al., 1995). We collected the mantle olivine samples represented by xenoliths, xenocrysts and phenocrysts from kimberlite pipes of Yakutian kimberlite province (Udachnaya, Obnazennaya, Mir, Slyudyanka, Vtorogodniza and Kievlyanka). Olivine samples are classified as xenoliths, xenocrysts and phenocrysts on the base of petrographic examination. Phenocrysts are olivine single grains disintegrated from groundmass of kimberlite rock. Xenoliths are fragments of adjacent mantle and crustal rocks trapped by kimberlite magma and transported from the depths during kimberlite eruption. Xenocrysts are olivine singlegrains disintegrated from adjacent rocks, either mantle or crustal, and lifted from the depths during kimberlite eruption. Xenoliths are of several cm in size. Xenocrysts are less than 1 cm in size and comparable by size with phenocrysts. The samples were studied with optical microscopy, FTIR and TEM.

H2O Content in the Olivine Samples The highest H2O content in the mantle olivine samples from Yakutian kimberlites was measured as 400–420 wt. ppm of H2O (Koch-Muller et al., 2006; Matsyuk & Langer, 2004). H2O contents in xenoliths from Udachnaya and Obnazennaya pipes vary between 14 and 246 wt. ppm (Table 1). Present FTIR study on the H2O content in xenoliths from Udachnaya and Obnazennaya, together with previous Infrared spectroscopic data on xenocrystic and phenocrystic olivine samples (Matsyuk and Langer, 2004; Koch-Muller et al., 2006) show the wide variation of

30


the H2O content in the olivine samples, such as between 0–3 and 400 wt. ppm.of H2O. The results are shown in histograms (Figure 1). Summarized FTIR data on the H2O content in the all studied mantle olivine samples including xenoliths, xenocrysts and phenocrysts from Yakutian kimberlites (Matsyuk and Langer, 2004; Koch-Müller et al., 2006; present study) are represented at histogram (Figure 1a). The same data is plotted individually for phenocrysts (Figure 1b) and xenoliths together with xenocrysts (Figure 1c). For comparison, the H2O contents in mantle-derived olivine megacrysts from the Monastery kimberlite, South Africa (Bell et al., 2004) are plotted on the histogram in Figure 1d. The most frequent values of H2O in olivine samples from Monastery kimberlite are 150–200 wt. ppm of H2O. Table 1. Water contents in olivine from mantle xenoliths

Ob-152

Obnazennaya pipe

Garnet lerzolite

H2O content, wt. ppm 246

Ob-105

″

Garnet lerzolite

14–65

Ob-174

″

Garnet lerzolite

143(15)

Ob-312

″

Garnet lerzolite

45

U-47/76

Udachnaya pipe

18

9206

″

9.5 cm in size megacrystalline . diamond-bearing xenolith. Harzburgite-dunite 5.2 cm in size coarse-grained xenolith. Garnet-free harzburgite

Xenolith sample

Locality

Xenolith description

12

12

8

8

4

4

0

0 0

100

200 300 H2O, wt. ppm

400

0

100

(A)

200 H2O, wt. ppm

(B) Figure 1. Continued on next page.

29

300

400

H2O in the Mantle: From Fluid to High-Pressure Hydrous Silicates 12

31

10

8

8 6

4

4 2

0

0 0

100

200 H2O, wt. ppm

300

(C)

400

0

50 100 150 200 250 H2O, wt. ppmÐ2Ùá öåþ ççüÐ2Ùá öåþ ççü

300

(D)

Figure 1. Histograms of the H2O contents in mantle olivine samples from kimberlites. a – c: The FTIR data for Yakutiyan olivine samples from Udachnaya, Obnazennaya, Mir, Vtorogodnitza, Kievlyanka and Sluydyanka kimberlites (present data; Matsyuk & Langer, 2004; Koch-Müller et al., 2006); a – xenoliths, xenocrysts and phenocrysts all together; b – phenocrysts; c – xenoliths and xenocrysts; d – Xenoliths from Monastery kimberlite, South Africa (Bell et al., 2003).

Extrinsic H2O in Olivine Samples TEM examination of the olivine samples (Khisina et al., 2001; Khisina & Wirth, 2002; Khisina et al., 2008) revealed the OH- segregation resulted in nano-heterogeneity of several types: (i) nanoinclusions; (ii) lamellar precipitates; (iii) veins developed along healed microcracks. All kinds of heterogeneity are associated with deformation slip bands in the samples understudy (Khisina et al., 2008). (i) Nanoinclusions were observed in both xenoliths and xenocrysts. They are several tens of nanometers in size and have a shape of pseudohexagonal negative crystals. The nanoinclusions are often arranged in arrays along [100], [011], [101] and [-101] crystallographic directions of the olivine host. The phase constituents of nanoinclusions were identified from TEM data (Wirth & Khisina, 1998; Khisina et al., 2001; Khisina & Wirth, 2002; Khisina et al., 2008) as 10Å-Phase Mg3Si4O10(OH)2.nH2O, where n = 0.65, 1.0, and 2.0, and hydrous olivine (MgH2SiO4).n(Mg2SiO4), both of them represent high-pressure DHMS phases (Bauer & Sclar, 1981; Khisina & Wirth, 2002; Churakov et al., 2003). 10Å-Phase is a typical constituent of nanoinclusions in xenocrysts, while hydrous olivine is more common for nanoinclusions in xenoliths. Both 10Å-Phase and hydrous olivine are replaced often by a low-pressure assemblage of serpentine Mg3Si2O5(OH)4 + talc Mg3Si4O10(OH)2. High-pressure phases in nanoinclusions as well as their replacement products are strictly aligned relative to the crystallographic directions of the olivine matrics, with aol ║ ahy ║ c10Å ║ ctc ║ cserp (hy is abbreviation of hydrous

32


Figure 2. Array of nanoinclusions in olivine xenocryst from Udachnaya kimberlite. Nanoinclusions are composed of the 10Å-Phase in the middle part of the inclusions, and filled by H2O fluid in the regions bordered the adjacent olivine (white areas of nanoinclusions).

Figure 3. Healed microcrack filled by 10Å-Phase in olivine xenolite sample from Udachnaya kimberlite.

olivine), which is indicative of the topotaxic character of these intergrowths. A characteristic feature of nanoinclusions in xenocrystic olivine samples is the presence of voids unfilled with solid material (Figure 2). Solid phase fills the equatorial area of the inclusions parallel to the (100) plane of the olivine host, and voids in nanoinclusions are observed at the polar areas bordering the olivine matrix (Figure


33

2). The observations led to the conclusion that these voids are not artifacts and were not produced during sample preparation. According to TEM observations, the nanoinclusions are not connected with the surface of the crystals through channels, which could have served as pathways for the transport of H2O from the external medium during the formation of inclusions. (ii) (100) Lamellar precipitates of hydrous olivine, several nanometers in thickness were observed in xenolithic olivine (Khisina et al., 2001; Khisina & Wirth, 2002; Churakov et al., 2003). Veins filled by 10Å-Phase + talc, were observed in xenolithic olivine (Figure 3). These veins were developed along (100) healed microcracks.

Discussion Olivine as Water Storage in the Mantle Histogram of the all H2O content measurements for olivine samples, as summarized for xenoliths, xenocrysts and phenocrysts together (Figure 1a) reveals two maximums, at 0–50 wt. ppm of H2O and 200–250 wt. ppm of H2O. The same maximums are pronounced at the histogram represented the H2O contents in xenoliths and xenocrysts together (Figure 1c), whereas only one maximum at 200–250 wt. ppm of H2O is observed at the histogram represented the H2O data for phenocrystic olivine (Figure 1b). Histograms at Figure 1b,c show that the maximum between 0–50 wt. ppm of H2O corresponds to the most frequent H2O contents in xenoliths and xenocrysts, while the maximum between 200–250 wt. ppm of H2O corresponds to the most frequent H2O values in phenocrysts. We suggest that the most frequent H2O contents correspond to the initial H2O contents incorporated by the olivine samples during crystallization. Hence, the initial H2O content in olivine derived from adjacent and kimberlite rocks has been different. We conclude that compared to xenolithic and xenocrystic samples, the olivine phenocrysts have incorporated more amount of water during crystallization. According to the experimental data on the P-f(H2O)–dependence of the OHsolubility in olivine (Kohlstedt et al, 1996), this could mean that the trapped fragments of adjacent and kimberlite rocks have been crystallized either at different depths or from different melts. On the basis of the data by Kohlstedt (1996) the pressure conditions of olivine crystallization from water-saturated melt can be estimated as < 1 GPa for xenocrysts and xenoliths and as 4–4.5 GPa for phenocrysts. The pressure of 4–4.5 GPa estimated as the pressure of crystallization of phenocrysts corresponds to the depth of 120–135 km that is consistent with the upper mantle depths and is in good agreement with estimations of the depths of kimberlite formation as between 120 and 230 km (Pochilenko et al., 1993). The pressure < 1 GPa estimated as the pressure of crystallization of mantle olivine samples represented by xenoliths and xenocrysts, is less than the lowest pressures in the mantle and corresponds to the depth of <30 km that is consistent with crustal depths; this is not compatible with petrographic identification of these samples as derived from mantle rocks. Therefore the difference between mantle xenoliths (and xenocrysts) on the one hand, and phenocrysts on the other hand, in respect of initial water contents can not be explained by different depths of their crystallization. Consequently, this led us to the conclusion that the

34


mantle adjacent rocks and the kimberlites have been crystallized from different melts, watersaturated and water depleted, respectively.

Post-Crystallization H2O Behavior in Olivine Post-crystallization H2O behavior in the olivine samples is pronounced mostly in xenoliths and xenocrysts and manifests itself in the following processes. (i) Secondary protonation of olivine samples derived from adjacent mantle rocks. Secondary protonation of the olivine derived from adjacent mantle rocks could take place at the contact of adjacent rocks and the parent kimberlite magma chamber. Interaction of adjacent rocks with water-saturated parent kimberlite melt resulted in H2O enrichment up to 150–200 wt. ppm in olivine (Figure 1). (ii) Internal re-distribution of H2O within the olivine host resulted in nucleation of nanoinclusions consisted of high-pressure DHMS phases represented by10Å-Phase and hydrous olivine, with the H2O fluid separation from the solid inside of the nanoinclusions. (iii) Subsequent replacement of DHMS by low-pressure assemblage of serpentine + talc in nanoinclusions. (iv) Olivine hydration due to interaction of adjacent rocks with the H2O fluid collected in kimberlite magma chamber; this resulted in the olivine replacement by 10Å-Phase along microcracks; (v) Subsequent dehydration of the 10Å-Phase developed along micrcracks in olivine.

H2O Fluid in Kimberlite Melt During a crystallization of early generation of olivine from the parent water-saturated kimberlite melt in the magma chamber the separation of H2O fluid from the melt occurred that led to the internal pressure increase in the kimberlite magma chamber. The bulk H2O contents in the olivine phenocrystic samples vary between150 and 300 wt. ppm about the most frequent content as 200–250 wt. ppm of H2O. This variations may indicate that the pressure in kimberlite magma chamber has been oscillated with time by magnitude between at least of 3 GPa and 5.2 GPa because of the alternating processes of pressure increasing due to the H2O fluid separation and accumulation in the chamber followed then by decompression during chamber expansion and H2O fluid liberation. The pressure oscillation with time took place in kimberlite chamber prior to explosion, and resulted in the deformation of olivine and subsequent appearance of DHMS nanoinclusions associated with deformation slip bands in olivine. The alteration of adjacent rocks took place at the contact with the kimberlite magma chamber before the explosion and eruption of kimberlite, and resulted in (i) secondary protonation of olivine due to interaction with water-saturated kimberlite melt and (ii) metasomatic hydration of olivine resulted in the olivine replacement by the 10Å-Phase due to interaction of the olivine with H2O fluid. The reaction of hydration of olivine may be written as 4(Mg,Fe)2SiO4 + 3H2O(fluid) = Mg3Si4O14H6 + 5(Fe,Mg)O

(1)


35

The kimberlite explosion happened because the internal pressure increased as the H2O fluid separated from the melt inside of the kimberlite magma chamber. This event resulted when the pressure corresponded to the highest H2O content in olivine phenocrysts measured as 400–420 wt. ppm (Figure 1) attained at least 6.5 GPa, as it may be estimated from the pressure dependence on the H2O content in olivine (Kohlstedt et al., 1996). The subsequent eruption and rapid lifting of the mantle material from the depth of 120-135 km was accompanied by destruction of adjacent rocks through the flow path. The pressure increasing inside the chamber prior to explosion followed then by rapid decompression during eruption, when the 10Å-Phase substituted for talc Mg3Si4O10(OH)2 (low-pressure phase) with H2O fluid liberation: Mg3Si4O14H6 = Mg3Si4O10(OH)2 + H2O (fluid)↑

(2)

According to Chinnery et al. (1999), dehydration of 10Å-Phase (reaction 2) proceeds at pressures lower than 3.5–5.2 GPa at T <700°C.

OH-Bearing Nanoinclusions and Intracrystalline H2O Fluid The nucleation of OH-bearing nanoinclusions could be initiated by deformation of olivine that is evident from the similar arrangement of arrays of nanoinclusions and the optically visible slip bands (Khisina et al., 2008), both along common crystallographic directions in the olivine host. The slip bands in olivine could be produced by the internal pressure progressively increasing in the kimberlite magma chamber due to separation of the H2O fluid from the magma melt. Slip bands are zones of high density of dislocations in a crystal and, consequently, contain relatively high concentration of OH-bearing point defects. Nanoinclusions composed of DHMS phases could be nucleated within the former slip bands under subsequent decompression and cooling. Further, the DHMS phases were substituted by a low-pressure serpentine + talc assemblage. The question is what was a driving force for the exsolution of high-pressure hydrous silicate phases from the host olivine that occurs within slip bands. If it is assumed that the process of self-purification of olivine from water took a place under decompression, then how to explain that this process resulted in creation of high-pressure phases? Are they in equilibrium with the olivine host? In order to explain the mechanism and reactions of formation and further transformation of the primary high-pressure phases, 10Å-Phase and hydrous olivine, to the low-pressure serpentine + talc assemblage, we need to understand the nature of voids in the nanoinclusions (Figure 2), the source of H2O, and the mechanism of formation of nanoinclusions themselves. TEM observations show the topotaxic relationships of the olivine/10Å-Phase intergrowth, the exact Mg/Si stoichiometric ratio and the lack of channels connecting the inclusions with the grain surface. Therefore we propose that no fluid infiltration occurred, and the nanoinclusions observed in olivine samples might have formed due to internal isochemical process within a crystal. Consequently, a crystal should have been water saturated prior to the nucleation of inclusions. We suggest that nanoinclusions in olivine have been formed at the post-crystallization stage under the pressure oscillation in the kimberlite chamber and the formation of the nanoinclusions proceed through the diffusion and segregation of intrinsic OH-bearing point defects, i.e., without any infiltration from outside. Diffusion experiments

36


have shown that hydrogen can rapidly diffuse through olivine at temperatures >800°C (Kohlstedt & Mackwell, 1998). The diffusion process could result in an aggregation of the OH- and associated metal vacancies within the OH-saturated slip bands of the olivine host. We consider the formation of OH-bearing nanoinclusions in olivine as a process of selfpurification of olivine from the OH-bearing defects by the coupled diffusion mechanism Mg2+ ═ {vMg,2H+}.With the assumption that the cation composition of the inclusion corresponds to the stoichiometry of the 10Å Phase (Mg/Si = 3:4), the reaction of the nanoinclusion formation by unmixing of OH-bearing olivine solid solution is the following one: OH-olivine (s.s.) = [stoichiometric olivine](matrix) + [10Å-Phase + H2O(fluid)](inclusion) (3a) or 5y[Mg2-xvxSiO4H2x] = (5y-4yx)[Mg2SiO4](matrix) + yx[Mg3Si4O14H6 + 2H2O](inclusion) (3b) Taking into account the variations of the 10Å-Phase chemical composition with respect to H2O content (Bauer & Sclar, 1981; Chinnery et al., 1999), the bracketed part of the rightside of the reaction (3b) should be more correctly presented as Mg3Si4O10(OH)2.nH2O + (4n)H2O, where n is 0.65; 1.0; 2.0. Reaction (3) occurs in a closed volume limited by the size of the inclusion. The plausibility of the proposed model for the nanoinclusion formation can be checked by the criterion of volume conservation, i.e., in reaction (3) the volume of reactants must be equal to the volume of products. The volume of the inclusion can be expressed as a function of the molar volume of olivine as 4yx[Vu.c.(Ol)/Z].N, and the volume occupied by the 10Å-Phase formed in reaction (3) is yx[Vu.c.(10Å-Phase)/Z].N, where N is Avogadro’s number, Vu.c. is a unit cell volume and Z is a number of chemical formula units per unit cell. The difference of these volumes is ΔV = 52yxN under normal P-T-conditions. With the assumption that the volume ΔV is filled with the H2O fluid released owing to reaction (3), we calculated the density of the fluid by the equation ρ = M/Vmol, where Vmol is the molar volume per one molecule of H2O in the inclusion, which equals ΔV/2yx in accordance with

Figure 4. Fluid inclusion in olivine xenolith (Bullfontain kimberlite, South Africa). 10Å-Phase is observed as a shell of the inclusion at the contact between the inclusion and olivine matrix.


37

reaction (3), and M is the formula weight of H2O (M=18). For n of 0.65, 1.0 and 2.0 in the chemical formula of the 10Å-Phase, the number of formula units of H2O fluid is 3.35, 3.0, and 2.0, and the calculated fluid density ρ is approximately 1.9, 1.7, and 1.1 g/cm3, respectively. The value ρ=1.1 g/cm3 corresponding to the highest H2O content in the 10ÅPhase (n=2) is most consistent with the density of water under normal P-T-conditions (ρH2O = 1.0 g/cm3), which allows consideration of the proposed model of inclusion formation as rather realistic at n=2. It should be mentioned here that the 10Å-Phase together with H2O fluid was also observed in the shell-rim of fluid inclusion occurring in the mantle olivine sample from Bullfontain kimberlite (South Africa) (Figure 4). In this case, the 10Å-Phase was most likely created due to recrystallization of the trapped fluid inclusion.

Conclusion The collected data show that xenoliths, xenocrysts and phenocrysts from kimberlite represent specimens of the so-called “wet” olivine; xenoliths and xenocrysts have a number of common features, such as frequently occurring deformation slip bands and the OH-bearing nanoinclusions that are filled by high- and low-pressure hydrous magnesium silicates together with the H2O fluid. The observations show that the 10Å-Phase is closely related to olivine in both kimberlite and adjacent rocks and, therefore, the 10Å-Phase could be considered ubiquitous nanomineral of kimberlites marking at about 4–5 GPa a certain stage of the kimberlite process. On the other hand, xenoliths and xenocrysts differ from phenocrysts by the water bulk content, as well as by the mechanism of the 10Å-Phase formation. On the basis of collected FTIR data, the xenoliths and xenocrysts on one hand, and phenocrysts on the other hand, are considered crystallized from different melts, such as water-depleted magma and watersaturated kimberlite melt, correspondingly. The pressure regime inside the kimberlite magma chamber was controlled by H2O fluid, with pressure oscillation with time between of at least 3 GPa and 5.2 GPa due to alternating processes of pressure increasing during the H2O fluid separation and accumulation in the chamber followed then by decompression due to chamber expansion and H2O fluid liberation. The released H2O fluid participates in metasomatic hydration of the olivine-bearing adjacent mantle rocks, resulting in the olivine replacement for the 10Å-Phase.

Reviewed by Prof. O. Lukanin Institute of Geochemictry and Analytical Chemistry, Russian Academy of Sciences, Kosygin st. 19, 119991 Moscow Russia

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References [1] Bai, Q., and Kohlstedt, D.L. (1993): Effects of chemical environment on the solubility and incorporation mechanism for hydrogen in olivine. Phys. Chem. Minerals, 19, 460471. [2] Bauer, J.F., and Sclar, C.B. (1981): The 10 Å Phase in the system MgO-SiO2-H2O. Am. Mineral., 66, 576-585. [3] Bell, D.R., Rossman, G.R., Maldener, J., Endisch, D., and Rauch, F. (2003): Hydroxide in olivine: a quantitative determination of the absolute amount and calibration of the IRSpectrum. J. Geophys. Res., 108, B2:2105-2113. [4] Beran, A., and Putnis, A. (1983): A model of the OH positions in olivine derived from infrared-spectroscopic investigations. Phys. Chem. Minerals, 9, 57-60. [5] Beran A., and Libowitzky E. (2006). Water in natural mantle minerals II: olivine, garnet and accessory minerals. In: Water in nominally anhydrous minerals. (H.Klepper and J.R.Smyth ed.) Rev. in Mineralogy and Geochemistry, Geochemical Soc. and Mineralogical Soc. of America, v.62, p.169-191. [6] Chinnery N.J., Pawley, A.R., and Clark S.M. (1999): In situ observation of 10 Å Phase from talc + H2O at mantle pressures and temperatures. Science, 286, 940-942. [7] Churakov, S.V., Khisina, N.R., Urusov, V.S., and Wirth, R. (2003): First-principles study of (MgH2SiO4).n(Mg2SiO4) hydrous olivine structures. 1. Crystal structure modelling of hydrous olivine Hy-2a (MgH2SiO4).3(Mg2SiO4). Phys. Chem. Minerals, 30, 1-11. [8] Kamenetsky M.B., Sobolev A.V., and Kamenetsky V.S. (2004) Kimberlite melts rich in alkali chlorides and carbonates: a potent metasomatic agent in the mantle. Geology, 32, 10, 845 – 848. [9] Katayama I., Karato S-i., and Brandon M. Evidence of high water content in the deep upper mantle inferred from deformation microstructures. (2005): Geology, 33, 613-616. [10] Khisina, N.R., and Wirth, R. (2002): Hydrous olivine (Mg1-yFe2+y)2-xvxSiO4H2x – a new DHMS phase of variable composition observed as sized-sized precipitations in mantle olivine: Phys. Chem. Minerals, 29, 98-11. [11] Khisina, N.R., Wirth, R., Andrut, M., and Ukhanov, A.V. (2001): Extrinsic and intrinsic mode of hydrogen occurrence in natural olivines: FTIR and TEM investigation. Phys. Chem. Minerals, 28, 291-301. [12] Khisina N.R., and Wirth R. (2008) Nanoinclusions of high-pressure hydrous silicate, Mg3Si4O10(OH)2.nH2O (10Å-Phase), in mantle olivine: mechanisms of formation and transformation. Geochemistry International, 46, 4, 319 – 327. [13] Khisina N.R., Wirth R., Matsyuk S., and Koch-Müller M. (2008) Microstructures and OH-bearing nano-inclusions in “wet” olivine xenocrysts from Udachnaya kimberlite. Eur. J. Mineral. (in press), [14] Koch-Müller, M., Matsyuk, S.S., Rhede, D., Wirth, R., and Khisina, N. (2006): Hydroxyl in mantle olivine xenocrysts from the Udachnaya kimberlite pipe. Phys. Chem. Minerals, 33, 276-287. [15] Kohlstedt, D.L., Keppler, H., and Rubie, D.C. (1996): Solubility of water in the α, β and γ phases of (Mg,Fe)2SiO4. Contrib. Mineral. Petrol., 123, 345-357.


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[16] Kurosawa, M., Yurimoto, H., and Sueno, S. (1997): Patterns in the hydrogen and trace element compositions of mantle olivines. Phys. Chem. Minerals, 24, 385-395. [17] Lemaire C., Kohn S.C., and Brooker R.A. (2004): The effect of silica activity on the incorporation mechanisms of water in synthetic forsterite: a polarized infrared spectroscopic study. Contrib. Mineral. Petrol. 147, 48-57. [18] Matsyuk, S.S. (1995): Olivine macrocrysts from kimberlites of Yakutiya. Mineralogicheski journal, 17, 3, 44-57. (in Russian). [19] Matsyuk, S.S., and Langer, K. (2004): Hydroxyl in olivines from mantle xenoliths in kimberlites of the Siberian platform. Contrib. Mineral. Petrol., 147, 413-437. [20] Miller, G.H., Rossman, G.R., and Harlow, G.E. (1987): The natural occurrence of hydroxide in olivine. Phys. Chem. Minerals, 14, 461-472. [21] Peslier, A.N., and Luhr, J.F. (2006): Hydrogen loss from olivines in mantle xenoliths from Simcoe (USA) and Mexico: mafic alkalic magma ascent rates and water budget of the sub-continental lithosphere. Earth and Planetary Science Letters, 242, 302-319. [22] Pokhilenko, N.P., Sobolev, N.V., Boyd, F.R., Pearson, D.G., and Shimizu, N. (1993): Megacrystalline pyrope peridotites in the lithosphere of the Siberian Platform: mineralogy, geochemical peculiarities and the problem of their origin. Geol. Geophys., 34/1, 50-62. (in Russian). [23] Prewitt, C.T., and Downs, R.T. (1999): High-pressure crystal chemistry. in J. Hemley (Ed), Rev. Mineral., vol. 37, pp. 301-312. [24] Sobolev, V.S., Dobretsov, N.L., and Sobolev N.V. (1972): Classification of deep seated xenoliths and the type of the upper mantle. Geol. Geophys., 12, 37-42 (in Russian). [25] Sobolev, N.V. (1974): Deep-seated inclusions in kimberlites and the problem of the composition of the upper mantle. Nauka, Novosibirsk. 264 pp. (in Russian). [26] Sobolev A.V., and Chaussidon M. (1996) H2O concentrations in primary melts from supra-subduction zones and mid-ocean ridges: Implications for H2O storage and recycling in the mantle. Earth and Planetary Sci. Lett., 137, 45 – 55. [27] Thompson, A.B. (1992): Water in the Earth upper mantle. Nature, 358, 295-302. [28] Ukhanov, A.V., Ryabchikov, I.D., and Kharkiv, A.D. (1988): Lithospheric mantle of the Yakutiya kimberlite province . Nauka, Moscow, 286 pp. [29] Wirth, R., and Khisina, N.R. (1998): OH-bearing crystalline inclusions in olivine from kimberlitic peridotite (Udachnaya-East, Yakutiya). Suppl. EOS Trans., 79,45, T32B-17. [30] Zhao, Y.H., Ginsberg, S.B., and Kohlstedt, D.L. (2004): Solubility of hydrogen in olivine: dependence on temperature and iron content. Contrib. Mineral. Petrol., 147, 155-161.



Chapter 3

ON THE NUMERICAL SIMULATION OF TURBULENCE MODULATION IN TWO-PHASE FLOWS K. Mohanarangam and J.Y. Tu* School of Aerospace, Mechanical and Manufacturing Engineering RMIT University, Vic. 3083, Australia

Abstract With the increase of computational power, computational modelling of two-phase flow problems using computational fluid dynamics (CFD) techniques is gradually becoming attractive in the engineering field. The major aim of this book chapter is to investigate the Turbulence Modulation (TM) of dilute two phase flows. Various density regimes of the twophase flows have been investigated in this paper, namely the dilute Gas-Particle (GP) flow, Liquid-Particle (LP) flow and also the Liquid-Air (LA) flows. While the density is quite high for the dispersed phase flow for the gas-particle flow, the density ratio is almost the same for the liquid particle flow, while for the liquid-air flow the density is quite high for the carrier phase flow. The study of all these density regimes gives a clear picture of how the carrier phase behaves in the presence of the dispersed phases, which ultimately leads to better design and safety of many two-phase flow equipments and processes. In order to carry out this approach, an Eulerian-Eulerian Two-Fluid model, with additional source terms to account for the presence of the dispersed phase in the turbulence equations has been employed for particulate flows, whereas Population Balance (PB) have been employed to study the bubbly flows. For the dilute gas-particle flows, particle-turbulence interaction over a backward-facing step geometry was numerically investigated. Two different particle classes with same Stokes number and varied particle Reynolds number are considered in this study. A detailed study into the turbulent behaviour of dilute particulate flow under the influence of two carrier phases namely gas and liquid was also been carried out behind a sudden expansion geometry. The major endeavour of the study is to ascertain the response of the particles within the carrier (gas or liquid) phase. The main aim prompting the current study is the density difference between the carrier and the dispersed phase. While the ratio is quite high in terms of the dispersed phase for the gas-particle flows, the ratio is far more less in terms of the liquidparticle flows. Numerical simulations were carried out for both these classes of flows and *

E-mail address: [email protected]. Tel: +61-3-99256191. Fax: +61-3-99256108. Mailing address: Prof Jiyuan Tu, School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, PO Box 71, Bundoora Vic 3083, AUSTRALIA

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K. Mohanarangam and J.Y. Tu their results were validated against their respective sets of experimental data. For the LiquidAir flows the phenomenon of drag reduction by the injection of micro-bubbles into turbulent boundary layer has been investigated using an Eulerian-Eulerian two-fluid model. Two variants namely the Inhomogeneous and MUSIG (MUltiple-SIze-Group) based on Population balance models are investigated. The simulated results were benchmarked against the experimental findings and also against other numerical studies explaining the various aspects of drag reduction. For the two Reynolds number cases considered, the buoyancy with the plate on the bottom configuration is investigated, as from the experiments it is seen that buoyancy seem to play a role in the drag reduction. The under predictions of the MUSIG model at low flow rates was investigated and reported, their predictions seem to fair better with the decrease of the break-up tendency among the micro-bubbles.

Introduction The basics of two-phase flows have received interdisciplinary attention from chemical, mechanical and industrial engineers for decades as they play a major role in a variety of industrial and design processes. Among these dilute particulate flows are encountered in a variety of industrial (Kolaitis & Founti, 2002) and natural processes irrespective of their carrier phase being gas or liquid. Particles constantly interact with the gas in industries through sand blasting equipments (Qianpu et al., 2005), pneumatic transport equipments (Gil et al., 2002), and also play a major role in the safe operation of the power plants (Tu, 1999), gas turbine engines (Awatef et al., 2005) and helicopters. In chemical industries they appear as reactants and catalysts, thereby controlling the order and the fate of the chemical reaction. There are even found in nature as dust dispersed in the room and as pollen released from the plants carried away by the wind. Lately there has been continued interest in these classes of flows in Bio-medical applications, to study the dust deposition patterns in realistic human nasal airway (Inthavong et al., 2006) and also to aid better delivery of the medications into the human nose. In industrial and engineering applications a better understanding of the physics of these flows will not only lead to better operating efficiency by improved equipment designs but also increase the longevity accompanied with lower maintenance costs and better operational improvements. Flows with particles amid liquid are an important class of two-phase flows classified under as slurry flows, whose flow systems are representative of many mineral processing operations and also provide useful operation correlations for such processes. They form an important class of flows encompassing pneumatic conveying system, turbines and machineries operating in particulate-laden environments. These flows provide a useful tool in the simulation of sprays in industrial and natural processes, as they have comparable phasedensity ratios. Comparable densities are of particular interest, since all the effects of interphase momentum transfer are important (Parthasarathy and Faeth, 1987). They also serve as a good test of methods to predict particle motion in turbulent environments (Parthasarathy and Faeth, 1987) as they exhibit high relative turbulence intensities for particle motion, which influence particle drag properties (Clift et al., 1978). Over the past few decades there have also been inter disciplinary research to study the mechanism of drag reduction with a mission to adopt them as tangible practice for a wide range of applications using micro-bubbles. While surfactant and polymer injections along the turbulent boundary layer, contribute a major portion towards this endeavour. Nonetheless, they are limited in terms of realistic applications due to their inherent ability to cause

On the Numerical Simulation of Turbulence Modulation in Two-Phase Flows

43

environmental pollution and there by causing the destruction of the oceans flora and fauna. Drag reduction by the injection of micro-bubbles along the boundary layer seems to be a realistic approach because fixing the dispersed phase as air for drag reduction devices may sound quite pragmatic; due to its wide spread availability unlike polymers or surfactants. Although there may be some overhead losses in placing them along the boundary layer, they are still undoubtedly the cheapest source, with no requirement of large storage space unlike the use of other low density inert (environment friendly) gases to form micro-bubbles. With the increase in the computational power and efficiency, computational fluid dynamics (CFD) have taken a centre point in offering effective solutions to not only single phase flows, but also for the simulation of wide range of two-phase flows viz., gas-particle, liquid-particle and liquid-gas flows. They not only offer a cheap solution, but also help scientists and engineers probe into places prohibitive by experimental methods or hostile environments unsuitable for human life forms. Numerical simulation of these two-phase flows lay broadly into two major categories namely the Eulerian-Eulerain two-fluid model approach and the Eulerian-Lagrangian particle tracking approach. In the Lagrangian particle tracking approach, each and every particle is tracked, thereby providing a detailed behaviour of their trajectories, velocities, bounce back angles and other parameters. Although they encompass a great deal of information within the domain, they prove to be rather cumbersome for multi-dimensional problems for the same reason being not able to track rather large number of particles given the computational power such as to obtain a good statistical information of the dispersed phase. Besides this, there is the problem of representing turbulent interactions between two phases (two-way coupling), which necessitates the need to fully understand the interactions of particles with individual vortices (Fessler and Eaton, 1997). Eulerian-Eulerian approach constituted by Anderson and Jackson (1967), Ishii (1975), regard the carrier and the dispersed phases as two interacting fluids with momentum and energy exchange between them. One major advantage of using the Eulerian approach is that the well-proven numerical procedures for single-phase flows can be directly extended to the secondary phase with the effects of turbulent interactions between the two fluids, lately there are considerations of extending the Eulerian two-fluid model by adopting the large eddy simulation (LES) approach (Pandya and Mashayek, 2002). Shirolkar et al (1996) in their paper stated that Eulerian models have problems to account for the particle history effects as they do not re-trace the motion of individual particles together; they also suffer from continuum assumption problems with respect to particles, as the particles equilibrate with neither local fluid nor each other when flowing through the flow field, in addition, crossing trajectories become more pronounced as particle inertia increases and Eulerian methods may become less accurate with increasing Stokes number, so a priori and rudimentary Lagrangian calculations should always be performed to check its validity. The above argument may be true for dispersed phase flows that contain solid particles that does not undergo any shape changes, the same was used to simulate the bubbly flows not time long, however with the advances in CFD in relation to bubble dynamics, it is envisioned that the constant bubble size may only be valid for problems where the dispersed phase does not undergo deformation. Directly adopting the model for flow problems, wherein dispersed phase undergoes constant change in the shape, may introduce substantial error into the final predictions. The MUSIG model was thereby developed to circumvent the above short-coming (Lo, 1996). Theoretically speaking, the model works in a manner in which it resolves the

44

K. Mohanarangam and J.Y. Tu

bubble size mechanistically while working within a range specified groups for the dispersed phase. However, each group possess an individual equation to be solved, which increases the turn around time for each simulation, there by making the parametric study of the problem computationally expensive. Numerical modelling of these classes of flows poses a problem, considering the diversity of two-phase flows, wherein the addition of a dispersed phase to a flow greatly increases the parameter space of the problem. In addition to the complexity involved in tracking the dispersed phase, their mere presence can drastically change the characteristics of the flow itself (Fessler and Eaton, 1995). The turbulent dispersion of these dispersed phase particles/bubbles within the carrier phase can be studied by two orders, one by presuming that the presence of the dispersed phase does not have any effect on the turbulent carrier flow field which is considered to be ‘one way coupling’ and the second by considering a feedback of the dispersed phase into the carrier flow field, in addition to the dispersed phase getting affected by the carrier flow field, this is considered to be ‘two-way coupling’, which is most of the times the precise phenomena taking place in the real world. Turbulence Modulation (TM) which re-defines the carrier phase both at the velocity and at the turbulence level in the presence of dispersed phase is crucial in the design of engineering applications. However, this study is paralysed owing to the complexities of the flows and limitations of the instruments. A nearly homogenous flow like liquid-particle flow can circumvent this problem, wherein all turbulence properties are attributed due to the relative motion of the particles; thereby any change felt due to the dispersed phase on the carrier phase is a direct result of only the TM phenomenon (Parthasarathy and Faeth, 1990). This phenomenon have been exploited by many experimental researchers (Parthasarathy and Faeth, 1987; Parthasarathy and Faeth, 1990; Alajbegovic et al, 1994; Rashidi et al, 1990; Sato and Hishida, 1996; Ishima et al, 2007; Borowsky and Wei, 2007; Righetti and Romano, 2007) to not only investigate, study and understand the basic features of TM but also to aid in the better formulation of numerical models. A number of previous studies have examined particle response for gas-particle flows in a sudden expansion flow experimentally (Ruck and Makiola, 1988; Hishida and Maeda, 1999; Fessler and Eaton, 1997). Whereas for the liquidparticle flows, although there have been studies in channel flow geometries, but their publication is limited for a sudden expansion geometry except for Founti & Klipfel (1998). Reynolds-averaged Navier-Stokes (RANS) equations are one of the well known numerical approaches to predict the Turbulence Modulation (TM) of the carrier phase in many industrial flow applications. They stem out as a consequence of time average to yield the constitutive mean flow equations, however the turbulent stresses which arise as a direct consequence should be modeled with some degree of approximation. Specification of the turbulent eddy viscosity serves as the turbulent closure and the k-ε turbulence formulation specifies this by solving two additional transport equations for turbulent kinetic energy and the eddy dissipation. This being the case for single phase flows, for dilute non-reacting particulate flows, these pose a problem in the sense the momentum of the carrier phase undergoes a phenomenal change due to the presence of particles, this is also reflected in terms of TM of the carrier phase. The major aim of this article is to investigate the various density regimes of the twophase flows and to study the effects of the dispersed phase onto the carrier phase at the velocity and at the turbulence level (Turbulence Modulation), for the dilute gas-particle flow, liquid-particle flow and also the air-liquid flow. While the density is quite high for the


45

dispersed phase flow for the gas-particle flow, the density ratio is almost the same for the liquid particle flow, while for the air-liquid flow the density is quite high for the carrier phase flow. The study of all these density regimes gives a clear picture of how the carrier phase behaves in the presence of the dispersed phases, which ultimately leads to better design and safety of many two-phase flow equipments.

Conservation Equations 3.1. Gas-Particle and Liquid-Particle Flows The modified Eulerian two-fluid model developed by Tu and Fletcher (1995) and Tu (1997) is employed for the simulation of gas-particle and liquid-particle flows, which basically considers the carrier and the dispersed phases as two interpenetrating continua. Hereby, a two way coupling is achieved between the dispersed and the carrier phases. The underlying assumptions employed in the current study are: 1) The particulate phase is dilute and consists of mono disperse spherical particles. 2) For such a dilute flow, the gas volume fraction is approximated by unity. 3) The viscous stress and the pressure of the particulate phase are negligible. 4) The flow field is isothermal.

3.1.1. Governing Equations for Carrier Phase Modeling The governing equations in Cartesian form for steady, mean turbulent gas flow are obtained by Favre averaging the instantaneous continuity and momentum equations

∂ ( ρ g u ig ) = 0 ∂ xi

(3.1)

∂p ∂ ∂ ∂ i ∂ ( ρ g u gj uig )= - g + ( ρgν gl ( ρ u'gj u'ig ) − FDi ug ) − ∂ xj ∂ xi ∂ x j ∂ xj ∂ xj g

(3.2)

Eq. (3.1) and (3.2) respectively are the continuity and momentum equation of the carrier gas phase, where ρ g , u g , u ' g and p g are the bulk density, mean velocity, fluctuating velocity and mean pressure of the gas phase, respectively. νgl is the laminar viscosity of the gas phase. FDi is the Favre-averaged aerodynamic drag force due to the slip velocity between the two phases and is given by

F Di = ρ

f ( u ig - u ip ) p

tp

where the correction factor f is selected according to Schuh et al (1989)

(3.3)

46


⎧ 1+ 0.15 Re 0p.687 ⎪⎪ f = ⎨ 0.914 Re 0p.282 + 0.0135 Re p ⎪ 0.0167 Re p ⎩⎪

0 < Re p ≤ 200 200 < Re p ≤ 2500

(3.4)

2500 < Re p

with the particle response or relaxation time given by t p =

ρ s d 2p /(18 ρ gν gl ) , wherein dp is

the diameter of the particle.

3.1.2. Governing Equations for Particulate Phase Modeling After Favre averaging, the steady form of the governing equations for the particulate phase is ∂ ( ρ p u ip ) = 0 ∂ xi

(3.5)

∂ ∂ ( ρ p u pj uip ) = ( ρ p u ' pj u 'ip ) + FGi + FDi + FWMi ∂ xj ∂ xj

(3.6)

where ρ p , u p and u ' p are the bulk density, mean and fluctuating velocity of the particulate phase, respectively. In equation (3.6), there are three additional terms representing the gravity force, aerodynamic drag force, and the wall-momentum transfer force due to particle-wall collisions, respectively. The gravity force is FGi = ρ p g , where g is the gravitational acceleration.

3.1.3. Turbulence Modeling for Carrier Phase For the carrier gas phase, which uses an eddy-viscosity model, the Reynolds stresses are given by

ρ g u' u' = − ρ gν gt ( i g

j g

∂u gi ∂x j

+

∂u gj ∂x i

)+

2 ρ g k g δ ij 3

(3.7)

where νgt is the turbulent or ‘eddy’ viscosity of the gas phase, which is computed by ν gt = C μ ( k 2g / ε g ) . The kinetic energy of the turbulence, kg and its dissipation rate, εg is governed by separate transport equations. The RNG theory, models the kg and εg transport equations (3.8) & (3.9) respectively by taking into account the particulate turbulence modulation, in which α is the inverse Prandtl number.

∂k g ∂ ∂ ( ρ g u gj k g ) = (αρ gν gt ) + Pkg − ρ g ε g + S k ∂x j ∂x j ∂x j

(3.8)


∂ε g ε g ∂ ∂ ( ρgu gj ε g ) = (αρgν gt ) + (C P − C ρ ε ) − ρ g R + Sε ∂x j k g ε 1 kg ε 2 g g ∂x j ∂x j

47 (3.9)

The rate of strain term R in the εg-equation is expressed as

R=

C μ η3 (1 − η η0 ) ε 2g , kg 1 + βη3

η=

kg εg

(2Sij2 )1/ 2 ,

i j 1 ⎛ ∂ug ∂ug ⎞ ⎟ Sij = ⎜⎜ + 2 ⎝ ∂x j ∂xi ⎟⎠

(3.10)

where β = 0.015, η0 = 4.38. The major endeavour of including this term is to take into account the effects of rapid strain rate along with the streamline curvature, which in many cases the standard k-ε turbulence model fails to predict. The constants in the turbulent transport equations are given by α = 1.3929, Cμ = 0.0845, Cε1 = 1.42 and Cε2 = 1.68 as per the RNG theory (Yakhot & Orszag, 1986). For the confined two-phase flow, the effects of the particulate phase on the turbulence of the gas phase are taken into account through the additional terms Sk and Sε in the kg and εg equations which arise from the correlation term given by

S k = (− u 'ig F ' Di ) = −

2f ρ p (k g − k gp ) tp

(3.11)

in the kg equation and

S ε = − 2ν gl

∂ u' gi ∂ F ' Di ∂x j ∂x j

=−

2f ρ p ( ε g − ε gp ) tp

(3.12)

in the εg equation, where kgp and εgp will be presented in the next following section discussing the particulate turbulence modeling.

3.1.4. Turbulence Modeling for the Dispersed Phase The transport equation governing the particulate turbulent fluctuating energy can be written as follows:

ν ∂k ∂ ∂ ( ρ p u pj k p ) = ( ρ p pt p ) + Pkp − I gp ∂x j ∂x j σ p ∂x j

(3.13)

The turbulence production Pkp of the particulate phase is given by

Pkp = ρ pν pt (

∂u ip ∂u pj ∂u ip 2 ∂u k ∂u i + ) − ρ pδ ij (k p + ν pt p ) p ∂x j ∂xi ∂xk 3 ∂ xk ∂ x k

and the turbulence interaction between two phases Igp is given by

(3.14)

48


I gp = Here k gp = 1 u 'i u 'i g

2

p

2f ρ p (k p − k gp ) tp

(3.15)

is the turbulence kinetic energy interaction between two phases.

The transport equation for the gas-particle covariance

u'

i i u' g p

can be again derived (Tu,

1997), to obtain the transport equation governing the gas-particle correlation which is given by

ν ν ∂k ∂ ∂ [ ρ p (ugj + u pj )kgp ] = ρ p ( gt + pt ) gp + Pgp − ρ pε gp − IIgp ∂x j ∂x j σ g σ p ∂x j

(3.16)

where the turbulence production by the mean velocity gradients of two phases is Pgp = {ρ p (ν gt

∂u gi ∂u pj 2 ∂u kp ∂u kp ∂u gi ∂u ip 1 + ν pt ) − ρ pδ ij k gp − ρ pδ ij (ν gt + ν pt )}( + ) (3.17) ∂x j ∂xi ∂xk ∂xk ∂x j ∂x j 3 3

The interaction term between the two phases takes the form

II gp =

f ρ p [( 1 + m ) 2 k gp − 2 k g − m 2 k p ] 2t p

(3.18)

here m is the mass ratio of the particle to the gas, m=ρp/ρg. The dissipation term due to the gas viscous effect is modeled by

ε

gp

= ε

g

exp( − B ε t p

ε

g

kg

)

(3.19)

where Bε=0.4. The turbulent eddy viscosity of the particulate phase, νpt, is defined in a similar way as the gas phase as:

ν pt =

2 k p t pt = l pt 3

2 kp 3

(3.20)

The turbulent characteristic length of the particulate phase is modeled by l pt = min( l ′pt , D s ) where l ′pt is given by

l ′pt =

l gt 2

(1 + cos 2 θ ) exp[ − B gp

| u 'r | sign ( k g − k p )] | u 'g |

(3.21)

where θ is the angle between the velocity of the particle and the velocity of the gas to account for the crossing trajectories effect (Huang et al, 1993). Bgp is an experimentally determined


49

constant, which takes a value of 0.01. Ds is the characteristic length of the system and provides a limit to the characteristic length of the particulate phase. The relative fluctuating velocity is given by

u 'r = u ' g − u ' p

(3.22)

and 2

2

| u 'r |= u ' g − 2u ' g u ' p + u ' p =

2 ( k g − 2 k gp + k p ) 3

(3.23)

The solution process of the above stated numerical formulation can be depicted in the form of a flow chart as shown in figure 3.1. Initial flow setup

B

A

Solve turbulence equations for gas-particle coupling (kgp, egp)

Solve gas-phase momentum equations

Solve pressure correction for the gas phase

Two-way coupling terms

Solve turbulence equations for the particle phase (kp)

Solve turbulence equations for the gas phase (kg, eg)

Update primitive variables (concentration, viscosity)

Solve particle-phase momentum equations

Overall convergence

Solve particle concentration equation

No

B

Yes Stop

A

Figure 3.1. Solution procedure for Eulerian two-fluid model.

3.2. Liquid-Air Flows (Micro-bubble) 3.2.1. Inhomogeneous Two-Fluid Model 3.2.1.1. Mass Conservation Numerical simulations presented in this paper are based on the two-fluid model using Eulerian-Eulerian approach. The liquid phase is treated as continuum while the gas phase (bubbles) is considered as dispersed phase (ANSYS, 2006). Under isothermal flow condition,

50


with no interfacial mass transfer, the continuity equation of the two-phases with reference to Ishii (1975) and Drew and Lahey (1979) can be written as:

∂ ( ρi α i ) K + ∇ ⋅ ( ρi α i u i ) = 0 ∂t

(3.24)

K

where α, ρ and u is the void fraction, density and velocity of each phase. The subscripts i = l or g denotes the liquid or gas phase. 3.2.1.2. Momentum Conservation The momentum equation for the two-phase can be expressed as follow:

K ∂( ρi αi ui ) KK K K K T + ∇ ⋅ ( ρi αi ui ui ) = −αi ∇P + αi ρi g + ∇ ⋅ αi μie (∇ui + (∇ui ) ) + Flg (3.25) ∂t

[

]

On the right hand side of Eq. (3.25), Flg represents the total interfacial force calculated K with averaged variables, g is the gravity acceleration vector and P is the pressure. The term Flg represents the inter-phase momentum transfer between gas and liquid due to the drag force resulted from shear and drag which is modelled according to Ishii and Zuber (1979) as:

1 K K K K Flg = − Fgl = C D aif ρl u g − ul (u g − u l ) 8 where C D is the drag coefficient which can be evaluated by correlation of several distinct Reynolds number regions for individual bubbles proposed by Ishii and Zuber (1979). 3.2.1.3. Interfacial Area Density In Eq. (3.25), interfacial momentum transfer due to the drag force is directly dependent on the contact surface area between the two phases and is characterized by the interfacial area per unit volume between gas and liquid phase, named as the interfacial area density aif. Based on the particle model, assuming that liquid phase is continuous and the gas phase is dispersed, the interfacial area per unit volume is then calculated based on the Sauter mean bubble diameter dg given by

aif =

6α g* dg

where

⎧max(α g , α min ) if (α g < α max ) ⎪ α =⎨ 1−α g ⎪max(1 − α α max , α min ) if (α g > α max ) max ⎩ * g


51

The non-dimensional inter-phase transfer coefficients can be correlated in terms of the particle Reynolds number and is given by

Re lg =

ρl U g − U l d g μl

where µl is the viscosity of the liquid phase.

3.2.2. MUSIG Model To account for non-uniform bubble size distribution, the MUSIG model employs multiple discrete bubble size groups to represent the population balance of bubbles. Assuming each bubble class travel at the same mean algebraic velocity, individual number density of bubble class i based on Kumar and Ramkrishna (1996a) can be expressed as:

⎛ ⎞ ∂ni K + ∇ ⋅ (u g ni ) = ⎜⎜ ∑ R j ⎟⎟ ∂t ⎝ j ⎠i where

(∑ R ) j

j

i

(3.26)

represents the net change in the number density distribution due to

coalescence and break-up processes. The discrete bubble class between bubble volumes vi and vi +1 is represented by the centre point of a fixed non-uniform volume distributed grid interval. The interaction term

(∑ R ) = (P j

j

i

C

+ PB − DC − DB ) contains the source rate of

PC , PB , DC and DB , which are, respectively, the production rates due to coalescence and break-up and the death rate due to coalescence and break-up of bubbles. 3.2.2.1. MUSIG Break-up Rate The production and death rate of bubbles due to the turbulent induced breakage is formulated as:

PB =

∑ Ω (v j : vi )n j N

j =i +1

N

DB = Ωi ni with Ωi = ∑ Ωki

(3.27)

k =1

Here, the break-up rate of bubbles of volume v j into volume vi is modelled according to the model developed by Luo and Svendsen (1996). The model is developed based on the assumption of bubble binary break-up under isotropic turbulence situation. The major difference is the daughter size distribution which has been taken account using a stochastic breakage volume fraction fBV. By incorporating the increase coefficient of surface area, cf =

52

K. Mohanarangam and J.Y. Tu 2/3

[ f BV +(1-fBV)2/3-1], into the breakage efficient, the break-up rate of bubbles can be obtained as:

Ω (v j : vi )

⎛ ε ⎞ = FB C ⎜ 2 ⎟ ⎜d ⎟ (1 − α g )n j ⎝ j ⎠ where

1/ 3 1

∫

ξ min

(1 + ξ )2 × exp⎛⎜ − ξ

⎜ ⎝

11 / 3

12c f σ βρl ε

2/3

d

ξ

5/3

11 / 3

⎞ ⎟dξ ⎟ ⎠

ξ = λ / d j is the size ratio between an eddy and a particle in the inertial sub-range

and consequently

ξ min = λmin / d j and C and β are determined, respectively, from

fundamental consideration of drops or bubbles breakage in turbulent dispersion systems to be 0.923 and 2.0. 3.2.2.2. MUSIG Coalescence Rate The number density of individual bubble groups governed by coalescence can be expressed as:

PC =

1 i i ∑∑η jki χ ij ni n j 2 k =1 l =1

η jki =

(ν j + ν k ) − ν i −1 /(ν i − ν i −1 )

if ν i −1 < ν j + ν k < ν i

ν i +1 − (ν j + ν k ) /(ν i +1 − ν i )

if ν i < ν j + ν k < ν i +1

0

otherwise

N

DC = ∑ χ ij ni n j j =1

From the physical point of view, bubble coalescence occurs via collision of two bubbles which may be caused by wake entrainment, random turbulence and buoyancy. However, only turbulence random collision is considered in the present study as all bubbles are assumed to be spherical (wake entrainment becomes negligible). Furthermore, as all bubbles travel at the same velocity in the MUSIG model, buoyancy effect is also eliminated. The coalescence rate considering turbulent collision taken from Prince and Blanch (1990) can be expressed as:

χ ij = FC

π

[d 4

i

+dj

] (u 2

2 ti

+ u tj2

)

0.5

⎛ t ij exp⎜ − ⎜ τ ⎝ ij

where τ ij is the contact time for two bubbles given by ( d ij / 2)

2/3

⎞ ⎟ ⎟ ⎠

/ ε 1 / 3 and t ij is the time

required for two bubbles to coalesce having diameter di and dj estimated to be

[(dij / 2)3 ρl / 16σ ]0.5 ln(h0 / h f ) . The equivalent diameter dij is calculated as suggested by


53

−1

Chesters and Hoffman (1982): (d ij = ( 2 / d i + 2 / d j ) . According to Prince and Blanch (1990), for air-water systems, experiments have determined the initial film thickness ho and −4

−8

critical film thickness hf at which rupture occurs as 1 × 10 and 1 × 10 m respectively. The turbulent velocity ut in the inertial subrange of isotropic turbulence (Rotta, 1972) is given by: u t =

2ε 1 / 3 d 1 / 3 .

Numerical Procedure All the transport equations are discretized using a finite volume formulation in a generalized coordinate space, with metric information expressed in terms of area vectors. The equations are solved on a nonstaggered grid system, wherein all primitive variables are stored at the centroids of the mass control volumes. Third-order QUICK scheme is used to approximate the convective terms, while second-order accurate central difference scheme is adopted for the diffusion terms. The velocity correction is realized to satisfy continuity through SIMPLE algorithm, which couples velocity and pressure. At the inlet boundary the particulate phase velocity is taken to be the same as the gas velocity. The concentration of the particulate phase is set to be uniform at the inlet. At the outlet the zero streamwise gradients are used for all variables. The wall boundary conditions are based on the model of Tu and Fletcher (1995). All the governing equations for both the carrier and dispersed phases are solved sequentially at each iteration, the solution process is started by solving the momentum equations for the gas phase followed by the pressure-correction through the continuity equation, turbulence equations for the gas phase, are solved in succession. While the solution process for the particle phase starts by the solution of momentum equations followed by the concentration then gas-particle turbulence interaction to reflect the two-way coupling, the process ends by the solution of turbulence equation for the particulate phase. At each global iteration, each equation is iterated, typically 3 to 5 times, using a strongly implicit procedure (SIP).The above solution process is marched towards a steady state and is repeated until a converged solution is obtained.

Numerical Predictions Gas –Particle Flow 4.1. Code Verification In this section the code is validated for mean streamwise velocities and fluctuations for both the carrier and dispersed phases against the benchmark experimental data of Fessler and Eaton (1995). This task is undertaken to verify the fact that these two classes of particles, which share the same Stokes number but varied particle Reynolds number can be handled by the code. Figure 4.1 show the backward facing step geometry used in this study, which is similar to the one used in the experiments of Fessler and Eaton (1995), which has got a step height (h) of 26.7mm. As the span wise z-direction perpendicular to the paper is much larger

54


than the y-direction used in the experiments, the flow is considered to be essentially twodimensional. The backward facing step has an expansion ratio of 5:3. The Reynolds number over the step works out to be 18,400 calculated based on the centerline velocity and step height (h). The independency of grid over the converged solution was checked by refining the mesh system through doubling the number of grid points along the streamwise and the lateral directions. Simulations results revealed that the difference of the reattachment length between the two mesh schemes is less than 3%.

H

y

H=40mm h=26.7mm

x h

x=35h Figure 4.1. Backward facing step geometry.

4.1.1. Mean Streamwise Velocities The mean streamwise velocities for the gas phase have been shown in Figure 4.2 for various stations along the backward-facing step geometry, it can be generally seen that there is fairly good agreement with experimental findings of Fessler and Eaton (1995). This is then followed by the streamwise velocities for the two classes of particles considered in this study, whose properties are tabulated in Table 1. The broad varying characteristics of different particle sizes and material properties can be unified by a single dimensionless parameter; it is also used to quantify the particles responsivity to fluid motions and this non-dimensional parameter is the Stokes number (St) and is given by the ratio of particle relaxation time to time that of the appropriate fluid time scale, St = tp/ts. In choosing the appropriate fluid time scale, the reattachment length has not been considered as it varies due to addition of particles and is not constant in this study, rather a constant length scale of five step heights, which is in accordance to the reattachment length is used. The resulting time scale is given by ts=5h/Uo, in accordance with the experiments of Fessler and Eaton (1995). Table 4.1 Properties of the dispersed phase particles Nominal Diameter(μm)

150

70

Material

Glass

Copper

Density(kg/m3)

2500

8800

Stokes Number (St)

7.4

7.1

Particle Reynolds number (Rep)

9.0

4.0

On the Numerical Simulation of Turbulence Modulation in Two-Phase Flows 0

u/Ub 1.5

2.50

Height (y/h)

2.00 x/h=

x/h=

x/h=

x/h=

x/h=9

x/h=1

x/h=1

1.50 1.00 0.50 0.00 Distance along the step (x/h) ○ Experimental

Numerical

Figure 4.2. Mean streamwise gas velocities.

0

u/Ub 1.5

2.50

Height (y/h)

2.00 1.50

x/h=

x/h=5

x/h=

x/h=9

1.00 0.50 0.00

Distance along the step (x/h) ○ Experimental

Numerical

Figure 4.3.a. Streamwise mean velocity for 70μm copper particles.

x/h=1

55

56


The significance of the Stokes number is that a particle with a small Stokes number (St<<1) are found to be in near velocity equilibrium with the surrounding carrier fluid, there by making them extremely responsive to fluid velocity fluctuations, in fact Laser Doppler velocimetry (LDV) takes advantage of this to deduce local fluid velocity from the measured velocity of these very small Stokes number particles, however on the other hand for a larger Stokes number (St>>1) particles are found be no longer in equilibrium with the surrounding fluid phase as they are unresponsive to fluid velocity fluctuations and they will pass unaffected through eddies and other flow structures. Figures 4.3a & 4.3b shows the mean streamwise velocity profiles for the two particle classes considered in this study, it can be seen that there is generally a fairly good agreement with the experimental results. From the particle mean velocity graphs of the carrier and dispersed phases it can be inferred, that the particle streamwise velocity at the first station x/h=2 is lower than the corresponding gas velocities, this is in lines with the fully developed channel flow reaching the step as described in the experiments of Kulick et al (1994), wherein the particles at the channel centerline have lower streamwise velocities than that of the fluid as a result of cross-stream mixing. However the gas velocity lags behind the particle velocities aft of the sudden expansion as the particles inertia makes them slower to respond to the adverse pressure gradient than the fluid. 0

u/Ub 1.5

2.50

Height (y/h)

2.00 1.50

x/h=

x/h=

x/h=7

x/h=9

x/h=1

1.00 0.50 0.00 Distance along the step (x/h) ○ Experimental

Numerical

Figure 4.3.b. Streamwise mean velocity for 150μm glass particles.

4.1.2. Mean Streamwise Fluctuations The code has been further validated for mean streamwise fluctuations and Figure 4.4 shows the mean streamwise fluctuations for the gas phase against the experimental data. It is seen that there is a general under prediction of the simulated data with the experimental


57

results and this is more pronounced towards the lower wall for a height of up to y/h≤2, however the pattern of the simulated results have been found to be in tune with its experimental counterpart. 0

u'/Ub

0.2

2.50

Height (y/h)

2.00 x/h=

x/h

x/h=

x/h

x/h=1

1.50

1.00

0.50

0.00 Distance along the step (x/h) Figure 4.4. Fluctuating streamwise gas velocities particles.

0

u'/Ub 0.3

2.50

Height (y/h)

2.00

x/h=2

x/h=5

x/h=7

x/h=9

x/h=12

1.50 1.00 0.50 0.00

Distance along the step (x/h) ○ Experimental

Numerical

Figure 4.5.a. Fluctuating streamwise particle velocities for 70μm copper particles.

58

K. Mohanarangam and J.Y. Tu 0

u'/Ub

0.3

2.50

Height (y/h)

2.00 x/h=5

x/h=

x/h=7

x/h=9

1.50

x/h=1


Numerical

Figure 4.5.b. Fluctuating streamwise particle velocities for 150μm glass particles.

Figures 4.5a & 4.5b show the streamwise fluctuating particle velocities for the two different classes of particles considered, there has been a minor under-prediction until stations y/h ≤ 1, over all a fairly good agreement can be observed. It can be also seen that for y/h > 1.5, the particle fluctuating velocities are considerably larger than those of the fluid. This again is in accordance with channel flow inlet conditions, where the particles have higher fluctuating velocities than those of the fluid owing to cross-stream mixing. All the experimental results used for comparison of particle fluctuating velocities correspond to maximum mass loadings of particles as reported in the experiments of Fessler & Eaton (1999).

4.2. Results and Discussion 4.2.1. Turbulence Modulation (TM) The plot depicted in the following sections to represent the Turbulent Modulation (TM) of the carrier gas phase is given by the ratio of the laden flow r.m.s streamwise velocity to the unladen r.m.s streamwise velocity. These plots signify that any turbulence modulation felt in the carrier phase is reflected as an exit of the ratio from unity. 4.2.1.1. Analysis of Experimental Data Plots 4.6a and 4.6b depict the experimental data as obtained from the experiments of Fessler and Eaton (1995). Figures 4.6a&b represent the mean streamwise particle velocities


59

and fluctuations respectively for the two sets of dispersed phase particles i.e., copper and glass particles. It can be well seen that the mean velocities and to a similar extent the fluctuations for these two classes seem to behave analogous to each other. However from figures 4.7a-c, which shows the experimental TM for the carrier phase in the presence of the dispersed phase at the same mass loading of 40% seem to behave in contrary to the above findings. In all these stations considered in this study the glass particle seems to attenuate the carrier phase more than copper particles. It is also be seen at the station x/h=2, the attenuation 0

u/Ub

1.5

2.50 x/h=2

x/h=5

x/h=7

x/h=9

Height (y/h)

2.00

1.50

1.00

0.50

0.00 Distance along the step (x/h)

70μm copper

150μm glass

Figure 4.6.a. Experimental mean streamwise particle velocity.

of the turbulence for glass particle is totally opposite in relation to copper particle for location y/h>1.25. At station x/h=7 for certain regions along the height of the step the turbulence attenuation for the two particle classes seems to be in phase, whereas before and after this small region of unison the glass particle seem to attenuate more than the copper particles. At the station x/h=14, it can be clearly seen, that there is a uniform degree of difference in attenuation all along the step, this is more attributed to the uniform distribution of the particles seen along the step. The maximum turbulence attenuation can be seen for the 150μm particles, which up to 35% as reported by Fessler and Eaton (1999). From the above analysis, it is quite clear that particles with the same Stokes number modulate the carrier phase turbulence in a totally different fashion. This makes us conclude that although Stokes number can be generalized to account for mean values of velocity and fluctuations, it cannot be generalized when it comes to TM, in which case something more than Stokes number is required to define one’s particle response to surrounding carrier phase turbulence either to attenuate or enhance it.

60


u'/Ub

0.3

2.50 x/h=

x/h=

x/h=

x/h=7

Height (y/h)

2.00

1.50

1.00

0.50

0.00 Distance along the step (x/h) Figure 4.6.b. Experimental fluctuating streamwise particle velocity. 1.10

Turbulence Modulation (TM)

x/h=2 1.00

0.90

0.80

0.70

0.60 0.00

0.50

1.00

70μm copper

1.50 Height (y/h)

2.00

150μm glass

Figure 4.7.a. Experimental Turbulence Modulation at x/h=2.

2.50


1.10

Turbulence Modulation (TM)

x/h= 1.00

0.90

0.80

0.70

0.60 0.00

0.50

1.00 1.50 Height (y/h)

2.00

2.50

Figure 4.7.b. Experimental Turbulence Modulation at x/h=7.

1.10 Turbulence Modulation (TM)

x/h=14

1.00

0.90

0.80

0.70

0.60 0.00

0.50 70μm copper

1.00 1.50 Height (y/h)

2.00 150μm glass

Figure 4.7.c. Experimental Turbulence Modulation at x/h=14.

2.50

61

62


4.2.2. TM & (Particle Number Density) PND Results

12.00

1.20

10.00

1.00

8.00

0.80

6.00

0.60

4.00

0.40

2.00

0.20

0.00 0.00 ●Experimental-TM

0.50

1.00 1.50 Height (y/h)

2.00

Numerical-TM ▲Experimental-PND

Turbulence modification (TM)

Particle Number density (PND)

In this section we have tried to derive an understanding for the Turbulence Modulation (TM) of the carrier gas phase in the presence of particles using our turbulent formulation, along with its corresponding PND results for the two classes we have considered in this study. The simulated results of the above two parameters are plotted along side the experimental findings of Fessler and Eaton (1995). The experimental values for the modulation are plotted with error bars, as significant scatter are apparent in these plots, thereby making any variations on the order of ±5% insignificant (Fessler & Eaton, 1995).

0.00 2.50

Numerical-PND

Figure 4.8.a. Turbulence Modulation & Particle Number Density for 70μm copper particles at x/h=2.

The plots 4.8a-c shows the combined numerical and experimental results of TM (secondary axis) and PND (primary axis) for three sections along the step viz. x/H=2, 7 &14 for copper particles. It can be noted at station x/h=2 just aft of the step, for y/h<1 there exist very few particles, but however the experimental TM shows a wavy behavior in comparison to the simulated results which seem to behave in unison with the PND, however aft of this section, the simulated values compare well with the experimental data. In the middle of the step at station x/h=7, there is generally a good comparison of the experimental and simulated values for both the TM and PND. At the exit (x/h=14) however, there seems to be general under prediction of TM for y/h>1.5, while the PND seem to vary from under predicting to over predicting the experimental data. Figures 4.9a-c shows plots of the glass particles for the same three sections along the step. At section x/h=2, the simulated values seem to over predict the experimental data for y/h>1, however this distinct behavior of maximum attenuation as reported by Fessler and Eaton (1995) occurs here, this under prediction is not quite in terms with the PND results which seem to show a uniform distribution. A fairly good


63

12.00

1.20

10.00

1.00

8.00

0.80

6.00

0.60 `

4.00

0.40

2.00

0.20


0.50

1.00 1.50 Height (y/h)

2.00




agreement is with both turbulence modulation and PND can be seen for stations x/h=7 and 14.

0.00 2.50 Numerical-PND

Figure 4.8.b. Turbulence Modulation & Particle Number Density for 70μm copper particles at x/h=7.

It is generally seen that for the two classes of particles considered, there exist considerably more particles in the region y/h>1 for locations x/h=2 and 7, and the particles exhibit a uniform distribution by the time it reaches the location x/h=14 due to its uniform spreading action, but despite a uniform distribution the turbulence attenuation is still small for y/h<1 at the x/h=14, and this is attributed to the non-effectiveness of the particles to large scale vortices which are reported to exist downstream of the single phase backward facing step (Le et al., 1997). From the PND results for the two classes of particles, both simulated and experimental plots show that there exits very few particles in the region y/h<1 before the re-attachment point and a significant spreading has taken place by, at locations x/h=9 and 14, considering this behavior along side with the turbulence attenuation one can state that lack of turbulence modulation is not simple due to the absence of particles but clearly a difference on response of the turbulence in a specific region due to the presence of particles (Fessler & Eaton, 1995) and this has been explicitly seen in our simulated values using our turbulence formulation mentioned in the previous section.


1.20

10.00

1.00

8.00

0.80

6.00

0.60

4.00

0.40

2.00

0.20

0.00

0.00


12.00

0.00

●Experimental-TM

0.50

1.00 1.50 Height (y/h)

2.00



64

2.50

Numerical-PND

Figure 4.8.c. Turbulence Modulation & Particle Number Density for 70μm copper particles at x/h=14.

4.2.3. Effect of Particle Reynolds Number on TM In this section, we have tried to explain the phenomenon of the carrier phase TM by particles in relation to particle Reynolds number. Looking back at Table 4.1, it is seen that for the glass particles the Rep is almost 125% more than for the copper particles and also considering the size of the particles which is around 114% more for the glass particles in comparison to the copper particles. The presented results on the degree of carrier phase turbulence attenuation also seem to throw some interesting pattern especially in regards to copper and glass particles, which almost share the same Stokes number. Based on this similarity one would expect that their response to the turbulence be the same, although this fact seems to be quite in lines with the particle mean and fluctuating velocities but their behavior in regards to turbulence modulation paint a totally different picture, which makes us conclude that the turbulence attenuation has a direct link of the particles onto the turbulence rather a derived effect on the mean flow modifications (Fessler and Eaton, 1999). And this attenuation cannot follow suit with increasing loading and particle size as there will be an increase in the turbulence as reported by Hetsroni (1989), it is also interesting to note from the same author that apart from the Stokes number, particle Reynolds number plays an important role in the systematic behavior of the particles, the same has been reported from the experimental findings of Fessler and Eaton (1995), this is in conformity to our simulated results wherein the two classes of copper and glass behave differently although their particle Stokes number remain the same.

12.00

1.20

10.00

1.00

8.00

0.80

6.00

0.60

4.00

0.40

2.00

0.20

0.00 0.00

0.50

1.00

1.50

2.00

65

Turbulence Modification(TM)

Particle Number density(PND)


0.00 2.50

Height(y/h) ●Experimental-TM


Numerical-PND

Figure 4.9.a. Turbulence Modulation & Particle Number Density for 150μm glass particles at x/h=2.

It is known from previous studies that there is a physical increase in the carrier phase turbulence and this is attributed to the wake formation behind these large particles similar to the ones behind the cylinders encountered in single phase flows. Experimental studies from Eaton et al (1999) state that for Rep around 10, the modulation is caused due to the strong asymmetric wake distortion extending many particle diameters downstream, whereas for Rep=1, the flow distortion is highly localized to a nearly symmetric region extending only a few particle diameters. Taking this into consideration one would expect a significant enhancement of the carrier phase turbulence for the glass particles but in contrast significant attenuation have taken place, the maximum attenuation is noted at location x/h=2 and the attenuation of glass particles is roughly about 44% more than its counterpart copper particles. This behavior of the glass particle to attenuate rather than enhance the turbulence of the carrier phase can be explained by the fact that, due to strong asymmetric wake, the distortion caused by this class of particles is dissipated more easily within the carrier gas phase, this is in sharp contrast to the parallel experiments run by Sato et al (2000) using water as the carrier phase fluid, although similar Stokes number were achieved, much higher Rep was realized, this in total showed significant vortices behind the particles and also an enhancement in the carrier phase turbulence, which is primarily caused due to the different density ratios between the carrier and dispersed phases, which makes one to justify that Stokes number alone will not be able to describe the parameter space of the dispersed two-phase flow problems, Rep should also be taken into consideration.

K. Mohanarangam and J.Y. Tu 1.20

10.00

1.00

8.00

0.80

6.00

0.60

4.00

0.40

2.00

0.20


12.00

0.00 0.00

0.50

1.00 1.50 Height(y/h)

2.00


66

0.00 2.50

1.20

10.00

1.00

Particle Number density(PND)

12.00

8.00

0.80

6.00

0.60

4.00

0.40

2.00

0.20


0.50

1.00 1.50 Height(y/h)

2.00



Figure 4.9.b. Turbulence Modulation & Particle Number Density for 150μm glass particles at x/h=7.

0.00 2.50 Numerical-PND

Figure 4.9.c. Turbulence Modulation & Particle Number Density for 150μm glass particles at x/h=14.

Liquid–Particle Flow This section describes the numerical investigation of the turbulent gas-particle flow over a backward facing step, using the Eulerian-Eulerian model as described in the previous section. The numerical procedure is almost the same as of section 3.1. However, two


67

backward-facing step geometries are utilized in this study one to validate the GP flow and the other for the LP flow. Figure 5.1a shows the backward facing step geometry, which is similar to the one used in the experiments of Fessler and Eaton (1995), comprising of a step height (h) of 26.7mm. As the span wise z-direction perpendicular to the paper is much larger than the y-direction used in the experiments, the flow is considered to be essentially twodimensional. The backward facing step has an expansion ratio of 5:3. The Reynolds number over the step works out to be 18,400 calculated based on the centerline velocity and step height (h). The experimental set up of Founti and Klipfel (1998) consisted of a pipe flow with a sudden expansion ratio of 1:2, with a step height of 25.5mm, as depicted in figure 5.1b, working at a Reynolds number of 28,000. The summary of the flow conditions along with the properties of the dispersed phase particles used in this study are summarized in Table 5.1.

H

y x

H=40mm h=26.7mm h x=35h

Figure 5.1.a. Backward facing step geometry (Fessler & Eaton; 1995).

y x

H

H=25.5mm h=25.5mm

h x=35h Figure 5.1.b. Backward facing step geometry (Founti & Klipfel; 1998).

Table 5.1. Flow properties of carrier and dispersed phases for LP & GP flows Parameters Reynolds number (Reh) Geometry Continuous Phase Mass loading Particle Density Particle Diameter Phase-density ratio

Gas-Particle (GP) flow Fessler & Eaton (1995)

Liquid-Particle (LP) flow Founti & Klipfel (1998)

18,700 BFS Air 20% 2500 150 micron

28,000 Pipe (BFS) Diesel Oil 15% 2500 450 micron

2137:1

3.0:1

68


5.1. Analysis of Experimental Data In this section, the experimental data for mean velocities and fluctuations are analysed, in order to understand the particle behaviour in relation to its carrier phase namely the gas and the liquid. For this purpose, three sections were selected aft of the sudden expansion one near the step, another almost at the middle and the other farther away from the step nearing the exit of the geometry. An important, dimensionless scaling parameter in defining on how the fluid-particle behave with the flow field is the Stokes number (St), which is given by the ratio of the particle relaxation time to a time characteristic of the fluid motion, i.e., St = tp/ts. This determines the kinetic equilibrium of the particles with the surrounding liquid. In choosing, the fluid time scale ts, amidst the complexity of having two different geometries with different expansion ratios and also with the re-attachment length varying with the addition of the particles, the fluid time scales were determined by ts=5h/Uo in lieu with the experimental conditions of Fessler and Eaton (1997). A small stokes number (St << 1) signifies that the particles are in near velocity equilibrium with the carrier fluid. For larger stokes number (St >> 1) particles are no longer in equilibrium with the surrounding fluid phase, which will be exemplified in the later sections. Based on the above definition, the Stokes number for the GP and the LP flow examined in our study worked out to be 14.2 and 0.59 respectively. Figures 5.2a-c shows the mean streamwise velocities of the liquid and particle phase flows as presented from the experiments of Founti & Klipfel (1998). It can be seen at section x/h=0.7, the particles seem to exhibit a higher negative velocity for a section of y/h<1, after this height the particles seem to surpass the liquid velocities for the section y/h>1, while for a small region at the proximity of the step they seem to exhibit a homogeneous behaviour. At section x/h=7.8, which is almost the middle section, the liquid phase have a higher negative velocity than that of the particulate phase, this behaviour seems to follow for a height of up to y/h=1, after which the both the phases seem to behave in unison. The final section is x/h= 15.7, herein again the liquid seems to exhibit a higher velocity than that of the particles, with unified flow at the top. Figures 5.3a-c shows the mean velocities of the GP flow as obtained from the experiments of Fessler and Eaton (1999). At section x/h=2, just aft of the step the particle velocities seem to lag behind the gas velocities, whereas at section x/h=7 in the middle of the geometry, the particle velocities seem to ‘catch up’ with that of the gas phase and their velocities are more or less the same, however at the exit of the backward-facing step geometry (x/h=14), a clear marked difference in velocities is observed, wherein the particles seem to overtake the gas due to its inertial. From the two sets of experimental results outlined above, one could observe that at near the inlet sections, where the recirculation is quite predominant for both the cases, particles seem to lag behind the gas for the GP flows, where as the particles seem to exhibit a higher inertial with respect to the LP flows, with particles leading throughout the height of the step. Overall, with respect to the magnitude of the mean velocities, the particle seem to exhibit more or less a change in the pattern from ‘lead’ to ‘lag’, as we proceed along the step for LP flows, whereas the particles seem to exhibit the opposite pattern from ‘lag’ to ‘lead’ for the GP flows along the step. The presence of a clear shear layer just aft of the step is quite prominent for the


69

particles at sections all along the step for LP flow, which in single phase is the major source of turbulence generation due to shear. In order to understand this behaviour better we now turn to the mean streamwise fluctuations for these two kinds of flows. Figures 5.4a-c shows the fluctuating velocities of LP flows and it can be seen that at the entry the particles seem to ‘lag’ behind the liquid for a height of y/h>1, while the showing an increase with respect to liquid in the lower part. At the middle section considered, for a height of about y/h>1 they exhibit a homogenous flow behaviour, whereas at the lower part the particles again seem to exceed its liquid counterpart. Whereas near the exit, both the continuous and dispersed phases have almost the same pattern prompting to the fact that the particle ‘catch up’ with the liquid phase, mimicking a homogenous flow pattern. Figures 5.5a-c show the fluctuation results for the GP flow, at the entry section x/h=2, it can be seen that the particulate phase has a higher fluctuation in comparison to the gas phase, while at the middle section x/h=7, the gas phase seem to catch up with the particulate phase, while at the exit at x/h=14, it is observed that both the dispersed and the continuous phase seem to fluctuate in unison. From the above experimental results of the fluctuation, it can be ascertained that the particle ‘lag’ behind with respect to the continuous phase in terms of LP flows, whereas they ‘lead’ in terms of the gas-particle flows.

2.00 1.75 x/h=0.7 1.50

y/h

1.25 1.00 0.75 0.50 0.25 0.00 -0.20 0.00 0.20 0.40 0.60 0.80 1.00 u/Uo Liquid

Particle

Figure 5.2.a. Experimental mean streamwise velocities at x/h=0.7 for liquid-particle flow.

70


2.00 1.75

x/h=7.8

1.50

y/h

1.25 1.00 0.75 0.50 0.25 0.00 -0.20 0.00

0.20

0.40 0.60 u/Uo

0.80

1.00

Figure 5.2.b. Experimental mean streamwise velocities at x/h=7.8 for liquid-particle flow.

2.00 1.75 x/h=15.7

1.50

y/h

1.25 1.00 0.75 0.50 0.25 0.00 -0.20 0.00 0.20 0.40 0.60 0.80 1.00 u/Uo Liquid

Particle

Figure 5.2.c. Experimental mean streamwise velocities at x/h=15.7 for liquid-particle flow.

On the Numerical Simulation of Turbulence Modulation in Two-Phase Flows 2.50

2.00

x/h=2

y/h

1.50

1.00

0.50

0.00 -0.25

0.00

0.25 0.50 u/Uo

0.75

1.00

Figure 5.3.a. Experimental mean streamwise velocities at x/h=2 for gas-particle flow. 2.50

2.00 x/h=7

y/h

1.50

1.00

0.50

0.00 0.00

0.25

Gas

0.50 u/Uo

0.75

1.00

Particle

Figure 5.3.b. Experimental mean streamwise velocities at x/h=7 for gas-particle flow.

71

72


2.50

2.00 x/h=14

y/h

1.50

1.00

0.50

0.00 0.00

0.20

0.40

Gas

0.60 u/Uo

0.80

1.00

Particle

Figure 5.3.c. Experimental mean streamwise velocities at x/h=14 for gas-particle flow.

2.00 1.75 1.50

x/h=0.7

y/h

1.25 1.00 0.75 0.50 0.25 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 u'/Uo

Liquid

Particle

Figure 5.4.a. Experimental mean fluctuating velocities at x/h=0.7 for liquid-particle flow.


2.00 1.75 1.50

x/h=7.8

y/h

1.25 1.00 0.75 0.50 0.25 0.00 0.02

0.04

0.06

0.08 0.10 u'/Uo

0.12

0.14

0.16

Figure 5.4.b. Experimental mean fluctuating velocities at x/h=7.8 for liquid-particle flow.

2.00 1.75

x/h=15.7

1.50

y/h

1.25 1.00 0.75 0.50 0.25 0.00 0.02 0.04

0.06 0.08 0.10 0.12 u'/Uo

Liquid

0.14 0.16

Particle

Figure 5.4.c. Experimental mean fluctuating velocities at x/h=15.7 for liquid-particle flow.

73

74


2.50

2.00

x/h=2

y/h

1.50

1.00

0.50

0.00 0.00

0.05

0.10 u'/Uo

0.15

0.20

Figure 5.5.a. Experimental mean fluctuating velocities at x/h=2 for gas-particle flow. 2.50

x/h=7

2.00

y/h

1.50

1.00

0.50

0.00 0.00

0.05

Gas

0.10 u'/Uo

0.15

0.20

Particle

Figure 5.5.b. Experimental mean fluctuating velocities at x/h=7 for gas-particle flow.


75

2.50

2.00

x/h=14

y/h

1.50

1.00

0.50

0.00 0.00

0.05

0.10 u'/Uo

Gas

0.15

0.20

Particle

Figure 5.5.c. Experimental mean fluctuating velocities at x/h=14 for gas-particle flow.

5.2. Numerical Code Validation In this section the code is validated for mean streamwise velocities and fluctuations for both the carrier and dispersed phases against the benchmark experimental data of Fessler and Eaton (1995) for GP flow and the experimental data of Founti and Klipfel (1998) for the LP flow. This task is undertaken to verify the fact that particulate flows with two varied carrier phases can be handled by the code. The ability of the numerical code to replicate the experimental results of GP (Fessler and Eaton, 1999) and LP (Founti and Klipfel, 1998) flows are tested. Figure 5.6a shows the numerical findings of single phase (Diesel oil) mean velocities against the experimental data, although the overall behaviour is replicated numerically there have been some under prediction for a height of y/h>1 for mid-section of the geometry, while a minor over prediction is felt along the entire height at section x/h=15.7. Figure 5.6b shows the fluctuating liquid velocities along the step compared against the experimental findings, there have been some minor under prediction for a height of y/h<1 at some sections, the majority show a good comparison with the experimental data. Figure 5.6c and figure 5.6d depicts the experimental and numerical comparison of particle mean and fluctuating velocities and it can be seen that overall numerical results have a good agreement with the experimental data.

76


0

u/U

1.5

2.00

Height (y/h)

1.50

1.00

0.50

x/h=0

x/h=5

x/h=7

x/h=11

x/h=15


Figure 5.6.a. Axial liquid velocities along the step for LP flows.

0

u'/Uo

0.

2.00

Height (y/h)

1.50

1.00

0.50

x/h=0.7

x/h=5.9

x/h=7.

x/h=11.

x/h=15.


○ Experimental

Numerical

Figure 5.6.b. Fluctuating axial liquid velocities along the step for LP flows.

On the Numerical Simulation of Turbulence Modulation in Two-Phase Flows 0

u/Uo

1.5

2.00

1.00

0.50

x/h=0.7

x/h=5.9

x/h=7.8

x/h=11.8

x/h=15.7


Figure 5.6.c. Axial particle velocities along the step for LP flows. 0

u'/U

0.3

2.00

1.50

Height (y/h)

Height (y/h)

1.50

1.00

0.50 x/h=0.

x/h=5.

x/h=7.

x/h=11.

x/h=15.


○ Experimental

Numerical

Figure 5.6.d. Fluctuating axial particle velocities along the step for LP flows.

77

78


u/U o

1.5

2.50

2.00

Height (y/h)

x/h=2

x/h=5

x/h=7

x/h=9

x/h=14

1.50

1.00

0.50

0.00 D istance along the step (x/h)

Figure 5.7.a. Streamwise gas velocities along the step for GP flows. 0

u'/Uo

0.2

2.50

2.00

Height (y/h)

x/h=

x/h=

x/h=

x/h

x/h=1

1.50

1.00

0.50


○ Experimental

Numerical

Figure 5.7.b. Fluctuating streamwise gas velocities along the step for GP flows.


0

u/Uo 1.5

2.50

Height (y/h)

2.00 x/h=

1.50

x/h=

x/h=

x/h=9

x/h=1 4

1.00

0.50

0.00 Distance along the step (x/h) Figure 5.7.c. Streamwise mean velocity for 150μm glass particles. 0

u'/Uo

0.3

2.50

Height (y/h)

2.00 1.50

x/h=2

x/h=5

x/h=7

x/h=9

x/h=1


Numerical

Figure 5.7.d. Fluctuating streamwise particle velocities for 150μm glass particles.

79

80


With the LP flow showing satisfactory agreement, the code is further validated to substantiate the numerical findings of GP flows against the experimental data of Fessler and Eaton (1999). Figure 5.7a shows the numerical comparison against the experimental findings and it can be seen that a fairly good agreement have been obtained between the experimental and its numerical counterpart, minor under prediction have been observed within a height of y/h<1 for the first two sections considered along the step. Figure 5.7b shows the mean streamwise fluctuating velocities along the step and it can be there is a general under prediction along the length of the step with some minor over prediction for a height of y/h<1 at section x/h=2. But however, the trend as seen from the numerical simulation is in lines with the experimental data. Figure 5.7c shows the comparison of the mean streamwise particle velocities for the 150µm glass particle against its experimental counterpart, it can be seen that there is a good agreement between the experimental and the numerical findings all along the step. The simulated streamwise fluctuating velocities are compared against the experimental findings in figure 5.7d and it can be seen that a fairly good agreement is felt along various sections of the backward-facing step geometry. From the comparisons of the GP flows, it is worth while to note, that mean velocities of the particles are lower at x/h=2 at the entry of the step than the gas phase, this is similar to the fully developed channel flow before the step, as reported in the experiments of Kulick et al. (1994), wherein the particles at the channel centreline exhibit lower streamwise velocities than that of the fluid as a result of cross-stream mixing. However, the gas velocity lags behind the particles aft of the step as particle inertia is slower to respond to the adverse pressure gradient than that of the fluid. At the fluctuation level, the particles exhibit higher values than the gas which again is attributed to the cross-stream mixing (Kulick et al., 1994).

5.3. Results and Discussion 5.4.1. Particle Response- Mean Velocity Level With the numerical code apt to replicate the experimental findings of the GP and LP flows, we now embark on a mission to study the response of the particles to the surrounding carrier phases. In order to proceed with this endeavour, four different Stokes numbers, by invariably changing the particle response time have been chosen for both the GP and the LP flows. For this numerical experiment the step geometry of Fessler and Eaton (1995) has been adopted, while the inlet velocity for the carrier and the particulate phase corresponds to that of the experimental conditions of Founti and Klipfel (1998). The four different Stokes number chosen corresponds to 0.05, 0.5, 2.0 and 6.0. Particles with Stokes number 0.05 acts as tracers to the carrier phase and are widely used by experimentalists for PIV/LDA study. The Stokes number of 0.5 corresponds to unveil realistically the response of particles for St≤1. Stokes number of 2.0 was chosen such as to fall within the range of overshoot phenomena (Chein and Chung, 1987; Hishida et al., 1992; Ishima et al., 1993), wherein the particles is said to disperse more readily than the fluid, while higher Stokes number of 6.0 is to study the independent responsitivity of the particles in relation to the two different carrier phases namely the air and diesel oil. In order to represent the results more qualitatively 12 interrogation points made up of a matrix of three sections (x/h=2, 7 & 14) along the length of the step and four sections along the height of the step (y/h=0.5, 1.0, 1.5, 2.0) have been considered.


81

2.50

2.00

St=0.05 St=0.50 St=2.00 St=6.00

y/h

1.50

1.00

0.50

0.00

0

0.2

0.4

0.6

0.8

up/U o Figure 5.8.a. Mean streamwise particle velocities for varying Stokes number for LP Flows. 2.50

St=0.05 St=0.50 St=2.00 St=6.00

2.00

y/h

1.50

1.00

0.50

0.00

0

0.005

0.01

0.015

0.02

0.025

0.03

TKP/U 2o Figure 5.8.b. Fluctuating streamwise particle velocities for varying Stokes number for GP flows.

Figure 5.8a shows the mean streamwise particle velocities along a section for the LP flows for varying Stokes number, the monitoring stations along the height of the step has been depicted with the help lines running with constant y/h values. Figure 5.8b also shows the monitoring

82


stations of particle fluctuation for the GP flows along varying Stokes number. From both the figures shown here, the mean velocities as well as the fluctuations increase with a corresponding increase in Stokes number for LP and GP flow respectively.

1.20 x/h=2; y/h=0.5 1.00

up/Uo

0.80 0.60 0.40 0.20 0.00

1.00

2.00

3.00 St

4.00

5.00

6.00

Figure 5.9.a. Mean streamwise particle velocities for varying Stokes number along the height of the step for x/h=2 & y/h=0.5.

1.20 x/h=2; y/h=1.0 1.00

up/U o

0.80 0.60 0.40 0.20 0.00

1.00

2.00

3.00 St

4.00

5.00

6.00

Figure 5.9.b. Mean streamwise particle velocities for varying Stokes number along the height of the step for x/h=2 & y/h=1.0.


83

1.20 x/h=2; y/h=1.5

up/Uo

1.00 0.80 0.60 0.40 0.20 0.00

1.00

2.00

3.00 St

4.00

5.00

6.00

Figure 5.9.c. Mean streamwise particle velocities for varying Stokes number along the height of the step for x/h=2 & & y/h=1.5.

1.20 x/h=2; y/h=2.0 1.00

up/Uo

0.80 0.60 0.40 0.20 0.00

1.00

2.00

3.00 St

4.00

5.00

6.00

Figure 5.9.d. Mean streamwise particle velocities for varying Stokes number along the height of the step for x/h=2 & y/h=2.0.

In this part of the results and discussion section the response of the particles to the mean flow with two different carrier mediums are investigated in relation to varying Stokes number. Three sections along the step and four along the height of the step have been investigated. Figures 5.9a-d shows the plot of the normalized mean particle velocities for x/h=2 section along varying Stokes number across the step height. The solid lines with circles in the figure depict the particle behaviour in GP flow, while the broken lines with squares depict again the particle behaviour but in LP flow. It can be seen from the figures that the mean particle velocities increase with a corresponding increase in its Stokes number for both

84


the carrier phases with the maximum particulate velocity occurring for the LP flows as one progresses along the height. There is along a steady increase in the particulate velocities along the height of the step for up to y/h=1.5 after which there is a meagre drop for y/h=2.0 where in the wall boundary conditions try to retard the flow. The particle velocities for the GP tend to overtake its counterpart LP flow at section y/h=2.0 but however not for the maximum Stokes number considered in our study. 0.80 x/h=7; y/h=0.5 0.70

uup/Uo p/Uo

0.60 0.50 0.40 0.30 0.00

1.00

2.00

3.00 St

4.00

5.00

6.00

Figure 5.10.a. Mean streamwise particle velocities for varying Stokes number along the height of the step for x/h=7 & y/h=0.5. 0.80 x/h=7; y/h=1.0

up/Uo

0.70 0.60 0.50 0.40 0.30 0.00

1.00

2.00

3.00 St

4.00

5.00

6.00



85

0.80 x/h=7; y/h=1.5

up/U o

0.70 0.60 0.50 0.40 0.30 0.00

1.00

2.00

3.00 St

4.00

5.00

6.00

Figure 5.10.c. Mean streamwise particle velocities for varying Stokes number along the height of the step for x/h=7 & y/h=1.5.

0.80 x/h=7; y/h=2.0

up/Uo

0.70 0.60 0.50 0.40 0.30 0.00

1.00

2.00

3.00 St

4.00

5.00

6.00


Figures 5.10a-d shows the mean particulate velocities at section x/h=7 along the height of the step geometry, it can also be seen here that the velocities keep increasing for a height of about y/h=1.5 after which there is a small drop due to the wall interference. However in this section of the step geometry, it can seen that the particulate velocities for the GP flows most of the time exceed than that of the particles in the LP except for section y/h=0.5. This is attributed to the fact that the GP flow particles are free to move in a less restricted

86


environment like air while particles in the LP flow move in a highly viscous environment, which fundamentally restricts it motion.

0.70

up/Uo

x/h=14; y/h=0.5

0.60

0.50

0.40 0.00

1.00

2.00

3.00 4.00 St

5.00

6.00

Figure 5.11.a. Mean streamwise particle velocities for varying Stokes number along the height of the step for x/h=14 & y/h=0.5.

0.70 x/h=14; y/h=1.0

up/Uo

0.60

0.50

0.40 0.00

1.00

2.00

3.00 4.00 St

5.00

6.00



87

0.70 x/h=14; y/h=1.5

up/Uo

0.60

0.50

0.40 0.00

1.00

2.00

3.00 4.00 St

5.00

6.00

Figure 5.11.c. Mean streamwise particle velocities for varying Stokes number along the height of the step for x/h=14 & y/h=1.5.

0.70 x/h=14; y/h=2.0

up/Uo

0.60

0.50

0.40 0.00

1.00

2.00

3.00 4.00 St

5.00

6.00


Figures 5.11a-d shows the plot of particle velocities for the section x/h=14 along the step, here also it can be seen that similar trend corresponding to two previous sections along the step is felt, with the increase in the particle velocities along the height and also with the particles in GP flow exceeding than that of the LP flow. For all the three x/h sections considered along the length of the step it can be seen that there is decrease in the particle velocities, which is attributed to the fact that particle losing their momentum as they travel along the step of the geometry.

88


5.4.2. Particle Response-Turbulence Level In this part of the results and discussion section, particles turbulent behaviour is investigated for two variants of carrier phase flows namely the GP and the LP flows. Here again three sections along the length of the step and four along its height have been considered. Figures 5.12a-d shows the turbulent kinetic energy of the particles normalized by its free stream velocity at section x/h=2 for four different heights and it can be seen that for all sections considered there has been an increase in the particle kinetic energy with a subsequent increase in the Stokes number for the GP flows but however for the LP flows there has been a decrease with subsequent increase in Stokes number. It can also be seen along the height of the step a decrease in the particulate kinetic energy is pronounced for both GP as well as LP flows. Figures 5.13a-d shows the particulate turbulent kinetic energy for the section x/h=7 of the step. While there have been a steady increase in the kinetic energy with a rise in the Stokes number for the GP flows, they seem to work in the opposite fashion for the LP flows which show a decrease with increase in Stokes number. It can also be seen that the magnitude of the kinetic energy is a fold less than at the section x/h=2 and that also a decrease is felt with the increase in the height of the step for the corresponding Stokes number. Figures 5.14a-d also shows a similar pattern for the section x/h=14 at the farther end near the exit of the geometry. The magnitude of the particulate turbulent kinetic energy is lesser than the previous section of x/h=7 and x/h=2. However, the turbulent kinetic energy of the particles seems to decrease with the subsequent increase in the Stokes number for LP flows.

0.050 x/h=2; y/h=0.5

TKP/Uo

2

0.040

0.030

0.020

0.010

0.000 0.00

1.00

2.00

3.00

4.00

5.00

6.00

St Figure 5.12.a. Fluctuating streamwise particle velocities for varying Stokes number along the height of the step for x/h=2 & y/h=0.5.


89

0.050 x/h=2; y/h=1.0

TKP/U o

2

0.040

0.030

0.020

0.010

0.000 0.00

1.00

2.00

3.00

4.00

5.00

6.00

St Figure 5.12.b. Fluctuating streamwise particle velocities for varying Stokes number along the height of the step for x/h=2 & y/h=1.0.

0.050 x/h=2; y/h=1.5

TKP/Uo

2

0.040

0.030

0.020

0.010

0.000 0.00

1.00

2.00

3.00

4.00

5.00

6.00

St Figure 5.12.c. Fluctuating streamwise particle velocities for varying Stokes number along the height of the step for x/h=2 & y/h=1.5.

90


0.050 x/h=2; y/h=2.0

TKP/Uo

2

0.040

0.030

0.020

0.010

0.000 0.00

1.00

2.00

3.00

4.00

5.00

6.00

St Figure 5.12.d. Fluctuating streamwise particle velocities for varying Stokes number along the height of the step for x/h=2 & y/h=2.0.

0.030 x/h=7; y/h=0.5 0.025

TKP/U o

2

0.020 0.015 0.010 0.005 0.000 0.00

1.00

2.00

3.00

4.00

5.00

6.00



91

0.030 x/h=7; y/h=1.0 0.025

TKP/Uo

2

0.020 0.015 0.010 0.005 0.000 0.00

1.00

2.00

3.00

4.00

5.00

6.00


0.030 x/h=7; y/h=1.5 0.025

TKP/Uo

2

0.020 0.015 0.010 0.005 0.000 0.00

1.00

2.00

3.00

4.00

5.00

6.00


92


0.030 x/h=7; y/h=2.0

0.025

TKP/Uo

2

0.020 0.015 0.010 0.005 0.000 0.00

1.00

2.00

3.00 St St

4.00

5.00

6.00

Figure 5.13.d. Fluctuating streamwise particle velocities for varying Stokes number along the height of the step for x/h=7 & y/h=2.0.

0.020 x/h=14; y/h=0.5

TKP/Uo

2

0.015

0.010

0.005

0.000 0.00

1.00

2.00

3.00

4.00

5.00

6.00



93

0.020 x/h=14; y/h=1.0

TKP/Uo

2

0.015

0.010

0.005

0.000 0.00

1.00

2.00

3.00

4.00

5.00

6.00


0.020 x/h=14; y/h=1.5

TKP/Uo

2

0.015

0.010

0.005

0.000 0.00

1.00

2.00

3.00

4.00

5.00

6.00


94


0.020 x/h=14; y/h=2.0

TKP/Uo

2

0.015

0.010

0.005

0.000 0.00

1.00

2.00

3.00

4.00

5.00

6.00

St Figure 5.14.d. Fluctuating streamwise particle velocities for varying Stokes number along the height of the step for x/h=14 & y/h=2.0.

5.4.3. Summary of Particulate Responsitivity The above two sections which outlined the particle response at the mean velocity level and at the turbulence level shows that the mean velocity of the particles increase with a subsequent increase in the Stokes number for both the carrier phases namely the gas and the liquid. The mean particulate velocity not only increase with the Stokes number but is also higher than its corresponding carrier phase velocities for the three Stokes number viz St=0.5, 2.0 & 6.0 considered in our study. This is quite in lines with the recent experimental data of Ishima et al (2007) and the phenomenon is explained with the help of the particle terminal velocity which gives a rough approximation as a percentage of how much the particle velocity exceeds the carrier phase velocity. The other reason for the particle velocity to lead the carrier phase is the attribute of the particulate phase to respond slowly to the adverse pressure gradient dominant in shear flow geometries like back-ward facing step, in lieu to the carrier phase. The particle turbulent kinetic energy plots for both the GP and the LP flows depict the response of the particles at the turbulence level, across these plots it can be summarized that while there is a steady increase in the particulate turbulence for the GP flows with successive increase in Stokes number, with some sections showing even a 100% increase between the minimum and the maximum Stokes number considered. However, for the LP flows, the magnitude of the increase in the particulate turbulence across the increasing Stokes number is not as characteristic as its counterpart. Across the same sections for LP flows the majority of the trend shows a decrease after which they more of less remain a constant. Previous studies of liquid-particle flows in vertical channel (Ishima et al., 2007; Borowsky & Wei, 2007)


95

shows that with the increase in the Stokes number there is usually a corresponding increase in the particulate turbulence, however the flow considered in this study is a shear flow geometry, which basically depicts a totally different flow feature unlike the simple channel geometry. In these lines even the Turbulence Modulation (TM) of the shear flow geometry for the well established gas-particle flows using the same set of experimental data employed in this study does not seem to well correspond with the models been employed and formulated for the vertical channel flows (Mohanrangam & Tu, 2007). Thereby, given the complexity of the problem much deeper understanding and experimental data may be required to ascertain the same. The readers are also advised that the two sets of experimental data (Fessler & Eaton, 1997; Founti & Klipfel, 1998) used in his study were not alike in all respects within the flow field with parity only in carrier phases. However, for the current set of results obtained the particle response to the turbulence for the LP flow in comparison to the GP flow may be explained in terms of the carrier phase employed to study the particle response. Firstly, the density and the viscosity used to study LP flow are approximately 709 and 257 times higher than that of the GP flow, which basically prohibits the fluctuating motion of the particle. Another rationale being, in regions of strong mean velocity gradient, the streamwise particle fluctuating velocities are determined more by the mean gradient than by the actual response of the particles to turbulent fluctuations. In the absence of the same the particle velocity fluctuations tend to be lower than the fluid velocity fluctuations as noted from the experiments of Fessler and Eaton (1995), it was also stated by the same authors that in their experiments wall-normal fluctuating particle velocities were lower that fluid, which was attributed to the large inertia of the particles making them unresponsive to many of the fluid motions. From these conclusions the eventual decrease in the particle fluctuation is more or less attributed to the decrease in the velocity gradient with a corresponding increase in Stokes number. The cross-stream mixing, which attributes towards higher particle fluctuating velocities in GP flow may be prohibitive in LP flow considering the elevated density and viscosity of LP flows.

Air-Liquid Flows In the modelling of micro-bubble laden Air_liquid flows, two sets of governing equations for momentum were solved. The generic CFD code ANSYS CFX 11 (ANSYS, 2006) was employed as a platform for two-fluid flow computation. The built-in Inhomogeneous and MUSIG models had been adopted for our numerical simulations. Figure 6.1a shows the schematic diagram of the numerical model used in our computations. Numerical simulations were performed with a velocity inlet and a pressure outlet, on the left and right side of the 2D computational domain respectively. The top wall is modelled as a friction free boundary condition, wherein the height of the computational domain reflects only half the height of the original test section. The bottom part of the domain has been divided into three distinct sections, section 1 & 3 were modelled as no-slip walls for liquid and free-slip for microbubbles, emulating the experimental boundary conditions. The section 2 is specified as the inlet boundary condition for our gas inlet imitating the experimental conditions of gas injection thought the porous plate.

96

K. Mohanarangam and J.Y. Tu Traction-free opening Velocity Inlet

Pressure outlet

y x

Section 2

Section 1

0.280m

0.178m

0.057m

Section 3

0.254m

Figure 6.1.a. Schematic diagram of the numerical model.

A uniform liquid velocity was specified at the inlet of the test section, different gas flow rates were specified along the section 2 of the computational domain, the free stream velocities and the gas injection rates used in the simulations are summarized in Table 1. An area permeability of 0.3 which lies in line with the sintered metal used in the experiments and also employed in the numerical work of Kunz et al. (2003) is used all along section 2 for gas injection purposes. At the outlet, a relative averaged static pressure of zero was specified. For all flow conditions, reliable convergence were achieved within 2500 iterations when the RMS (root mean square) pressure residual dropped below 1.0 × 10-7. A fixed physical time scale of 0.002s is adopted for all steady state simulations. Table 6.1. Input boundary conditions for the computational model Air flow rate Qa(m3/s)

Case Q0-V9.6 (Cfo) Q1-V9.6 Q2-V9.6 Q3-V9.6 Q4-V9.6 Q5-V9.6 Q0-14.2(Cfo) Q1-V14.2 Q2-V14.2 Q3-V14.2 Q4-V14.2 Q5-V14.2

0 0.001 0.0015 0.002 0.0025 0.003 0 0.001 0.0015 0.002 0.0025 0.003

Section 1

Water free stream velocity (m/s) 9.6 9.6 9.6 9.6 9.6 9.6 14.2 14.2 14.2 14.2 14.2 14.2

ReL based on the total plate length 7.66 x 106 7.66 x 106 7.66 x 106 7.66 x 106 7.66 x 106 7.66 x 106 1.13 x 107 1.13 x 107 1.13 x 107 1.13 x 107 1.13 x 107 1.13 x 107

Section 2

Figure 6.1.b. Computational grid used for computations.

Section 3


97

In handling turbulent micro-bubble flow, unlike single phase fluid flow problem, no standard turbulence model has been customized for two-phase (liquid-air) flow. Nevertheless, numerical investigation revealed that standard k-ε model predicted an unrealistically high gas void fraction peak close to wall (Frank et al., 2004, Cheung et al., 2006). The k-ω based Shear Stress Transport (SST) model by Menter (1994) provided more realistic prediction of void fraction close to wall. 30.00

25.00

U

+

20.00

15.00 +

+

+

+

U =y ; U = (1/k)Ln(Ey ) Numerical

10.00

5.00

0.00 1

10

100 y+

1000

10000

100000

Figure 6.2. Comparison of simulated boundary layer velocity profile with the standard law of curves for U∞=14.2m/s.

The SST model is a hybrid version of the k-ε and k-ω models with a specific blending function. Instead of using empirical wall function to bridge the wall and the far-away turbulent flow, the k-ω model solves the two turbulence scalars right up to the wall boundary. This approach eliminates errors arising from empirical wall function and thus provides better prediction at the near wall region. The SST model is thereby employed in the present study. Moreover, to account for the effect of bubbles on liquid turbulence, the Sato’s bubble-induced turbulent viscosity model (Sato et al., 1981) has been adopted as well. Figure 6.1b shows the mesh distribution within the computational model, wherein a nonuniform orthogonal mesh with 151x101 grid points was spanned over the whole computation domain. Wall-normal clustering was used in order to resolve the boundary layer and the height of the two closest cells next to the walls was designed to be y+≤1, the height of which can be approximated from the logarithmic law

98


U ∞ 1 ⎛ uτ L ⎞ 1 = ln⎜ ⎟+B− uτ κ ⎝ υ ⎠ κ =

⎛ U L uτ ln⎜⎜ ∞ κ ⎝ υ U∞ 1

⎞ 1 ⎟⎟ + B − κ ⎠

Using the relation

U ∞ ⎛⎜ 2 ⎞⎟ = ⎜C ⎟ uτ ⎝ f ⎠

1/ 2

(3.28)

and substituting it to the above equation, we get

⎛ 2 ⎜ ⎜C ⎝ f

⎞ ⎟ ⎟ ⎠

1/ 2

⎡ ⎛Cf = ln ⎢Re ∞ ⎜⎜ κ ⎢ ⎝ 2 ⎣ 1

⎞ ⎟⎟ ⎠

1/ 2

⎤ 1 ⎥+B− κ ⎥⎦

(3.29)

The skin friction co-efficient (Cf) can be solved from the above equation (3.29) by substituting values of B=5.0 and κ=0.41. uτ can be solved from the equation (3.28) and the distance of the first grid point from the wall can be solved from the equation

y first

y +υ = uτ

6.1. Results and Discussion In order to better understand the drag reduction phenomenon and to better comprehend the various mechanisms behind it, a prognostic approach has been carried out in throughout this section starting from the single phase and then to the air-liquid micro-bubble flows, wherein at each juncture our numerical outcomes have been verified and validated against well established experimental and numerical findings. Before investigating the physical phenomenon of the drag reduction, it is very crucial to ascertain adequate grid spacing has been generated to resolve the inner, buffer and the outer boundary layer. Figure 6.2 shows the single phase boundary layer velocity profile of the outlet compared against the standard law of the wall curves for the maximum free stream velocity of 14.2 m/s. The excellent agreement with the standard curve clearly confirmed that sufficient mesh resolution has been spanned throughout the boundary layer to capture the associated velocity gradient. The 9.6 m/s case had the similar y+ values and hence been not shown here for brevity.


99

6.1.1. Experimental Validation (Inhomogeneous Model) With the mesh spacing sufficiently fine enough to resolve the turbulent boundary layer attention is then focussed on the micro-bubbles drag reduction mechanism. Figure 6.3 shows the comparison of the predicted skin friction ratios of the two-fluid inhomogeneous model against the experimental data of Madavan et al. (1984) along various gas injection rates (Q1Q5) for both the freestream velocities of 9.6 and 14.2m/. Herein, Cf & Cfo are the skin-friction co-efficients with and without the gas injection respectively. The skin-friction co-efficient throughout our numerical study have been obtained by averaging out the entire flat plate of ‘section 3’. It can be observed from the figure that satisfactory agreement was obtained using the inhomogeneous two-fluid model for both the free stream velocities considered in our study. 1.00

14.2m/s (Expt) 14.2m/s (Num) 9.6m/s (Expt) 9.6m/s (Num)

C f /C fo

0.90

0.80

0.70

0.60

0.50 0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

Qa

Figure 6.3. Comparison of skin friction co-efficient with the experimental findings for U∞=14.2m/s & 9.6m/s.

Readers are advised that these results have been compared treating that the experimental errors encountered during various process of measurements are nil (error percentages of the experimental data). However, hypothetical they may seem, the numerical predictions were in remarkable agreement with the measurements. It can be seen that there is always an increase in the drag reduction (DR) or a decrease in the Cf/Cfo ratio with a corresponding increase in the gas flow rates for both the free stream velocities considered in our simulation, which again is congruent to the previous micro-bubble studies. In all the simulations presented, buoyancy was included as a necessary force, as one could find a marked change from the experimental results with the plate on ‘Top’ and ‘Bottom’ configuration.

100


0.025

Q1-V14.2 0.020

Q2-V14.2 Q3-V14.2 Q4-V14.2 Q5-V14.2

Y (m)

0.015

0.010

0.005

0.000 6.00

8.00

10.00

12.00

14.00

16.00

Streamwise liquid velocity (m/s)

Figure 6.4.a. Velocity profiles for varying gas injection rates for free stream velocity U∞ = 14.2m/s. 1.00 0.95 0.90 0.85

ul/uul

0.80 0.75 0.70 Q1-V14.2

0.65

Q2-V14.2 Q3-V14.2

0.60

Q4-V14.2 0.55

Q5-V14.2

0.50 1

10

y

+

100

1000

Figure 6.4.b. Change in the mean flow velocity for the carrier phase along the boundary layer for U∞ = 14.2m/s.


101

6.1.2. Investigation of Mechanisms of Drag Reduction With the skin friction co-efficients showing fairly good comparison for the two-fluid inhomogeneous model, it can be further investigated to study the various mechanisms of drag reduction. To begin with the mean streamwise velocities of the carrier phase are scrutinised. Figure 6.4a shows the mean streamwise liquid velocity profiles along varying gas injection rates for the high Reynolds number case of 14.2m/s. As depicted, a clearly marked change in the velocity profile can be seen with a subsequent increase in the gas flow injection rates, which is certainly in relation to the large amount of micro-bubbles present along the boundary layer as shown from the void fraction profiles in 6.7b. The streamwise velocities reveal that with the increase in the gas flow rates, there is a subsequent and a gradual increase in the streamwise velocities in the outer layer. In contrast, a close investigation of the velocities within the buffer and inner layer shows an opposite phenomenon. In figure 6.4b, the velocity profiles of the liquid phase for the micro-bubbles laden flows have been normalized with the corresponding single phase liquid velocities along the length of the boundary layer, where by any velocity change felt in the carrier liquid phase is reflected as an exit of the ratio from unity, from the figure it can be revealed, that there is a marked decrease in the mean streamwise velocities with a subsequent increase of the gas injection rates. It can also be seen, that the flow undergoes a maximum reduction in the mean velocity of about 40% for the highest gas flow rate, while it is quite nominal and about 13% for the lowest gas injection rate, this trend keeps increasing until a y+ value of 100, aftermath of which there is a spike for the largest of the three gas flow rates and then a downward trend follows. 0.06 Q0-V14.2 Normal liquid velocity (m/s)

0.05

Q1-V14.2 Q2-V14.2 Q3-V14.2

0.04

Q4-V14.2 Q5-V14.2

0.03

0.02

0.01

0.00 1

10

y+

100

1000

Figure 6.4.c. Liquid normal velocity for the carrier phase along the boundary layer for U∞ = 14.2m/s.

102


These findings reported above are in lines with the DNS findings of Ferrante and Elghobashi (2004), wherein the presence of micro-bubbles in the turbulent boundary layer results in a local positive divergence of the fluid velocity, ∇ • U > 0 , creating a positive mean velocity normal to (and away from) the wall which in turn, reduces the mean streamwise velocity and displaces the quasi-streamwise longitudinal vortical structures away from the wall. The shifting of the vortical structures away from the wall indicates that the ‘sweep’ and ‘ejection’ events (Robinson, 1991), which are located respectively at the downward and upward sides of these longitudinal vortical structures, are moved farther away from the wall, thereby reducing the intensity of wall streaks along the wall and consequently decreasing the skin-friction. It was also reported that there is shift with respect to the location of peak Reynolds stress production away from the wall, thus reducing the production rate of turbulence kinetic energy and enstrophy.

Water flow U∞

Outlet

y

x

Figure 6.4.d. Air void fraction contour plot for Q5-V14.2.

Figure 6.4c shows the plot of water normal velocity through varying gas injection rates, it can be seen that there is generally an increasing trend in the velocities and then a decrease which is followed by a maximum peak, and this is in direct relation to the loss incurred by the flow along the streamwise direction, across varying gas injection rates. There is also a sudden spike in the normal velocities within a y+ range of 60-120. However, the onset of the increase and the occurrence of the maximum differ in accordance to the gas injection rates. For the


103

three higher injection rates (Q3-Q5) the location of the start of sudden increase and the occurrence of maximum seems to occur more or less in unison, but their magnitude of maximum normal velocities differ wide apart. While for the lower gas injection rates (Q1 & Q2) the location and the magnitude are more distinct and separated wide apart. It can also be seen that the unladen wall normal velocity is quite smaller in lieu with the laden normal velocities. This can be further confirmed from the contour plot of the air void fraction along the boundary layer as shown in figure 6.4d, for the highest air flow rate considered in our study, where there is a small layer of water, which is immediately followed on the top by a thick layer of air and then followed again by water, herein due to the inherent presence of the micro-bubbles in the middle section, caused an upward shift in the water normal velocities.

40.00 +

30.00 25.00 U+

+

+

+

U =y ; U = (1/k)ln(Ey ) Q1-V14.2 Q2-V14.2 Q3-V14.2 Q4-V14.2 Q5-V14.2

35.00

20.00 15.00 10.00 5.00 0.00 1

10

100 y+

1000

10000

100000

Figure 6.5. Change in the boundary layer for varying gas flow rates for U∞ = 14.2m/s.

Figure 6.5 shows the plot of non-dimensional streamwise velocity profiles along the boundary layer for varying gas flow rates at the middle of ‘section 3’. The presence of the micro bubbles can be felt for a y+≥10, where in there is a gradual thickening of the viscous zone with an upward shift of the logarithmic region, while the inner layer seems more or less unaltered. With these findings, it can be ascertained that the important aspect in achieving drag reductions is the accumulation of the micro bubbles within a critical zone in the buffer layer. This is in lieu with the experimental findings of Villafuerte & Hassan (2006), whereby high drag reductions were reported due the accumulation of the micro bubbles within a range of 15 ≥ y+ ≤ 30.

6.1.3. Turbulence Modulation (TM) The plot depicted in the figure 6.6 demonstrates the turbulent modulation (TM) of the liquid phase in the presence of the micro-bubbles and is given by the ratio of the micro-

104


bubble laden flow r.m.s streamwise velocity to the unladen r.m.s streamwise velocity. These plots signify that any TM felt in the carrier liquid phase is reflected as an exit of the ratio from unity. It can be seen from the plot, across various gas injection rates a marked attenuation is felt up to a distance along the boundary layer and then a subsequent increase, which is attributed towards the turbulence enhancement of the liquid phase. It is also worthwhile to note that the flow has a tendency to attenuate more for higher gas flow rates. On the other hand there is a turbulence augmentation effect pronounced more in the outer layer of the boundary. The marked attenuation felt for a small distance from the wall is attributed to the presence of a thin lining of liquid all along the wall (as explained above). However, in order to explain the augmentation of the turbulence felt within the boundary, the “bubble-repelling” and the “bubble-rising” events observed from the experiments of Y.Murai et al. (2006) is used. From their findings, it is outlined that the vertical rise velocity of the bubble or the “bubble-rising” event towards the wall is only about 5% of the streamwise velocity, after which the bubble reaches equilibrium with its surroundings and starts its journey back through the “bubble-repelling” event away from the wall, but however this downward journey accounts for 25% of the streamwise velocity. 2.00 1.80

Q1-V14.2 Q2-V14.2

1.60

Q3-V14.2

1.40

Q5-V14.2

u'/u'l

Q4-V14.2

1.20 1.00 0.80 0.60 0.40 0.20 0.00 1.00

10.00

y

+

100.00

1000.00

Figure 6.6. Turbulence Modulation (TM) along the boundary layer for U∞ = 14.2m/s.

Although the aforementioned experimental observations refer to individual bubble motion which can only be tracked numerically using Lagrangian approach, the phenomenon of turbulence augmentation in the carrier phase is taken care in our simulation through the SATO (Sato et al; 1981) model, which accounts for the additional viscosity generated through the bubble slip velocity, wherein the vortices are formed behind the bubbles by their motion, thereby causing an increase in the turbulence levels in the outer layer of the boundary. It can


105

also be seen that this turbulence enhancement is more pronounced in the outer layer of the boundary, while most part of the inner and the buffer layer experiences attenuation. 0.80 Q1-V9.6

0.70

Q2-V9.6 Volume fraction of air

0.60

Q3-V9.6 Q4-V9.6

0.50

Q5-V9.6 0.40 0.30 0.20 0.10 0.00 0.00

500.00

1000.00 + y

1500.00

2000.00

Figure 6.7.a. Volume fraction of air along the outlet for U∞ = 9.6m/s.

Figure 6.7a shows the void fraction profiles for the dispersed phase for the freestream velocity of 9.6m/s, along the outlet plane of the geometry, it can be seen that there is a sharp increase in the void fraction for a y+ value of about 200 for the maximum flow rate, with the maximum occurring there, where as from figure 6.7b, it can be seen that the void fraction profiles for the dispersed phase occurring in a slightly different pattern with the maximum occurring at a distance around y+ =150. This occurrence of maximum void fraction can be best used to explain the degree of drag reduction between the two Reynolds numbers considered in our study. For the 9.6m/s case where the maximum void fraction occurs at a distance higher than that of the 14.2m/s, a higher degree of drag reduction is seen and this is attributed to the fact that the bubbles can rise and distribute themselves within the boundary layer there by expanding the boundary layer (buffer and outer layer) and thus causing a greater drag reduction, but however for the high Reynolds number 14.2m/s case the bubbles have lower residence time to re-distribute themselves within the boundary layer there by causing a smaller drag reduction and consequently a higher Cf/Cfo value (Kodama et al. 2000). From figures 6.7a and 6.7b, it can be seen that with the increase in the gas flow rates the void fraction of the injected air increases for all the cases and later flatten out due to the dispersion and also the washing down effect of the continuous phase. From the close examination of the graphs it can be seen that there exists a thin film of liquid covering all over the wall irrespective of the injected gas flow rates, these finding are in accordance to the experimental findings of Murai et al. (2007), who later concluded that this liquid film thickness gradually becomes thinner from the front to the rear part of the bubble such that the skin friction varies along the coordinate. From the work of Tinse et al. (2003) it can be deduced that thinner this

106


liquid film becomes the lower the shear stress the film has. This has also been verified from our numerical findings that the skin friction ratio decreases with the increase in the gas flow due to the inherent thinning of the liquid film adjacent to the wall. 0.80 0.70

Volume fraction

0.60 0.50 0.40

Q1-V14.2 Q2-V14.2 Q3-V14.2

0.30

Q4-V14.2 Q5-V14.2

0.20 0.10 0.00 0.00

500.00

1000.00 y

+

1500.00

2000.00

2500.00

Figure 6.7.b. Volume fraction of air along the outlet for U∞ = 14.2m/s.

6.1.3. Effect of Bubble Coalescence and Break-up in Drag Reduction In the light of the aforementioned results, one should notice that a prescribed bubble diameter has been specified throughout all numerical simulations for Inhomogeneous twofluid model. Although encouraging results have been presented in previous sections, the value of bubble diameter (i.e. 500 µm) unfortunately at best served as a fair engineering estimation which is calibrated against experimental data based on trail-and-error without solid physical interpretations. As discussed before, the constant diameter assumption may introduce numerical errors if the bubble coalescence and break-up become dominant in the problem, especially when the air injection rate is considerably high. In attempting to overcome this problem, the MUSIG model is introduced into the simulation allowing bubble diameter to be evaluated mechanistically using the coalescence and breakage kernels. Figure 6.8a shows the numerical comparison against its experimental counterpart for a free stream velocity of 9.6m/s, the MUSIG model in addition to Inhomogeneous model is used for comparison. Herein the MUSIG model had been specified 10 groups of bubbles, diameters ranging from 100µm-1000µm. As depicted in the figure, at high flow rates (i.e. Q4 and Q5), MUSIG model gives the best agreement while the inhomogeneous model tends to slightly over-predict the skin friction coefficients. However, using the default coefficients for


107

break-up and coalescence models serious under-prediction has been observed for the MUSIG model for low gas flow rates (i.e. Q1-Q3). Figure 6.8b shows the similar comparison of experimental data with higher free-stream velocity of 14.2 m/s. Analogy to the previous result, predictions of the MUSIG model appear marginally superior to the inhomogeneous at high gas injection rates (i.e. Q4 & Q5), while considerably under predictions have shown for lower gas injection rates. 1.00 Expt

0.90

Cf/Cfo

Two-Fluid-Inhomogenous MUSIG

0.80

0.70

0.60

0.50 0.0000

9.6m/s

0.0005

0.0010

0.0015 0.0020 0.0025 Flow rate of Air (Qa)

0.0030

0.0035

0.0040

Figure 6.8.a. Comparison of computed skin-friction co-efficient Inhomogeneous & MUSIG models U∞= 9.6m/s. 1.0 Expt 0.9

Two-Fluid-Inhomogenous

C f/C fo

MUSIG 0.8

0.7

0.6 U=14.2m/s 0.5 0.0000

0.0005

0.0010

0.0015 0.0020 Flow rate of Air (Qa)

0.0025

0.0030

0.0035

Figure 6.8.b. Comparison of computed plate drag co-efficient Inhomogeneous & MUSIG models U∞= 14.2m/s.

108


One possible reason attributed to the under-prediction of the MUSIG model for low gas injection rates could be the over-estimation of bubble break-up rate which sequentially introduced more small bubbles into calculations. These additional small bubbles were thereby dispersed with the boundary layer caused greater drag reduction on the surface. It should be emphasized that default model parameters of the MUSIG have been adopted directly in the above numerical investigation. These parameters were calibrated with bubbly flow condition where isotopic turbulence was assumed. At high air injection rate, such assumption may be quite close the physical behaviour as higher turbulence modulation has been introduced by the presence of bubbles. However, it may become invalid for low flow rates. In essence, it is well known that these model parameters may vary from cases to cases which should be recalibrated for particular flow condition. 1.00

Expt

0.90

MUSIG Breakup co-efficient=1.0 MUSIG Breakup co-efficient=0.05

Cf/Cfo

0.80

0.70

0.60

0.50 0.0000

9.6m/s

0.0005

0.0010

0.0015 0.0020 0.0025 Flow rate of Air (Qa)

0.0030

0.0035

0.0040

Figure 6.9.a. Comparison of skin friction co-efficients with different break-up co-efficients for U∞= 9.6m/s.

Based on the above argument, also serves as a confirmation of the above observations, another set of simulations have been carried out with the break up coefficient deliberately decreased 20 times to minimize the resultant bubble break-up rate. Figure 6.9a shows the corresponding predictions of the skin-friction coefficient plots for the free stream velocity of 9.6m/s. Compared with the default MUSIG model, the predicted results were generally in satisfactory agreement with measurements across varying flow rates, while a considerably improvements have been obtained for the lower air injection rates. Similar observations can also be found for the freestream velocity of 14.2m/s showing in Figure 6.9b. The effect of the reduced break-up rate can be exemplified by a closer visualization of the predicted bubble size distributions of the two numerical results. Figure 6.10a shows the predicted bubble size distribution at the outlet obtained from the default and the re-calibrated MUSIG models. With the default break-up coefficient, in both free-stream velocities, the relatively high volume fraction of small bubble clearly demonstrated that the default model


109

tends to create additional small bubble via the dominating break-up mechanism. In contrast, by limiting the break-up rate, bubble coalescence overcomes the break-up mode forming relatively higher volume fraction for the larger bubbles. 1.0

Expt

0.9

MUSIG breakup co-efficient=1.0 MUSIG breakup co-efficient=0.05 C f /C fo

0.8

0.7

0.6

14.2 m/s

0.5 0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0.0040

Flow rate of Air (Qa)

Figure 6.9.b. Comparison of skin friction co-efficients with different break-up co-efficients for U∞= 14.2m/s. 8.0% 7.0%

Volume fraction

6.0% MUSIG breakup co-efficient=0.05 5.0%

MUSIG breakup co-efficient=1.0

4.0% 3.0% 2.0% 1.0% 0.0% 100

200

300

400

500 600 d (µm)

700

800

Figure 6.10.a. Bubble diameter distribution function for Q1-V9.6.

900

1000

110


In the present study, the deduction of break-up coefficient arose only as an engineering estimation. In fact, as the air injection rates are considerably low, interactions between bubbles are relatively insignificant compared with that for the high injection rates. It is thereby unsurprising to re-calibrate the model constants for obtaining a “better” comparison. Although revealing the short-comings of the current kernels is certainly one of the findings, this drawback of the model should be circumvented by refining the model assumption and the mechanism which unfortunately is left far beyond the focus of this paper. Directing back to the theme of current work, one could easily state that good predictions of the skin-friction coefficients can be obtained by specifying a proper bubble size for each simulation. Nevertheless, one should also be reminded that bubble sizes may change significantly which is impossible to be represented by a fixed average value. This problem is further exacerbated; if rigorous bubble interactions are involved at high air injection rate. In practical micro-bubble problems, it is more easily to acquire the range of bubble size rather than exact bubble diameter. The MUSIG model which tailored to resolve the bubble size distribution mechanistically within a given range of bubble size appeared as the best candidate to resolve the physics embedded in micro-bubble drag reduction problems.

10.0% 9.0% MUSIG breakup co-efficient=0.05

8.0%

MUSIG breakup co-efficient=1.0

Volume fraction

7.0% 6.0% 5.0% 4.0% 3.0% 2.0% 1.0% 0.0% 100

200

300

400

500

600 d (µm)

700

800

900

1000

Figure 6.10.b. Bubble diameter distribution function for Q2-V14.2.

Conclusion In this article outlined above, a lot of work was undertaken numerically to study the behavior of two-phase turbulent flows of varying density regimes viz., Gas-Particle, LiquidParticle and Liquid-Air flows. In addition to the carrier and the dispersed phases mean and turbulent behavior, Turbulence Modulation (TM) has also been investigated. It is given by the


111

change in the carrier phase amidst the dispersed phase or the effect of the dispersed phase on to the carrier phase at the turbulence level. For the Gas-Particle flow, the particle-turbulence two-phase flow interaction has been successfully investigated with the Eulerian two-fluid model. The numerical code has been validated against the experimental results of Fessler and Eaton (1999) for mean velocity and the fluctuating velocities for both carrier and the dispersed phases. Two classes of particles sharing the same Stokes number, but different particle Reynolds number has also been investigated in this study. The majority of the results agree well with the experimental data; however there have been some minor discrepancies felt at the proximity of the experimental results. The Turbulence Modulation (TM) of the carrier phase for these two classes of particles have been studied along the three sections that is near the inlet (x/h=2), in the mid-section (x/h=7) and just aft of the exit (x/h=14). It can be concluded that even though the 70μm copper and 150μm glass particles share the same Stokes number, their behavior seems to be quite different, which suggests that Stokes number alone does not characterize the particle behavior, thereby making particle Reynolds number an important parameter in classifying the way the particles behave. Particles response to turbulent GP (Gas-Particle) and LP (Liquid-Particle) flow, behind a turbulent backward-facing step geometry have also been successfully analysed and simulated numerically using an Eulerian two-fluid model. A significant amount of work was undertaken to provide an in-depth understanding of the particle response, amidst turbulent flow conditions for two different carrier phases namely the gas and the liquid (diesel oil). From the two sets of experimental data, at the mean velocity level, the particles seem to ‘lead’ and later ‘catch up’ with the carrier phase for the LP flow, whereas they ‘lag’ behind and later ‘lead’ for the GP flow. While at the turbulence level, the particles seem to ‘lag’ and then ‘catch up’ for the LP flow while they ‘lag’ and phenomenally ‘lead’ for the GP flow. The detailed study and also the numerical diagnosis undertaken for turbulent particulate flows with two different carrier phases, to study the particle response both at the mean velocity and at the turbulence level, behind a shear flow sudden expansion geometry is quite unique and one of its kind, as there is no current published work dealing with the analysis and numerical validation of the same. Numerically the code was validated against the benchmark experimental data of Fessler and Eaton (1997) for GP and the experimental data of Founti & Klipfel (1998) for the LP flows. The numerical results revealed good agreement with the experimental data. From there the code was further used to investigate Stokes number effect on the two different carrier phases both at the mean velocity and at the turbulence level, for this exercise the experimental geometry of the GP flow and the inlet conditions of the LP flow were used. In order to present the results in a more methodical manner, 12 points consisting of a matrix of three sections along the length of the step and four along the height of the step were used to study the Stokes number effect on the two types of flows. At the mean velocity level the particles seem to move faster that the carrier phases for both the GP and the LP flow. However, at the particle fluctuation level, although the GP flow show an escalation with the increase in the Stokes number, the same feature seem to be absent in the LP flow, wherein the particle fluctuation seem to decrease and almost flatten out with the increase in its Stokes number. The main reason for this behaviour is the difference in the physical characteristics of the carrier phase namely the liquid, which is far denser than the

112


gas, this eventually changes the cross-stream and the mean gradient behaviour, which is shown to cause elevated particle fluctuations in the GP flow. For the Liquid-Air flows, the turbulent micro-bubble laden flow has been investigated with the help of two numerical models namely the two-fluid Inhomogeneous and MUSIG models, for two different free-stream velocities. Inhomogeneous model, which uses a fixed bubble diameter, shows a very good comparison of the skin-friction co-efficients with the experiment. This model is further probed to study the various physical phenomenon’s causing the drag reduction along the boundary layer, firstly it was observed that there is drop in the mean streamwise water velocities with a subsequent increase in the normal along varying gas injection rates. Secondly, the presence of the micro-bubbles caused turbulence attenuation for some distance along the boundary layer and later an augmentation was felt due to the shedding of the vortices behind the bubbles. Thirdly, the peak of the void fractions seem to differ in relation to the degree of drag reduction along the two free-stream velocities considered in our study. However, with respect to the drag reduction caused due to the presence of micro-bubbles in the turbulent boundary layer MUSIG model seem to show good predictions for higher gas flow rates while under predicting for lower gas flow rates. This poor prediction of the model at low flow rates was investigated to be the dominating break-up phenomenon that was taking place within the flow. This was done in order to represent the actual flow condition, where by groups of bubbles of varying bubble sizes are found within the boundary layer. Thereby allowing the MUSIG model to resolve the bubble size distribution mechanistically within a given range of bubble size and feeding it back to the Inhomogeneous model appeared as the best candidate to resolve the hidden physics in micro-bubble induced drag reduction problems.

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Righetti, M, & Romano, GP 2007, ‘Particle- fluid interactions in a plane near-wall turbulent flow’, Journal of Fluid Mechanics, Vol. 505, pp. 93-121. Robinson, S.K., 1991. Coherent motions in the turbulent boundary layer. Annual Review of Fluid Mechanics 23, 601-639. Ruck, B, & Makiola, B 1988, ‘Particle Dispersion in a Single-Sided Backward-Facing Step Flow’, International Journal of Multiphase Flow, Vol. 14, pp. 787-800. Sato, Y, Sadatomi, M, Sekoguchi, K 1981, ‘Momentum and heat transfer in two-phase bubbly flow – I’, International Journal of Multiphase Flow, Vol. 7, pp. 167-178. Sato, Y, Hishida, K 1996, ‘Transport process of Turbulence energy in particle-laden turbulent flow’, International Journal of Heat & Fluid Flow, Vol. 17, pp. 202-210. Sato, Y, Fukuichi, V, & Hishida, K 2000, ‘Effect of Inter-Particle Spacing on Turbulence Modulation by Lagrangian PIV’, International Journal of Heat and Fluid Flow, Vol. 21, pp. 554-561. Schuh, MJ, Schuler, CA, & Humphrey JAC 1989, ‘Numerical calculation of particle-laden gas flows past tubes’, AIChE Journal, Vol. 35, pp. 466-480. Shirolkar, JS, Coimbra, CFM, McQuay, Q 1996, ‘Fundamental Aspects of modeling turbulent Particle Dispersion in Dilute Flows’, Progress in Energy Combustion and Science, Vol. 22, pp. 363–399. Tisne, P, Aloui, F, & Doubliez, L 2003. ‘Analysis of wall shear stress in wet foam flow using the electrochemical method’, International Journal of Multiphase Flow, Vol. 29, pp. 841–854. Tu, JY, & Fletcher, CAJ 1995, ‘Numerical computation of turbulent gas-solid particle flow in a 90o bend’, AIChE Journal, Vol. 41, pp. 2187-2197. Tu, JY 1997. ‘Computational of turbulent two-phase flow on overlapped grids’, Numerical Heat Transfer Part B.; Vol. 32, pp. 175–195. Villafuerte, JO, Hassan, YA 2006, ‘Investigation of microbubble boundary layer using particle tracking velocimetry’, Journal of Fluids Engineering, Vol. 129, pp. 66-79. Yakhot, V, & Orszag, SA 1986, ‘Renormalization group analysis of turbulence I. basic theory’, Journal of Scientific Computing, Vol. 1, pp. 3-51.



Chapter 4

A REVIEW OF POPULATION BALANCE MODELLING FOR MULTIPHASE FLOWS: APPROACHES, APPLICATIONS AND FUTURE ASPECTS Sherman C.P. Cheung1, G.H. Yeoh2 and J.Y. Tu1* 1

School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, Victoria 3083, Australia 2 Australian Nuclear Science and Technology Organisation (ANSTO), PMB 1, Menai, NSW 2234, Australia

Abstract Population balance modelling is of significant importance in many scientific and industrial instances such as: fluidizations, precipitation, particles formation in aerosols, bubbly and droplet flows and so on. In population balance modelling, the solution of the population balance equation (PBE) records the number of entities in dispersed phase that always governs the overall behaviour of the practical system under consideration. For the majority of cases, the solution evolves dynamically according to the “birth” and “death” processes of which it is tightly coupled with the system operation condition. The implementation of PBE in conjunction with the Computational Fluid Dynamics (CFD) is thereby becoming ever a crucial consideration in multiphase flow simulations. Nevertheless, the inherent integrodifferential form of the PBE poses tremendous difficulties on its solution procedures where analytical solutions are rare and impossible to be achieved. In this article, we present a review of the state-of-the-art population balance modelling techniques that have been adopted to describe the phenomenological nature of dispersed phase in multiphase problems. The main focus of the review can be broadly classified into three categories: (i) Numerical approaches or solution algorithms of the PBE; (ii) Applications of the PBE in practical gas-liquid multiphase problems and (iii) Possible aspects of the future development in population balance modelling. For the first category, details of solution algorithms based on both method of moment (MOM) and discrete class method (CM) that have been proposed in the literature are provided. Advantages and drawbacks of both approaches are also discussed from the theoretical and practical viewpoints. For the second category, applications of existing *

E-mail address: [email protected]. Phone no.:+61-3-9925 6191. Fax no.:+61-3-9925 6108. Corresponding Author: Prof. Jiyuan Tu, SAMME, RMIT University, Bundoora, Melbourne, Victoria 3083, Australia

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Sherman C.P. Cheung, G.H. Yeoh and J.Y. Tu population balance models in practical multiphase problems that have been proposed in the literature are summarized. Selected existing mathematical closures for modelling the “birth” and “death” rate of bubbles in gas-liquid flows are introduced. Particular attention is devoted to assess the capability of some selected models in predicting bubbly flow conditions through detail validation studies against experimental data. These studies demonstrate that good agreement can be achieved by the present model by comparing the predicted results against measured data with regards to the radial distribution of void fraction, Sauter mean bubble diameter, interfacial area concentration and liquid axial velocity. Finally, weaknesses and limitations of the existing models are revealed are suggestions for further development are discussed. Emerging topics for future population balance studies are provided as to complete the aspect of population balance modelling.

Keywords: Population balance; Computational Fluid Dynamics; bubbly flow.

1. Introduction Particles embedded within flow structures are featured in a wide diversity of industrial systems, such as gas-solid dispersion in combustors, catalytic reactions in fluidized beds, liquid-liquid dispersion in stirring tanks, microbial processes in bioreactors, and gas-liquid heat and mass transfer in bubble column reactors. In most cases, these particles (regardless whether they are inherently presence within the system or deliberately introduce into the system) are often the dominant factor affecting the behaviour of the systems. Such mounting industrial interests have certainly stimulated numerous scientific and engineering studies attempting to synthesize the behaviour of the population of particles and its dynamical evolution subject to the system environments in which has resulted in a widely adopted concept known as Population Balance. The population balance of any system is a record for the number of particles, which may be solid particles, liquid nuclei, bubbles or, variables (in mathematical terms) whose presence or occurrence governs the overall behaviour of the system under consideration. In most of the systems under concern, the record of these particles is dynamically depended on the “birth” and “death” processes that terminate existing particles and create new particles within a finite or defined space. Mathematically, dependent variables of these particles may exist in two different coordinates: namely “internal” and “external” coordinates (Ramkrishna and Mahoney, 2002). The external coordinates refer to the spatial location of each particle which is governed by its motion due to convection and diffusion flow behaviour. On the other hand, internal coordinates concerns the internal properties of particles such as: size, surface area, composition and so on. Figure 1 shows an example of the internal and external coordinates involve in the population balance for gas-solid particles flows. From a modelling perspective such as demonstrated in Figure 1, enormous challenges remain in fully resolving the associated nucleation, growth and agglomeration processes of particles within the internal coordinates and flow motions of external coordinates which are subjected to interfacial momentum transfer and turbulence modulation between gas and solid phases. Owing to the significant advancement of computer hardware and increasing computing power over the past decades, Direct Numerical Simulations (DNS), which attempt to resolve the whole spectrum of possible turbulent length scales in the flow, provide the propensity of describing the complex flow structures within the external coordinates (Biswas et al., 2005; Lu et al., 2006). Nevertheless, practical multiphase flows that are encountered in

A Review of Population Balance Modelling for Multiphase Flows

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natural and technological systems generally contain millions of particles that are simultaneously varying along the internal coordinates. Hence, the feasibility of DNS in resolving such flows is still far beyond the capacity of existing computer resources. The population balance approach, which records the number of particles as an averaged function, has shown to be a more promising way in handling the flow complexity because of its comparatively lower computational requirements. It is envisaged that the next stages of multiphase flow modelling in research and in practice would most probably concentrate on the development of more definitive efficient algorithms for solving the population balance equation (PBE).

Figure 1. An exmple of the internal and external coordinates of population balance for gas-solid particle flows.

The development of population balance model has a long standing history. Back to the end of 18th century, the Boltzmann equation, devised by Ludwig Boltzmann, could be regarded as the first population balance equation which can be expressed in terms of statistical distribution of molecules or particles in a state space. Nonetheless, the derivation of a generic population balance concept was actually initiated from the middle of 19th century. In 1960s, Hulburt and Katz (1964) and Pandolph and Larson (1964), based on the statistical mechanics and continuum mechanical framework respectively, presented the population balance concept to solve particle size variation due to nucleation, growth and agglomeration processes. A series of research development were thereafter presented by Fredrickson et al. (1967), Ramkrishna and Borwanker (1973) and Ramkrishna (1979, 1985) where the treatment of population balance equations were successfully generalized with various internal coordinates. A number of textbooks mainly concerning the population balance of

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Sherman C.P. Cheung, G.H. Yeoh and J.Y. Tu

aerocolloidal systems have also been published (Hidy and Brock, 1970; Pandis and Seinfeld, 1998; Friedlander, 2000). The flexibility and capability of population balance in solving practical engineering problems has not been fully exposed, until recently, where Ramkrishna (2000) wrote a textbook focusing on the generic issues of population balance for various applications. Although the concept of population balance has been formulated over many decades, implementation of population balance modelling was only realized until very recent times. Such dramatic breakthrough was made possible by the rapid development of computational fluid dynamics (CFD) and in-situ experimental measuring techniques. The availability of CFD softwares certainly facilitated a solid foundation in obtaining useful PBE solutions. With the field information provided by the CFD framework, external variables of the PBE can be easily acquired by decoupling the equation from external coordinates which can then enabled solution algorithms to be developed within internal coordinates. The capacity to measure particle sizes or other population balance variables from experiment is also of significant importance. These experimental data not only allow the knowledge of particle sizes and their evolution within systems to be realized but also provide a scientific basis for model calibrations and validations. In view of current developments of the state-of-the-art, this paper aims to further exploit the methodology of population balance modelling for multiphase flows from several aspects. First, it attempts to elucidate the implementation of population balance in conjunction with CFD techniques in handling practical industrial problems. Second, it seeks to reviews the solution algorithms of the PBE that have been proposed in literatures and discuss the advantages and drawbacks of each from the theoretical and practical viewpoints. Third, it demonstrates the model’s capability in predicting gas-liquid bubbly flows with or without heat and mass transfer through rigorous validation studies against various experimental data. Finally, it seeks to outline the fundamental weakness and limitations of current model development. Particular focus will be centred on the possible directions for further development.

2. Population Balance Approaches 2.1. Population Balance Equation The foundation development of the PBE stems from the consideration of the Boltzman equation. Such equation is generally expressed in an integrodifferential form describing the particle size distribution (PSD) as follow:

1 ξ ∂f (ξ , r , t ) + ∇ ⋅ (u (ξ , r , t ) f (ξ , r , t ) ) = ∫ a(ξ − ξ ′, ξ ′) f (ξ − ξ ′, t ) f (ξ ′, t )dξ ′ 2 0 ∂t ∞

− f (ξ , t ) ∫ a(ξ − ξ ′, ξ ′) f (ξ ′, t )dξ ′ 0

∞

+ ∫ γ (ξ ′)b(ξ ′) p(ξ / ξ ′) f (ξ ′, t )ds − b(ξ ) f (ξ , t ) ξ

(1)


121

where f (ξ , r , t ) is the particle size distribution dependent on the internal space vector ξ , whose components could be characteristics dimensions, surface area, volume and so on. r and t are the external variables representing the spatial position vector and physical time in external coordinate respectively. u (ξ , r , t ) is velocity vector in external space. On the RHS, the first and second terms denote birth and death rate of particle of space vector ξ due to merging processes, such as: coalescence or agglomeration processes; the third and fourth terms account for the birth and death rate caused by the breakage processes respectively. a(ξ , ξ ′) is the coalescence or agglomeration rate between particles of size ξ and ξ ′ .

γ (ξ ′) is the number of fragments/daughter particles generated from the breakage of a particle of size ξ ′ and p(ξ / ξ ′) represents the probability density function for a particle of size ξ to be generated by breakage of a particle of size ξ ′ . Conversely, b(ξ ) is the breakage rate of particles of size ξ .

Owing to the complex phenomenological nature of particle dynamics, analytical solutions only exist in very few cases of which coalescence and breakage kernels are substantially simplified (Scott, 1968; McCoy and Madras, 2003). Driven by practical interest, numerical approaches have been developed to solve the PBEs. The most common methods are Monte Carlo methods, Method of Moments and Class Methods. Theoretical speaking, Monte Carlo methods, which solve the PBE based on statistical ensemble approach (Domilovskii et al., 1979; Liffman, 1992; Debry et al., 2003; Maisels et al., 2004), are attractive in contrast to other methods. The main advantage of the method is the flexibility and accuracy to track particle changes in multidimensional systems. Nonetheless, as the accuracy of the Monte Carlo method is directly proportion to number of simulations particles, extensive computational time is normally required. Furthermore, incorporating the method into conventional CFD program is also not straightforward which greatly degraded its applicability for industrial problems. Because of their relevance in CFD applications, numerical approaches developed for Method of Moments and Class Methods are then discussed.

2.2. Method of Moments (MOM) Approach The method of moments (MOM), first introduced by Hulburt and Katz (1964), has been considered as one of the many promising approaches in viably attaining practical solutions to the PBE. The basic idea behind MOM centres in the transformation of the problem into lower-order of moments of the size distribution. The moments of the particle size distribution are defined as: ∞

m( k ) (t ) = ∫ f (ξ , t )ξ k dξ 0

(2)

From the above equation, the first few moments give important statistical descriptions on the population which can be related directly to some physical quantities. In the case space vector ξ represents the volume of particle, the zero order moment (k = 0) represents the total

122


number density of population and the fraction moment, k = 1/3 and k = 2/3 gives information on the mean diameter and mean surface area respectively. The primary advantage of MOM is its numerical economy which condenses the problem substantially by tracking the evolution of limited number of moments (Frenklach, 2002). This becomes a critical value in modeling complex industrial systems when particle dynamics is coupled with already time-consuming calculations of turbulence multiphase flows. Another significance of the MOM is that it does not suffer from truncation errors in the PSD approximation. Mathematically, the transformation from the PSD space to the space of moments is rigorous. Unfortunately, throughout the transformation process, fraction moments, representing mean diameter or surface area, are normally involved posing serious closure problem (Frenklach and Harris, 1987). In order to overcome the closure problem, in the early development of MOM, Frencklach and his co-workers (Frenklach and Wang, 1991; Markatou et al., 1993; Frenklach and Wang, 1994) proposed an interpolative scheme to determine the fraction moment from integer moments – namely Method of moments with interpolative closure (MOMIC).

2.2.1. Quadrature Method of Moments Another different approach for computing the moment is to approximate the integrals in Eq. (1) using numerical quadrature scheme – the quadrature method of moment (QMOM) as suggested by McGraw (1997). In the QMOM, instead of space transformation, Gaussian quadrature closure is adopted to approximate the PSD by a finite set of Dirac’s delta functions as follow: M

f (ξ , t ) ≈ ∑ N iδ (ξ − xi )

(3)

i =1

where Ni represents the number density or weight of the ith class consists of all particles per unit volume with a pivot size or abscissa, xi . A graphical representation of the QMOM in approximating the PSD is depicted in Figure 2. Although the numerical quadrature approach suffers from truncation errors, it successfully eliminates the problem of fraction moment which special closure is usually required. The closure of the method is then brought down to solving 2M unknowns, xi and Ni. A number of approaches in the specific evaluation of the quadrature abscissas and weights have been proposed. McGraw (1997) first introduced the product-difference (PD) algorithm formulated by Gordon (1968) for solving monovariate problem. Nonetheless, as point out by Dorao et al. (2006, 2008), the PD algorithm is a numerical ill-conditioned method for computing the Gauss quadrature rule (Lambin and Gaspard, 1982). Comprehensive derivation of the PD algorithm can be found in Bove (2005). In general, the computation of the quadrature rule is unstable and sensitive to small errors, especially if large number of moments is used. Later, McGraw and Wright (2003) derived the Jacobian Matrix Transformation (JMT) for multi-component population which avoids the instability induced by the PD algorithm. Very recently, Grosch et al. (2007) proposed a generalized framework for various QMOM approaches and evaluated different QMOM formulations in terms of


123

numerical quadrature and dynamics simulation. Several studies have also been carried out validating the method against different gas-solid particle problems (Barrett and Webb, 1998; Marchisio et al., 2003a,b,c). Encouraging results obtained thus far clearly demonstrated its usefulness in solving monovariate problems and its potential fusing within Computational Fluid Dynamics (CFD) simulations. One of the main limitations of the QMOM is that moments are adopted to represent the PSD, each moment is “convected” in the same phase velocity which is apparently non-physical, especially for gas-liquid flow where bubble could be deformed and travel in different trajectory.

Figure 2. Graphical presentations of the Quadrature Method of Moments (QMOM).

2.2.2. Direct Quadrature Method of Moments (DQMOM) With the aim to solve multi-dimensional problems, Marchisio and Fox (2005) extended the method by developing the direct quadrature method of moment (DQMOM) where the quadrature abscissas and weights are formulated as transport equations. The main idea of the method is to keep track the primitive variables appearing in the quadrature approximation, instead of moments of the PSD. As a result, the evaluation of the abscissas and weights are solved obtained using matrix operations. Substitute Eq. (3) into Eq. (1) and after some mathematical manipulations, transport equations for weights and abscissas are given by:

∂N i (r , t ) + ∇ ⋅ (u (r , t ) N i (r , t ) ) = ai ∂t

(4)

∂ζ i (r , t ) + ∇ ⋅ (u (r , t ) ζ i (r , t ) ) = bi ∂t

(5)

124


where ζ i = N i xi is the weighted abscissas and the terms ai and bi are related to the “birth” and “death” rate of population which forms 2M linear equations of which unknowns can be evaluated via matrix inversion:

Aα = d

(6)

where the 2M × 2M coefficient matrix A= [A1 A 2 ] is given by:

⎡ ⎢ ⎢ A1 = ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎢ ⎢ A2 = ⎢ ⎢ ⎢ ⎢⎣

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(7)

⎤ ⎥ 1 ⎥ ⎥ 2 xM ⎥ # ⎥ (2 M − 1) xM2 M −2 ⎥⎦

(8)

1

"

1

0

"

0

− x12 #

"

− x M2 #

% "

2 M −1 1

2(1 − M ) x 0

"

1 2 x1

"

2(1 − M ) x M2 M −1 0

"

#

% "

2 M −2 1

(2 M − x) x

The 2M vector of unknowns α is defined by:

⎡a ⎤

α = [a1 " aM b1 " bM ]T = ⎢ ⎥ b ⎣ ⎦

(9)

and the source on the RHS is:

[

d = S0 " S 2 M −1

]

T

(10)

The source term for the kth moment S k is defined by: ∞

S k (r , t ) = ∫ ξ k S (ξ , r , t )dξ 0

(11)

One attractive feature of the DQMOM is that it permits weights and abscissas to be varied within the state space according to PSD evolution. Furthermore, different travelling velocity can be also incorporated into transport equations allowing the flexibility to solve poly-dispersed flows where weights and abscissas travel indifferent flow fields (Ervin and Tryggvason, 1997; Bothe et al., 2006). In summary, the MOM represents a rather sound


125

mathematical approach and an elegant tool of solving the PBE with limited computational burden. Such approach with no doubt is an emerging technique for solving PBE, due to the considerably short development history, thorough validation studies comparing model predictions against experimental data are however outstanding. It can be concluded that the DQMOM or other moment methods still require further assessments and validations for various multiphase flow problems.

2.3. Class Method (CM) Approach Instead of inferring the PSD to derivative variables (i.e. moments), the class method (CM) which directly simulate its main characteristic using primitive variable (i.e. particle number density) has received greater attention due to its rather straightforward implementation within CFD software packages. In the method of discrete classes, the continuous size range of particles is discretized into a series number of discrete size classes. For each class, a scalar (number density of particles) equation is solved to accommodate the population changes caused by intra/inter-group particle coalescence and breakage. The particle size distribution is thereby approximated as follow: M

f (ξ , t ) ≈ ∑ N iδ (ξ − xi )

(12)

i =1

The expression is exactly the same with QMOM in Eq. (3), however, the groups (or abscissas) of class methods are fixed and aligned continuously in the state space. A graphical representation of the CM in approximating the PSD is depicted in Figure 3.

Figure 3. Graphical presentations of Class Methods (CM).

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2.3.1. Average Quantities Approach The simplest approach in class methods is adopted an averaged quantity to represent the overall changes of the particle population. Kocamustafaogullari and Ishii (1995) first derived an interfacial area concentration (IAC) transport equation for tracking the interfacial area between gas and liquid phase in bubbly flow problems. They concluded that the governing factor of the interfacial transfer mechanisms is strongly dominated by the interfacial area concentration. Modelling of the interfacial area concentration is therefore essential. In addition, as the population of particle is represented by a single average scalar, such average quantity approach requires very limited computational time in solving the PBE, which provides an attractive feature for practical engineering problems. Following their study, Ishii and his co-workers preformed extended the capability of their (IAC) transport model to simulate different bubbly flow regime in different flow conditions (Wu et al, 1998; Hibiki and Ishii, 2002; Fu et al., 2002a,b; Sun et al., 2004a,b). A series of experimental studies covering a wide range of flow conditions have been carried out in order to provide a solid foundation for their model development and calibration. Recently, similar modelling approach has been also adopted by Yao and Morel (2004) with the attempt of better improving the bubble coalescence and breakage kernels. On the other hand, equivalent to the formulation of the interfacial area transport equation, an Average Bubble Number Density (ABND) equation has been proposed very recently in our previous studies (Yeoh and Tu, 2006; Cheung et al., 2007).

2.3.2. MUltiple SIze Group (MUSIG) model Besides the proposed the average quantity approach, a more sophisticated model, namely homogeneous MUltiple-SIze-Group (MUSIG) model which first introduced by Lo (1996) is becoming widely adopted. Research studies based on the by Pochorecki et al. (2001), Olmos et al. (2001), Frank et al. (2004), Yeoh and Tu (2005) and Cheung et al. (2007) typified the application of MUSIG model in bubbly flow simulations. In the MUSIG model, the continuous particle size distribution (PSD) function is approximated by M number size fractions; the mass conversation of each size fractions are balanced by the inter-fraction mass transfer due to the mechanisms of particle coalescence/agglomeration and breakage processes. The overall PSD evolution can then be explicitly resolved via source terms within the transport equations. Snayal et al. (2005) examined and compared the CM and QMOM in a two-dimensional bubbly column simulation; both methods were found to yield very similar results. The CM solution has been found to be independent of the resolution of the internal coordinate if sufficient number of classes were adopted. Computationally speaking, as the number of transport equations depends on the number of group adopted, the MUSIG model requires more computational time and resources than the MOM to achieve stable and accurate numerical predictions. For typical bubbly flow simulations, the model requires around 10 groups to yield accurate results.


127

Figure 4. Schemetic diagram of the homogeneous and imhomogeneous MUSIG models.

Nonetheless, unlike the QMOM, CM provides the feasibility of accounting different bubble shapes and travelling gas velocities. The inhomogeneous MUSIG model developed by Krepper et al. (2005), which consisted of sub-dividing the dispersed phase into N number of velocity fields, demonstrated the practicability of such an extension. This flexibility represents a robust feature for multiphase flows modelling, especially for bubbly flow simulations where bubbles may deform into different shapes. Figure 4 shows the concept of the inhomogeneous MUSIG in comparison to homogeneous MUSIG. Useful information on the implementation and application of the inhomogeneous MUSIG model can be found in Shi et al. (2004) and Krepper et al. (2007). In spite of the sacrifices being made to computational efficiency, the extra computational effort will rapidly diminish due to foreseeable advancement of computer technology; the class method should therefore suffice as the preferred approach in tackling more complex multiphase flows. The formulation of the MUSIG model originates from the discretised PSE is given by:

∂ni K + ∇ ⋅ (ui ni ) = (∑ R )i − ( R ph ) i ∂t

(13)

To ensure overall mass conservation for all poly-dispersed vapour phases, the above bubble number density equation for the inhomogeneous MUSIG model can be re-expressed in terms of the volume fraction and size fraction of the bubble size class i, i ∈ [1, M j ] , of

velocity group j, j ∈ [1, N ] according to:

∂ρ j α j f i ∂t

K + ∇ ⋅ (ρ g α j f i u j ) = S j ,i − mi ( R ph ) i

with additional relations and constraints: Mj

N

N

j =1

j =1 i =1

α g = ∑ α j = ∑∑ α ij ;

Mj

α j = ∑α i i =1

(14)

128

Sherman C.P. Cheung, G.H. Yeoh and J.Y. Tu Mj

∑f

α g + αl = 1 ;

i =1

i

=1

(15)

where mi is the mass fraction of the particular size group I and R ph is the mass transfer rate due to phase changes which will be discussed in the coming sections. On the right hand side of Eq. (14) , the term S j ,i = mi

(∑ R ) = (P

C

i

+ PB − DC − DB )

represents the net mass transfer rate of the bubble class i resulting from the source of PC ,

PB , DC and DB , which are the production rates due to coalescence and breakage and the death rate due to coalescence and breakage of particles respectively. They can be formulated as:

PC =

η jki =

1 i i ∑∑η jki χ ij ni n j 2 k =1 l =1

(ν j + ν k ) − ν i −1 /(ν i − ν i −1 )

if ν i −1 < ν j + ν k < ν i

ν i +1 − (ν j + ν k ) /(ν i +1 − ν i )

if ν i < ν j + ν k < ν i +1

0

otherwise N

DC = ∑ χ ij ni n j j =1

PB =

∑ Ω (v N

j =i +1

D B = Ωi n i

j

: v i )n j

with Ωi =

N

∑Ω k =1

ki

(16)

with Mj

S j = ∑ S j ,i and i =1

N

∑S j =1

j

=0

(17)

3. Population Balance Modelling for Bubbly Flows In the previous section, approaches of population balance modelling have been reviewed. One should now realize the broad application of population balance modelling in practical engineering problems. In essence, population balance has been applied in numerous multiphase flow systems. For example: predicting the soot formation rate from incomplete


129

combustion processes (Frenklach and Wang, 1994; Zucca et al., 2007); solving the turbulent reacting flow in fluidized bed (Lakatos et al., 2008; Khan et al., 2007; Fan et al., 2004); evaluating the bubble side distribution and interfacial area in bubble column (Sha et al., 2006; Borel et al., 2006; Jia et al., 2007). In the following sections, to exemplify the capability of population balance modelling in conjunction with computational fluid dynamics framework, numerical simulations based on the MUSIG model of the CM were performed to predict the evolution of bubble size in both isothermal and heat and mass transfer conditions. Predictions were validated against experimental data measured by Hibiki et al. (2001), Yun et al. (1997) and Lee et al. (2002). Before presenting the methodology of the numerical study, phenomenological discussion and background of bubbly flows is firstly provided below.

Figure 5. Schematic of the physical phenomenon embedded in (a) isothermal bubbly flows and (b) subcooled boiling flows.

3.1. Isothermal Bubbly Flows For bubbly flows, the dynamic evolution of the PSD is governed predominantly by the bubble mechanistic behaviours such as bubble coalescence and bubble breakage. Figure 5a illustrates the physical characteristic of a typical isothermal bubbly flow. Owing to the positive lift force created by the lateral velocity gradient, small bubbles (under 5.5 mm for air-water flows) being injected at the bottom have a tendency to migrate towards the channel walls thereby increasing the bubble number density near the wall region. The net effect of

130


bubble coalescence encourages the formation of larger bubbles downstream. Larger bubbles (above 5.5mm), driven by the negative lift force, will move towards to the centre core of the channel of which they will further coalesce with other bubbles to yield distorted/cap bubbles. These mechanisms strongly govern the distribution of bubble size and void fraction of the gas phase within the bulk liquid; appropriate sub-models (or kernels) in describing the bubble coalescence and bubble breakage are essential to the proper modelling of bubbly flows. The Prince and Blanch (1990) coalescence and breakage kernels are widely applied for the bubbly flow simulations. Various models focusing towards modifying the coalescence frequency and daughter bubble distribution due to breakage have also been proposed such as those from Chester et al. (1982), Luo and Svendsen (1996), Lehr et al. (2001), Hagesaether et al. (2002), Wang et al. (2003, 2005, 2006) and Andersson and Andersson (2006). Chen et al. (2005) compared various kernels using a two-dimensional two-fluid model and showed that although the mechanistic considerations were different in each model, the predicted results were found

Figure 6. Schematic drawings illustrating (a) the mechanism of bubble departing, sliding and lifting off from a vertical heated surface and (b) the area of influence due to bubble growth and sliding referring to Eq. (26).


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to be rather similar across the broad range of models. A number of research studies have also been performed applying the aforementioned kernels to simulate size distribution evolution in bubbly turbulent pipe flows (Yeoh and Tu, 2004; Sha et al., 2006; Borel et al., 2006; Jia et al., 2007). All these works were nonetheless limited in solving isothermal bubbly turbulent coflow problems with low superficial gas velocity where there was no significant formation of cap/slug bubbles. It should be noted that the adoption of the above kernels remain debatable for modelling the bubble dynamics beyond bubbly flow regime, especially for high superficial gas velocity and flows involving complex inter-phase exchange of heat and mass transfer. Subcooled boiling flow belongs to another special category of bubbly turbulent pipe flow which embraces the complex dynamic interactions of bubble coalescence and bubble breakage in the bulk flow as well as the presence of heat and mass transfer occurring in the vicinity of the heated wall due to nucleation and condensation. Heterogeneous bubble nucleation occurs naturally within small pits and cavities on the heated surface designated as nucleation sites as stipulated in Figure 5b, which is in contrast to having the bubbles being injected externally at the bottom of the flow as illustrated in Figure 5a for isothermal bubbly turbulent pipe flow. These nucleation sites, activated by external heat, act as a continuous source generating relatively small bubbles along the vertical wall of the channel. The presence of a heated wall represents the fundamental difference between isothermal bubbly and subcooled boiling flows, where the former has a constant bubble injection rate to govern the overall void fraction of gas phase, but the latter has a variable bubble nucleation rate which is subjected mainly to the heat transfer and phase changing phenomena. Furthermore, experimental observations have confirmed that the vapour bubbles, driving by external forces, have a tendency to travel a short distance away from the nucleation sites, gradually increasing in size, before lifting off into the bulk subcooled liquid (Klausner et al., 1993). Figure 6 shows a schematic illustration of the bubble motion and its area of influence on the heater surface. Such bubble motion not only alters the mode of heat transfer on the surface, but also governs the departure and lift-off diameter of bubbles, which in turn also influences the bubble distribution in the bulk liquid. In isothermal flow, coalescence prevails in the channel core where the turbulent dissipation rate is relatively low. Conversely, bubbles tend to decrease in size for subcooled boiling flow as a result of increasing condensation away from the heated walls since the temperature in the bulk liquid remains below the saturation temperature limit. This subcooling effect is a well-known phenomenon confirmed by various experiments (Gopinath et al., 2002). Subject to this effect, such flows will certainly yield a broader range of bubble sizes and possibly even amplifying greater dynamical changes of the bubble size distribution when compared to isothermal bubbly turbulent pipe flow. Given the challenging task of modelling the sophisticated phenomena, empirical equations (Anglart and Nylund, 1996) were inevitably employed to determine the bubble diameter in the gas phase. In order to make progress, a modified version of the MUSIG model incorporating nucleation at the heated wall and condensation in the subcooled liquid were developed to better resolve the problem (Yeoh and Tu, 2005; Yeoh and Tu, 2006). Although some empirical equations were still retained to determine the bubble nucleation and detachment, the potential of adopting population balance approach in subcooled boiling flow demonstrated considerable success in aptly predicting the bubble Sauter diameter distribution of the gas bubbles for vertical boiling flows.

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4. Mathematical Models Referring back to the formulation of PBE, one should notice that the left-hand side of the equation denotes the time and spatial variations of the PSD which depends on the external variables. By incorporating the PBE within CFD solver, external variables can be obtained. In this section, governing equations of the two fluid model and its associated model for handling interfacial momentum and mass transfer are introduced.

4.1. Two-Fluid Model The three-dimensional two-fluid model solves the ensemble-averaged of mass, momentum and energy transport equations governing each phase. Denoting the liquid as the continuum phase (αl) and the vapour (i.e. bubbles) as disperse phase (αg), these equations can be written as:

Continuity Equation of Liquid Phase

∂ρl αl K + ∇ ⋅ ( ρl αl ul ) = Γlg ∂t

(18)

Continuity Equation of Vapour Phase

∂ρg α g f i ∂t

K + ∇ ⋅ (ρg α g f iu g ) = Si − f i Γlg

(19)

Momentum Equation of Liquid Phase

K ∂ρl α l u l KK K + ∇ ⋅ ( ρl α l u l u l ) = −α l ∇P + α l ρ l g ∂t K K K K + ∇[α l μle (∇ul + (∇ul )T )] + (Γlgu g − Γglul ) + Flg

(20)

Momentum Equation of Vapour Phase

K ∂ρ g α g u g ∂t

K K K + ∇ ⋅ (ρ g α g u g u g ) = −α g ∇P + α g ρ g g

(21)

K K K K + ∇[α g μ ge (∇u g + (∇u g )T )] + (Γglul − Γlgu g ) + Fgl

Energy Equation of Liquid Phase

∂ρl αl H l K + ∇ ⋅ ( ρl αl ul H l ) = ∇[α l λel (∇Tl )] + (Γgl H l − Γlg H g ) ∂t

(22)


133

Energy Equation of Vapour Phase

∂ρ g α g H g ∂t

K + ∇ ⋅ (ρ g α g u g H g ) = ∇[α g λeg (∇Tg )] + (Γgl H l − Γlg H g )

(23)

On the right-hand side of equation (4), Si represents the additional source terms due to coalescence and breakage. For isothermal bubbly turbulent pipe flows, it should be noted that the mass transfer rate Γlg and Γgl are essentially zero. The total interfacial force Flg appearing in equation (5) is formulated according to appropriate consideration of different sub-forces affecting the interface between each phase. For the liquid phase, the total interfacial force is given by:

Flg = Flgdrag + Flglift + Flglubrication + Flgdispersion

(24)

The sub-forces appearing on the right hand side of equation (9) are: drag force, lift force, wall lubrication force and turbulent dispersion force. More detail descriptions of these subforces can be found in Anglart and Nylund (1996). Note that for the gas phase, Fgl = - Flg. The interfacial mass transfer rate due to condensation in the bulk subcooled liquid in equation (4) can be expressed by:

Γlg =

h aif (Tsat − Tl ) h fg

(25)

Here, h indicates the inter-phase heat transfer coefficient which is correlated in terms of the Nusselt number (Tu and Yeoh, 2002). The wall generation rate for the vapour is modelled in a mechanistic manner by considering the total mass of bubbles detaching from the heated surface as:

Γgl =

Qe h fg + C pl (Tsat − Tl )

(26)

Here, Qe refers as the heat transfer rate due to evaporation. For subcooled boiling flows, the wall nucleation rate is accounted in equation (4) as a specified boundary condition apportioned to the discrete bubble class based on the size of the bubble lift-off diameter, which is evaluated from the improved wall heat partition model. The term f i Γlg represents the mass transfer due to condensation. The gas void fraction along with the scalar size fraction fi are related to the number density of the discrete bubble ith class ni (similarly to the jth class nj) as α g f i = niν i .

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4.1.1. Turbulence Modelling for Two-Fluid Model In handling bubble induced turbulent flow, unlike single phase fluid flow problem, no standard turbulence model is tailored for multiphase flow. For simplicity, the standard k-ε model has been employed with encouraging results in early studies (Schwarz and Turner, 1988; Davidson, 1990). Nonetheless, based on our previous study (Cheung et al, 2006), the Menter’s (1994) k-ω based Shear Stress Transport (SST) model were found superior to the standard k-ε model. Similar observations have been also reported by Frank et al. (2004). Based on their bubbly flow validation study, they discovered that standard k-ε model predicted an unrealistically high gas void fraction peak close to wall. Interestingly, they also found that the two turbulence models behaved very similar by reducing the inlet gas void fraction to a negligible value. This could be attributed to a more realistic prediction of turbulent dissipation close to wall provided by the k-ω formulation. It revealed that further development should be focused on multiphase flow turbulence modelling in order to better understand or improve the existing models. The SST model is a hybrid version of the k-ε and k-ω models with a specific blending function. Instead of using empirical wall function to bridge the wall and the far-away turbulent flow, it solves the two turbulence scalars (i.e. k and ω) explicitly down to the wall boundary. The ensemble-averaged transport equations of the SST model are given as:

μ ⎞ ⎛ ∂ρl αl kl K + ∇ ⋅ ( ρl αl ul kl ) = ∇ ⋅ ⎜⎜ αl ( μl + t ,l )∇kl ⎟⎟ + α l Pk ,l − ρl β ′klωl σ k3 ∂t ⎠ ⎝ μ ⎞ ⎛ ∂ρl αlωl K + ∇ ⋅ ( ρl αl ulωl ) = ∇ ⋅ ⎜⎜ αl ( μl + t ,l )∇ωl ⎟⎟ σω3 ∂t ⎠ ⎝ − 2 ρ1α l (1 − F1 )

∂kl ∂ωl ω 2 + α l γ 3 l Pk ,l − ρl β 3ωl kl σ ω 2ωl ∂x j ∂x j 1

(27)

where σ k 3 , σ ω 3 , γ 3 and β 3 are the model constants which are evaluated based on the blending function F1. The shear induced turbulent viscosity μts,l is given by:

μts ,l =

ρa1kl , max(a1ωl , SF2 )

S = 2 S ij S ij

(28)

The success of SST model hinges on the use of blending functions of F1 and F2 which govern the crossover point between the k-ω and k-ε models. The blending functions are given by:

⎡ ⎛ kl 500 μl ⎞⎟ 4 ρl kl ⎤ F1 = tanh(Φ14 ) , Φ1 = min ⎢max⎜ , , ⎥ ⎜ 0.09ωl d n ρlωl d n2 ⎟ Dω+σ ω 2 d n2 ⎥ ⎠ ⎝ ⎦ ⎣⎢


135

⎛ kl 500μl ⎞⎟ , F2 = tanh(Φ 22 ) , Φ 2 = max⎜ ⎜ 0.09ωl d n ρlωl d n2 ⎟ ⎝ ⎠

(29)

Here, default values of model constants were adopted. More detail descriptions of these model constants can be found in Menter (1994). In addition, to account the effect of bubbles on liquid turbulence, the Sato’s bubble-induced turbulent viscosity model was also employed (Sato et al., 1981). The turbulent viscosity of liquid phase is therefore given by:

μt ,l = μts ,l + μtd ,l

(30)

and the particle induced turbulence can be expressed as:

K

K

μ td ,l = C μp ρ lα g DS U g − U l

(31)

For the gas phase, dispersed phase zero equation model was adopted and the turbulent viscosity of gas phase can be obtained as:

μt , g =

ρ g μ t ,l ρl σ g

(32)

where σ g is the turbulent Prandtl number of the gas phase.

4.2. Coalescence and Breakage Kernels 4.2.1. Bubble Coalescence and Breakage Kernels Bubble breakage rate of volume v j into volume vi is modelled according to Luo and Svendsen (1996), which is based on the assumption of bubble binary breakage under isotropic turbulence situation. The daughter size distribution is accounted using a stochastic breakage 2/3

volume fraction f BV . Denoting the increase coefficient of surface area as cf = [ f BV +(1-

f BV )2/3-1], the breakage rate can be obtained as: ⎛ ε ⎞ Ω (v j : vi ) = FBC ⎜ 2 ⎟ ⎜d ⎟ (1 − αg )n j ⎝ j⎠

1/ 3

where

1

∫ ξ

min

(1 + ξ )2 × exp⎛⎜ − ξ

11 / 3

⎜ ⎝

12c f σ βρl ε

2/3

d

ξ

5 / 3 11 / 3

⎞ ⎟⎟dξ ⎠

(33)

ξ = λ / d j is the size ratio between an eddy and a particle in the inertial sub-range and

consequently ξ min = λ min / d j and C and β are determined from fundamental consideration of

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drops or bubbles breakage in turbulent dispersion systems to be 0.923 and 2.0. FB is the breakage calibration factor. Bubble coalescence occurs via collision of two bubbles which may be caused by wake entrainment, turbulence random collision and buoyancy. Only turbulence random collision is considered in the present study as all bubbles are assumed to be of spherical shape (wake entrainment becomes negligible). The coalescence rate considering the turbulent collision taken from Prince and Blanch (1990) can be expressed as:

χ ij = FC

π

[d 4

i

+dj

] (u 2

2 ti

+ u tj2

)

0.5

⎛ t ij exp⎜ − ⎜ τ ⎝ ij

where τ ij is the contact time for two bubbles given by ( d ij / 2)

2/3

⎞ ⎟ ⎟ ⎠

(34)

/ ε 1 / 3 and t ij is the time

required for two bubbles to coalesce having diameter di and dj estimated to be

[(d ij / 2) 3 ρ l / 16σ ]0.5 ln(h0 / h f ) . The equivalent diameter dij is calculated as suggested by −1

Chesters and Hoffman (1982): ( d ij = ( 2 / d i + 2 / d j ) . According to Prince and Blanch (1990), experiments have determined the initial film thickness ho = 1 × 10

−4

m and critical

−8

film thickness hf = 1 × 10 m at which rupture for air-water systems. The turbulent velocity ut in the inertial sub-range of isotropic turbulence (Rotta, 1972) which is given by: u t =

2ε 1 / 3 d 1 / 3 . FC is the coalescence calibration factor.

4.2.2. Bubble Source and Sink Due to Phase Change in Subcooled Boiling Flows The term ( R ph ) i in Eq. (14) constitutes the essential formulation of the source/sink rate for the phase change processes associated with subcooled boiling flow. Considering the condensation of bubbles, the bubble condensation rate in a control volume for each bubble class can be determined from:

φCOND = −

dR ni AB dt VB

(35)

The bubble condensation velocity (Gopinath et al., 2002) is obtained from:

dR h(Tsat − Tl ) = dt ρ g h fg

(36)

Substituting Eq. (21) into Eq. (20), and given that the bubble surface are AB and volume VB which are based on Sauter mean bubble diameter, Eq. (20) can be rearranged to yield:


( R ph ) i = φCOND = −

1 ⎡ haif (Tsat − Tl ) ⎤ ⎢ ⎥ ni ρ gα i ⎣⎢ h fg ⎦⎥

137

(37)

At the heated surface, bubbles at the nucleation sites are formed through the evaporation processes. The bubble nucleation rate from these sites can be expressed as:

φWN =

Nafξ H AC

(38)

where Na , f, ξ H and AC is the active nucleation site density, the bubble generation frequency from the nucleation sites, the heated perimeter, and the cross-sectional area of the boiling channel, respectively. Since the bubble nucleation process is taken to occur only at the heated surface, this heated wall nucleation rate is treated as a specified boundary condition to Eq. (14) apportioned to the discrete bubble class i, based on the bubble lift-off diameter determined from the mechanistic wall heat partition model.

4.3. Mechanistic Wall Heat Partition Model To determine the bubble generation frequency and lift-off diameter for the boundary condition of the MUSIG model, an improved wall heat partition model based on a mechanistic approach is proposed (Yeoh et al., 2008) and briefly discussed in this section. With the presence of convective force or buoyancy force acting upon a vertically orientated boiling flow as depicted in Figure 6a, vapour bubble departs from its nucleation site, slides along the heating surface and continues to grow downstream until it lifts off from the surface (Klausner et al., 1993). The motion of the travelling bubble affects the heat transfer at the heated wall according to two mechanisms: (i) the latent heat transfer due to micro-layer evaporation and (ii) transient conduction as the disrupted thermal boundary layer reforms during the waiting period (i.e. incipience of the next bubble at the same nucleation site).

4.3.1. Transient Conduction Due to Bubble Motion Transient conduction occurs in regions at the point of inception and in regions being swept by sliding bubbles. For a stationary bubble, the heat flux is given by:

Qtc = 2

+2

k l ρ l C pl

πt w k l ρ l C pl

πt w

⎛ πDd2 (Ts − Tl ) R f N a ⎜⎜ K 4 ⎝ ⎛ πDd2 (Ts − Tl ) R f N a ⎜⎜ ⎝ 4

⎞ ⎟⎟t w f ⎠

⎞ ⎟⎟(1 − t w f ) ⎠

(39)

where Dd is the bubble departure diameter, Ts is the temperature of the heater surface and Tl is the temperature of the liquid. Eq. (24) indicates that some fraction of the nucleation sites will

138


undergo transient conduction while the remaining will be in the growth period. For a sliding bubble, the heat flux due to transient conduction that takes place during the sliding phase and the area occupied by the sliding bubble at any instant of time is given by:

Qtcsl = 2 +2

k l ρ l C pl

πt w k l ρ l C pl

πt w

(Ts − Tl ) R f N a l s KDt w f ⎛ πD 2 ⎞ ⎟⎟(1 − t w f ) (Ts − Tl ) R f N a ft sl ⎜⎜ ⎝ 4 ⎠

(40)

where the average bubble diameter D is given by D = (Dd + Dl ) 2 and Dl is the bubble lift-off diameter. In this study, a value of 1.8 is assumed for the area of influence, K (Judd and Hwang, 1976). The reduction factor Rf appearing in equations (24) and (25) depicts the ratio of the actual number of bubbles lifting off per unit area of the heater surface to the number of active nucleation sites per unit area, viz., R f = 1 (l s / s ) where ls is the sliding distance and s is the spacing between nucleation sites. In the present study, the spacing between nucleation sites is approximated as s = 1

N a (Basu et al., 2005). The active

nucleation site density, Na, is expressed by N a = [185(Tw − Tsat )]

1.805

(Končar et al., 2004).

The factor Rf is obtained alongside with the sliding distance evaluated from the force balance model described below.

4.3.2. Force Convection for Single Phase Component Forced convection always prevails at all times in areas of the heater surface that are not influenced by the stationary and sliding bubbles (see also in Figure 6b). The fraction of the heater area for stationary and sliding bubbles is given by:

⎡ ⎛ πDd2 1 − Aq = 1 − R f ⎢ N a ⎜⎜ K 4 ⎢⎣ ⎝

⎞ ⎛ πD 2 ⎟⎟t w f + N a ⎜ d ⎜ 4 ⎠ ⎝

⎛ πDd2 + N a l s KDt w f + N a ft sl ⎜⎜ ⎝ 4

⎞ ⎟⎟(1 − t w f ) ⎠

⎤ ⎞ ⎟⎟(1 − t w f )⎥ ⎠ ⎦⎥

(41)

The heat flux due to forced convection can be obtained according to the definition of local Stanton number St for turbulent convection is:

Qc = Stρ l C pl u l (1 − Aq )(Tw − Tl ) where ul is the adjacent liquid velocity.

(42)


139

4.3.3. Latent Heat Due to Vapour Evaporation Processes The heat flux attributed to vapour generation is given by the energy carried away by the bubbles lifting off from the heated surface. It also represents the energy of vaporization whereby the bubble size of Dl is produced, which is expressed as:

⎛ πDl3 ⎞ ⎟ ρ g h fg Qe = R f N a f ⎜⎜ ⎟ 6 ⎝ ⎠

(43)

The total wall heat flux Qw is the combination of the following heat flux components: Qw = Qe + Qtc + Qtcsl + Qc. Mechanistic Approach for Bubble Frequency Evaluation The bubble nucleation rate in Eq. (23) requires the knowledge of the bubble frequency (f). Within the wall partition model, the bubble frequency is determined by a mechanistic approach based on the description of an ebullition cycle in nucleate boiling, which is formulated as:

f =

1 tg + tw

(44)

The waiting period (tw) and the growth period of vapour bubbles (tg) is derived from the transient conduction and force balance model respectively. When transient conduction occurs, the boundary layer gets disrupted and cold liquid comes in contact with the heated wall. Assuming that the heat capacity of the heater wall ρsCpsδs is very small, the conduction process can be modelled by considering one-dimensional transient heat conduction into a semi-infinite medium with the liquid at a temperature Tl and the heater surface at a temperature Ts. The wall heat flux can be approximated by:

Qw = where δl(=

k l (Ts − Tl )

δl

(45)

πηt ) is the thickness of the thermal boundary layer. If the temperature profile

inside this layer is taken to be linear (Hsu and Graham, 1976), it can be expressed as:

Tb = Tw −

(Ts − Tl ) x

δl

(46)

where x is the normal distance from the wall. Based on the criterion of the incipience of boiling from a bubble site inside the thermal boundary layer, the bubble internal temperature for a nucleus site (cavity) with radius rc is

140


Tb = Tsat −

2σTsat at x = C1 rc C 2 rc h fg ρ g

(47)

where C1 = (1 + cos θ ) / sin θ and C 2 = 1 / sin θ . The angle θ represents the bubble contact angle. By substituting Eq. (32) into Eq. (31), the waiting time tw can be obtained as

⎤ (Ts − Tl )C1 rc 1 ⎡ t = tw = ⎢ ⎥ πη ⎢⎣ (Tw − Tsat ) − 2σTsat / C 2 ρ g h fg rc ⎥⎦

2

(48)

The cavity radius rc can be determined by applying Hsu’s criteria and tangency condition of equations (32) and (33), viz., 2

2

⎡ C1C 2 ρ g h fg rc2 ⎤ (Ts − Tl ) 2 ⎡ k l ⎤ (Ts − Tl ) 2 t=⎢ =⎢ ⎥ ⎥ πη πη ⎢⎣ 2σTsat ⎥⎦ ⎣ Qw ⎦

(49)

From the above equation,

⎡ 2σTsat k l ⎤ rc = F ⎢ ⎥ ⎢⎣ ρ g h fg Qw ⎥⎦

1/ 2

(50)

where,

⎛ 1 F = ⎜⎜ ⎝ C1C 2

⎞ ⎟⎟ ⎠

1/ 2

⎛ sin 2 θ ⎞ ⎟⎟ = ⎜⎜ + 1 cos θ ⎝ ⎠

1/ 2

(51)

According to Basu et al. (2005), the factor F indicates the degree of flooding of the available cavity size and the wettability of the surface. If the contact angle θ → 0, all the cavities will be flooded. Alternatively, as θ → 90o, F → 1, all the cavities will not be flooded (i.e. they contain traces of gas or vapour).

4.3.4. Force Balance Model and the Bubble Growth Time The bubble growth time is correlated to its lift-off diameter which depends on various forces acting on the bubble in the directions parallel and normal to a vertical heating surface. Figure 7 illustrates the forces acting on the bubble in the x-direction and y-direction which can be formulated according to the studies performed by Klausner et al. (1993) and Zeng et al. (1993):.

ΣFx =Fsx + Fdux + FsL + Fh + Fcp

(52)


141

ΣFy =Fsy + Fduy + Fqs + Fb

(53)

and

where Fs is the surface tension force, Fdu is the unsteady drag due to asymmetrical growth of the bubble and the dynamic effect of the unsteady liquid such as the history force and the added mass force, FsL is the shear lift force, Fh is the force due to the hydrodynamic pressure, Fcp is the contact pressure force accounting for the bubble being in contact with a solid rather than being surrounded by liquid, Fqs is the quasi steady-drag in the flow direction, and Fb is the buoyancy force. In addition, g indicates the gravitational acceleration; α, β and θi are the advancing, receding and inclination angles respectively; dw is the surface/bubble contact diameter; and d is the vapour bubble diameter at the wall.

Figure 7. Schematic drawings illustrating the forces balance of a growing vapour bubble attached to the heated surface.

The forces acting in the x-direction can be estimated from:

Fsx = −d wσ

FsL =

π α −β

[cos β − cos α ] ; Fdux = − Fdu cos θ

πd 1 9 C L ρ l ΔU 2πr 2 ; Fh = ρ l ΔU 2 w 2 4 4

2

142


Fcp =

πd w2 2σ 4

rr

(54)

The forces acting in the x-direction can be estimated from:

Fsy = − d wσ

π (α − β ) [sin α + sin β ] ; Fduy = − Fdu sin θ π − (α − β ) 2 2

4 Fqs = 6C D μ l ΔUπr ; Fb = π r 3 (ρ l − ρ g )g 3

(55)

From the various forces described along the x-direction and y-direction, r is the bubble radius, ΔU is the relative velocity between the bubble centre of mass and liquid, CD and CL are the respective drag and shear lift coefficients and rr is the curvature radius of the bubble at the reference point on the surface x = 0, which is rr ∼ 5r (Klausner et al., 1993). The growth force Fdu is modelled by considering a hemispherical bubble expanding in an inviscid liquid, which is given by Zeng et al. (1993) as:

⎞ ⎛3 Fdu = ρlπr 2 ⎜ Cs r 2 + rr⎟ ⎠ ⎝2

(56)

where ( ˙ ) indicates differentiation with respect to time. The constant Cs is taken to be 20/3 (Zeng et al., 1993). In estimating the growth force, additional information on the bubble growth rate is required. As in Zeng et al. (1993), a diffusion controlled bubble growth solution by Zuber (1961) is adopted:

r (t ) =

2b

π

Ja ηt ; Ja =

ρ l C pl ΔTsat kl ;η= ρ g h fg ρ l C pl

(57)

where Ja is the Jakob number, η is the liquid thermal diffusivity and b is an empirical constant that is intended to account for the asphericity of the bubble. For the range of heat fluxes investigated in this investigation, b is taken to be 0.21 based on a similar subcooled boiling study performed by Steiner et al. (2006), which has been experimentally verified through their in-house measurements with water as the working fluid. While a vapour bubble remains attached to the heated wall, the sum of the parallel and normal forces must satisfy the following conditions: ΣFx = 0 and ΣFy = 0. For a sliding bubble case, the former establishes the bubble departure diameter (Dd) while the latter yields the bubble lift-off diameter (Dl). The growth period tg appearing in Eq. (29) can be readily evaluated based on the availability of the bubble size at departure from its nucleation site through Eq. (42). Details of the present wall partition model can also be found in literature (Yeoh et al., 2008) and the references therein.


143

4.4. Experimental Details Brief discussions of the experimental setup for the isothermal and subcooled bubbly turbulent pipe flows are provided below. Isothermal bubbly turbulent pipe flow experiments have been performed at the ThermalHydraulics and Reactor Safety Laboratory in Purdue University (2001). The test section comprised of an acrylic round pipe with an inner diameter D = 50.8 mm and a length of 3061 mm. Temperature of the apparatus was kept at a constant temperature (i.e. 20oC) within a deviation of ±0.2oC controlled by a heat exchanger installed in a water reservoir. Local flow measurements using the double sensor and hotfilm anemometer probes were performed at three axial (height) locations of z/D = 6.0, 30.3 and 53.5 and 15 radial locations of r/R = 0 to 0.95. Experiments at a range of superficial liquid velocities jf and superficial gas velocities jg were performed covering most bubbly flow regions including finely dispersed bubbly flow and bubbly-to-slug transition flow regions. A series of subcooled boiling experiments have been performed by Yun et al. (1997) and Lee et al. (2002). The experimental setup consisted of a vertical concentric annulus with an inner rod of 19 mm outer diameter uniformly heated by a 54 kW DC power supply. This heated section comprised of a 1.67 m long Inconel 625 tube with a 1.5 mm wall thickness filled with magnesium oxide powder insulation. The outer wall comprised of two stainless steel tubes with 37.5 mm inner diameter. Demineralised water was used as the working fluid. Local gas phase parameters such as radial distribution of the void fraction, bubble frequency and bubble velocity were measured by a two-conductivity probe method located 1.61 m downstream of the beginning of the heated section. The bubble Sauter diameters (assuming spherical bubbles) were determined through the IAC, calculated using the measured bubble velocity spectrum and bubble frequency.

4.5. Numerical Details For isothermal gas-liquid bubbly turbulent pipe flow, the generic CFD code ANSYSCFX 11 (2006) was utilised to handle the two sets of equations governing conservation of mass and momentum. Numerical simulations were performed on a 60o radial sector of the pipe with symmetry boundary conditions imposed at the end vertical sides. At the test section inlet, uniformly distributed superficial liquid and gas velocities, void fraction and bubble size were specified. Details of the boundary conditions for different flow conditions are summarized in Table 1. At the pipe outlet, a relative averaged static pressure of zero was specified. A threedimensional mesh containing hexagonal elements was generated resulting in a total of 108,000 elements over the entire pipe domain. For all flow conditions, bubble size in the range of 0-10 mm was discretised into 10 bubble classes as tabulated in Table 2. The prescribed 10 size groups were further divided into two velocity fields for the inhomogeneous MUSIG model to predict the transition of bubbly-slug flow. Reliable convergence were achieved within 2500 iterations when the RMS (root mean square) pressure residual dropped below 1.0 × 10-7. A fixed physical time scale of 0.002s was adopted for all steady state simulations.

144


Table 1. Bubbly flow conditions and its inlet boundary conditions employed in the present study Superficial gas velocity,

Superficial liquid velocity, j f (m/s) Hibiki et al. (2001) experiment

j g (m/s) Bubbly flow Regime

0.491 [ αg

0.0556

z / D = 0.0

[ DS

(%)]

z / D =0.0

[ DS

z / D = 0.0

(%)]

z / D = 0.0

-

[10.0] [2.5]

-

0.0473

0.1130

0.242

[5.0] [2.5]

[10.0] [2.5]

[20.0] [2.5]

(mm)]

0.986 [ αg

Transition Regime

(mm)]

Table 2. Diameter of each discrete bubble class for MUSIG model Class No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Central Class Diameter di (mm) Isothermal Subcooled Boiling 0.5 0.45 1.5 0.94 2.5 1.47 3.5 2.02 4.5 2.58 5.5 3.14 6.5 3.71 7.5 4.27 8.5 4.83 9.5 5.40 5.96 6.53 7.10 7.66 8.23

For subcooled boiling flow, numerical solutions were obtained from the two sets of transport equations governing not only mass and momentum but also energy using the generic CFD code CFX 4.4. A total number 15 bubble classes were specified for the dispersed phases in the homogeneous MUSIG model (see Table 2). Similar to the isothermal flow simulations, only one quarter of the annulus geometry was modelled due to the uniform prescription of the heat flux at the inner wall. A body-fitted conformal system was employed to generate the three-dimensional mesh within the annular channel resulting in a total of 13 (radial) × 30


145

(axial) × 3 (circumference) control volumes. A standard k-ε model was applied for both phases while additional turbulent viscosity induced by bubbles was included using Sato’s model95. Since wall function was used in the present study to bridge the wall and the fully turbulent region away from heater surface, the normal distance between the wall and the first node in the bulk liquid should be such that the corresponding x+ was greater than 30. Grid independence was examined. In the mean parameters considered, further grid refinement did not reveal significant changes to the two-phase flow parameters. Convergence was achieved within 1500 iterations when the mass residual dropped below 1.0 × 10-7. Experimental conditions used for comparison with the simulated results are tabulated in Table 3. Table 3. Subcooled boiling flow conditions measured by Yun et al. (1997) adopted in the present numerical study Case

Pinlet [Mpa]

Tinlet [°C]

Tsub (inlet) [°C]

Qw [kW/m2]

G [kg/m2s]

L1 L2 L3

0.143 0.137 0.143

96.9 94.9 92.1

13.4 13.8 17.9

152.9 197.2 251.5

474.0 714.4 1059.2

5. Results and Discussion 5.1. Isothermal Bubbly Turbulent Pipe Flow Results Preliminary numerical simulations performed in this study over a range of superficial gas and liquid velocities within the bubbly flow regime have consistently resulted in the prediction of larger than expected bubble sizes. However, these results were found to clearly contradict the measurements performed by Hibiki et al. (2001) as well as some experimental observations (Bukur et al., 1996; George et al., 2000). One plausible explanation for this discrepancy could possibly be the error embedded in the turbulent dissipation rate prediction (Bertola et al., 2003) as a consequence of the turbulence model being applied contributing in turn to excessively high coalescence rates in the MUSIG model. As reported in Chen et al. (2005), similar observations also confirmed the high coalescence rates that were experienced in their bubble column study. They argued that the local coalescence rate should be reduced by an order of magnitude lower than the local breakage rate by a factor of about 10. Olmos et al. (2001) further demonstrated the need to prescribe suitable calibration factor in order to aptly predict the bubble size distributions in their bubble column flow configuration. A value of 0.075 was assumed for the coalescence calibration factor. According to similar arguments stipulated above, the coalescence and breakage calibration factors (i.e. FC and FB), were set as 0.05 and 1.0 throughout all intended simulations of isothermal flow conditions (Cheung et al., 2007b).

5.1.1. Local Distributions of Void Fraction snd Interfacial Gas Velocity Depending on the range of superficial velocities of gas and liquid experienced in the bubbly flow regime, the phase patterns of the vapour void fraction can be broadly categorized

146


into four types of distributions: “wall peak”, “intermediate peak”, “core peak” and “transition” (Serizawa an Kataoka, 1988). Figure 8 compares the gas void fraction profiles obtained from the homogeneous MUSIG model with the experimental data measured at the location of z/D = 53.5 (i.e. close to the exit of channel) for three different bubbly turbulent pipe flow conditions. The high void fractions close to wall proximity typically characterised the “wall peak” behaviour. Its phenomenological establishment can be best described by the balance between the positive lift force that acted to impel the bubbles away from the central core of the flow channel and the opposite effect being imposed by the lubrication force preventing the bubbles from being obliterated at the channel walls. In spite of similar trends predicted, the predicted void fraction peaks, on closer examination, appeared to be leaning more towards the channel wall in contrast to the actual bubble distributions observed during experiments. Against the assessment on other wall lubrication models by Frank et al. (2004) and Tomiyama (1998), a much lower than expected wall force determined via Antal et al. (1991) model purported to be the most probable cause of the discrepancy. As aforementioned, the two-phase turbulence modelling that could affect the magnitude of the turbulence induced dispersion force could also contribute to the additional modelling uncertainty.

Figure 8. Predicted radial void fraction distribution and experimental data of Hibiki et al.(2001) at measuring station of z/D =53.5


147

Figure 9. Predicted interfacial gas velocity distribution and experimental data of Hibiki et al. (2001) at measuring station of z/D =53.5.

Figure 9 illustrates the local interfacial gas velocity distributions at the measuring station of z/D = 53.5, close to the outlet of the pipe. Unlike in single phase flows, the presence of bubbles has the tendency to enhance the liquid flow turbulence intensity (Serizawa et al., 1990; Hibiki et al., 2001). Additional turbulence being experienced at the core flattened the liquid velocity profile as expected. Through the interfacial momentum transfer, such effects are brought down to the gas phase yielding similar interfacial gas velocity profiles. In general, the interfacial gas velocity profiles for all flow conditions were found to be in good agreement with measurements, especially at the channel core. In essence, the prediction of the local bubble sizes evaluated by the coalescence and breakage kernels of the MUSIG model has a coupling effect on the phases velocities. In better determining the Sauter mean bubble diameter, a more accurate description of the interfacial forces between the two phases should surface in more enhanced liquid/gas velocity predictions.

148


5.1.2. Local Distributions of Sauter Mean Bubble Diameter, Interfacial Area Concentration and the Evolution of Bubble Size Distribution Figure 10 illustrates the predicted and measured Sauter mean bubble diameter radial profiles at the location of z/D = 53.5. As measured by Hibiki et al. (2001), the Sauter mean bubble diameter profiles remained roughly unchanged throughout the whole channel. Overall, bubble size changes were found mainly due to the bubble expansion caused by the static pressure variation along the axial direction. As demonstrated by the good agreement between the predicted and the measured bubble diameters at the channel core, the ensemble bubble expansion effect was adequately captured by the two-fluid approach via the ideal gas assumption. Near the wall region, slightly larger bubbles were formed due to the tendency of small bubbles migrating towards the wall thereby creating higher concentration of bubbles, increasing the likelihood of possible bubble coalescence. Remarkable agreement with the measurement at the wall region clearly illustrated the enlargement of bubble size due to bubble coalescence and bubble breakage successfully represented by the kernels in the MUSIG model. As the Sauter mean bubble diameter is closely related to the interfacial momentum forces (i.e. drag and lift forces), appropriate evaluation of the bubble diameter is thus crucial for the prediction of interfacial gas velocities.

Figure 10. Predicted Sauter mean bubble diameter distribution and experimental data of Hibiki et al. (2001) at measuring station of z/D =53.5.


149

Figure 11. Predicted size fraction of each bubble classes and its evolution along radial direction at the measuring station of z/D =53.5.

The population balance of bubbles within the bubbly turbulent pipe flow can be further exemplified by tracking the evolution of bubble size distribution at the measuring station. Figure 11 shows the development of the size fraction of each bubble classes along the radial direction at the location of z/D = 53.5. Since the turbulence intensity is relatively low at the channel core (i.e. r/R = 0.05), bubble sizes remained unaffected owning to the insignificant bubble coalescence and bubble breakage rates. With increasing number of bubbles driven by the lift force and the rising turbulence intensity within the boundary layer towards the channel walls (i.e. r/R=0.8 and 0.95), bubble coalescence and bubble breakage became increasingly noticeable forming larger bubbles and re-distributed the BSD to higher bubble classes. Such behaviour was amplified especially for cases with relatively higher void fraction (e.g. Figure 8a,c). Based on the assumption where the bubbles are spherical in shape, the local Interfacial Area Concentration (IAC) profiles may be determined based on the relation between the local void fraction and Sauter mean bubble diameter according to a if = 6α g / D s . The measured and predicted local interfacial area concentration profiles for the respective flow condition are shown in Figure 12. Similar to the void fraction profiles, the IAC predictions at the measuring station yielded good agreement against the measured profiles. The discrepancy between predicted results and measured data could be attributed to the error introduced by the void fraction prediction owing to the uncertainties embedded within the turbulence and wall lubrication models. Another possible cause could be the invalid assumption of spherical bubbles to aptly resolve the

150


gas-liquid flow where large distorted bubbles prominently featured at high superficial velocities (i.e. =0.986m/s).

Figure 12. Predicted Interfacial Area Concentration (IAC) distribution and experimental data of Hibiki et al. (2001) at measuring station of z/D =53.5.

5.2. Subcooled Boiling Flow Results Increasing complexity of numerical simulations is now further elaborated by considering the bubble dynamics in conjunction with the heat and mass transfer processes typified by a subcooled boiling flow. For the local cases of L1, L2 and L3, the measured and predicted radial profiles of the Sauter mean bubble diameter, vapour void fraction and interfacial area concentration located at the measuring plane 1.61 m downstream of the beginning of the heated section are discussed below. In all the figures, the dimensionless parameter (r-Ri)/(RoRi) = 1 indicates the inner surface of the unheated flow channel wall while (r-Ri)/(Ro-Ri) = 0 indicates the surface of the heating rod in the channel.


151

Figure 13. Prediction Sauter mean bubble diameter distribution and experimental data at the measuring station.

5.2.1. Local Distributions of Void Fraction, Sauter Bubble Diameter and Interfacial Area Concentration Figure 13 illustrates the predicted Sauter mean bubble diameter profiles at the measuring plane of the heated annular channel. Experimental data and observations (Bonjour and Lallemand et al., 2001; Lee et al., 2002) suggested that vapour bubbles, relatively small when detached from the heated surface, have the tendency of significantly colliding with other detached bubbles at the downstream and subsequently forming bigger bubbles via coalescence. For all three cases, the bubble size changes were found to be adequately predicted by the modified MUSIG model. Observed consistent trends between the predicted and measured Sauter mean bubble diameter reflected the measure of the modified MUSIG model in aptly capturing the bubble coalescence especially in the vicinity of the heated wall. The development of bubbles in this region stemmed from the evaporation process occurring at the heated wall and forces acting on the vapour bubbles determining the bubble size at departure or lift-off.

152


Table 4. Predicted heat partitions, bubble departure and lift-off diameter of subcooled boiling flow conditions Case L1

Case L2

Case L3

0%

0%

0%

Qtc (W/m )

2.51%

4.56%

6.42%

2

55.07

61.25%

65.58%

42.42%

34.19%

28.00%

Dd (mm)

0.56

0.58

0.57

Dl (mm)

1.45

1.31

1.20

2

Measuring location

Qc (W/m ) 2

Qtcsl (W/m ) 2

Qe (W/m )

Table 4 illustrates the various contributing heat flux components and the associated bubble departure and lift-off diameters evaluated by the improved heat partition model. On the basis of the force balance model, the bubble departure diameter were predicted with a size of approximately 0.56~0.58 mm while the lift-off diameters were found to range from 1.2~1.45 mm. The ratio between the bubble lift-off diameter and bubble departure diameter was thus ascertained to be between 2 and 3, which incidentally closely corresponded to experimental observations of Basu et al. (2005). Surface quenching due to sliding bubbles and evaporation were found to be the dominant modes of heat transfer governing the heat partition model. The former highlighted the prevalence of bubble sliding motions on the surface significantly altering the rate of heat transfer and subsequently the resultant vapour generation rate. Away from the heated wall, bubbles entering the bulk subcooled liquid were condensed due to the subcooling effect. Predicted trends of the Sauter mean diameter profiles clearly showed the gradual collapse of the bubbles from the channel centre to the outer unheated wall. The over-prediction of the bubble diameters for cases L1 and L2 could be attributed by the under-estimation of the subcooling condensation, which was determined by an empirical correlation based on the Nusselt number description. Although the measured bubble sizes near the heated wall were found to agree rather well with the measured data confirming to certain extend the appropriate estimation of the bubble lift-off diameters, a closer examination of the local void fraction profiles at the measuring station in Figure 14 indicated a less than satisfactory prediction of the void fraction near the heated surface where they were either over- or under-predicted as exemplified in cases L1 and L3. The void fraction distribution in case L2 compared nonetheless reasonably well with measurement. This discrepancy could be attributed to the uncertainties within the heat partition model in specifically evaluating the vapour generation rate. In the quest of reducing the application of empirical correlations, the consideration of the active nucleation site density in the present study still depended on the use of an appropriate relationship, which could be sensitive to the flow conditions. The significance of active wall nucleation site density linking to the prediction of the IAC has also been reported in Hibiki and Ishii43. Nevertheless, as the population of cavities may vary significantly between materials and cannot be measured directly, an adequate expression of the active nucleation site density covering a wide range of flow conditions remains outstanding and more concerted research is required. The IAC profiles as shown in


153

Figure 15 also exhibited similar trends with the void fraction distributions plotted in Figure 14.

Figure 14. Predicted radial void fraction distribution and experimental data at the measuring station.

5.2.2. Evolution of Bubble Size Distribution and Bubble Generation Rate Due to Coalescence, Breakage and Condensation Figure 16 shows the bubble size distribution expressed in terms of interfacial area concentration of individual bubbles classes along radial direction for the case L3 at the measuring station describes the bubble dynamics caused by coalescence, breakage and condensation in subcooled boiling flows. Significant vapour bubbles represented from bubble class 3 in the vicinity of the heated wall essentially indicated the size of the bubble lift-off diameter which coalesced with downstream/neighbouring bubbles forming larger void fraction peaks as indicated by bubble classes of 7 and 9. Owing to the high shear stress within the boundary layer, some bubbles are affected by turbulent impact due to breakage resulting in the formation of smaller bubbles as evidenced by the significant distributions indicated

154


within bubble classes 1 and 2. Away from the heated wall, the condensation process dominated in reducing the void fraction of each bubble classes and eventually collapsing majority of the bubbles beyond the position (r-Ri)/(Ro-Ri)=0.6. The net generation rate due to coalescence and breakage and condensation rate of selected bubble classes are depicted in Figure 17. Close to the wall region, the highest generation rate corresponding to the peak value observed in Figure 16 is represented by bubble class 7; substantial generation rate was also found for bubble class 3 at the same region. While the coalescence of bubbles was seen to be governed mainly by bubble classes 3 and 7, bubble classes 3 and 12 also contributed to the condensation process due to their considerably high number density and interfacial area. These two figures aptly demonstrated the mechanisms of coalescence, breakage and condensation in the modified MUSIG model affecting the thermo-mechanical and hydrodynamics processes within the subcooled boiling flow.

Figure 15. Predicted interfacial gas velocity distribution and experimental data at the measuring station.


155

Figure 16. Predicted IAC of each bubbles class along radial direction for the case L3 at the measuring station.

Figure 17. Predicted net bubble generation rate due to coalescence and breakage and condensation rate of selected bubble classes of modified MUSIG model.

156


5.3. Limitations and Shortcomings of Existing Models Encouraging predictions by the homogeneous MUSIG and modified MUSIG models clearly demonstrated their viable applications in resolving isothermal bubbly turbulent pipe flow and subcooled boiling conditions. Nevertheless, the flow cases that have been investigated from above generally possessed only weak bubble-bubble interactions and narrow bubble size distributions. Specifically, the limitation involved solving a single velocity field for all bubble classes. To circumvent the problem, the inhomogeneous MUSIG model (Krepper et al., 2005), which divides the gaseous dispersed phase into N number of velocity groups, was further assessed to evaluate its feasibility in handling bubbly-to-slug transition flow regime. Transitional flow condition with liquid superficial velocity =0.986m/s and gas superficial velocity =0.242m/s as measured by Hibiki et al. (2001) was investigated for the application of both the homogeneous and inhomogeneous MUSIG models. Similar to the bubbly turbulent pipe flow simulations presented in previous section, the same boundary conditions were specified for both models where details can be referred in Table 1. Bubble size in the range of 0-10 mm discretized into 10 sub-size groups as tabulated in Table 2 was similarly considered. For the inhomogeneous model, two velocity fields were solved representing the travelling speed of small and big bubbles. The first five bubble classes (range of 0 – 5 mm) were assigned to the first velocity field while the remaining bubble classes (range of 5 – 10 mm) were assigned to the second velocity field. Sensitivity studies on the increasing resolution greater than two velocity fields were also performed. With regards to the mean parameters investigated, negligible differences were nonetheless found. The measured and predicted local radical void fraction, Sauter mean bubble diameter, IAC and gas velocity distribution at the measuring station of z/D=53.5 are illustrated in Figure 18. Comparing the predicted Sauter mean diameters, the inhomogeneous MUSIG model was found to yield comparatively better prediction when compared against the measured data. This could be attributed to the merit of splitting the bubble velocity with two independent fields which facilitated the model to re-capture the separation of small and big bubbles caused by different lift force actuation. Nevertheless, notable discrepancies were found when comparing against other variables (i.e. void fraction, gas velocity and IAC) against the experimental measurements. As depicted in Figure 18b, void fractions of both models were obviously over-predicted at the channel core but under-predicted at the wall region, which resulted in unsatisfactory IAC predictions (see Figure 18c). Interestingly enough, the consideration of multiple velocity fields in the inhomogeneous MUSIG model did not contribute to the desired expected improvements when comparing the gas velocity predictions against those of the homogeneous MUSIG model (see Figure 18d). This could be possibly due to the interfacial force models which have been developed principally for isolated bubbles rather than on a swarm or cluster of bubbles. Direct applications of these models for high void fraction conditions, where bubbles are closely packed, become questionable and introduce uncertainties in the model calculations. Such findings have also been reported lately in the experimental work by Simonnet et al. (2007). Based on their measurements, they concluded that the aspiration of bubbles in the wake of the leading ones became dominant if the void fraction exceeded the critical value 15%. This caused a sharp increase of the relative velocity of bubbles and significantly altered the associated drag coefficient. Interfacial forces models based on empirical correlations of isolated spherical


157

bubbles have also been found to be not appropriate (Jakobsen, 2001; Behzadi et al., 2004) for high void fraction conditions, As high void fraction (i.e. 20%) condition has been simulated here, the ambiguity of the interfacial drag forces could plausibly be the main source of error. Furthermore, the wall lubrication force could be under-predicted by the Antal’s model (1991) which could represent another source of error in calculating lateral interfacial forces (Lucas et al., 2007).

Figure 18. Predicted local radical void fraction, Sauter mean bubble diameter, IAC and gas velocity distribution of transition bubbly-to-slug flow condition by homogenous and inhomogeneous MUSIG model.

On the other hand, coalescence due to wake entrainment and breakage of large bubbles caused by surface instability may prevail beyond the critical void fraction limit. The existing kernels which only featured coalescence due to random collision and breakage due to turbulent impact for spherical bubbles have to be extended to account for additional bubble mechanistic behaviours for cap/slug bubbles. Several papers in the literature have attempted to deal with this problem by the development of a two-group interfacial area transport equation (Fu et al., 2002a,b; Sun et al., 2004a,b). Encouraging results attained thus far not

158


only suggested the feasibility of the proposed approach but more importantly the prevalence of large bubble mechanisms that are substantially different from spherical bubble interactions in turbulent gas-liquid flows.

6. Conclusions and Further Developments In previous section, a complete three-dimensional two-fluid model coupled with the class method of population balance approach was presented to handle the complex hydrodynamics and thermo-mechanical processes of various bubbly turbulent pipe flow conditions. Numerical study of isothermal bubbly turbulent pipe flows in a vertical pipe was first studied in order to confine the complexity of solely modelling the bubble coalescence and bubble breakage mechanisms. The homogenous MUSIG model which assumed all bubbles travelling with the same velocity was applied. Comparison of the predicted results was made against the measurements of Hibiki et al. (2001). Overall, the homogeneous MUSIG model yielded good agreement for the local radial distributions of void fraction, interfacial area concentration, Sauter mean bubble diameter and gas and liquid velocities against measurements. Numerical results also clearly showed that the range of bubbles sizes existed in the gas-liquid flows required substantial resolution. Numerical results obtained through this study clearly demonstrated the competence of the MUSIG model and the robustness of the bubble coalescence and bubble breakage kernels in accommodating the interactions of finely dispersed bubbles within isothermal bubbly turbulent pipe flows. The potential of the population balance approach was further exploited in modelling subcooled boiling flows. Such flows by nature are inherently complex since they simultaneously embrace all complex flow hydrodynamics, bubble coalescence and bubble breakage accompanied by heat and mass transfer processes in the bulk flow, and various surface heat flux characterisations. An improved heat partition model which mechanistically determined the bubble departure and lift-off diameter was also presented as a closure for the problem in order to determine the appropriate evaporation rate of the nucleation process. Numerical results validated against the experiment data of Yun et al. (1997) and Lee et al. (2002) for low-pressure subcooled boiling annular channel flows showed good agreement for the local Sauter mean bubble diameter, void fraction, IAC profiles. Detailed vapour size distribution and its corresponding generation rates were probed via the size fractions of the modified MUSIG model in order to better envisage the condensation effect in conjunction with the occurrence of bubble coalescence and bubble breakage within the gas-liquid flows. In addition to the homogeneous MUSIG model, inhomogeneous MUSIG model was also applied to investigate the feasibility of application in handling transition bubbly-to-slug flow. It was observed that the inhomogeneous MUSIG gave better prediction of the Sauter mean bubble diameter distributions when compared to the homogeneous model. The more complicated inhomogeneous MUSIG model provided the premise of better capturing the different effects of the lift force acting on the small and large bubbles. Less encouraging results were however ascertained between the predicted and measured void fraction, interfacial gas velocity and IAC profiles. From the above numerical study, as an example of population balance modelling, some important numerical issues have been demonstrated. Firstly, it exposed the limitation of the existing interfacial forces models. Most of the existing interfacial forces models were


159

developed and calibrated from isolated single particle which may not be strictly applicable if particles are closely packed. This is due to the fact that particle motion may be influenced by neighbouring particles resulting different momentum transfer. According to the recent experimental study by Simonnet et al. (2007), significant decrement of gas bubble drag coefficient was found if the gas void fraction exceeded 15%. In fact, very recently, some research studies have been performed attempting to improve the existing interfacial drag and lift force models (Hibiki and Ishii, 2007; Marbrouk et al., 2007; Liu et al, 2008) Secondly, it unveiled the constraint of the current coalescence and breakage kernels. For bubbly simulations presented in this review, coalescence and breakage kernels were derived based on the spherical bubble assumption. This assumption limits the kernels to be only applicable to bubbly flow regime where bubbles are in spherical shape. This also explains why less encouraging results were obtained when bubbly-to-slug flow was considered. In bubbly-to-slug flows, subject to the balance of the surface tension and surrounding fluid motion, large bubbles may be deformed into cap or taylor bubbles. Thus, coalescence due to wake entrainment may become significant which unfortunately was not modelled in current adopted kernels. Unfortunately, fundamental knowledge of bubble coalescence and breakage mechanisms still remain elusive forming a bottleneck for the development of more robust population balance kernels. In essence, not only for gas-liquid systems, similar problems and challenges are also prevalent in other PBE applications; for example: soot formation prediction in combustion system (Frenklach and Wang, 1994). It is certainly that substantial research works would be centred in kernels development in near future. Finally, although encouraging results had been obtained from the MUSIG model based on CM, one should be also reminded that QMOM or other moment methods which represent a rather sound mathematical approach and an elegant tool of solving the PBE with limited computational burden could be alternatively considered for modelling practical multiphase flow systems for future investigative studies.

Notation aif Ac Aq C1 , C2 CD CL Cp Cs Cv d dw D Dc,DB Db Dd Dl

= = = = = = = = = = = = = = = =

interfacial area concentration cross sectional area of the boiling channel fraction of heater area occupied by bubbles constants defined in equation (32) drag coefficient shear lift coefficient specific heat constant defined in equation (41) Acceleration coefficient vapor bubble diameter at heated surface surface/bubble contact diameter average bubble diameter bubble death rate due to coalescence and breakage departing bubble diameter bubble departure diameter bubble lift-off diameter

160


Ds f

= =

Sauter mean bubble diameter bubble generation frequency

fi

=

scalar size fraction of the ith discrete bubble classes f i = α i / α g

F Fc,FB Fb Fcp Fdu Fh Fqs Fs FsL Flg Fgl

= = = = = = = = = = = =

degree of surface cavity flooding calibration factors for coalescence and breakage buoyancy force contact pressure force unsteady drag force due to asymmetrical growth of the bubble force due to the hydrodynamic pressure Quasi-steady-drag force surface tension force shear lift force action of interfacial forces from vapor on liquid action of interfacial forces from liquid on vapor

Flglift

=

lift force

Flglub rication

=

wall lubrication force

Flgdispersion

=

turbulent dispersion force

g

K g

= =

gravitational constant gravitational vector

G Gs h h0, hf hfg H Ja k K hfg ls ni Na P Pk Pc,PB Qw Qc Qe Qtc Qtcsl r

= = = = = = = = = = = = = = = = = = = = = =

mass flux dimensionless shear rate interfacial heat transfer coefficient initial and critical film thickness latent heat enthalpy Jakob number thermal or turbulent kinetic energy projected area of bubble latent heat of vaporization sliding distance number density of the discrete bubble ith class active nucleation site density pressure turbulent kinetic energy production term bubble production rate due to coalescence and breakage wall heat flux heat transfer due to forced convection heat transfer due to evaporation heat transfer (transient conduction) due to stationary bubble heat transfer (transient conduction) due to sliding bubble bubble radius at heated wall or flow spacing within annular

Flgdrag

drag force


161

rc rr R Re Rf

= = = = =

Rph Ri Ro s Si Sij St t tg tsl tw T ΔT u

= = = = = = = = = = = = = = = = = = = =

channel cavity radius at heated surface curvature radius of the bubble at heated surface source/sink term due to coalescence and breakage bubble Reynolds number ratio of the actual number of bubbles lifting off to the number of active nucleation sites source/sink term due to phase change radius of inner heated wall radius of outer unheated wall spacing between nucleation sites additional source terms due to coalescence and breakage tensor of shear stress Stanton number time bubble growth period bubble sliding period bubble waiting period temperature difference in temperature velocity velocity vector friction velocity specific volume of discrete bubble ith class cartesian coordinate along x non-dimensional normal distance from heated wall cartesian coordinate along y

φWN

= = = = = = = =

advancing angle vapor void fraction liquid void fraction receding angle coalescence rate Thermal boundary layer thickness turbulent dissipation rate bubble nucleation rate

φCOND

=

bubble condensation rate

η

λ λe

= = =

thermal diffusivity or coalescence volume matrix size of an eddy

μ θ

= =

K u

uτ vi x x+ y Greek Letters α αg αl β

χ

δl ε

effective viscosity viscosity bubble contact angle

162


θi ρ σ

= = = =

inclination angle density surface tension/Prandtl number bubble contact time

= = =

turbulent frequency breakage rate size ratio between an eddy and a particle in the inertial subrange

=

heated perimeter of boiling channel

Γlg Γgl

= =

interfacial mass transfer from vapor to liquid interfacial mass transfer from liquid to vapor

Subsripts axial g inlet l local s t sat sub sup w

= = = = = = = = = = =

axial distribution vapor channel entrance liquid local distribution surface heater turbulent saturation subcooled superheated heated surface wall

τ ij ω

Ω

ξ ξH

Acknowledgment The financial support provided by the Australian Research Council (ARC project ID DP0877734) is gratefully acknowledged.

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[5]

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Chapter 5

NUMERICAL ANALYSIS OF HEAT TRANSFER AND FLUID FLOW FOR THREE-DIMENSIONAL HORIZONTAL ANNULI WITH OPEN ENDS Chun-Lang Yeh* Department of Aeronautical Engineering, National Formosa University, Huwei, Yunlin 632, Taiwan, R.O.C.

Abstract Study of the heat transfer and fluid flow inside concentric or eccentric annuli can be applied in many engineering fields, e.g. solar energy collection, fire protection, underground conduit, heat dissipation for electrical equipment, etc. In the past few decades, these studies were concentrated in two-dimensional research and were mostly devoted to the investigation of the effects of convective heat transfer. However, in practical situation, this problem should be three-dimensional, except for the vertical concentric annuli which could be modeled as twodimensional (axisymmetric). In addition, the effects of heat conduction and radiation should not be neglected unless the outer cylinder is adiabatic and the temperature of the flow field is sufficiently low. As the author knows, none of the open literature is devoted to the investigation of the conjugated heat transfer of convection, conduction and radiation for this problem. The author has worked in industrial piping design area and is experienced in this field. The author has also employed three-dimensional body-fitted coordinate system associated with zonal grid method to analyze the natural convective heat transfer and fluid flow inside three-dimensional horizontal concentric or eccentric annuli with open ends. Owing to its broad application in practical engineering problems, this chapter is devoted to a detailed discussion of the simulation method for the heat transfer and fluid flow inside threedimensional horizontal concentric or eccentric annuli with open ends. Two illustrative problems are exhibited to demonstrate its practical applications.

Keywords：three-dimensional horizontal concentric or eccentric annuli with open ends, conjugated heat transfer of convection, conduction and radiation, body-fitted coordinate system, zonal grid method. *

E-mail address: [email protected]. Tel. No.：886-5-6315527, Fax No.：886-5-6312415.

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Nomenclature Cp

specific heat capacity

Dh

hydraulic diameter

Ec

V* Eckert number , = * * C pref q w* Dh* / k ref

Fr

Froude number , = V /

2

*

g

g * Dh*

gravitational acceleration jk

g

contravariant metric tensor

h J k

convective heat transfer coefficient Jacobian thermal conductivity

Pr

Prandtl number , = υ / α

p

hydraulic pressure , =

q

q

*

*

* p * + ρ ref g * y* * ρ ref V*

2

heat flux j

curvilinear coordinate

RΦ

source term

Ra

Rayleigh number , = g

Ra o

modified Rayleigh number , = g

T

temperature

(u,v,w)

physical velocity , = ( u , v , w ) / V

Vj V*

contravariant velocity

(x,y,z)

Cartesian coordinates , = ( x , y , z ) / Dh

*

3

β * Dh* ΔT * / υ *α *

*

*

*

4

β * Dh* q w* / υ *α * k *

*

*

characteristic velocity, = α / Dh *

*

*

*

*

*

α β

thermal diffusivity； also radiation absorptivity thermal expansion coefficient

ΓΦ ε

diffusion coefficient radiation emissivity

θ

non-dimensionalized temperature , =

μ υ ρ

viscosity kinematic viscosity density , =

* ρ * / ρ ref

T * − Tref* q w* Dh* / k *

Numerical Analysis of Heat Transfer and Fluid Flow… σ

Φ

φ

173

Stefan-Boltzmann constant energy dissipation term； also dependent variable azimuthal angle

Subscripts i nb o P ref w

inner cylinder neighboring grid points outer cylinder main grid point reference state（ at atmospheric pressure and room temperature） wall

Superscripts ▬ *

averaged quantity dimensional quantity

Introduction Heat transfer and fluid flow inside concentric or eccentric annuli can be found in numerous engineering applications, e.g. solar energy collection, fire protection, underground conduit, heat dissipation for electrical equipment, etc. In the past few decades, these studies were concentrated in two-dimensional research and were mostly devoted to the investigation of the effects of convective heat transfer. However, in practical situation, this problem should be three-dimensional, except for the vertical concentric annuli which could be modeled as two-dimensional (axisymmetric); in addition, the effects of heat conduction and radiation should not be neglected unless the outer cylinder is adiabatic and the temperature of the flow field is very low. As the author knows, none of the open literature is devoted to the investigation of the conjugated heat transfer of convection, conduction and radiation for this problem. Numerous theoretical and experimental studies on natural convection in horizontal concentric or eccentric annuli have been conducted. In most of these studies, a twodimensional model was used in which the annuli were assumed to be infinitely long and coupled with thermal boundary conditions on the cylinder surfaces specified as either with two constant wall temperatures or one with constant wall temperature while the other with constant wall heat flux (including adiabatic surface) [1-56]. Some representative studies are listed below. 1. Gju et al. [2,3] conducted experiments for the natural convection in horizontal concentric or eccentric annuli. The inner and outer cylinders were both kept at constant temperature. The instability and transition of the heat transfer and fluid flow were investigated. It is found that for a concentric annulus of inner/outer diameter ratio 2.36, chaos occur near Rayleigh number between 0.9×105 and 3.37×105.

174

Chun-Lang Yeh 2. Kuehn and Goldstein [7-9] also performed experimental investigations for the natural convection in horizontal concentric or eccentric annuli. The inner and outer cylinders were both kept at constant temperature. The instability and transition of heat transfer and fluid flow were investigated. It is found that for a concentric annulus of inner/outer diameter ratio 2.6, transition from laminar to turbulent regimes occurs near Rayleigh number of 4×106. 3. Vafai et al. [11-20] studied numerically the two- and three-dimensional natural convection in horizontal concentric annuli. The heat transfer and fluid flow under different inner/outer diameter ratios and Rayleigh numbers were investigated. The scope of discussion covers laminar and turbulent regimes as well as steady and transient flows. In respect of the fluid flow field, the interaction of primary and second flows was examined. With respect to the thermal field, the distribution of Nusselt number was analyzed. In addition, the influence of geometry on the heat transfer and fluid flow was also discussed. 4. Yoo et al. [21-27] performed two-dimensional simulation of natural or mixed convection in horizontal concentric annuli. The outer cylinder was kept at constant temperature while the inner cylinder was kept at constant temperature or heat flux. The heat transfer and fluid flow were analyzed for different Rayleigh numbers, Reynolds numbers, Prandtl numbers and inner/outer diameter ratios. The scope of discussion covers the transition of laminar to turbulent regimes as well as the phenomenon of chaos. 5. Shu et al. [28-33] studied numerically the two-dimensional natural convection in horizontal concentric or eccentric annuli. The outer cylinder is a square duct kept at a lower temperature while the inner cylinder is a circular one kept at a higher temperature. The heat transfer and fluid flow were investigated for different Rayleigh numbers, Reynolds numbers, Prandtl numbers, inner/outer diameter ratios and eccentricity of inner/outer cylinders. In respect of the thermal field, the distribution of Nusselt number was analyzed. 6. Mujumdar et al. [34-36] investigated numerically the two-dimensional natural convection and phase change in horizontal concentric annuli. The outer cylinder is a square or circular duct while the inner cylinder is a circular or square one. The heat transfer and fluid flow were examined for different Rayleigh numbers and heating rates on the inner or outer cylinders. 7. El-Shaarawi et al.[37-39] studied numerically the laminar mixed convection in horizontal concentric annuli with non-uniform circumferential heating. Secondary flow and Nusselt number were analyzed in their study. They also investigated the transient conjugated natural convection and conduction in open-ended vertical concentric annuli, which could be modeled as two-dimensional flow because of its axisymmetry. The authors also analyzed the free convection in vertical eccentric annuli with a uniformly heated boundary, which has to be simulated by threedimensional model due to the eccentricity. 8. Mota et al. [40-42] simulated the two-dimensional natural convection in horizontal concentric or eccentric annuli of elliptic or circular cross sections. The heat transfer and fluid flow were analyzed for different inner/outer diameter ratios and eccentricity of inner/outer pipes. They found that the eccentric elliptic annulus has the lowest heat loss.

Numerical Analysis of Heat Transfer and Fluid Flow…

175

9. Char et al. [43-45] investigated numerically the two-dimensional natural or mixed convection in horizontal concentric or eccentric annuli. The outer cylinder was kept at a lower temperature while the inner cylinder was rotating and kept at a higher temperature. Low Reynolds number turbulence model was adopted to account for the turbulence effects. The heat transfer and fluid flow were analyzed for different Rayleigh numbers, Reynolds numbers and inner/outer diameter ratios. 10. Ho et al. [46-48] performed experimental and numerical investigations for the twodimensional natural convection in horizontal concentric or eccentric annuli. The outer cylinder was kept at constant temperature while the inner cylinder was kept at constant heat flux. The heat transfer and fluid flow were investigated for different Rayleigh numbers, Prandtl numbers and eccentricity of inner/outer cylinders. They found that the heat and fluid flow were affected mainly by the Rayleigh number and eccentricity of inner/outer cylinders, while the Prandtl number had only minor effect on this flow. 11. Mizushima et al. [49,50] studied the instability and transition of two-dimensional natural convection in horizontal concentric annuli. The outer cylinder was kept at a lower temperature while the inner cylinder was at a higher temperature. The heat transfer and fluid flow were analyzed for different Prandtl numbers. The authors also investigated the linear stability of the flow. 12. Hamad et al. [51,52] conducted experiments for the natural convection in inclined annuli with closed ends. The inner cylinder was kept at constant heat flux. The heat transfer and fluid flow were investigated for different Rayleigh numbers, Nusselt numbers, inclined angles and inner/outer diameter ratios. They found that the heat and fluid flow were affected mainly by the Rayleigh number and inner/outer diameter ratios, while the inclined angle had only minor effect on this flow. 13. Raghavarao and Sanyasiraju [53,54] studied numerically the two-dimensional natural convection in horizontal concentric or eccentric annuli. The inner and outer cylinders were both kept at constant temperature. The heat transfer and fluid flow were investigated for different inner/outer diameter ratios and eccentricity of inner/outer cylinders. 14. Choi and Kim [55,56] investigated numerically the three-dimensional linear stability of mixed-convective flow between rotating horizontal concentric cylinders. The inner cylinder was rotating and kept at constant heat flux. The authors found that the heating of the inner cylinder delays the formation of Taylor vortices when the rotating effect is more pronounced than the buoyancy effect. On the other hand, when rotation and buoyancy are comparable, the flow becomes unstable. This instability is caused mainly by the buoyancy effect. From the above literature review, it can be seen that there have been very few threedimensional investigations of the heat and fluid flow in concentric or eccentric annuli between two horizontal cylinders with open ends, except with the configuration of cavity type [1,15,16]. Owing to the broad application in practical engineering problems, this chapter is devoted to a detailed discussion of the simulation method for the heat transfer and fluid flow inside three-dimensional horizontal concentric or eccentric annuli with open ends, with the aim to get a more thorough understanding of this problem.

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Chun-Lang Yeh

Mathematical Model For convenience of illustration, take the underground conduit of electrical power cable as an example. In this section, two situations are discussed. One is a purely natural convective flow and the other is a conjugated heat and fluid flow which involves conduction, convection and radiation.

2.1. Natural Convective Flow Heat is generated from the electrical resistance of the power cable and the heat dissipation process in the annulus relies on the natural convection heat transfer from both open ends of the conduit, which penetrate onto the manhole surfaces. The heat dissipation rate is determined from the ventilation stemmed from natural convection and will affect the lifetime of the power cable. Owing to the symmetric nature of the flow field with respect to the two open ends and to a vertical plane crossing the center of the cylinders, the computational domain is schematically shown in Figure1. Selected transverse and longitudinal sections are illustrated in Figure 2.

(a) Figure 1. Continued on next page.


177

(b) Figure 1. Illustration of the computational domain and zonal grid distribution - (a) overall view (b) zoom-in view near the interface of the two zones.

Unlike the vertical annuli in which the fully developed thermal boundary conditions may be achievable[39], the boundary conditions at the open ends of the present problem are much more troublesome. Although this plane is named as the outlet (see Figure 1) in consideration of the heat dissipation route in the conduit, it consists of inflow (fresh flowing fluid) and outflow (heated flowing fluid) at the same plane. An approach used in the simulations of the non-cavity type, buoyancy-induced flows is the “zonal grid” approach[57-60] which extends the computational domain outside the outlet plane so that the boundary conditions can be reasonably specified with the ambient flow properties. Typical examples can be found in Refs.59 and 60. Although this approach requires enormous computations for threedimensional problems, it provides more reliable results among existing approaches. In this work, the “zonal grid” approach is adopted to resolve the problem of the outlet boundary conditions. In the following discussion, two different cases are discussed. The first is a numerical investigation of the three-dimensional natural convection inside a horizontal concentric annulus with specified wall temperature or heat flux[59] and the second is a threedimensional natural convection inside horizontal concentric or eccentric annuli with specified wall heat flux[60]. The boundary conditions for the first problem as illustrated schematically in Figure 1 are stated as follows. Either adiabatic or isothermal condition is given on the outer cylinder surface, while on the inner cylinder a constant heat flux is given. For the isothermal condition, the outer cylinder surface is maintained at Tref（ i.e. 300 K） . No slip condition is given to all the three components of the velocity on the outer and inner cylinder surfaces. Since no flow crosses the circumferentially (or longitudinally) symmetric plane, the angular (or axial)

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Chun-Lang Yeh

velocity vanishes on that plane. The angular (or axial) derivatives of the remaining velocity components and temperature also vanish on the circumferentially (or longitudinally) symmetric plane.

Figure 2. Illustration of the selected - (a) transverse sections for eccentric annulus, (b) transverse sections for concentric annulus, and (c) longitudinal sections.

The boundary conditions for the second problem are stated as follows. The adiabatic condition is given on the outer cylinder surface, while on the inner cylinder a constant heat flux, which in practical situation is resulted from the heat generation of the power cable due to electrical resistance, is given. No slip condition is given to all the three components of the velocity on the outer and inner cylinder surfaces. Since no flow crosses the circumferentially (or longitudinally) symmetric plane, the angular (or axial) velocity vanishes on that plane. The angular (or axial) derivatives of the remaining velocity components and temperature also vanish on the circumferentially (or longitudinally) symmetric plane.


179

Because the thermal boundary conditions at the inner cylinder surfaces are specified in terms of heat fluxes instead of temperatures in the above two example problems, a modified Rayleigh number is defined as follows.

Ra = o

* g * β ref Dh*3 * * υ ref α ref

* (q w* Dh* / k ref )

(1)

As pointed out by Kuehn and Goldstein [8], an onset of transition from laminar to turbulent regimes starts near Ra = 4 × 10

6

for gases in a concentric annulus of

Do / Di = 2.6 . Moreover, a study by Labonia and Gju [3] indicated that chaotic flows were observed in the range of 0.9 × 10 ≤ Ra ≤ 3.37 × 10 5

5

for a concentric annulus of

Do / Di = 2.36 . However, their conclusions were drawn from the experimental observations and based upon the cases associated with the constant temperature differences between the outer and inner cylinders. Kumar [6] made a numerical investigation using a two-dimensional model for an infinitely long, horizontal, concentric annulus where the inner cylinder was specified by a constant heat flux and the outer cylinder was isothermally cooled. He found that the critical Ra above which the numerical results failed to converge were 3.1 × 10 and o

5

3 × 10 6 at Do / Di = 1.5 and 2.6, respectively. Kumar also pointed out that it was hard to judge whether the flow would become oscillatory or three-dimensional beyond the critical

Ra o for a given ratio of Do / Di . The present example problems, which are essentially three-dimensional with the medium of air（ Pr = 0.7 ） , encounter the same convergence difficulty beyond Ra = O (10 ) , which may imply an onset of transition from steady o

7

laminar to chaotic or even turbulent flows. Here, two cases of Rayleigh numbers,

Ra o = 10 5 and Ra o = 10 6 , are calculated and demonstrated. The flow pattern of interest here necessitates the solution of three-dimensional fully elliptic type of partial differential equations, which describe the natural convective flow field. Considering the steady state flow situation, the governing equations in Cartesian coordinates read：

∂ (ρu ) + ∂ (ρv ) + ∂ (ρw ) = 0 ∂x ∂y ∂z

∂ (ρuu ) + ∂ (ρvu ) + ∂ (ρwu ) = − ∂p + ∂x ∂x ∂y ∂z ⎡ ∂ 2 u ∂ 2 u ∂ 2 u 1 ∂ ⎛ ∂u ∂v ∂w ⎞⎤ ⎜ + ⎟⎥ + Pr⋅ ⎢ 2 + 2 + 2 + 3 ∂x ⎜⎝ ∂x ∂y ∂z ⎟⎠⎦ ∂y ∂z ⎣ ∂x

(2)

(3)

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Chun-Lang Yeh

∂ (ρuv ) + ∂ (ρvv ) + ∂ (ρwv ) = − ∂p + ∂y ∂x ∂y ∂z ⎡ ∂ 2 v ∂ 2 v ∂ 2 v 1 ∂ ⎛ ∂u ∂v ∂w ⎞⎤ 1 − ρ ⎜ + ⎟⎥ + + Pr⋅ ⎢ 2 + 2 + 2 + 3 ∂y ⎜⎝ ∂x ∂y ∂z ⎟⎠⎦ Fr 2 ∂y ∂z ⎣ ∂x ∂ (ρuw) + ∂ (ρvw) + ∂ (ρww) = − ∂p + ∂z ∂x ∂y ∂z ⎡ ∂ 2 w ∂ 2 w ∂ 2 w 1 ∂ ⎛ ∂u ∂v ∂w ⎞⎤ ⎜ + ⎟⎥ + Pr⋅ ⎢ 2 + 2 + 2 + 3 ∂z ⎜⎝ ∂x ∂y ∂z ⎟⎠⎦ ∂y ∂z ⎣ ∂x 2 2 2 ∂ (ρuθ ) + ∂ (ρvθ ) + ∂ (ρwθ ) = ∂ θ2 + ∂ θ2 + ∂ θ2 + ∂z ∂y ∂x ∂x ∂y ∂z

⎛ ∂p ∂p ∂p ⎞ + v + w ⎟⎟ + Pr⋅ Ec ⋅ Φ − Π Ec ⋅ ⎜⎜ u ∂y ∂z ⎠ ⎝ ∂x

(4)

(5)

(6)

where

⎡⎛ ∂u ⎞ 2 ⎛ ∂v ⎞ 2 ⎛ ∂w ⎞ 2 ⎤ ⎛ ∂u ∂v ⎞ 2 ⎛ ∂u ∂w ⎞ 2 ⎛ ∂v ∂w ⎞ 2 ⎟⎟ Φ = 2⎢⎜ ⎟ + ⎜⎜ ⎟⎟ + ⎜ ⎟ ⎥ + ⎜⎜ + ⎟⎟ + ⎜ + ⎟ + ⎜⎜ + ⎢⎣⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎝ ∂z ⎠ ⎥⎦ ⎝ ∂y ∂x ⎠ ⎝ ∂z ∂x ⎠ ⎝ ∂z ∂y ⎠ (7) 2 2 ⎛ ∂u ∂v ∂w ⎞ ⎟ + − ⎜⎜ + 3 ⎝ ∂x ∂y ∂z ⎟⎠ and

Π = μ *V * g * / q w*

(8)

Note that the Boussinesq approximation usually made in the formulation of the natural convection is not adopted here and the density is determined using the ideal gas law. The reason of using the ideal gas law rather than Boussinesq approximation for density determination is that the Boussinesq approximation is not valid in case of high temperature difference, which may arise when the Rayleigh number exceeds a critical value.

2.2. Conjugated Heat and Fluid Flow In section 2.1, the effects of heat conduction and radiation are neglected. This assumption is valid only if the outer cylinder（concrete conduit） is adiabatic and the temperature of the flow field is low enough. A practical underground power cable placed in a concrete conduit can be schematically shown as Figure 3. To give a preliminary estimate for the influence of the conduction and radiation heat transfer on this problem, the two-dimensional simplified


181

model shown in Figure 4 is analyzed. Consider conjugated conduction and convection heat transfer first. solar beam

concrete conduit

soil

power cable

Figure 3. Illustration of a practical underground power cable placed in a concrete conduit.

soil

concrete conduit 2cm

power cable

r2

r 1= do /2 di /2 air

Figure 4. Illustration of a two-dimensional simplified model for the underground power cable placed in a concrete conduit.

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Chun-Lang Yeh

Assume that the concrete conduit is in thermal equilibrium with the surrounding soil, diameter of the power cable is di=0.1m, the inner radius of the concrete conduit is r1(=do/2)=0.1m，and its thickness is r2-r1=2cm, then the heat transfer（which includes both conduction and convection）from the soil to the air is[61]

q = U ( PΔx)

(Tsoil − Tm | x ) − (Tsoil − Tm | x + Δx ) , Tsoil − Tm | x ln Tsoil − Tm | x + Δx

where the conjugated heat transfer coefficient, U , is

U =

1 r1

(9)

r 1 + ln 2 hair k concrete r1

kconcrete=0.7W/m．K is the thermal conductivity of the concrete conduit. From Figure 4 of Ref.7, the Nusselt number Nu=

ho Dh ≅ 10 . Then, the convection k

heat transfer coefficient for the air inside the annulus is

ho = 10k / Dh = 10 × 2.7 ×10 −2 / 0.1 = 2.7 W / m 2 ⋅ K

(10)

where

Dh = d o − d i = 0.2 − 0.1 = 0.1 m; thermal conductivity for air

k air = 2.7 × 10 −2 W / m ⋅ K . The conjugated heat transfer

coefficient, U , can then be found by substituting

U =

The

error

caused

ho

from Eq.(10) into hair in Eq.(9).

1 = 2.52 W / m 2 ⋅ K 1 0.1 0.12 + ln 2.7 0.7 0.1 by

the

neglect

of

conduction

heat

transfer

then

is

ho − U 2.7 − 2.52 = = 7 .1% U 2.52 The above analysis can give us a preliminary estimate of the influence of the conduction heat transfer on this problem. In fact, it can be observed from Eq.(9) that the conjugated heat


183

transfer coefficient, U , is closely connected with the outer/inner radius ratio（r2/r1）of the concrete conduit, which is related to the thickness of the concrete conduit. For a thin concrete conduit, r2/r1 approaches unity and ln(r2/r1) is very small. Under such situation, the conduction heat transfer can be neglected. However, in practical engineering problems, the concrete conduit can not be too thin because of the high pressure it sustained（due to the soil for a land cable or the sea water for a submarine cable）. Therefore, the conduction heat transfer should be taken into account in practical engineering applications. To illustrate the influence of the radiation heat transfer on this problem, the twodimensional simplified model（Figure 4）is analyzed again. Assume the surface temperature of the power cable to be Ti , radiation emissivity ε i ≅ 0.88 （for rubber）, and the inner surface temperature of the concrete conduit to be To , radiation emissivity ε o ≅ 0.91 （for concrete）. Also assume that the surface of the power cable and the inner surface of the concrete conduit are gray surfaces, i.e. radiation emissivity and absorptivity are equal（ ε = α ）. Then, the net radiation heat transfer on the surface of the power cable can be estimated as

qr

''

× Api = ε oσTo Fo→i Apo − ε iσTi Fi→o Api 4

cable

4

where the view factors

Fo→i =

A pi

，

A po

Fi →o = 1 ， σ = 5.67 × 10 −8 W / m 2 ⋅ K 4

Then

qr Ti To

where

can be written as

''

= σTo [ε o − ε i ( 4

cable

Ti To + ΔT ΔT = = 1+ To To To

From the author’s previous studies[59,60], Δ T and

ΔT ≈ 12 K

, To ≈ 316 K

for

Ti 4 ) ] To .

≈ 1K , To ≈ 301K

for

Ra o = 10 5 ,

Ra o = 10 6 , where the modified Rayleigh

o

number（ Ra ）is defined in Eq.(1).

ΔT This yields To

4

4

⎛ Ti ⎞ ⎛ ΔT ⎞ ΔT ⎟ ⎜ ⎟ = + ≈ + 1 1 4 << 1 ⇒ ⎜ . ⎜T ⎟ ⎜ To ⎟⎠ To ⎝ o⎠ ⎝ o 5 When Ra = 10 , the net radiation heat transfer from the surface of the power cable

then is

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Chun-Lang Yeh

qr

'' cable

and when Ra

qr

o

'' cable

= 5.67 × 10 −8 × 3014 × [0.91 − 0.88(1 + 4 ×

1 )] ≈ 8.5 W/m2 301

(11)

= 10 6 , the net radiation heat transfer from the surface of the power cable is = 5.67 × 10 −8 × 316 4 × [0.91 − 0.88(1 + 4 ×

12 )] ≈ −58.6 W/m2 (12) 316

On the other hand, from the definition of the modified Rayleigh number（Eq.(1)）, the heat transfer from the surface of the power cable is

q = (k * w

*

where Dh * g * β ref

υ

* 2 ref

* ref

* * υ ref α ref 1 Ra o o * * / D ) * * *3 Ra = (k ref / Dh ) * * g β ref Dh g β ref Pr⋅ Dh* 3 * 2 υ ref * h

= 0.1 m, k ref * = 2.7 × 10 −2 W / m ⋅ K

for air , Pr = 0.711, and

= 13.2 × 10 7 ( 1 / m 3 ⋅ K ) .

Therefore, when Ra

o

= 10 5 , the heat transfer from the surface of the power cable is q w* = 0.29 W / m 2

and when Ra

o

(13)

= 10 6 , the heat transfer from the surface of the power cable is

q w* = 2.9 W / m 2

(14)

Comparisons of Eq.(11) with (13) and Eq.(12) with (14) reveal that the radiation heat transfer plays an important role in this problem. Although the above analysis is only a two-dimensional simplified analysis, it can give us a preliminary view of the influence of the conduction and radiation heat transfer on this problem. The analysis shows that conduction and radiation heat transfer should not be neglected unless the outer cylinder is adiabatic and the temperature of the flow field is very low. In respect of the conjugated heat transfer of convection and conduction, Ha and Jung[62] investigated numerically the three-dimensional conjugated heat transfer of conduction and natural convection in a differentially heated cubic enclosure with a heat-generating cubic conducting body. Méndez and Treviño[63] made a numerical study for the conjugated


185

conduction and natural convection heat transfer along a thin vertical plate with non-uniform internal heat generation. Liu et al.[64] performed a two-dimensional numerical analysis of the coupled conduction-convection problem for an underground rectangular duct containing three insulated cables. Du and Bilgen[65] studied numerically the coupling effect of the wall conduction with natural convection in a rectangular enclosure. Kim and Viskanta[66] analyzed numerically the effects of wall conductance on the natural convection in differently oriented square cavities. A common feature of the above studies for the conjugated conduction-convection heat transfer is to consider the individual heat transfer mode in each medium and to keep continuities of the temperature and heat flux at the interfaces of any two different media. Take the underground power cable as an example, the heat transfer modes include conduction of the power cable (solid), convection of the air inside the concrete conduit (gas), conduction of the concrete conduit (solid) and conduction of the soil (solid). Heat transfer modes in each medium have to be solved separately. Continuities of the temperature and heat flux at the interfaces of any two different media provide the necessary boundary conditions. With respect to the influence of the radiation heat transfer [67-73], the zone model and Monte Carlo method are generally recognized as more accurate ones. However, these two methods require enormous computing time and storage. In addition, they are not of differential type and therefore are more cumbersome to incorporate with the differential equations of fluid motion and energy. Perhaps the most commonly used radiation model for engineering application is the flux model. Recently, the finite volume method （FVM）and the discrete ordinates method（DOM）have become more and more popular because of their success in simulating irregular geometries [69-72]. In using the FVM and DOM, the simulation domain is divided into small blocks. This is quite similar to the concept of the finite volume method in computational fluid dynamics. The DOM needs a quadrature set which has a crucial influence on the accuracy of the method. On the other hand, the FVM is less restricted and is better for the conservation of radiation heat transfer. As pointed out earlier, an onset of transition from laminar to turbulent regimes starts near

Ra = 10 7 . Among the existing turbulence simulation methods, Direct Numerical Simulation（ DNS） and Large Eddy Simulation（ LES） can simulate the actual physics of turbulence more accurately. However, using them for routinely designing and analyzing engineering problems can not be achieved nowadays. DNS is limited to low Reynolds number flows because of the large computer resources needed. Although LES reduces the requirement to some extent; however, much more sophisticated sub-grid scale models and near-wall treatments are required for many flows of interest. A recent research of the LES was conducted by Worthy and Rubini [74] who studied LES stress and scalar flux sub-grid scale models in the context of buoyant jets. Reynolds stress model (RSM) is essentially more realistic than the Eddy viscosity model (EVM) because it involves modeled transport equations for all of the Reynolds stresses. Yilbas et al. [75] employed the RSM to examine jet impingement onto a hole with a constant wall temperature and analyzed the influence of the hole wall temperatures and jet velocities. In their study, the Nusselt number ratio (ratio of the Nusselt number predicted to the Nusselt number obtained for a fully developed turbulent flow based on the hole entrance Reynolds number) was computed and the mass flow ratio (ratio of mass flow rate through the hole to mass flow emanating from the nozzle) was determined. On the other hand, even though the EVM assumes a crude relation between the turbulence

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Chun-Lang Yeh

quantity（ Reynolds stress） and the mean flow quantity（ strain rate） , turbulence models based on this concept, such as the standard k-ε model, have been widely used for industrial CFD calculations due to numerical stability and low cost. However, it has long been recognized that the standard k-ε model has defects in predicting turbulent flows with large streamline curvature and rotational effects. Consequently, the more sophisticated twoequation models employ the Low Reynolds Number（ LRN） Model, the Non-Linear Eddy Viscosity Model（ NEVM） or the Algebraic Reynolds Stress Model（ ARSM） to improve the suitability of the standard k-ε model. Brescianini and Delichatsios [76] examined several versions of the k-ε turbulence model as to their suitability for computational fluid dynamics modeling of free turbulent jets and buoyant plumes. Yang and Ma [77] investigated the predictive performance of linear and nonlinear eddy-viscosity turbulence models for a confined swirling coaxial jet. Abdon and Sundén [78] carried out an investigation of a single round unconfined impinging air jet under different flow and geometrical conditions to assess the performance of linear and nonlinear two-equation turbulence models. Wei et al. [79] proposed and formulated an algebraic turbulent mass flux model (AFM), which properly accounts for swirl–turbulence interactions, and incorporated this model with an algebraic Reynolds stress model proposed previously to simulate the swirling turbulent flow and mixing in a combustor with helium/air jet and swirling air stream. In the author’s previous study[80], the performances of three LEVM, one ARSM and one DRSM（ Differential Reynolds Stress Model） turbulence models are evaluated for the simulation of the plainorifice atomizer and the pressure-swirl atomizer flows. These models include the standard k-ε model[81], Launder-Sharma’s LRN k-ε model[82], Nagano-Hishida’s LRN k-ε model[83], Gatski-Speziale’s ARSM model[84], and Randriamampianina-Schiestel-Wilson’s DRSM model[85]. Adequate evaluations can help users to select a suitable turbulence model for their applications.

Numerical Method The governing equations stated in last section can be cast into the following general form, which permits a single algorithm to be applied.

⎞ ⎛ ∂ (ρv j Φ ) = ∂ ⎜⎜ Γ Φ ∂Φ ⎟⎟ + R Φ ∂x j ∂x j ⎝ ∂x j ⎠

(15)

To facilitate the handling of complex geometry of the present problem, the body-fitted coordinate system is used to transform the physical domain into a computational domain, which is in a rectangular coordinate system with uniform control volumes. Transformation of Eq.(15) to the body-fitted coordinates leads to

∂ ∂ JρV j Φ = j ∂q j ∂q

(

)

⎛ jk Φ ∂Φ ⎜⎜ Jg Γ ∂q k ⎝

⎞ ⎟⎟ + JR Φ ⎠

(16)

Numerical Analysis of Heat Transfer and Fluid Flow… j

where q ： curvilinear coordinates（ J： Jacobian； ≡ xξ yη zζ

187

ξ ,η , ζ ）

+ xη yζ zξ + xζ yξ zη − xζ yη zξ − xξ yζ zη − xη yξ zζ

V j ： contravariant velocity（ U,V,W）

U=

1 [ u ( yη zζ − yζ zη ) + v( xζ zη − xη zζ ) + w( xη yζ − xζ yη ) J

V=

1 [ u( yζ zξ − yξ zζ ) + v( xξ zζ − xζ zξ ) + w( xζ yξ − xξ yζ ) J

]

W=

1 [ u( yξ zη − yη zξ ) + v( xη zξ − xξ zη ) + w( xξ yη − xη yξ ) J

]

]

g jk ： metric tensor

g 11 =

1 [ ( yη zζ − yζ zη ) 2 + ( xζ zη − xη zζ ) 2 + ( xη yζ − xζ yη ) 2 2 J

g 22 =

1 [ ( yζ zξ − yξ zζ ) 2 + ( xξ zζ − xζ zξ ) 2 + ( xζ yξ − xξ yζ ) 2 J2

g 33 =

1 [ ( yξ zη − yη zξ ) 2 + ( xη zξ − xξ zη ) 2 + ( xξ yη − xη yξ ) 2 2 J

]

] ]

1 [ ( yη zζ − yζ zη )( yζ zξ − yξ zζ ) + ( xζ zη − xη zζ )( xξ zζ − xζ zξ ) J2 + ( xη yζ − xζ yη )( xζ yξ − xξ yζ ) ] g 12 = g 21 =

1 [ ( yη zζ − yζ zη )( yξ zη − yη zξ ) + ( xζ zη − xη zζ )( xη zξ − xξ zη ) J2 + ( xη yζ − xζ yη )( xξ yη − xη yξ ) ] g 13 = g 31 =

1 [ ( yζ zξ − yξ zζ )( yξ zη − yη zξ ) + ( xξ z ζ − xζ zξ )( xη zξ − xξ zη ) J2 + ( xζ yξ − xξ yζ )( xξ yη − xη yξ ) ] g 23 = g 32 =

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Chun-Lang Yeh

The grid layout is constructed by connecting the grid points in each transverse plane, which are generated by solving the two-dimensional elliptic type of partial differential equations governing the distribution of the grid points [86]. Numerical calculation of Eq.(16) is performed using the control-volume based finite difference procedure. The discretized governing equations are solved on a non-staggered grid system in association with the SIMPLEC algorithm [87] and QUICK scheme [88]. In the use of the “zonal grid” approach[57-60], the computational domain is extended outside the outlet plane and is divided into two sub-domains of Zones I and II, as schematically illustrated in Figure 1. Zone I and Zone II are overlapped by three layers of grid points. The outer cylinder is flush with the adiabatic solid wall while the inner cylinder extends to the free boundary. The boundary conditions for Zone I have been stated in “MATHEMATICAL MODEL” and will not be repeated here. The boundary conditions of Zone II are as follows. On the surface of the inner cylinder, the same boundary conditions as specified for Zone I are used. On the adiabatic solid wall, the no-slip condition and adiabatic wall are specified. The condition of zero normal gradients is met on the symmetric plane except for the normal velocity component, which vanishes naturally. On the free boundaries, the normal gradients of all the dependent variables except for the temperature are set to be zero. The temperature conditions at the free boundaries are specified as follows： When the flow at any such boundary is leaving the domain, the normal temperature gradient is taken as zero. However, when the flow comes into the domain, its temperature is assigned to that of the ambient. The treatment of the interface of the two zones follows the overlapping grid method [57]. Starting with guessed values, solutions in Zone I together with the overlapped region are updated by one sweep of iteration. The updated solutions at the outlet plane of Zone I are then interpolated by bilinear interpolation as the boundary conditions of Zone II. The solutions in Zone II are then updated by one sweep of iteration and are used, in turn, to interpolate the values in the overlapped region, which provide the new boundary conditions for the resolutions of Zone I together with the overlapped region. This completes a full solution cycle. In each solution cycle, continuity of the dependent variables and conservation of fluxes are preserved across the interface. In fact, this is the key to the success of the approach. The solution cycle is repeated until the convergence criterion is satisfied. The convergence criterion is described below. The general form of the discretized governing equations can be written as

AP Φ P = ∑ Anb Φ nb + b Φ

(17)

nb

Define

BiΦ = AP Φ P − ∑ Anb Φ nb − b Φ

(18)

nb

H iΦ = AP Φ P +

∑A

nb

nb

Φ nb + b Φ

(19)


λΦ =

max(BiΦ ) 1 ∑ H iΦ N i

189

(20)

where i is an arbitrary grid point in the computational domain and N is the total number of grid points. When λ ≤ 1.0 × 10 the iteration process is convergent. Φ

−4

for each dependent variable Φ in both the two zones,

Illustrative Problems In this section, two illustrative problems are discussed. The first is a numerical investigation of the three-dimensional natural convection inside a horizontal concentric annulus with specified wall temperature or heat flux[59] and the second is a threedimensional natural convection inside horizontal concentric or eccentric annuli with specified wall heat flux[60]. Illustration of the computational domain and zonal grid distribution are schematically shown in Figure 1. The inner and outer diameters of Zone I are 0.1m and 0.2m, respectively, and its length is 3m (equivalent to 30Dh* ’s). Zone II is constructed by a cube with side length of 20 Dh* ’s to assure its boundary conditions being reasonably specified by the ambient properties. Numerical tests reveal that the maximum difference in mass inflow rate at the outlet plane between 51 × 51 × 101（ radial by angular by axial） and 61 × 61 × 121 grid meshes for Zone I, while the corresponding grid meshes for zone II are 21 × 41 × 21（ x by y by z） and 31 × 61 × 31, respectively, is less than 0.5﹪. Therefore, the former set of grid mesh is adopted in the present work. Figure 2 illustrates the section positions of the configuration being studied.

4.1. Natural Convection inside a Horizontal Concentric Annulus with Specified Wall Temperature or Heat Flux For the first illustrative problem, both the adiabatic and isothermal conditions for the outer cylinder surface are examined for the conditions of ro / ri = 2 and Rao=106 to investigate the natural convection heat dissipation inside the conduit. Figure 5 shows the velocity vector plots at selected transverse and longitudinal sections for the examined cases. Note that the results for sections A-A, B-B and D-D are the vertical projections of the flow field and therefore the ordinate is y, while the result for section C-C is the horizontal projection of the flow field and therefore the ordinate is x. The flow patterns for the two cases are rather different. As observed from the velocity vector plots on the longitudinal sections, there appear secondary flows in all the six longitudinal sections for both cases. The secondary flow of the adiabatic case is stronger than that of the isothermal case. Further downstream, the secondary flows evolve into counter-rotating recirculation zones for the adiabatic case. Another interesting phenomenon can be observed from the velocity vector plots on the transverse sections. For both cases, the inflow paths are clearly observed in the portion below the inner cylinder, whereas the outflow paths are in the portion above. Such flow patterns result obviously from the buoyancy effect, which can be observed in thermal

190

Chun-Lang Yeh

plume phenomena. This can also be seen from the velocity vector plots at a distance slightly outside the open end (i.e. in Zone II) shown in Figure 6, in which the entrainment effect caused by the upward motion of the buoyant flows can also be seen. adiabatic

Section A-A

1.5 1

1000V

y

0.5 0 -0.5 0

10

z

20

30

20

30

20

30

20

30

Section B-B

1.2

y

1.1 1 0.9 0

10

z

Section C-C

1

x

0.8 0.6 0

10

z

Section D-D 0.1 y

0 -0.1 -0.2

0

10

z

z=6

z=12

1.5

1.5

1

1

y

y

y

0.5

0.5

0.5

0

0

0

z=0

1.5

500V

*

1

-0.5 0

0.5 x 1

-0.5 0

0.5

x 1

-0.5 0

z=18

z=24

z=30

1.5

1.5

1.5

1

1

1

y

y

y

0.5

0.5

0.5

0

0

0

-0.5 0

0.5 x 1

-0.5 0

0.5 x 1

0.5 x 1

Figure 5. Continued on next page.

-0.5 0

0.5 x 1

*


isothermal

191

Section A-A

1.5 1

1000V *

y

0.5 0 -0.5 0

10

z

20

30

20

30

20

30

20

30

Section B-B

1.2

y

1.1 1 0.9 0

10

z

Section C-C

1

x

0.8 0.6 0

10

z

Section D-D 0.1 y

0 -0.1 -0.2

0

10

z

z=6

z=12

1.5

1.5

1

1

y

y

y

0.5

0.5

0.5

0

0

0

z=0

1.5

1000V *

1

-0.5 0

0.5 x 1

-0.5 0

0.5

x 1

-0.5 0

z=18

z=24

z=30

1.5

1.5

1.5

1

1

1

y

y

y

0.5

0.5

0.5

0

0

0

-0.5 0

0.5 x 1

-0.5 0

0.5 x 1

0.5 x 1

-0.5 0

0.5 x

Figure 5. Velocity vector plots at selected transverse and longitudinal sections.

1

192

Chun-Lang Yeh adiabatic

isothermal

1000V *

4

4

2

2

y

y

0

0

-2 0

1

-2 0

2

x

1

2

x

Figure 6. Velocity vector plots near the end of the annulus (z=30.6).

adiabatic z=0

z=12

1.5

1.5

18

7

y

6

0.5 5

4

0

3

Level 9 8 7 6 5 4 3 2 1

T(K) 315 314 313 312 311 310 309 308 307

5

15 y

0.5 3

0

0.5

-0.5 0

x 1

z=24

z=30

1.5

1.5

16

4

0.5

3

0 1 -0.5 0

2

0.5 x 1

Level 6 5 4 3 2 1

T(K) 314 312 310 308 306 304

T(K) 315 313 311 309 307 305

Level 6 5 4 3 2 1

T(K) 314 312 310 308 306 304

5

y

4

0.5

3 2

0 -0.5 0

Level 6 5 4 3 2 1

0.5 x 1

16

5

y

2

1

2

-0.5 0

4

1 0.5 x 1



193

isothermal z=0

z=12

1.5

1.5 3 4

4 15

15 0.5 1

0

y

0.5

-0.5 0

0.5 x 1

z=24

z=30

1.5

1.5

3

4

y

Level 6 5 4 3 2 1

2

0.5 1

0

T(K) 302.5 302 301.5 301 300.5 300

T(K) 302.5 302 301.5 301 300.5 300

Level 6 5 4 3 2 1

T(K) 302.5 302 301.5 301 300.5 300

3

y

0.5 1

0 -0.5 0

0.5 x 1

Level 6 5 4 3 2 1

0.5 x 1

4 15

15

-0.5 0

1

0

2

-0.5 0

T(K) 302.5 302 301.5 301 300.5 300

2

Level 6 5 4 3 2 1

2

y

3

0.5 x 1

Figure 7. Temperature contours at selected longitudinal sections.

adiabatic

isothermal

6

6 1

300 K

Level 8 7 6 5 4 3 2 1

2 2

4

3

1

7 5

0

-2 0

1

2

x

3

4

y

T(K) 308 307 306 305 304 303 302 301

2 9 3 2

1

3

y

1

2

300 K

4

0

4

-2 0

1

Level 12 11 10 9 8 7 6 5 4 3 2 1

2

T(K) 303 302.75 302.5 302.25 302 301.75 301.5 301.25 301 300.75 300.5 300.25

3

4

x

Figure 8. Temperature contours near the end of the annulus (z=30.6).

194

Chun-Lang Yeh adiabatic

T(K)

316

z=0 z=12 z=24 z=30

314

312

310

308

306

0

60

120

180

120

180

φ( o)

isothermal

T(K)

302.5

z=0 z=12 z=24 z=30

302

301.5

301

0

60

φ( o)

Figure 9. Azimuthal distributions of the inner cylinder surface temperatures at four longitudinal sections.

Figure 7 displays the temperature contours at four selected longitudinal sections along the annulus in Zone I (refer to Figure 2 for the section positions), while Figure 8 shows the temperature contours at a distance slightly outside the open end for the examined cases. It is observed that higher temperature regions around the inner cylinder surface locate in the upper portion. This can also be seen from the azimuthal temperature distributions along the inner cylinder surface shown in Figure 9. As pointed out in the above discussion of the flow pattern on the transverse sections, the (hot) outflow is observed in the portions above the inner cylinder, whereas the (cold) inflow in the portions below. The temperature contours at a position slightly outside the annulus, as shown in Figure 8, consistently reflect the above observation.


195

Figure 9 shows the azimuthal temperature distributions along the inner cylinder surface at four selected longitudinal sections. It is clearly seen that the highest temperature occurs right at the top of the inner cylinder (i.e.

φ = 0 0 ). Also from the figure it is observed that the inner

cylinder surface temperatures decrease toward the outlet plane for the adiabatic case, while remains relatively constant for the isothermal case. In addition, the variation of the surface temperatures is smaller for the isothermal case. The maximum inner cylinder surface temperature of the isothermal case is lower than that of the adiabatic case by about 11 K.

4.2. Natural Convection inside Horizontal Concentric or Eccentric Annuli with Specified Wall Heat Flux For

the

second

illustrative

problem,

two

cases

of

modified

Rayleigh

number, Ra = 10 and Ra = 10 , are examined. However, the temperature variation in o

5

o

6

the entire flow field for the case of Ra = 10 is not obvious. Therefore, only the case of o

5

Ra o = 10 6 , which may lead to more distinct temperature variations on the inner cylinder surface, is demonstrated here. Figure 10 shows the velocity vector plots at selected transverse and longitudinal sections for the concentric and eccentric annuli. The flow patterns for the two cases are rather different. As observed from the velocity vector plots on the longitudinal sections, there appear secondary flows in all the six longitudinal sections for both cases. eccentric case

Section A-A

1

0.8 0.6

1000V *

0.4 0.2 00

10

20

30

20

30

20

30

20

30

Section B-B 0.75 0.5 0.25 0 0

10

Section C-C 0.5 0 -0.5 0

10

Section D-D 0.25 0 -0.25 -0.5 -0.75 0

10


196

Chun-Lang Yeh

z=6

z=12

1

1

0.5

0.5

0.5

y

y

y

z=0 1000V

1

*

0

0

0

-0.5

-0.5

-0.5

-1 0

-1 0

0.5 x 1

0.5

-1 0

x 1

z=24

z=18

z=30

1

1

1

0.5

0.5

0.5

y

y

y

0

0

0

-0.5

-0.5

-0.5

-1 0

-1 0

0.5 x 1

concentric case

0.5 x 1

-1 0

0.5 x 1

0.5 x 1

Section A-A

1.5

1

1000V *

0.5 0 -0.5 0

10

20

30

20

30

20

30

20

30

Section B-B

1.2 1.1 1 0.9 0

10

Section C-C

1 0.8 0.6 0

10

Section D-D 0.1 0 -0.1 -0.2

0

10



z=6

z=12

1.5

1.5

1

1

y

y

y

0.5

0.5

0.5

0

0

0

z=0 1000V

1.5

*

1

-0.5 0

0.5 x 1

-0.5 0

0.5

x 1

-0.5 0

z=18

z=24

z=30

1.5

1.5

1.5

1

1 y

y

0.5

0.5

0.5

0

0

0

0.5 x 1

-0.5 0

0.5 x 1

1

y

-0.5 0

197

0.5 x 1

-0.5 0

0.5 x 1

Figure 10. Velocity vector plots at selected transverse and longitudinal sections.

Further downstream, the secondary flows evolve into counter-rotating recirculation zones. Another interesting phenomenon can be observed from the velocity vector plots on the transverse sections. For the concentric case, the inflow paths are clearly observed in the portion below the inner cylinder, whereas the outflow paths are in the portion above. For the eccentric case, since the inner cylinder sits on the bottom of the outer cylinder, the inflow and outflow paths coexist in each transverse plane. Such flow patterns result obviously from the buoyancy effect that can be observed in thermal plume phenomena. This can also be seen from the velocity vector plots at a distance slightly outside the open end (i.e. in Zone II) shown in Figure 11, in which the entrainment effect caused by the upward motion of the buoyant flows can also be seen. Another point to be drawn from Figure 10 is that the axial flow motion is significant and cannot be neglected. Therefore the adoption of a threedimensional formulation is necessary. Figure 12 displays the temperature contours at four selected longitudinal sections along the annulus in Zone I (see Figure 2 for the section positions), while Figure 13 shows the temperature contours at a distance slightly outside the open end for the examined configurations. It is observed that higher temperature regions around the inner cylinder surface locate in the lower portions of the annulus for the eccentric annulus, whereas in the upper portion for the concentric annulus. This can also be seen from the azimuthal temperature distributions along the inner cylinder surface shown in Figure 14. The temperature contours at a position slightly outside the annulus, as shown in Figure 13, consistently reflect the corresponding flow pattern at a distance slightly outside the open ends shown in Figure 11.

198

Chun-Lang Yeh

Figure 11. Velocity vector plots near the end of the annulus (z=30.6).



Figure 12. Temperature contours at selected longitudinal sections.

Figure 13. Temperature contours near the end of the annulus (z=30.6).

199

200

Chun-Lang Yeh

Figure 14. Azimuthal distributions of the inner cylinder surface temperatures at four transverse sections.

Figure 14 shows the azimuthal temperature distributions along the inner cylinder surface at four selected longitudinal sections. It is clearly seen that the highest temperatures occur right at the bottom of the inner cylinder (i.e.

φ = 180 0 ) for the eccentric annulus, whereas

Numerical Analysis of Heat Transfer and Fluid Flow… occur right at the top of the inner cylinder (i.e.

201

φ = 0 0 ) for the concentric annulus. Also from

the figure it is observed that the inner cylinder surface temperatures decrease toward the outlet plane. In addition, the variation of the surface temperatures is smaller for the case of concentric annulus. A more evenly distributed inner cylinder surface temperature is desirable as far as the lifetime of the inner cylinder is concerned. With the concentric annulus, about 30 K decrement of the maximum inner cylinder surface temperature can be achieved as compared to the eccentric annulus.

Conclusion In this chapter, the simulation method for the heat and fluid flow inside three-dimensional horizontal concentric or eccentric annuli with open ends is discussed. The simulation method discussed includes conjugated heat transfer model, turbulence model and zonal grid approach, which extends the outlet boundary from the open end of the conduit to a far enough outside position that can be reasonably specified with the ambient flow properties. Two illustrative problems are discussed. The first is a three-dimensional natural convection inside a horizontal concentric annulus with specified wall temperature or heat flux and the second is a three-dimensional natural convection inside horizontal concentric or eccentric annuli with specified wall heat flux. For the first illustrative problem, it is found that higher temperatures around the inner cylinder occur in the region near its top. The maximum inner cylinder surface temperature occurs right at the top of the inner cylinder. The inner cylinder surface temperatures decrease toward the outlet plane for the adiabatic case, while remains relatively constant for the isothermal case. The variation of the inner cylinder surface temperatures is smaller for the isothermal case, as compared to the adiabatic case. For the second illustrative problem, it is found that the eccentric annulus has a poorer natural convection heat dissipation rate, as compared to the concentric annulus. The inner cylinder surface temperatures decrease toward the outlet plane. It is also found that higher temperatures around the inner cylinder occur in the region near its bottom（ the contacting point of the inner and outer cylinders） for the eccentric annulus, whereas in the region near its top for the concentric annulus. The variation of the inner cylinder surface temperatures is smaller for the case of concentric annulus, as compared to the eccentric annulus. The maximum inner cylinder surface temperatures occur right at the bottom of the inner cylinder for the eccentric annulus, whereas right at the top of the inner cylinder for the concentric annulus.

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Chapter 6

CONVECTIVE HEAT TRANSFER IN THE THERMAL ENTRANCE REGION OF PARALLEL FLOW DOUBLE-PIPE HEAT EXCHANGERS FOR NON-NEWTONIAN FLUIDS Ryoichi Chiba1*, Masaaki Izumi2 and Yoshihiro Sugano3 Yamagata University, Jonan 4-3-16, Yonezawa 992-8510, Japan Ishinomaki Senshu University, Shinmito 1, Minamisakai, Ishinomaki 986-8580, Japan 3 Iwate University, Ueda 4-3-5, Morioka 020-8551, Japan 1

2

Abstract In this chapter, the conjugated Graetz problem in parallel flow double-pipe heat exchangers is analytically solved by an integral transform method—Vodicka’s method—and an analytical solution to the fluid temperatures varying along the radial and axial directions is obtained in a completely explicit form. Since the present study focuses on the range of a sufficiently large Péclet number, heat conduction along the axial direction is considered to be negligible. An important feature of the analytical method presented is that it permits arbitrary velocity distributions of the fluids as long as they are hydrodynamically fully developed. Numerical calculations are performed for the case in which a Newtonian fluid flows in the annulus of the double pipe, whereas a non-Newtonian fluid obeying a simple power law flows through the inner pipe. The numerical results demonstrate the effects of the thermal conductivity ratio of the fluids, Péclet number ratio and power-law index on the temperature distributions in the fluids and the amount of exchanged heat between the two fluids.

Key words: heat transfer, forced convection, heat exchanger, non-Newtonian fluid, analytical solution

*

E-mail address: [email protected]. Telephone: +81-238-26-3219. Fax: +81-238-26-3205

206

Ryoichi Chiba, Masaaki Izumi and Yoshihiro Sugano

Nomenclature a: radius of outer pipe, m b: radius of inner pipe, m B: radius ratio (= b/a) c: heat capacity, J/[kg·K] f: dimensionless total diffusivity g: expansion coefficient h: heat transfer coefficient, W/[m2·K] J0()/J1(): the first kind of Bessel functions of order zero/one k: overall heat transfer coefficient, W/[m2·K] K: constant in the convective term, Eq. (9) n: the number of partitions Nu: Nusselt number defined by Eqs. (25) and (26)

Pe : Péclet number ratio (= α u b /[α u ( a − b)] ) I

II

II

I

r: R: T: u:

radial coordinate, m heat capacity flow rate ratio of fluids defined by Eq. (C4) temperature, K fully developed velocity profile, m/s u : average velocity, m/s U: dimensionless velocity (= u / u ) w: mass flow rate, kg/s x: axial coordinate, m X: eigenfunction Y0()/Y1(): the second kind of Bessel functions of order zero/one α: thermal diffusivity, m2/s ε: heat exchanger effectiveness defined by Eq. (29) εH: eddy diffusivity of heat, m2/s γ: eigenvalue η: dimensionless radial coordinate (= r/a) λ: thermal conductivity, W/[m·K] λ : thermal conductivity ratio (= λII/λI) ν: power-law index

θ: dimensionless temperature (= (T − T0bII ) /(T0bI − T0bII ) ) ξ: dimensionless axial coordinate (= xα I /(a 2u I ) ) Ψ: the number of heat transfer units defined by Eq. (C3)

Subscripts 0: entrance b: bulk i: region number

Convective Heat Transfer in the Thermal Entrance Region...

207

l: eigenvalue number ∞: asymptotic

Superscripts I: fluid I II: fluid II

1. Introduction The usual double-pipe parallel flow heat exchanger consists of two concentric circular pipes, and fluids with different temperatures enter the annulus and inner pipe at the same side of the heat exchanger. As the fluids flow through their respective channels, heat is transferred from the hot fluid to the cold fluid. The traditional methods for predicting heat transfer in such situations are based on the assignment of heat transfer coefficients for each stream independently of the actual coupling of the boundary conditions [1]. The methods are employed under the two essential assumptions that are customarily made [2]: (i) the heat transfer coefficients are considered to be insensitive to the longitudinal distributions of both the heat flux and surface temperature, and (ii) they are taken to be uniform irrespective of the heat exchanger length. There is substantial evidence [3] that under turbulent flow conditions the abovementioned methods are acceptable for nonmetallic fluids because the local heat transfer coefficients are scarcely influenced by thermal boundary conditions in those cases, and the thermal entrance region usually covers a small part of the heat transfer length. On the other hand, under laminar flow conditions, which are encountered in low Reynolds number flow heat exchangers1, local heat transfer coefficients become very sensitive to the thermal boundary condition. In addition, the thermal entrance length can often be of the same order of magnitude as the heat exchanger length [2]. This leads to spatial variations in heat transfer coefficients in much of the heat transfer surfaces. Since the two streams are thermally coupled via the wall separating them, the varying heat transfer coefficients on both surfaces of the wall, i.e., overall heat transfer coefficient, cannot be defined a priori. In the case of spatially varying overall heat transfer coefficient, one has to solve the coupled forced convection heat transfer problem between heating and heated fluids—that is, the conjugated Graetz problem—for an accurate evaluation of the performance of the heat exchanger. Since the influence of coupling of the boundary conditions can be important in the thermal entrance regions, especially with laminar flow, the conjugated Graetz problem for parallel flow laminar heat exchangers with a relatively short heat transfer length has been analytically investigated. The earliest investigations on the heat transfer problems of parallel flow laminar heat exchangers were conducted by Stein [4, 5]. Subsequently, the same problem was treated by Mikhailov et al. [6] using a specialized version of the method for conjugated Graetz problems. Pagliarini et al. [2] theoretically investigated thermal interaction between the 1

Laminar or low Reynolds number turbulent flows are the consequences of either high viscosity fluids, compact flow passages (i.e., small hydraulic diameter), or low fluid velocities.

208


streams of laminar flow double-pipe heat exchangers, while considering axial heat conduction in the wall separating the fluids. Neto et al. [7] employed the generalized integral transform technique, deriving an analytical solution to a mixed lumped-differential formulation of double-pipe heat exchangers. Plaschko [8] used matched asymptotic expansions to obtain an approximate solution to the heat transfer in parallel flow heat exchangers with high Péclet numbers. Huang et al. [9] developed a control algorithm of heat flux imposed on the external surface of the outer pipe in order to obtain the desired thermal entrance length and fluid temperatures in parallel flow double-pipe heat exchangers. The problems of heat transfer in laminar co-current flow of two immiscible fluids, such as those studied in [10-14], are also governed by the same type of differential equations as those for parallel flow heat exchangers, and are therefore included in the conjugated Graetz problem. Bentwich et al. [10] and Nogueira et al. [13] analyzed the temperature distribution and heat transfer in core-annular two-phase liquid-liquid flow subject to constant wall temperature boundary condition. The heat transfer problem of core-annular laminar flow in a pipe with constant wall heat flux boundary condition was theoretically investigated by Leib et al. [11]. The same type of problem in a pipe with the third-kind boundary condition was studied by Su [14]. Davis et al. [12] examined three types of conjugated boundary value problems related to conjugated multiphase heat and mass transfer problems, and developed systematic procedures for their solutions. Simultaneous heat and mass transfer in internal gas flow in a duct whose walls are coated with a sublimable material [15] and the membrane gas absorption process [16] are also known to come under the category of conjugated Graetz problem. The above-cited papers considered fluids to be Newtonian. In view of the fact that heat exchange between Newtonian and non-Newtonian fluids is of importance in engineering applications as well as that between Newtonian fluids, it is desirable to develop a fully analytical method that is not restricted by the rheology characteristics of fluids. In this chapter, the conjugated Graetz problem in parallel flow double-pipe heat exchangers is analytically solved by an integral transform method—Vodicka’s method, and an analytical solution to the fluid temperatures varying along the radial and axial directions is obtained in a completely explicit form. Since the present study focuses on the range of sufficiently large Péclet number, heat conduction along the axial direction is considered to be negligible. An important feature of the analytical method presented is that it permits arbitrary velocity distributions, i.e., arbitrary rheology characteristics, of the fluids as long as they are hydrodynamically fully developed. Numerical calculations are performed for the case in which a Newtonian fluid flows in the annulus of the double pipe whereas a non-Newtonian fluid obeying a simple power law flows through the inner pipe. The numerical results demonstrate the effects of the thermal conductivity ratio of the fluids, Péclet number ratio and power-law index on the temperature distributions in the fluids and the amount of exchanged heat between the two fluids.


209

2. Theoretical Analysis 2.1. Analytical Model and Mathematical Formulation Figure 1 shows the physical model and coordinate system. Two fluids, fluid I and fluid II, flow concurrently with fully developed velocity distributions uI(r) and uII(r) in a double-pipe heat exchanger. The heat exchanger consists of two concentric thin pipes: outer pipe of radius a and inner pipe of radius b. While the fluid II flows through the inner pipe, the fluid I flows in the annulus between the pipe walls. The fluid temperatures at the entrance are T0I (r) and T0II (r). The surface at r = a is insulated and heat is exchanged between both fluids through the

inner pipe wall. r

Thin concentric pipes

u=uI(r)

λI, αI

I

T= T0 (r) u=uII(r)

λII, αII

II

T= T0 (r)

I

a

O

b

II x

Figure 1. Physical model and related cylindrical coordinate system for a parallel flow double-pipe heat exchanger.

The following assumptions are introduced: (i) physical properties are independent of temperature and are therefore constant, (ii) the heat resistance of the pipes is negligible, (iii) the axial heat conduction and turbulent axial diffusion are negligible, (iv) molecular and turbulent transport of momentum and heat are additive, (v) viscous dissipation and compression work are negligibly small, (vi) turbulent diffusion of heat can be described by available thermal eddy diffusivity expressions. The item (i) indicates that no natural convection occurs in the fluids. Since the effects of pipe wall conductivity on the heat exchange effectiveness are minor in parallel flow heat exchangers [2], the item (ii) is a reasonable assumption. The item (iii) is valid only for high Péclet number flow. In this case, the steady-state heat balance is expressed as follows [3]:

∂T m ( x, r ) 1 ∂ ⎧ m ∂T m ( x, r ) ⎫ m = u (r ) ⎨r[α + ε H (r )] ⎬ , m = I, II, r ∂r ⎩ ∂x ∂r ⎭ m

(1)

210


where α and εH denote the thermal diffusivity and eddy diffusivity of heat, respectively, and the superscript m indicates the fluid number. The boundary conditions and continuity condition at r = b are then

λI

T m (0, r ) = T0m (r ) , m = I, II,

(2)

∂T II ( x, 0) ∂T I ( x, a) = = 0, ∂r ∂r

(3)

∂T I ( x, b) ∂T II ( x, b) I II = λ II , T ( x, b ) = T ( x, b ) . ∂r ∂r

(4)

To simplify manipulations, we obtain the dimensionless forms of Eqs. (1)-(4):

⎧ ∂θ m m U ( η ) ⎪ ∂ξ 1 ∂ ⎡ m ∂θ m ⎤ ⎪ η f (η ) =⎨ ⎢ ⎥ η ∂η ⎣ ∂η ⎦ ⎪ m (1 − B) ∂θ m U (η ) Pe ⎪⎩ B ∂ξ

m=I ,

(5)

m = II

θ m (0,η ) = θ 0m (η ) , m = I, II,

(6)

∂θ II (ξ , 0) ∂θ I (ξ ,1) = = 0, ∂η ∂η

(7)

∂θ I (ξ , B) ∂θ II (ξ , B) I II =λ , θ (ξ , B ) = θ (ξ , B ) , ∂η ∂η

(8)

where we introduce the following dimensionless quantities: η = r / a , U m (η ) = u m (r ) / u m ;

m

=

I,

II,

Pe = α I u II b / [α II u I (a − b)] ,

B=b/a,

λ = λ II / λ I ,

θ m = (T m − T0bII ) / (T0bI − T0bII ) ; m = I, II, ξ = xα I / (a 2 u I ) , f m (η ) = 1 + ε Hm (r ) / α m ; m = I, II. Equation (5) is a partial differential equation with variable coefficients; therefore, it is very difficult to obtain the exact solution for arbitrary velocity profiles and total diffusivity distribution. In order to solve Eq. (5), we divide the annular and circular channels into nI and nII regions in the radial direction (η-axis direction), respectively, and approximate fm(η) and Um(η) as constants fi and Ui in each region [17]. In this case, the dimensionless energy equation in the ith region (i = 1, 2,..., nI+nII) is:


Ki

∂θ i ∂ 2θi 1 ∂θ i = + , ∂ξ ∂η 2 η ∂η

211

(9)

where

1 ≤ i ≤ n II ⎪⎧U Pe(1 − B) /( B ⋅ fi ) . Ki = ⎨ i U i / fi n II + 1 ≤ i ≤ n I + n II ⎪⎩ Note that the subscripts of θ, K, λ and η denote the region number, not the fluid number. The continuity conditions at the imaginary interfaces (including the real interface at η = B) and boundary conditions are expressed as:

λi ∂θi (ξ ,ηi ) ∂θi +1 (ξ ,ηi ) , θ i (ξ ,ηi ) = θi +1 (ξ ,ηi ) , = λi +1 ∂η ∂η

(10)

θi (0,η ) = θ 0m (η ) , if 1 ≤ i ≤ nII, then m = II, if nII+1 ≤ i ≤ nI+nII, then m = I,

∂θ1 (ξ , 0) ∂θ ( nI + nII ) (ξ ,1) = =0, ∂η ∂η

(11)

(12)

where ηi is the outer radius of the ith region, that is, ηnII = B and η( nI + nII ) = 1 . The initialboundary value problem expressed by Eqs. (9)-(12) is identical to that for transient heat conduction case in a composite medium consisting of nI+nII layers.

2.2. Derivation of Analytical Solution We employ Vodicka’s method [18] to derive an analytical solution of temperatures in the fluids and their related Nusselt numbers. Since it allows us to easily analyze the transient heat conduction in a composite medium with a large number of sub-regions, this method is applied to the analysis of temperature fields in a wide range of objects [17, 19]. Although it is possible, in principle, to analyze the problem under consideration using Laplace transform [20], the inverse transform is mathematically quite complicated for a large number of subregions. Therefore, it is the authors opinion that Vodicka’s method is to be preferred due to its advantage in facility. Note that the analytical treatment proposed by Mikhailov et al. [21] may be used as an alternative. Using Vodicka’s method, the solution to Eqs. (9)-(12) is assumed to be ∞

θi (ξ ,η ) = ∑ φl (ξ ) X il (η ) + θ ∞ , l =1

(13)

212


where θ∞ is the temperature of the fluids for ξ → ∞. An overall heat balance shows that with an infinite heat transfer area, the outlet temperature of each stream in parallel flow is

θ∞ =

1+ B . Peλ B + 1 + B

(14)

The eigenfunction X il (η ) is obtained as

X il (η ) = Ail J 0 ( K i γ lη ) + BilY0 ( K i γ lη ) ,

(15)

where J0() and Y0() are the first and second kind of Bessel functions of order 0, respectively. The constants Ail and Bil in Eq. (15) are determined from the conditions that are obtained by substituting Eqs. (13) and (15) into the continuity and boundary conditions, Eqs. (10) and (12). Consequently, the continuity conditions of the temperature field at the interfaces between neighboring regions can be fulfilled. The eigenvalues γl (l = 1, 2,...) are obtained from the condition under which all the Ail and Bil values are nonzero and are therefore positive roots of the following transcendental equation (for details, see the appendix A):

⎡const.⎤ E1 ⋅ E2 " E( nI + nII −1) ⋅ b ( nI + nII ) = ⎢ ⎥, ⎣ 0 ⎦

(16)

where

Ei = Ci ⋅ Di +1 , b ( nI + nII )

⎡ Y1 ( K I II γ l ) ⎤ n +n ⎥, =⎢ ⎢ − J1 ( K I II γ l ) ⎥ n +n ⎣ ⎦

⎡ −λi / λi +1 K i γ lY1 ( K i γ lηi ) −Y0 ( K i γ lηi ) ⎤ Ci = ⎢ ⎥, ⎢⎣ λi / λi +1 K i γ l J1 ( K i γ lηi ) J 0 ( K i γ lηi ) ⎥⎦ ⎡ ⎤ J 0 ( K i +1 γ lηi ) Y0 ( K i +1 γ lηi ) Di +1 = ⎢ ⎥, ⎢⎣ − K i +1 γ l J1 ( K i +1 γ lηi ) − K i +1 γ lY1 ( K i +1 γ lηi ) ⎥⎦

(17)

and J1() and Y1() are the first and second kind of Bessel functions of order 1, respectively. By substituting Eq. (13) into Eq. (11), the following equation is obtained: ∞

G (η ) = ∑ φl (0) X il (η ) = θ 0m (η ) − θ ∞ , m = I or II. l =1

(18)


213

The eigenfunction Xil(η) given by Eq. (15) has an orthognal relationship with discontinuous weight functions; it is expressed as follows (see the appendix B): ηi

n I + n II

∑ λ K ∫ ηX η i =1

i

i

il

i −1

⎧const. (l = k ) . (η ) X ik (η )dη = ⎨ (l ≠ k ) ⎩ 0

(19)

G(η) can be expanded into an infinite series of Xil(η) as follows: ∞

G (η ) = ∑ gl X il (η ) ,

(20)

l =1

where the expansion coefficient gl is given by ηi

n I + n II

gl =

∑ λ K ∫ ηG(η ) X η i

i =1

i

il

(η )dη

i −1

ηi

n I + n II

∑ λ K ∫ η[X η i =1

i

i

.

(21)

(η ) ] dη 2

il

i −1

Taking Eq. (15) into account, we substitute Eq. (13) into Eq. (9). This yields a first-order linear ordinary differential equation for φl(ξ) as follows:

dφl + γ l2φl = 0 . dξ

(22)

By solving Eq. (22) with the condition φl(0) = gl, which is obtained from the comparison between Eqs. (18) and (20), we obtain φl(ξ) as 2 l

φl (ξ ) = gl e−γ ξ .

(23)

Finaly, the dimensionless temperature in the ith region η ∈ [ηi−1, ηi] inside the doublepipe heat exchanger, θi(ξ, η), is derived as: ∞

θi (ξ ,η ) = ∑ gl e−γ ξ ⎡⎣ Ail J 0 ( K i γ lη ) + BilY0 ( K i γ lη ) ⎤⎦ + 2 l

l =1

1+ B . Peλ B + 1 + B

(24)

Local Nusselt numbers for outer and inner streams are defined in the usual manner as

Nu I =

2h I ( a − b)

λI

=

2( B − 1) ∂θ I θ η = B − θ bI ∂η

for the outer stream, η =B

(25)

214


Nu II =

2h IIb

λ II

=

2B ∂θ II θ η = B − θ bII ∂η

for the inner stream.

(26)

η =B

3. Numerical Calculation In this study, there is no analytical restriction on the form of the velocity profiles of the flowing fluids. Thus, as a numerical example, a double-pipe heat exchanger in which a Newtonian fluid flows in the annulus whereas a non-Newtonian fluid flows through the inner pipe is considered here. The non-Newtonian fluid is assumed to obey a power law, which can approximate the non-Newtonian viscosity of many types of fluids with good accuracy over a wide range of shear rates. The fluids have mutually different uniform temperatures at the entrance, that is, θ0I (η) = 1 and θ0II (η) = 0. Under laminar flow conditions, ε HI = ε HII = 0 and the fully developed velocity profile for each fluid is [1, 22]:

u I (η ) = 2u I

(1 − B 2 ) ln η + (η 2 − 1) ln B , ( B 2 − 1) − ( B 2 + 1) ln B

1+ν ⎡ ⎤ 3ν + 1 ⎢ ⎛ η ⎞ ν ⎥ , u (η ) = u 1− ⎜ ⎟ ν +1 ⎢ ⎝ B ⎠ ⎥ ⎣ ⎦ II

II

(27)

(28)

where ν is the power-law index. For values of ν less than unity, the behavior is pseudoplastic, whereas forν greater than unity the behavior is dilatant. For ν = 1, it reduces to Newton’s law of viscosity. In the numerical calculations, we radially divide the each channel into equal intervals with the number of partitions nI = nII = 20. Our ealier work [17] showed that the number of partitions 20 per a channel provides sufficiently accurate results. The number of terms in the infinite series in Eq. (24) is taken as 200. It should be noted that this value is used for the verification of sufficient convergence of the numerical results.

4. Results and Discussion To show the accuracy of the present analytical solution, we first compare local Nusselt numbers, bulk temperatures and wall temperatures to existing results [23, 24]. The values shown in Table 1 are in good agreement between the present analytical solution and the existing solutions: the similarity approach solution [23] and the eigenvalue solution given in [24].

Table 1. A comparison of the entrance region results obtained by the present analytical method and existing methods [23, 24] for a parallel flow double-pipe heat exchanger with ν = 1, Pe = 2/3,

λ = 2 and B = 0.5

x [23, 24]

ξ

θ BII

θ BII *

θ|η = B

θ|η = B*

NuI

NuI*

NuII

NuII*

0.001

0.00016667

0.010589

0.0103

0.43057

0.426

24.123

23.96

16.224

16.42

0.002

0.00033333

0.016574

0.0163

0.43345

0.431

19.215

19.15

12.892

12.98

0.01

0.0016667

0.047178

0.0470

0.44917

0.449

11.560

11.56

7.6210

7.64

0.02

0.0033333

0.073899

0.0737

0.46062

0.460

9.4119

9.42

6.1655

6.17

0.04

0.0066667

0.11529

0.115

0.47611

0.476

7.7722

7.78

5.0914

5.09

0.1

0.016667

0.20501

0.205

0.50591

0.506

6.2719

6.28

4.1998

4.20

0.2

0.033333

0.31095

0.311

0.54061

0.541

5.6245

5.63

3.9329

3.93

0.4

0.066667

0.45267

0.453

0.59330

0.594

5.3742

5.38

3.9267

3.92

0.6

0.10000

0.54102

0.541

0.62927

0.629

5.3390

5.35

3.9403

3.94

* Existing solutions, the data cited from [23] for x ≤ 0.01 and [24] for x ≥ 0.02 .

216


1

η= 0, 0.2, 0.4, 0.5, 0.6, 0.8, 1

Fluid I 0.8 ν = 0.3 3

θ

0.6 0.4

Inner pipe wall 0.2 Fluid II 0 10-4

10-3

10-2 ξ

10-1

100

(a)

1 Fluid I 0.8 ν = 0.3 3

θ

0.6

η= 0, 0.2, 0.4, 0.5, 0.6, 0.8, 1

0.4 Inner pipe wall 0.2 0 10-4

Fluid II 10-3

10-2 ξ (b)


10-1

100


217

1 Fluid I

η= 0, 0.2, 0.4, 0.5, 0.6, 0.8, 1

0.8 0.6 θ

ν = 0.3 3

0.4

Inner pipe wall

0.2 0 10-4

10-3

10-2 ξ

10-1

100

(c) Figure 2. Temperature variation along the streamwise direction at different radial locations with B = 0.5 and λ =10, for (a) Pe = 0.5, (b) Pe = 2 and (c) Pe = 10. 3 2.5 ν=3 1 0.3

UII

2 1.5 1 0.5 0

0

0.2

0.4

0.6

0.8

1

η/B

Figure 3. Velocity profiles for different power-law index values.

Figure 2 shows the axial distributions of dimensionless temperature θ at some radial locations. In this figure, the solid lines and dashed lines represent the cases for ν = 0.3 and ν = 3, respectively, for which fluid II exhibits the velocity profiles shown in Figure 3. It is observed that for a constant thermal conductivity ratio λ , a greater Péclet number ratio Pe produces a larger temperature variation in fluid I. Moreover, although we omit the graphical representation, for a constant Pe a greater λ causes a larger temperature variation in fluid I. These results can be predicted easily from Eq. (14). With regard to the effect of ν on the

218


temperature distributions, an increase in ν increases the values of θ in both fluids throughout the heat exchanger length except the vicinity of the inner pipe axis. Especially, the effect is pronounced at η = B, or the inner pipe wall, and it decreases with distance from the inner pipe wall. Not surprisingly, the temperatures of fluid II, a power-law fluid, are directly affected by ν. However, we can also observe a slight influence of ν on fluid I in the temperature distributions.

1 0.8 Pe = 0.1, 0.2, 0.5, 1, 2, 5, 10

θ|η=B

0.6 0.4 ν = 0.3 3

0.2 0 -4 10

10

-3

10

-2

10

-1

0

10

ξ (a)

1 ν = 0.3 3

0.8

θ|η=B

0.6 0.4 0.2 Pe = 0.1, 0.2, 0.5, 1, 2, 5, 10

0 -4 10

10

-3

10

-2

10

-1

ξ (b) Figure 4. Continued on next page.

10

0


219

1 Pe = 0.1, 0.2, 0.5, 1, 2, 5, 10

0.8

θ|η=B

0.6

ν = 0.3 3

0.4 0.2 0 -4 10

10

-3

10

-2

10

-1

10

0

ξ (c) Figure 4. Temperature variation along the streamwise direction at the inner pipe wall with B = 0.5, for (a) λ = 0.1, (b) λ = 1 and (c) λ = 10.

The temperatures of the inner pipe wall, which are the most susceptible to the effect of ν, are plotted against ξ in Figure 4. As λ becomes small, θ of the inner pipe wall increases. In addition, the larger the value of ν is, the higher θ always becomes. It is interesting that the variation behavior of θ along the ξ-axis direction depends on the value of Pe ; while θ of the inner pipe wall increases monotonically for a small Pe , it may have a maximal value for a large Pe , decreasing from a certain axial location. This is due to the following reasons: in the immediate vicinity of the entrance of the heat exchanger, large hot/cold heat flows from fluid I to fluid II because of great radial temperature gradients near the inner pipe wall. This leads to an increase in θ of the inner pipe wall. Additionally, since the temperature variation in fluid II becomes smaller than that in fluid I for a large Pe (see Figure 2-c), fluid I greatly varies its temperature over the full section of the annulus along the axial direction. In contrast, fluid II varies its temperature only in the immedeate vicinity of the inner pipe wall. Up to a downstream location, θ of the inner pipe wall remains elevated due to high heat flux passing through the wall. When the fluids reach downstream to some extent, the heat flux passing through the wall, i.e., radial temperature gradients decrease and meanwhile heat conduction in the negative η direction gradually increases in fluid II. As a result, the temperature of the inner pipe wall begins to decrease and then it converges at the asymptotic temperature given by Eq. (14). It is known that in parallel flow heat exchangers, the bulk temperatures of heating and heated fluids vary monotonically from each entrance temperature to the asymptotic one. However, local temperatures, especially those near the inner pipe wall, do not necessarily

220


vary in a monotone along the streamwise direction and may have a maximal value at a certain axial location. This phenomenon tends to occur in the case of large Péclet number ratio.

30 ν=3 1 0.3 Const. heat flux B.C. Const. temperature B.C.

25

NuI

20 15 10 5 0 10-4

10-3

10-2 ξ

10-1

100

(a)

30 ν=3 1 0.3 Const. heat flux B.C. Const. temperature B.C.

25

NuII

20 15 10 5 0 -4 10

10

-3

10 ξ

-2

10

-1

10

0

(b) Figure 5. Local Nusselt numbers of (a) outer stream and (b) inner stream for B = 0.5, λ = 1 and Pe = 1.


221

The local Nusselt numbers defined by Eqs. (25) and (26) are shown in Figure 5-(a) and (b), respectively. The discrete plots in Figure 5 indicate the local Nusselt numbers of single flows for a Newtonian fluid with constant temperature/constant heat flux boundary condition. The data are cited from [25] for flow in the pipe and [26, 27] for flow in the annulus. Figure 5 demonstrates that the local Nusselt number of the outer stream (fluid I) is little affected by ν. On the other hand, the Nusselt number of the inner stream (fluid II) depends on the value of ν, increasing with a decrease in ν. This trend can be also found in the case of single flow [28]. In summary, the local Nusselt number of each stream in parallel flow heat exchangers is determined by own rheology characteristics, irrespective of the rheology characteristics of counterpart. Next, we compare the curve for ν = 1 with the discrete plots in both streams. In the inner stream, the local Nusselt number obtained from the present numerical calculations falls somewhere in between the two local Nusselt numbers for single flow, being, on the whole, closer to the one for constant temperature boundary condition. On the other hand, in the outer stream, the Nusselt number obtained from the present numerical calculations is smaller than both Nusselt numbers for single flow; it is closer to the one for constant temperature boundary condition again. However, it is found difficult to accurately approximate local Nusselt numbers in parallel flow heat exchangers with those for constant temperature/heat flux boundary condition.

1 θIb

θIb, θIIb

0.8 ν = 0.3 3

0.6

0.4

θIIb

0.2

0 10-4

10-3

10-2

10-1 ξ

(a) Figure 6. Continued on next page.

100

222


1

θIb, θIIb

0.8 ν = 0.3 3

0.6

θIb

0.4

0.2

0 -5 10

θIIb -4

10

-3

10

-2

10

10

-1

10

0

ξ (b) Figure 6. Bulk temperatures of the heating and heated fluids with different power-law index values for B = 0.5, λ = 1 and (a) Pe = 1 and (b) Pe = 10.

Figure 6 shows the axial variations in the bulk temperatures of heating and heated fluids. Since a smaller ν leads to a larger Nusselt number in the inner stream, as shown in Figure 5(b), heat exchange between both fluids is found to be enhanced for ν = 0.3. From Figure 6-(a), maximum difference between the bulk temperatures for ν = 0.3 and 3 is calculated to be approximately 1% for

θ bI and 10% for θ bII , and from Figure 6-(b) approximately 7% for both

θ bI and θ bII . Figure 7 makes comparisons of the heat exchanger effectiveness1 ε defiined by

ε (ξ ) =

1 − θ bI (ξ ) , 1 − θ∞

(29)

between the value by the present analytical solution and that calculated by means of the bulk temperatures of the fluids and a constant overall heat transfer coefficient (measured by overall Nusselt number in the figure). θ∞ in Eq. (29) is already given by Eq. (14). For the derivation details of the latter heat exchanger effectiveness, q.v. the appendix C. Figure 7 demonstrates that it is impossible to approximate well the heat exchanger effectiveness throughout all 2

The heat exchanger effectiveness is defined as the ratio of the actual over-all rate of heat transfer to the maximum possible as computed for an exchanger with the same operating conditions but with infinite heat transfer area [29].


223

values of ξ with any constant overall heat transfer coefficients. Hence, in designing heat exchangers with laminar flow, i.e., low Reynolds number flow heat exchangers, it is crucial to consider streamwise variations in the overall heat transfer coefficient.

1

Present analytical solution Bulk temp. & const. overall Nu

0.8

5 4

0.6 ε

3

0.4 2 0.2 Overall Nu = 1 0 -4 10

10

-3

-2

-1

10

0

10

10

1

10

ξ

(a) 1

Present analytical solution Bulk temp. & const. overall Nu

0.8

10 8

0.6 ε

6

0.4 4 0.2 Overall Nu = 2 0 10-4

10-3

10-2

10-1

100

101

ξ

(b) Figure 7. Axial variation of heat exchanger effectiveness with B = 0.5, Pe = 1 and ν = 1, for (a) λ = 1 and (b) λ = 10.

224


5. Conclusion In this chapter, the conjugated Graetz problem in parallel flow double-pipe heat exchangers has been analytically solved by an integral transform method—Vodicka’s method—and an analytical solution to the fluid temperatures varying along the radial and axial directions has been obtained in a completely explicit form. An important feature of the analytical method presented is that it permits arbitrary velocity distributions of the fluids as long as they are hydrodynamically fully developed. Numerical calculations have been performed for the case in which a Newtonian fluid flows in the annulus of the double pipe, whereas a non-Newtonian fluid obeying a simple power law flows through the inner pipe. The numerical results have demonstrated the effects of the thermal conductivity ratio of the fluids, Péclet number ratio and power-law index, on the temperature distributions in the fluids and the amount of exchanged heat between the two fluids.

Acknowledgements The authors would like to thank Dr. S. Jian, Universidade Federal do Rio de Janeiro, for his useful comments. Thanks are also given to Prof. A. Haji-Sheikh, the University of Texas at Arlington, for helpful suggestions.

Reviewed by This manuscript was peer-reviewed by Dr. Su Jian, Universidade Federal do Rio de Janeiro, Brazil.

Appendix A: The Eigenvalue Equation The eigenfunctions Xil (i = 1, 2,..., nI+nII) must satisfy

X il′′ (η ) +

1

η

X il′ (η ) + K iγ l2 X il (η ) = 0 ,

(A1)

with the continuity and boundary conditions

X il (ηi ) = X (i +1)l (ηi ) ,

(A2)

λi X il′ (ηi ) = λi +1 X (′i +1) l (ηi ) ,

(A3)

X 1′l (0) = 0 ,

(A4)

X (′nI + nII ) l (1) = 0 ,

(A5)


225

where primes denote differentiation with respect to η. Substituting Eq. (15) into Eqs. (A2)(A5) yields ⎡ J 0′ ( K1 γ lη ) Y0′( K1 γ lη ) η =0 η =0 ⎢ ⎢ J ( K γ η ) Y ( K γ η ) 0 1 l 1 0 1 l 1 ⎢ ⎢ λ J ′ ( K γ η ) λ Y ′( K γ η ) 1 l 1 1 0 1 l 1 ⎢ 1 0 ⎢ 0 0 ⎢ # ⎢ ⎢ # ⎢ ⎢ # ⎢ " 0 ⎣

" 0⎤ ⎥ ⎥ − J 0 ( K 2 γ lη1 ) −Y0 ( K 2 γ lη1 ) 0 0⎥ −λ2 J 0′ ( K 2 γ lη1 ) −λ2Y0′( K 2 γ lη1 ) 0 0⎥ ⎥ − J 0 ( K 3 γ lη2 ) −Y0 ( K 3 γ lη2 ) #⎥ J 0 ( K 2 γ lη 2 ) Y0 ( K 2 γ lη2 ) ⎥ λ2 J 0′ ( K 2 γ lη2 ) λ2Y0′( K 2 γ lη 2 ) −λ3 J 0′ ( K3 γ lη2 ) −λ3Y0′( K3 γ lη2 ) 0 0 ⎥ % % % ⎥⎥ % % ⎥ ⎥ " " " ⎦ ⎧ A1l ⎫ ⎧0 ⎫ ⎪ B ⎪ ⎪ ⎪ 1l ⎪ ⎪ ⎪0 ⎪ (A6) ⎪ A2l ⎪ ⎪0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ B2l ⎪ ⎪0 ⎪ ⋅⎨ # ⎬ = ⎨ ⎬ ⎪ ⎪ ⎪# ⎪ ⎪ # ⎪ ⎪# ⎪ ⎪A ⎪ ⎪ ⎪ ⎪ ( nI + nII ) l ⎪ ⎪0 ⎪ ⎪B ⎪ ⎪0 ⎪ ⎩ ( nI + nII ) l ⎭ ⎩ ⎭ "

0

"

"

Since Eq. (A6) is a homogeneous system of equations, the determinant of the matrix must be zero to have a nontrivial solution for all Ail and Bil. However, the procedure for obtaining the determinant is complicated. Therefore, we perform the manipulation shown below. The boundary condition Eq. (A5) is expressed as:

G ⋅ a( nI + nII ) = 0 , where G = ⎡ J1 ( K nI + nII γ l )

⎣

Y1 ( K nI + nII γ l ) ⎤⎦ , a T(nI + nII ) = ⎡ A( nI + nII ) l ⎣

(A7)

B( nI + nII )l ⎤ and the ⎦

superscript T is the transpose operator. The continuity conditions Eqs. (A2) and (A3) are expressed as:

Fi ⋅ ai = Di +1 ⋅ ai +1 ,

(A8)

⎡ ⎤ J 0 ( K i γ lηi ) Y0 ( K i γ lηi ) Fi = ⎢ ⎥, ⎢⎣ −λi / λi +1 K i γ l J1 ( K i γ lηi ) −λi / λi +1 K i γ lY1 ( K i γ lηi ) ⎥⎦

(A9)

where

−1

and Di+1 is given in Eq. (17). With H i = Fi ⋅ Di +1 , Eq. (A8) can be written as

ai = H i ⋅ H i +1 " H nI + nII −1 ⋅ a ( nI + nII ) .

(A10)

226

Ryoichi Chiba, Masaaki Izumi and Yoshihiro Sugano From Eq. (A7), we obtain

⎡ J1 ( K nI + nII γ l ) ⎤ ⎥ A I II . a T(nI + nII ) = ⎢1 − Y1 ( K nI + nII γ l ) ⎥⎦ ( n + n )l ⎢⎣

(A11)

Since the constant B1l must be zero due to the nature of Bessel functions2,

a1T = [ A1l

0] . This leads to the following equation, considering Eqs. (A10) and (A11):

1 ⎡ ⎤ ⎢ ⎥ A ⎡ 1l ⎤ ⎢ 0 ⎥ = H1 ⋅ H 2 " H nI + nII −1 ⋅ ⎢ − J1 ( K nI + nII γ l ) ⎥ A( nI + nII ) l . ⎣ ⎦ ⎢ Y( K γ ) ⎥⎦ ⎢⎣ 1 n I + n II l ⎥

(A12)

In order that all the Ail and Bil (i = 1, 2,..., nI+nII) are nonzero, the eigenvalues γl must satisfy Eq. (A12) with A( nI + nII ) l ≠ 0 . Here one can write as Hi = Ci·Di+1/det Fi = Ei/det Fi (Ci and Ei are given in Eq. (17)), and therefore obtain Eq. (16).

Appendix B: Orthogonality of the Eigenfunctions The orthogonality of the eigenfunctions Xil (i = 1, 2,..., nI+nII) will now be established. Equation (A1) is written for l = j and then for l = k with j ≠ k. The equation for l = j is multiplied by ηXik, and the equation for l = k is multiplied by ηXij. The two resulting equations are subtracted, multiplied by λi, and then integrated between η = ηi−1 and η = ηi. The following results: ηi

ηi

′ ′ λi K i (γ − γ ) ∫ η X ij X ik dη = ⎡⎣λη i X ik (η ) X ij (η ) − λη i X ij (η ) X ik (η ) ⎤ ⎦η , 2 k

2 j

(B1)

i −1

ηi−1

where primes denote differentiation with respect to η. Taking the summation of both sides of Eq. (B1) for i = 1, 2,..., nI+nII, n I + n II

∑ λ K (γ i =1

i

i

2 k

ηi

n I + n II

ηi −1

i =1

− γ ) ∫ η X ij X ik dη = 2 j

∑ λ {η ⎡⎣ X i

i

ik

(ηi ) X ij′ (ηi ) − X ij (ηi ) X ik′ (ηi ) ⎤⎦

}

−ηi −1 ⎡⎣ X ik (ηi −1 ) X ij′ (ηi −1 ) − X ij (ηi −1 ) X ik′ (ηi −1 ) ⎤⎦ .

2

In other words, X1l (η) must have a finite value at η = 0.

(B2)


227

Equations (A2) and (A3) are written for l = j and then for l = k, respectively. The equation (A2) for l = j is multiplied by the equation (A3) for l = k at each side, and similarly the equation (A2) for l = k is multiplied by the equation (A3) for l = j. The two resulting equations make Eq. (B2) a simpler form: n I + n II

∑ λ K (γ i =1

i

i

ηi

2 k

− γ ) ∫ η X ij X ik dη = 2 j

ηi −1

λ( n + n ) ⎡⎣ X ( n + n ) k (1) X (′n + n ) j (1) − X ( n + n ) j (1) X (′n + n I

II

I

II

I

II

I

II

I

II

)k

(1) ⎤ . ⎦

(B3)

Considering Eq. (A5), Eq. (B3) reduces to ηi

n I + n II

∑ λ K ∫ ηX η i =1

i

i

ij

X ik dη = 0 .

(B4)

i −1

For the case of j = k = l, Eq. (A1) leads to a normalizing factor defined as the denominator of Eq. (21).

Appendix C: Heat Exchanger Effectiveness for a Constant Overall Heat Transfer Coefficient Ignoring the radial distributions of velocity and temperature of the fluids, we derive the heat exchanger effectiveness by means of their cross-sectional average values and a constant overall heat transfer coefficient. The bulk temperatures of the fluids at the axial location x in the parallel flow heat exchanger shown in Figure 1 can be written in dimensionless forms as [30]

θ bI (ξ ) =

1 + R ⋅ e − ( R +1)⋅Ψ (ξ ) , R +1

1 − θ bI (ξ ) θ (ξ ) = , R II b

(C1)

(C2)

where Ψ(ξ) and R are the number of heat transfer units and heat capacity flow rate ratio which occur frequently in heat exchanger analysis, respectively; they are given by

Ψ (ξ ) =

2 Nu k ⋅ 2π ⋅ b ⋅ x ξ, = II II w c Pe ⋅ λ (1 − B )

(C3)

228


R=

wII c II Pe ⋅ λ ⋅ B . = wI c I 1+ B

(C4)

Nu in Eq. (C3) is the overall Nusselt number based on the outer stream properties being defined as Nu = ka/λI and k denotes the usual overall heat transfer coefficient of the inner pipe wall. Consequently, the heat exchanger effectiveness ε expressed by Eq. (29) is obtained as a function of the heat exchanger length ξ as follows:

ε (ξ ) = 1 − e− ( R +1)⋅Ψ (ξ ) .

(C5)

References [1] Mikhailov, M. D., and Ozisik, M. N.: Unified Analysis and Solutions of Heat and Mass Diffusion, Dover Publications, New York (1994) [2] Pagliarini, G., and Barozzi, G. S.: Thermal Coupling in Laminar Flow Double-Pipe Heat Exchangers. Trans ASME Journal of Heat Transfer, 113, 526-534 (1991) [3] Stein, R. P.: Liquid Metal Heat Transfer. In: Irvine, T. F., Hartnett, J. P. (eds.) Advances in Heat Transfer, Vol. 3, Academic Press, New York, London (1966) [4] Stein, R. P.: Heat Transfer Coefficients in Liquid Metal Concurrent Flow Double Pipe Heat Exchangers. Chemical Engineering Progress Symposium Series, 59, 64-75 (1965) [5] Stein, R. P.: The Graetz Problem in Concurrent Flow Double Pipe Heat Exchangers. Chemical Engineering Progress Symposium Series, 59, 76-87 (1965) [6] Mikhailov, M. D., and Shishedjiev, B. K.: Coupled at boundary mass or heat transfer in entrance concurrent flow. International Journal of Heat and Mass Transfer, 19, 553-557 (1976) [7] Neto, F. S., and Cotta, R. M.: Lumped-differential analysis of concurrent flow doublepipe heat exchanger. Canadian Journal of Chemical Engineering, 70, 592-595 (1992) [8] Plaschko, P.: High Péclet number heat exchange between cocurrent streams. Archive of Applied Mechanics (Ingenieur Archiv), 70, 597-611 (2000) [9] Huang, C. H., and Yeh, C. Y.: An optimal control algorithm for entrance concurrent flow problems. International Journal of Heat and Mass Transfer, 46, 1013-1027 (2003) [10] Bentwich, M., and Sideman, S.: Temperature distribution and heat transfer in annular two-phase (liquid-liquid) flow. Canadian Journal of Chemical Engineering, 42, 9-13 (1964) [11] Leib, T. M., Fink, M., and Hasson, D.: Heat transfer in vertical annular laminar flow of two immiscible liquids. International Journal of Multiphase Flow, 3, 533-549 (1977) [12] Davis, E. J., Venkatesh, S.: The solution of conjugated multiphase heat and mass transfer problems. Chemical Engineering Science, 34, 775-787 (1979) [13] Nogueira, E., Cotta, R. M.: Heat transfer solutions in laminar co-current flow of immiscible liquids. Heat and Mass Transfer, 25, 361-367 (1990) [14] Su, J.: Exact Solution of Thermal Entry Problem in Laminar Core-annular Flow of Two Immiscible Liquids. Chemical Engineering Research and Design, 84, 1051-1058 (2006) [15] Sparrow, E. M., and Spalding, E. C.: Coupled laminar heat transfer and sublimation mass transfer in a duct. Trans ASME Journal of Heat Transfer, 90, 115-124 (1968)


229

[16] Wang, W. P., Lin, H. T., and Ho, C. D.: An analytical study of laminar co-current flow gas absorption through a parallel-plate gas-liquid membrane contactor. Journal of Membrane Science, 278, 181-189 (2006) [17] Chiba, R., Izumi, M., and Sugano, Y.: An analytical solution to non-axisymmetric heat transfer with viscous dissipation for non-Newtonian fluids in laminar forced flow. Archive of Applied Mechanics (Ingenieur Archiv), 78, 61-74 (2008) [18] Vodicka, V.: Linear heat conduction in laminated bodies (in German). Mathematische Nachrichten, 14, 47-55 (1955) [19] Chiba, R.: Stochastic heat conduction analysis of a functionally graded annular disc with spatially random heat transfer coefficients. Applied Mathematical Modelling, 33, 507523 (2009) [20] Carslaw, H. S., and Jaeger, J. C.: Conduction of Heat in Solids, 2nd Edition, Clarendon Press, Oxford (1986) [21] Mikhailov, M. D., Ozisik, M. N., and Vulchanov, N. L.: Diffusion in composite layers with automatic solution of the eigenvalue problem. International Journal of Heat and Mass Transfer, 26, 1131-1141 (1983) [22] Bird, R. B., Stewart, W. E., and Lightfoot, E. N.: Transport Phenomena, John Wiley & Sons Inc, New York (1960) [23] Gill, W. N., Porta, E. W., and Nunge, R. J.: Heat transfer in thermal entrance region of cocurrent flow heat exchangers with fully developed laminar flow. International Journal of Heat and Mass Transfer, 11, 1408-1412 (1968) [24] Nunge, R. J., and Gill, W. N.: An analytical study of laminar counterflow double-pipe heat exchangers. AIChE Journal, 12, 279-289 (1966) [25] Shah, R. K.: Thermal entry length solutions for the circular tube and parallel plates. 3rd National Heat Mass Transfer Conference, 1, HMT-11-75 (1975) [26] Kays, W. M., Lundberg, R. E., and Reynolds, W. C.: Heat transfer with laminar flow in concentric annuli with constant and variable wall temperature and heat flux. NASA Technical reports, AHT-2 (1961) [27] Worsoe-Schmidt, P. M.: Heat transfer in the thermal entrance region of circular tubes and annular passages with fully developed laminar flow. International Journal of Heat and Mass Transfer, 10, 541-551 (1967) [28] Cotta, R. M., and Ozisik, M. N.: Laminar forced convection of power-law nonNewtonian fluids inside ducts. Heat and Mass Transfer, 20, 211-218 (1986) [29] Stein, R. P., and Sastri, V. M. K.: A heat transfer analysis of heat exchangers with laminar tube-side and turbulent shell-side flows–A new heat exchanger Graetz problem. AIChE Symposium Series, 118, 81-89 (1972) [30] Quick, K.: Direct calculation of exchanger exit temperatures in cocurrent flow. Chemaical Engineering, 93, 92 (1986).



Chapter 7

NUMERICAL SIMULATION OF TURBULENT PIPE FLOW M. Ould-Rouis and A.A. Feiz Université Paris-Est, Laboratoire Modélisation et Simulation Multi Echelle, MSME, FRE3160 CNRS, 77454 Marne-la-Vallée, France

Abstract Many experimental and numerical studies have been devoted to turbulent pipe flows due to the number of applications in which theses flows govern heat or mass transfer processes: heat exchangers, agricultural spraying machines, gasoline engines, and gas turbines for examples. The simplest case of non-rotating pipe has been extensively studied experimentally and numerically. Most of pipe flow numerical simulations have studied stability and transition. Some Direct Numerical Simulations (DNS) have been performed, with a 3-D spectral code, or using mixed finite difference and spectral methods. There is few DNS of the turbulent rotating pipe flow in the literature. Investigations devoted to Large Eddy Simulations (LES) of turbulence pipe flow are very limited. With DNS and LES, one can derive more turbulence statistics and determine a well-resolved flow field which is a prerequisite for correct predictions of heat transfer. However, the turbulent pipe flows have not been so deeply studied through DNS and LES as the plane-channel flows, due to the peculiar numerical difficulties associated with the cylindrical coordinate system used for the numerical simulation of the pipe flows. This chapter presents Direct Numerical Simulations and Large Eddy Simulations of fully developed turbulent pipe flow in non-rotating and rotating cases. The governing equations are discretized on a staggered mesh in cylindrical coordinates. The numerical integration is performed by a finite difference scheme, second-order accurate in space and time. The time advancement employs a fractional step method. The aim of this study is to investigate the effects of the Reynolds number and of the rotation number on the turbulent flow characteristics. The mean velocity profiles and many turbulence statistics are compared to numerical and experimental data available in the literature, and reasonably good agreement is obtained. In particular, the results show that the axial velocity profile gradually approaches a laminar shape when increasing the rotation rate, due to the stability effect caused by the centrifugal force. Consequently, the friction factor decreases. The rotation of the wall has large effects on the root mean square (rms), these effects being more pronounced for the streamwise rms velocity. The influence of rotation is to reduce the Reynolds stress component 〈Vr'Vz'〉 and to increase the two other stresses 〈Vr'Vθ'〉 and 〈Vθ'Vz'〉. The effect of the Reynolds

232

M. Ould-Rouis and A.A. Feiz number on the rms of the axial velocity (〈Vz'2〉1/2) and the distributions of 〈Vr'Vz'〉 is evident, and it increases with an increase in the Reynolds number. On the other hand, the 〈Vr'Vθ'〉profiles appear to be nearly independent of the Reynolds number. The present DNS and LES predictions will be helpful for developing more accurate turbulence models for heat transfer and fluid flow in pipe flows.

Nomenclature Cf=2τw/ρUb2 Lz N Nθ, Nr, Nz r

R Reb=UbD/ν Reτ=uτD/ν Rep=UpD/ν Sij uτ Ub=Up/2 Up v’r,v’θ, v’z y y+=(1-r) uτ/ν z

friction coefficient length of the computational domain rotation number grids in θ, r and z directions dimensionless coordinate in radial direction scaled by the pipe radius pipe radius (m) Reynolds number based on bulk velocity Reynolds number based on friction velocity Reynolds number based on Up. rate of strain tensor shear stress velocity bulk velocity centreline streamwise velocity of the laminar Poiseuille flow fluctuating velocity components dimensionless distance from the wall, y=1-r distance from the wall in viscous wall units coordinate in axial direction

Greek letters R

δ*

defined by δ*(2R-δ*) = 2

∫

r(1-Vz(r)/Up)dr

0

Δ=(r Δr Δθ Δz)1/3 Δr Δθ Δz ν θ

characteristic gridspacing gridspacing in radial direction gridspacing in circumferential direction gridspacing in axial direction kinematic viscosity coordinate in circumferential direction R

θ∗

defined by θ*(2R-θ*) = 2

∫ 0

r Vz(r)/Up (1-Vz(r)/Up)dr

Numerical Simulation of Turbulent Pipe Flow

233

1. Introduction The turbulent circular pipe flow has attracted the interest of many investigators. The simplest case of non-rotating pipe has been extensively studied experimentally [Laufer (1954), Lawn (1971)] and numerically. Most of pipe flow numerical simulations have studied stability and transition [Itoh (1977), Patera and Orszag (1981)]. Some Direct Numerical Simulations (DNS) have been performed. Using mixed finite difference and spectral methods, Nikitin (1993) was able to obtain satisfactory agreement with experimental data, inside the Reynolds number range of 2250-5900. Unger et al. (1993) obtained excellent agreement with experiments, using a second order accurate finite difference method. They confirmed that pipe flow at low Reynolds number deviates from the universal logarithmic law. Zhang et al. (1994) reported simulation of low to moderate Reynolds number turbulent pipe flow obtained with a 3-D spectral code. Their initial results agree satisfactorily with both experiments and previous numerical simulations. Eggels et al. (1994a) have carried out DNS and experiments in order to investigate the differences between fully developed turbulent flow in an axisymmetric pipe and a plane channel geometry. Most of the statistics on fluctuating velocities appear to be less affected by the axisymmetric pipe geometry. When a flow is introduced to an axially rotating pipe, fluids are given a tangential component of velocity by the moving wall and the flow in the pipe exhibits a complicated three-dimensional nature. The high levels of turbulence and large shearing rates associated with swirling flows enhance the mixing process and provide a more homogeneous flow of fluids. The role of the swirl flow is of great importance for the overall performance of the gas turbine. Recently, the numerical simulation of turbulent rotating pipe flow has received some interest. Eggels et al. (1994b) used a DNS of the turbulent rotating pipe flow for moderate values of the rotation number. They confirmed numerically the drag reduction observed in experiments. Orlandi and Fatica (1997) have also performed DNS of the turbulent rotating pipe flow. Their investigation was devoted to the study of the range of the rotation number, N, not considered by Eggels et al. (1994b), that is the investigation of the flow field at high values of N (N ≤ 2) but not enough high to include re-laminarization, and to analyze the modifications of the near-wall vortical structures, for a more satisfactory explanation of the drag reduction. They showed that a degree of drag reduction is achieved in the numerical simulations just as in the experiments, and that the changes in turbulence statistics are due to the tilting of the near-wall streamwise vortical structures in the direction of rotation. The more recent study by Orlandi and Ebstein (2000) is an extension of the previous one. N has been increased up to 10. These authors have evaluated the budgets for the Reynolds stresses at high rotation rates. These budgets are useful to those interested in developing new oneclosure turbulence models for rotating flows. Using a modified mixing length theory, Kikuyama et al. (1983) conducted calculations of the flow in an axially rotating pipe in a region far downstream from its inlet section. They observed that when a turbulent level is introduced into the rotating pipe, a flow laminarization is set up through an increase in the rotational speed of the pipe while destabilizing effects occur when the flow is initially laminar. Malin and Younis (1997) used two Reynolds stress transport closures for modeling flow and heat transfer in fully developed axially rotating pipe flow. They showed that both models reproduce the observed influences of rotation. These include reduction in skin friction and wall heat transfer, suppression of radial turbulent

234

M. Ould-Rouis and A.A. Feiz

transport of heat and momentum, laminarization of the flow. An encouraging level of agreement between calculations and available findings of literature was generally found. Speziale et al. (2000) presented both the analysis and modeling of turbulent flow in an axially rotating pipe flow. A particular attention was paid to tracing the origin of each of the two central physical features: the rotationally dependent axial mean velocity and the rotationally dependent mean azimuthal or swirl velocity relative to the rotating pipe, in order to gain a better physical insight into this turbulent flow. It was shown that second-order closure models provide good description of this flow and can describe both these features fairly well. In all the previous numerical studies, only one Reynolds number has been considered. There is very limited literature on investigations devoted to Large Eddy Simulations (LES) of turbulence pipe flow. The first LES work on fully developed turbulent pipe flow is given by Unger and Friedrich (1991). LES have been applied to flows in complex geometries to a very limited extent. The major reasons for this are due to the need for describing the nontrivial geometry accurately whilst limiting the number of computational grid points. LES predictions on turbulent pipe flow with rotation are extremely rare. There are only three works which deal with the turbulent rotating pipe flow. In 1993, Eggels and Nieuwstadt performed Large Eddy Simulations (LES) for the fully developed turbulent flow inside an axially rotating pipe. The main objective of this study was to investigate the influences of pipe rotation on mean flow properties and mean velocity profiles as well as on the profiles of the Reynolds stress components. They showed that the mean flow properties are fairly well predicted by LES, especially about the reduction of wall friction, deformation of the mean axial velocity profile and parabolic distribution of the mean circumferential velocity. Yang and McGuirk (1999) reported LES of turbulent pipe flow for the rotating and non rotating cases. Their numerical results compare reasonably well with the experimental data. They confirmed the experimental observations that turbulence decreases with an increase in pipe rotation due to the stability effect of the centrifugal force. In the present study, DNS and LES of fully developed turbulent pipe flow are performed to report the effects of the rotation and Reynolds numbers on the flow characteristics. The direct simulations have been carried out at two Reynolds numbers, Re=4900 and Re=7400, for different rotation rates ranging from N = 0 to N= 18. To elucidate the impact of higher Reynolds and rotation numbers, large eddy simulations with dynamic model have been conducted for a Reynolds number up to 20600. The present paper is organized as follows: the mathematical formulation and the numerical methods (DNS and LES) are described in section 2. Section (3.1) presents DNS predictions of the turbulent pipe flow, and compares our computations to the results reported in the archival literature. Section (3.2) deals with the LES predictions of the turbulent pipe flow. The validation of DNS and LES approaches are achieved by comparing the predicted profiles and statistics to the numerical and experimental data available in the literature. The effects of the Reynolds and rotation numbers on different thermal statistics (mean velocity profiles, root mean square of fluctuating velocity components, Reynolds shear stresses, skewness and flatness factors, velocity and vorticity fields) are investigated and discussed in section 3. The conclusion in section 4 ends this work.


235

2. Governing Equations and Numerical Method 2.1. Direct Numerical Simulation The continuity and momentum equations governing 3D-incompressible turbulent flow are written in a cylindrical coordinate system, Figure 1, in terms of the variables qr=r.Vr, qθ=Vθ and qz=Vz, in order to avoid the singularity at the axis r=0. The dimensionless equations are obtained using Up, the centerline streamwise velocity of the laminar Poiseuille flow, and the pipe radius R as velocity and length scales respectively:

(1)

(2)

(3)

(4) where the stresses, τij=2Sij, are given by:

(5)

236


Figure 1. Sketch of control volumes.

A mean pressure gradient in the qz equation maintains a constant bulk velocity, Ub. The Reynolds number and the rotation rate are defined as Reb=UbD/ν and N=ΩR/Ub, respectively.

2.2. Large Eddy Simulation The governing equations for Large Eddy Simulation (LES) are given below. The overbar indicates the filtering of the instantaneous fields which leads to the resolved scale fields (the smaller scales of turbulent motion being removed by the spatial filtering approach). ∂qr ∂r

+

∂ qθ ∂θ

+r

∂q z ∂z

=0

2ˆ ⎡ ˆ ˆ ⎤ ∂ qθ q z 1 ∂ r qθ q r ∂ qθ qθ 1 ∂P 1 ⎢ 1 ∂ r τ rθ 1 ∂ τ θθ ∂ τ zθ ⎥ , + + + + N qr = − + + + 2 r ∂θ Re ⎢ r 2 ∂r r ∂θ ∂z ⎥ ∂t ∂r ∂θ ∂z r

∂ qθ

⎣

∂ qr ∂t

+

⎦

∂ τˆ ⎤ ∂ ⎛⎜ q r q r ⎞⎟ ∂ ⎛⎜ qθ q r ⎞⎟ ∂ q r q z ∂P 1 ⎡ ∂ rτˆrr ∂ τˆrθ rz − τˆ + + +r + + − qθ qθ = Nr qθ - r ⎢ ⎥, θθ ⎟ ⎜ ⎜ ⎟ Re ⎢ ∂r ∂r ∂θ ∂z r r ∂r ∂θ ∂z ⎥⎦ ⎣ ⎝ ⎠ ⎠ ⎝

∂ qz ∂t

+

ˆ ∂ τˆ ⎤ 1 ∂ q r q z 1 ∂ qθ q z ∂ q z q z 1 ⎡ 1 ∂ rτˆrz 1 ∂ τ zθ ∂P zz . + + =− + + + ⎢ ⎥ θ ∂ z Re r ∂ r r ∂ ∂ z r ∂r r ∂θ ∂z ⎢ ⎥

⎣

⎦

Numerical Simulation of Turbulent Pipe Flow The total stresses τîj = τ ij + τ ' ij are τîj = (1 + ν T Re) S ij where

237 the

strain

tensor

expressed by the variables qi is:

The eddy viscosity νT has different expressions according to the subgrid model used.

Smagorinsky Model In this model, the subgrid scale eddy viscosity is related to the deformation of the resolved velocity field as:

[

]

1/ 2 2 2 νT = (C Δ) S = (C Δ) 2 S ij S ij s s

In the present study, the Smagorinsky coefficient CS is set equal to 0.15. For a discussion on the value and the interpretation of this constant, we refer to Mason and Callen (1986). This subgrid model largely used in LES of isotropic turbulence produced good results. When it applied to inhomogeneous, and in particular to wall bounded flows, the constant was modified.

Dynamic Eddy Viscosity Model The dynamic model provides a methodology for determining an appropriate local value of the Smagorinsky coefficient. The model was proposed by Germano et al. (1991), with important modifications and extensions provided by Lilly (1992). In this model, the constant Cd is not given a priori, but is computed during the simulation from the flow variables. The turbulent viscosity is expressed using an eddy viscosity assumption as:

[

]

1/ 2 2 νT = C (Δ ) 2 Sij Sij d

but Cd is dynamically determined as follows.

238


~ Two different filter widths are introduced; the test filter Δ is larger than the computational filter Δ and it is applied to the momentum equations. Germano et al. (1991) derived an exact relationship between the subgrid scale stress tensors at the two different filter widths (Germano identity). Substitution of a Smagorinsky form |S| = 2 S ij S ij for the subgrid scale stress into Germano identity, along with some additional assumptions [Lilly (1992)], leads to the expression for the constants

C

d

=−

1 2 Δ2

L M ij ij

(1)

M M ij ij

where the second order tensors Lij and Mij are given as follows:

~ ~ ~

2 L = q i q j − q i q j = - 2C Δ M , ij d ij

M ij =

~~~

Δ2 Δ2

~

S S ij − S S ij

The constant could be positive or negative. The positive values are linked to energy flowing from large to small scales and the negative to energy going from small to large scales (backward energy transfer). The angled brackets in equation (1) denotes averages in the homogeneous direction. The governing equations are discretized on a staggered mesh in cylindrical coordinates. The numerical integration is performed by a finite difference scheme, second-order accurate in space and in time. The time-advancement employs a fractional-step method. A third-order Runge Kutta explicit scheme and a Crank-Nicolson implicit scheme are used to evaluate the non-linear and viscous terms respectively. Uniform computational grid and periodic boundary conditions are applied to the circumferential and axial directions. In the radial direction, nonuniform meshes specified by a hyperbolic tangent function are employed. On the pipe wall, the usual no-slip boundary condition is applied. For Reb = 4900 and Reb = 7400, we performed simulations on a pipe of length Lz=20R using a 65x39x65 grid in the θ-, r- and zdirections. We have investigated the influence of different grids on the accuracy of the solution. The finest grid (129x49x129) leads to well resolved simulations. However, the grid 65x39x65 seems to predict and capture all features of the flow although small differences occur between some of the statistics obtained with these two grids. Since the fine grid requires much larger CPU-time and storage requirements, we performed calculations on the 65x39x65 grid which gives a good compromise between the required CPU-time and accuracy. Tables 1 and 2 list the details of the grid resolutions used in the simulations for the other Reynolds numbers (Reb = 5300, Reb =10300 and Reb = 20600). The final statistics are accumulated by spatial averaging in the homogeneous streamwise and circumferential directions, and by time-averaging.


239

Table 1. Grids resolutions for LES (Dynamic model) Model

Dynamic Model (D. M.)

Reb

N 0 6 0 6

10300 20600

Reτ 628 607 1242 1218

(Lθ, Lr, Lz)

(2π, 1, 15)

Grid 128x95x128

128x129x256

Table 2. Grids resolutions for DNS Reb

5300

Reτ 360 360 334 231

N 0 2 6 18

(Lθ, Lr, Lz)

(2π, 1, 15)

Grid 256x257x256 256x257x256 256x257x256 128x257x256

3. Results and Discussion 3.1. Direct Numerical Simulations Mean Velocity Profiles The axial mean velocity profiles (mean axial velocity normalized by the bulk mean velocity Ub) are compared in Figure 2a with the DNS and experimental data by Eggels et al. (1994a) for a stationary pipe flow. The calculated profiles are in satisfactory agreement with these data. For both Reynolds numbers, the streamwise velocity increases near the centre and decreases near the wall when the pipe is rotating. An examination of the velocity profiles shows a gradual approach towards a parabolic shape (Poiseuille profile) when increasing N, and correspondingly the effect of turbulence suppression due to the pipe rotation becomes more and more noticeable. Rotation has thus a very marked influence on the damping of the turbulent motion and drag reduction. This is in agreement with the experiments conducted by Nishibori et al. (1987), Reich and Beer (1989) and Imao and Itoh (1996). At Re=7400, the computed velocities have lower values than those for Re=4900 in the central region of the pipe, whatever the rotation rate is. A similar observation has been reported in the experiments by Reich and Beer (1989). It appears thus that the Reynolds number dependence of the mean velocity profile decreases when the rotation rate increases. To better investigate the effect of rotation on the turbulent pipe flow, we conducted four DNS with a rotation rate changing gradually from zero to a limit value N = 18. For this value (N = 18), a tendency to the relaminarization of the flow is observed. Table 2 summarizes the grid resolutions. As shown in Figure 2b, the mean velocity profile is significantly affected by the Coriolis force. It is worth pointing that the extend of the near-wall region is reduced when N increases, in accordance with the experimental observation of Kikuyama et al. (1983).

240


Figure 2a. Axial mean velocity normalized by the bulk velocity Ub as function of the wall distance for 0 ≤ N ≤ 2.

Figure 2b. Axial mean velocity normalized by the bulk velocity Ub as function of the wall distance for N=0 (─), N=2 (---), N=6 (···) and N=18 (·─·).


241

Mean Axial Velocity Normalized on the Friction Velocity In Table 3, the mean flow parameters are compared to those of literature, for Reb ≈ 5300. The DNS calculations are first compared to the findings of Eggels et al. (1994a) for which the Reτ is close to our Reτ. There is a good agreement between the present predictions and the results of Eggels et al. (1994a). The slight difference between the data sets can be attributed to the difference in the grid resolutions between the two simulations. The axial mean velocity normalized by the friction velocity uτ versus the distance from the wall is shown in Figure 3a in wall units y+. This computed profile is consistent with the DNS profile of Eggels et al. (1994a). The validation of the present predictions has also been achieved by comparing our calculated kinetic energy, Figure 3b, and vorticity fluctuations, Figure 3c, to those of Eggels et al. (1994a). Table 3. Mean-flow parameters for Direct Numerical Simulation (DNS) compared to literature, at Reb = 5300 and N=0 DNS

Eggels et al. Loulou (1996) (1994b)

Westerweel et Westerweel et al. (1996) al. (1996) [PIV] [LDA]

Nθ

256

128

160

-

-

Nr

257

96

72

-

-

Nz

256

256

192

-

-

+

8.89

8.84

7.50

-

-

Δr+min

0.11

0.94

0.39

-

-

+

4.03

1.88

5.70

-

-

+

14.10

7.00

9.90

-

-

Rep

6954

6950

7248

7100

7200

Reb

5299

5300

5600

5450

5450

Re

362

360

380

366

371

Uc/u

19.19

19.31

19.12

19.38

19.39

Ub/u

14.63

14.73

14.77

14.88

14.68

Uc/Ub

1.31

1.31

1.29

1.30

1.32

Cf (x10-3)

9.35

9.22

9.16

9.03

9.28

*/R

0.127

0.127

0.121

0.124

0.130

θ*/R

0.069

0.068

0.066

0.068

0.071

H= */θ*

1.85

1.86

1.84

1.83

1.83

RΔθ Δr

Δz

max

max

242


Eggels et al, 1994a.

Figure 3a. Axial mean velocity normalized by the friction velocity uτ as function of the distance for N=0 and Reb=5300.: lignes (present DNS), symbols.


Figure 3b. Kinetic energy profile for N=0: lignes (present DNS), symbols.


243


Figure 3c. Vorticity fluctuations versus the wall distance for N=0 and Reb=5300.: lignes (present DNS), symbols.

Figure 3d. Axial mean velocity normalized by uτ: a comparison between the present DNS and the data of literature.

244


Figure 3e. Axial mean velocity normalized by uτ as function of the distance (in wall units) from the wall with Reb and N as parameters.

To ascertain the reliability and accuracy of the present numerical simulation, the present predictions are compared to the available results of literature at Re=7442 in the case of a stationary pipe: the experimental data of Eckelmann (1974) and Kim et al. (1987) for a channel flow. They are also compared to the LDA-mean flow distribution by Durst et al. (1995) for a turbulent stationary pipe flow. Figure 3d shows that all the velocity profiles meet fairly well with our numerical predictions. The satisfactory agreements confirm that the mean flow field is well predicted by the present numerical simulations. In Figure 3e, the solid line represents the universal velocity distributions in the viscous sublayer, in the buffer layer and in the inertial sublayer. The viscous sublayer is well resolved in the numerical simulations, yielding the linear velocity distribution Vz+=y+ for y+<5. The buffer region is also well predicted in accordance with the log-law Vz+=-3.05+5*ln y+. For the two Reynolds numbers considered, the agreement with the log-law at larger distances from the wall (y+>30) is less for the present DNS results, and also for the experimental data by Eggels et al. (1994a). When the pipe rotates, the differences between the computed mean velocity and the log-laws are due to the relaminarization of the flow when the rotation rate increases. Similar observations have been reported by Orlandi and Fatica (1997) and by Zhang et al. (1994). The reason is that the log-laws are not observed in the pipe flow for Reb ≤ 9600 (Reb=UbD/ν), in contrast to plane channel flows. Theoretically, the log-laws are only justified at large Reynolds numbers (Tennekes and Lumley, 1972). The variation of the velocity friction with the rotation number N is sketched in Table 4, for Reb = 5300. Here, each velocity friction uτ is normalized by that 0

of the non-rotating case, uτ . Note that the friction velocity diminishes with an augmentation in the rotation number, inducing a marqued reduction in the friction coefficient at the wall.


245

The friction factor decreases with an increase in the rotation number and this tendency becomes more remarkable for larger values of the Reynolds number. For the highest rotation number, N = 18, the reduction in the friction coefficient f is about 60%. For stationary pipe flows, the simulation predicts a friction factor close to the value evaluated by using the Blasius relation for Re=4900 (f = 0.3164 Re-1/4 = 0.0378). Table 4. Variation of the velocity friction with the rotation number at Reb = 5300. Reb

5300

N 0 2 6 18

u/ uτ0 1.00 1.00 0.93 0.64

Root Mean Square (rms) The radial distribution of the root mean square (rms) of the fluctuating velocities in a non-rotating pipe are plotted in Figure 4a, for Reb = 5300, along with DNS calculations of Eggels et al. (1994a). There is a satisfactory agreement between the present predictions and the measurements and DNS by Eggels et al. (1994a). The rms values of all velocity components are also depicted in Figures 4b-d and compared to the numerical computation of Kim et al. (1987), to the measured rms of Kreplin and Eckelmann (1979) obtained with a Xhot film probe and to the LDA measurements by Durst et al. (1995). The comparison between the present DNS results and the available experimental and numerical data of literature seems

Figure 4a. Root-mean-square profiles of azimuthal, radial and axial velocity components for N=0. Lines: Present DNS; Symbols (Eggels et al., 1994b): (Δ) Vr , rms , (O) Vθ , rms , ( ) Vz , rms.

246


to be better satisfactory. However, there are some slight differences. These differences are most likely caused by the coarse numerical resolution of our computations. For Reb = 4900, Figure 4b shows that the peak in the distribution of the streamwise rms is located at the inner edge of the buffer region while that of the normal and tangential components are located at the outer edge, as it can be seen in Figures 4c,d.

Figure 4b. Root-mean-square of the velocity fluctuations: axial velocity component, N=0.

Figure 4c. Root-mean-square of the velocity fluctuations: radial velocity component, N=0.


Figure 4d. Root-mean-square of the velocity fluctuations: tangential velocity component, N=0.

Figure 5a. Root-mean-square profiles of axial velocity with Reb and N as parameters (0≤N≤2).

247

248


Figure 5b. Root-mean-square profiles of velocity fluctuations for 0≤N≤18, Re=5300: <(Vz’)2>1/2 (─) , <(Vθ’)2>1/2 (---), <(Vr’)2>1/2 (···).

The rotation of the wall has large effects on the rms, these effects being more pronounced for the streamwise rms velocity. Similar observations have been reported by


249

Eggels et al. (1994a) and by Orlandi and Fatica (1997). For N=2 and N=1, the rms distributions are almost the same (Figure 5a). Figure 5b depicts the turbulence intensities normalized by the friction velocity for Reb = 5300 and higher rotation numbers. At a moderate rotation number (N = 1), the fluctuation levels of the three velocity components are increased in comparison to the non-rotating case, Figure 5b. From N ≥ 2, this tendency remains for the radial and azimuthal components. On the contrary, a significant reduction of the axial turbulence intensity is observed. The intensification of the radial and azimuthal turbulence intensities in the core region of the rotation pipe denotes that the turbulence there tends to be isotropic, especially at high rotation number. Near the wall (0 ≤ y ≤ 2), the azimuthal turbulence intensity exhibits remarkable peak values with increasing N, while the maximum of the axial fluctuations is considerably reduced (-10% for N = 2 and -30% for N = 18). From the three rms velocities, the effect of the Reynolds number on the rms of the axial velocity (〈Vz'2〉1/2) is evident, and it increases with an increase in the Reynolds number, Figure 5a. Similar observations are reported by Zhang et al. (1994). For the other rms velocities, 〈Vθ'2〉1/2 and 〈Vr'2〉1/2, this trend is observed in the core region of the flow.

Reynolds Shear Stresses Figure 6a shows the distributions of the Reynolds stress components 〈Vr'Vz'〉. The Reynolds stresses in a stationary pipe allow to check the accuracy of the simulations. The computed values of 〈Vr'Vz'〉 are compared with the measurements of Eggels et al. (1994a). The agreement between the present predictions and these results is satisfactory. The total shear stress (the sum of the turbulent and viscous stresses) is also plotted in this figure (solid line). As it can be seen the Reynolds shear stress dominates in the core region of the flow since the viscous shear stress is small. When the rotation rate increases, the decrease of the viscous stress in the core region leads to a decrease of 〈Vr'Vz'〉. On the other hand, the stress 〈Vr'Vz'〉 increases near the wall in accordance with the increase of the viscous stresses. For Re=7400, the values of 〈Vr'Vz'〉 are slightly larger than the corresponding values obtained for Re=4900, especially near the wall. The 〈Vr'Vθ'〉 and 〈Vθ'Vz'〉 stresses, which are zero in a stationary pipe flow, become non-zero when the pipe rotates. The influence of rotation is to reduce the 〈Vr'Vz'〉 shear stress and to increase the two other stresses 〈Vr'Vθ'〉 and 〈Vθ'Vz'〉 as it is shown in Figure 6b. Near the rotating wall, the high values of 〈Vθ'Vz'〉 are related to the tilting of the near wall vortical structures (Orlandi and Fatica, 1997). In the core region of the flow, the radial oscillations of 〈Vθ'Vz'〉 are due to the large-scale structures within this region. The behavior of 〈Vr'Vθ'〉 is almost linear in the central part of the pipe. The Reynolds number has larger effects on the component 〈Vθ'Vz'〉, these effects being more pronounced when the rotation rate increases. The largest variations occur in the central region of the pipe. At N=1, the distributions of 〈Vr'Vz'〉 for both Reynolds numbers are well separated, and the 〈Vr'Vz'〉-values are larger at Re=7400. On the other hand, the 〈Vr'Vθ'〉-profiles appear to be nearly independent of the Reynolds number.

250


Figure 6a. Distribution of the Reynolds shear stress component in wall units with Reb and N as parameters.

Figure 6b. Reynolds shear stresses distributions in wall units for N=1.


251

Velocity and Vorticity Fields : Structures and Statistics The instantaneous velocity and instantaneous vorticity fields are of great help in revealing the influence of the rotation effect on turbulence coherent structures. Figure 7a depicts the isosurface of the vorticity modulus (ω = 3) for various rotation numbers N. It clearly shows the impact of the rotation rate on the turbulence structures: • • •

the more and more flat shape of the isosurface near the wall, with increasing N, denotes a gradually reduction of the turbulence activity, As moving away from the wall, the augmentation in the rotation number induces a set of turbulence structures with strong vorticity developing up to the pipe centre, These vortical structures are inclined and better organized with increasing N.

Figure 7a. Isosurface of the vorticity modulus (ω=3) for 0≤N≤18.

The rotation seems to have a tendency to organize the flow near the pipe wall and in the core region, leading to a relaminarization of flow. A plot of a cross-sectional view of the streamwise vorticity field on an axial-azimuthal plane, for many rotation numbers, also exhibits the vertical structures in the pipe flow, Figure 7b. At moderate rotation rate (N = 2), the vertical structures concentrate in some regions of the flow, while the other regions appear rather quiet. The strong longitudinal expansion of

252


these regions denotes presence of elongated vorticity streaks, similar to those observed in experiments. The impact of rotation on the axial velocity fluctuation can be seen in Figure 7c. The patterns clearly illustrate a better organization of the pipe flow when N increases, and a more pronounced inclination of the axial velocity isosurface.

Figure 7b. Cross-sectional view of the streamwise vorticity field ans axial-azimuthal plane, for 0≤N≤18: N=0 (a), N=2 (b), N=6 (c), N=18 (d).

The rms values of all three vorticity components, for the four direct numerical simulations, are plotted in Figure 7d. Once again, one can clearly see the impact of the Coriolis force which gradually reduces the vorticity fluctuations. It is interesting to point out the increasing anisotropy of the vorticity fluctuations. Indeed, with increasing N, the axial vorticity fluctuations predominate those of the two other components.


Figure 7c. Isosurface of the axial velocity for 0≤N≤18.

Figure 7d. Continued on next page.

253

254


Figure 7d. Root mean square of the three vorticity components for 0≤N≤18: <(ωz’)2>1/2 (─) , <(ωθ’)2>1/2 (---), <(ωr’)2>1/2 (···).

High-Order Statistics Many experimental investigations of turbulent pipe flow at moderate Reynolds numbers have been published (Durst et al., 1995; Den Toonder, 1995; Westerweel et al., 1996). They used essentially Laser techniques to measure data: PIV (Particle Image Velocimetry) and LDA (Laser Doppler Anemometers). Figure 8 compares our DNS results for higher order statistics (skewness and flatness factors for axial velocity component) to the experimental data of Westerweel et al. (1996). There is a satisfactory agreement between the numerical and experimental profiles. However, the present DNS slightly overpredicts the peak of the axial component Vz’. Durst et al. (1995) reported similar observation when comparing their results with many DNS of literature. The skewness factors of the axial, radial and tangential velocity components (S1, S2 and S3, respectively) are compared in Figures 8c,d,e to the DNS of Kim et al. (1987), to the laserDoppler measurements of Niederschulte et al. (1990) for channel flow, and also to the LDA measurements of Durst et al. (1995) in the case of a stationary pipe. S3 is also compared to the LDV measurements of Karlsson and Johansson (1986) for a boundary layer flow. About S1 (Figure 8c) the trends observed in the present simulations, in the experimental data and DNS of literature are similar, with a positive skewness near the wall and a negative skewness factor approaching a value of –0,5 at larger y+ in the stationary pipe. The agreement with the experimental or numerical findings of the referenced literature is not so good, but still reasonable. Our predictions of S1 are somewhat overpredicted, especially near the wall. Figure 8d shows the variations of the skewness factor of S2 in wall region: the differences between the present predictions and the results of literature are larger than for S1. These discrepancies could be attributed to the too coarse grids used, but it should be also taken into


255

account that the experimental data are obscured by noises close to the wall, which lead to a smaller skewness factor. The predicted values of S3, are shown in Figure 8e: for a stationary pipe, almost zero-values are predicted over most of the pipe diameter because of symmetry. This is in good agreement with the data from DNS or measurements reported in the literature.


Figure 8. Skewness (a) and Flatness (b) factors for axial velocity component, N=0 and Reb=5300: lignes (present DNS), symbols.

Figure 8c. Skewness factors of the velocity fluctuations in the near-wall region: the streamwise velocity component with Reb and N as parameters.

256


Figure 8d. Skewness factors of the velocity fluctuations in the near-wall region: the normal-to-the-wall velocity component with Reb and N as parameters.

Figure 8e. Skewness factors of the velocity fluctuations in the near-wall region: the circumferential velocity component with Reb and N as parameters.


257

For a rotating pipe, the skewness factors S1 and S2 show large variations, especially in the near wall region. However S1 approaches the negative value of –0,5 at larger distances from the wall (y+ > 70), while S2 tends towards zero. These large variations of the skewnesses with the rotation rate are a further indication that changes of the orientation in the vortical structures occur near the wall (Orlandi and Fatica, 1997). When the pipe rotates, the skewness factor of the tangential velocity component, S3, is larger for N=1 near the wall. Its non zerovalues across the flow field suggest that the reflection symmetry is broken by rotation. In contrast to the skewness factor, the flatness factor of the axial velocity component, F1, is in better agreement with the results of literature as it can be seen in Figure 9a. However, higher flatness values are predicted at the wall region while the agreement is very good in the core region. The behavior of the radial velocity component, F2, shows similar trend (Figure 9b). The agreement is good with the data reported in previous studies for y+ > 25 (Durst et al., 1995; Niederschulte et al., 1990; Kim et al., 1987; Karlsson and Johansson, 1986). The flatness factor is slightly larger than 3, which is the value of flatness factor for a Gaussian distribution. However, near the wall, there is a strong increase of the flatness factor whereas the experimental data show a decreasing flatness factor as the wall is approached (Durst et al., 1995; Niederschulte et al., 1990). Clearly, noticeable discrepancies between the different results for the flatness factor of the radial velocity component are observed, but the origin of these discrepancies is not clarified. The distribution of the flatness factor of the tangential velocity component, F3, is plotted in Figure 9c. The agreement between the present simulation for stationary pipe flow and the results of literature is fairly good.

Figure 9a. Flatness factors of the velocity fluctuations in the near-wall region: the streamwise velocity component with Reb and N as parameters.

258


Figure 9b. Flatness factors of the velocity fluctuations in the near-wall region: the normal-to-the-wall velocity component with Reb and N as parameters.

Figure 9c. Flatness factors of the velocity fluctuations in the near-wall region: the circumferential velocity component.


259

In the case of a rotating pipe, Figures 9a,c show that large variations of the flatness factors F1, F2 and F3 occur near the rotating wall while they are almost unchanged in the remaining part of the pipe. This is an indication of the intermittent nature of the wall region. Durst et al. (1995) showed that the higher order-moments and turbulence intensity are highly interconnected, especially close to the wall. Therefore, we have examined the interrelations between these quantities over the cross-section of the pipe. We have plotted in Figure 10 the distribution of the turbulence intensity together with the distributions of the skewness and flatness factors of the streamwise velocity component. It can be seen that the increase of the turbulence intensity results from a decrease of the non-Gaussian behavior of the skewness and flatness factors. The points for the maximum intensity, zero skewness and minimum flatness coincide. This result agrees with the measurements of Durst et al. (1995).

Figure 10. Distributions of turbulent intensity, skewness and flatness factors of the axial velocity component.

3.2. Large Eddy Simulation This section is devoted to Large Eddy simulation (LES) of turbulent pipe flow. The objective of these LESs is to improve upon the description of such flow and to elucidate the effect of higher Reynolds numbers and higher rotation numbers on the field flow. The accuracy of the LES technique depends significantly on the ability of the subgrid-scale (SGS) model. Feiz (2005) examined the performance of two turbulence models: the classical algebraic eddy-viscosity model of Smagorinsky (S.M.) and the dynamic model (D.M.) for the range 556 ≤ Reτ ≤ 697. Details of the corresponding grid resolutions are listed in Table 5.

260


The results show that the flow field and its statistics are better predicted with the dynamic model. For example, Figure 11 compares our LES computations with the DNS data of Wagner et al. (2001). The reader can find more details in Feiz (2005). In the this section, we present some LES results obtained with the dynamic model. The motivation of these LES studies is to deal with the behaviour of the turbulent pipe flow in the rotating and non-rotating pipes for higher Reynolds numbers (Reτ > 1000), and to examine the effectiveness of the LES method for predicting such turbulent flows. Table 6 gives informations on the grid resolution for Reτ = 1242, using LES with the dynamic model. Table 7 lists the mean flow parameters of the present LES, along with the DNS data of Wagner et al. (2001), for Reb = 10300. The agreement between them is reasonable good. Table 5. Grids resolutions for LES (Smagorinsky and Dynamic models) at Reb = 10300 and N=0 Model S. M. D. M.

Reb

10300

Reτ 556 628

N

(Lθ, Lr, Lz)

Grid

0

(2π, 1, 15)

128x95x128

Table 6. Grids resolutions for LES (Dynamic model) at Reb = 20600 and N=0 Model D. M.

Reb 20600

N 0

Reτ 1242

(Lθ, Lr, Lz) (2π, 1, 15)

Grid 128x129x256

Table 7. Mean-flow parameters for Large Eddy Simulation (LES) and literature at Reb = 10300 and N=0

Nθ Nr Nz RΔθ+max Δr+min Δr+max Δz+ Rep Reb Reτ Uc/u Ub/u Uc/Ub Cf (x10-3) */R θ*/R H= */θ* Wagner et al., 2001.

LES 128 95 128 8.89 0.11 4.03 14.10 13181 10300 628 20.38 15.94 1.28 7.87 0.115 0.070 1.65

Wagner et al. (2001) 240 70 486 8.36 0.64 7.68 6.58 13210 10300 640 20.64 16.09 1.28 7.70 0.115 0.071 1.63


261

Figure 11. Axial mean velocity normalized by the friction velocity as function of the wall distance for Reb=10500 and N=0. Lignes (present LES): (a) Smagorinsky model, (b) dynamic model. Symboles: Wagner et al (2001).

The rms velocity fluctuations nondimensionalized by the friction velocity are given on Figures 12a,b,c for three Reynolds numbers Reb = 5300, Reb = 10300 and Reb = 20600. The predicted fluctuations in the wall-normal, Figure 12a, azimuthal, Figure 12b, and streamwise, Figure 12c, directions are similar to the experimental data of Westerweel et al. (1996). The fluctuations in the streamwise direction are more intense than those in the two other directions. The locations of the maxima for each velocity fluctuation are different: the maxima of azimuthal and radial turbulence intensities are located in the logarithmic region, while the maxima of the axial turbulence intensities occur in the buffer region. Furthermore, +

the peak in the axial velocity fluctuation is located at a distance in wall unit y ≈ 13,

262


Figure 12. Root mean square velocity fluctuations in wall units, for N=0 and Reb=5300 (─), Reb=10300 (---), Reb=20600 (···): (a) tangential component<(Vθ’)2>1/2/uτ; (b) radial component<(Vr’)2>1/2/uτ; (c) axial component <(Vz’)2>1/2/uτ

Figure 13. Reynolds shear stresses distributions in wall units for N=0 and Reb=5300 (─), Reb=10300 (--), Reb=20600 (···).


263

irrespective of the Reynolds number, i.e. the Reynolds number has no effect on the peak value of the axial velocity fluctuation. For the two other velocity components, the positions of the corresponding peaks shift farther off from the wall with increasing Reynolds number, suggesting that the maxima of and are not always located in the boundary layers. Similar to the behaviour of the rms of velocity fluctuations in the radial and azimuthal directions, the Reynolds shear stresses move away from the wall when Reb increases, Figure 13. Figures 14a,b depicts the distribution of the skewness and flatness factors of the axial velocity fluctuation near the wall region of a non-rotating pipe, for many Reynolds numbers. These distributions point out two interesting observations: • •

first, the skewness and flatness coefficients of the velocity fluctuations seem to be very sensitive to the variation of the Reynolds number, near the wall, in the core region, the skewness and flatness values for fully developed turbulent flow are not equal to those of a Gaussian distribution (S = 0 and F = 3). This implies, for the skewness of the radial velocity fluctuation (not represented here), that the turbulent energy is convected from the wall to towards the pipe axis.

Close to the pipe axis, the probability distribution of the axial fluctuations is asymmetric. The negative values of S1 indicate that the negative fluctuation of Vz’ are dominant in probability. In the vicinity of the wall, S1 becomes positive and is enhanced. This trend is more apparent with increasing Reb. Similarly, the profiles of the flatness coefficient of the axial velocity component increases near the wall with increasing Reb, suggesting that the axial turbulence fluctuation in the wall region become more intermittent with increasing Reynolds number.

Figure 14. Skewness (a) and Flatness (b) factors of axial velocity component for N=0 and Reb=5300 (─), Reb=10300 (---), Reb=20600 (···).

264


Figure 15. Effect of Re on the skewness and flatness factors of the axial velocity fluctuation for high N (N=6): Reb=5300 (─), Reb=10300 (---), Reb=20600 (···).

Figure 16. Effect of N on the skewness and flatness factors of the axial velocity fluctuation for high Reb (Reb=20600): N=0(─), N=6 (---).

Figures 15 a,b shows the effect of Reynolds number on the skewness and flatness factors of the axial velocity fluctuation, for high rotation rate (N = 6). It seems that the impact of the Reynolds number on the flatness factor is more important than that on the skewness factor. For the highest Reynolds number (Reb = 20600), the influence of the rotation number on the skewness factor is more pronounced than that on the flatness factor, Figures 16 a,b. In both


265

cases, the effect of Reb or N on the skewness and flatness coefficients are mainly restricted in the near wall region. The present LES predictions compare reasonably well with the results of literature. We can conclude that all phenomena in rotating and non-rotating turbulent pipe flow can be captured directly by LES with dynamic model.

4. Conclusion This chapter was devoted to DNS and LES of the fully developed turbulent flow in an axially rotating pipe for various Reynolds numbers and different rotation rates. Different statistical turbulence quantities including the mean and fluctuating velocity components, friction coefficient, the Reynolds shear stresses and higher order statistics are obtained and analyzed. An effort to reveal the effects of the Reynolds number and the rotation number on the turbulent pipe flow is sketched. The validation of the present DNS and LES has been achieved by comparing our predictions with some available results of literature. It is shown that the computed deformation of the mean axial velocity profile agrees with the deformation observed experimentally and that rotation has a very marked influence on the suppression of the turbulent motion and on the drag reduction. A tendency to the relaminarization of the flow is observed for the highest rotation number. The Reynolds number dependence of the mean velocity profile decreases when the rotation rate increases. The friction factor decreases with an increase in the rotation number and this tendency becomes more remarkable for larger values of the Reynolds number. The rms and Reynolds stresses profiles showed an encouraging level of agreement with the measured data and DNS results of literature. From N ≥ 2, a significant reduction of the axial turbulence intensity is observed, while the fluctuation levels of the radial and azimuthal components are increased denoting an isotropization of the in the core region of the rotation. The rotation also reduces the 〈Vr'Vz'〉 shear stress and increases the two other stresses 〈Vr'Vθ'〉 and 〈Vθ'Vz'〉. The Reynolds number has larger effects on the component 〈Vθ'Vz'〉, these effects being more pronounced when the rotation rate increases. On the other hand, the 〈Vr'Vθ'〉profiles appear to be nearly independent of the Reynolds number. The overall agreement between the predicted skewness and flatness factors and the results reported in the literature is satisfactory. For rotating pipe, the large variations of the skewness and flatness factors of the velocity components, in the near wall region denote the intermittent nature of the wall region and indicate that changes of the orientation in the vortical structures occur near the wall. For the highest rotation rate, the impact of the Reynolds number on the flatness factor is more important than that on the skewness factor. For the highest Reynolds number, the influence of the rotation number on the skewness factor is more pronounced than that on the flatness factor. In both cases, the effect of the Reynolds or rotation numbers on the skewness and flatness coefficients are mainly restricted in the near wall region. Visualizations of the instantaneous velocity and vorticity fields exhibit turbulence structures with strong vorticity developing up to the pipe centre. These vortical structures are inclined and better organized with increasing N. An interesting outcome of the present investigation is to establish databases of various turbulence statistics of the turbulent pipe flow at different Reynolds and rotation numbers. These databases will undoubtedly helpful for evaluating and developing turbulence models.

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References [1] Durst F. , Jovanovic J. and Sender J., 1995, J. Fluid Mech., 295, p. 305. [2] Eckelmann H., 1974, J. Fluid Mech., 65, p. 439. [3] Eggels, J. G. M. and Nieuwstadt, F. T. M., 1993, In Proc. 9th Symp. on Turbulent Shear Flows, Kyoto, Japan, p.310. [4] Eggels, J. G. M., Unger, F., Weiss, F., M. H., Westerweel, J., Adrian, R. J., Friedrich, R. and Nieuwstadt, F. T. M., 1994a, J. Fluid Mech., 268, p. 175. [5] Eggels, J. G. M., Boersma, B. J. and Nieuwstadt, F. T. M., 1994b, Preprint. [6] Feiz, A.A, 2006, Thèse de doctorat, Champs sur Marne. University of Paris-Est MarneLa-Vallée. [7] Germano, M., Piomelli, U. and Cabot, W. H., 1991, Phys. Fluids A, 3, p. 1760. [8] Imao, S. and Itoh, M., 1996, Int. J. Heat and Fluid Flow, 17, p. 444. [9] Itoh, N., 1977, J. Fluid Mech., 82, p. 469. [10] Karlsson R. I. and Johansson T. G., 1986, In Laser Anemometry in Fluid Mechanics III (Ed. R. J. Adrian), Lisbon, Portugal, p. 273. [11] Kikuyama , K., Murakami, M. and Nishibori, K., 1983, Bull. J.S.M.E., 26, p. 506. [12] Kim J., Moin P. and Moser R., 1987, J. Fluid Mech., 177, p. 133. [13] Kreplin H. and Eckelmann H., 1979, Phys. Fluids, 22, p. 1233. [14] Laufer, J., 1954, NACA Report 1174. [15] Lawn, C. J., 1971, J. Fluid Mech., 48, p. 477. [16] Lilly, D. K., 1992, Phys. Fluids A, 4, p. 633. [17] Loulou P., 1996, Thèse de doctorat, Stanford University, Department of Aeronautics and Astronautics. [18] Malin M. R. and Younis B. A., 1997, Int. Comm. Heat Mass Transfer, 24, p. 89. [19] Mason, P. J. and Callen, N. S., 1986, J. Fluid Mech., 162, p. 439. [20] Niederschulte N. A., Adrian R. J. and Hanratty T. J., 1990, Exps. Fluids, 9, p. 222. [21] Nikitin, N. V., 1993, In Bulletin of APS, Vol. 38, No. 12, p. 2311. [22] Nishibori, K., Kikuyama, K. and Murakami, M., 1987, JSME Intl J., 30, p. 255. [23] Orlandi, P. and Fatica, M., 1997, J. Fluid Mech., 143, p. 43. [24] Orlandi, P. and Ebstein, D., 2000, International Journal of Heat and Fluid Flow, 21, p. 499. [25] Patera, A. T. and Orszag, S. A., 1981, J. Fluid Mech., 112, p. 467. [26] Reich, G. and Beer, H., 1989, Intl J. Heat Mass Transfer, 32, p. 551. [27] Tennekes, H. and Lumley, J. L., 1972, “A First Course in Turbulence”, MIT Press. [28] Toonder Den J. M. J., 1995, Thèse de doctorat, Delft, University of Technology, Department of Aero- and Hydrodynamics. [29] Unger, F., Eggels, J. G. M., Friedrich, R. and Nieuwstadt, F. T. M., 1993, In Proc. 9th Symp. on Turbulent Shear Flows, Kyoto, Japan, pages 2/1/1-2/1/6. [30] Unger, F. and Friedrich, R., 1991, In Proc. 8th Symp. on Turbulent Shear Flows, Munich, Germany, pp. 19-3-1 – 19-3-6. [31] Wagner C., Huttl T. J. and Friedrich R., 2001, Computers & Fluids, 30, p. 581. [32] Westerweel J., Draad A. A., Hoeven van der J. G. Th. and Oord van J., 1996, Experiments in Fluids, 20, p. 165.


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[33] Yang, Z. and McGuirk, J.J., 1999, Proc. of Turbulence and Shear Flow Phenomena, USA, p. 863. [34] Zhang, Y., Gandhi, A., Tomboulides, A. G. and Orszag, S. A., 1994, AGARD Conf. Proc., 551, pp. 17.1-17.9.



Chapter 8

PIPE FLOW ANALYSIS OF URANIUM NUCLEAR HEATING WITH CONJUGATE HEAT TRANSFER G.H. Yeoh* and M.K.M. Ho Australian Nuclear Science and Technology Organisation (ANSTO), PMB 1, Menai,NSW 2234, Australia

Abstract The field of computational fluid dynamics (CFD) has evolved from an academic curiosity to a tool of practical importance. Applications of CFD have become increasingly important in nuclear engineering and science, where exacting standards of safety and reliability are paramount. The newly-commissioned Open Pool Australian Light-water (OPAL) research reactor at the Australian Nuclear Science and Technology Organisation (ANSTO) has been designed to irradiate uranium targets to produce molybdenum medical isotopes for diagnosis and radiotherapy. During the irradiation process, a vast amount of power is generated which requires efficient heat removal. The preferred method is by light-water forced convection cooling—essentially a study of complex pipe flows with coupled conjugate heat transfer. Feasibility investigation on the use of computational fluid dynamics methodologies into various pipe flow configurations for a variety of molybdenum targets and pipe geometries are detailed in this chapter. Such an undertaking has been met with a number of significant modeling challenges: firstly, the complexity of the geometry that needed to be modeled. Herein, challenges in grid generation are addressed by the creation of purpose-built bodyfitted and/or unstructured meshes to map the intricacies within the geometry in order to ensure numerical accuracy as well as computational efficiency in the solution of the predicted result. Secondly, various parts of the irradiation rig that are required to be specified as composite solid materials are defined to attain the correct heat transfer characteristics. Thirdly, the use of an appropriate turbulence model is deemed to be necessary for the correct description of the fluid and heat flow through the irradiation targets, since the heat removal is forced convection and the flow regime is fully turbulent, which further adds to the complexity of the solution. As complicated as the computational fluid dynamics modeling is, numerical modeling has significantly reduced the cost and lead time in the molybdenum-target design process, and such an approach would not have been possible without the continual improvement of computational power and hardware. This chapter also addresses the importance of *

E-mail address: [email protected], Phone no:+61-2-9717 3817, Fax no.:+61-2-9717 9263. Corresponding Author:Dr Guan Heng YEOH, B40, ANSTO, Private Mail Bag 1, Menai, NSW 2234, Australia.

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G.H. Yeoh and M.K.M. Ho experimental modeling to evaluate the design and numerical results of the velocity and flow paths generated by the numerical models. Predicted results have been found to agree well with experimental observations of pipe flows through transparent models and experimental measurements via the Laser Doppler Velocimetry instrument.

Introduction Australian Nuclear Science and Technology Organisation (ANSTO) is Australia’s national nuclear research centre, and it possesses two reactors: the long-serving and recentlydecommissioned High-Flux-Australia-Research (HIFAR) Reactor and the newlycommissioned Open Pool Australian Light-water (OPAL) research reactor. The main functions of both reactors are to produce radiopharmaceutical products for medicine, to generate neutron beams for scientific research and to irradiate silicon ingots for semiconductor applications. This paper aims to examine the state-of-the-art application of Computational Fluid Dynamics to obtain sensible solutions to three separate pipe flow cooling problems of the molybdenum-99 irradiation facility, of which target cans containing uranium are irradiated and subsequently processed for the production of medical isotopes. The diversity and reliability of the CFD methodology will be demonstrated by the degree of detail modeled in each study as well as by the consistency of agreement among simulation results and experimental validation data and other numerical forms of verification comparisons. Three case designs are examined. They are: 1. The ‘rocket-can’ design as used in the HIFAR reactor 2. The proposed ‘annular can’ replacement design for HIFAR 3. The current ‘Molybdenum-plate’ design used in the new OPAL reactor In simple terms, the study of reactor thermo-hydraulic characteristics is the evaluation of a reactor coolant system’s heat removal capacity. To this regard, it is an evaluation of flow through what is often a complex system of pipe-work containing various reactor components acting as blockages. Such complex systems are difficult to evaluate for pressure drop and flow velocity characteristics using standard correlation-based formula describing pipe flow. To complicate matters, the heat transfer characteristics of components are closely dependent on the unique flow characteristics of each specific pipe-flow system. In the past, such unique heat transfer characteristics had been evaluated by the use of scaled and prototype heattransfer rigs, but this has become increasingly and prohibitively expensive. With the advent of computers, many factors have made the direct simulation of pipe flows, by means of CFD, feasible. The exponential increase in computational power, and ever-increasing improvements in numerical methods and graphical post processing, met with a similar decrease in computational cost has made CFD a viable method to directly solve for such distinctive pipe flow systems.

Pipe Flow Analysis of Uranium Nuclear Heating with Conjugate Heat Transfer

271

Background Fluid dynamics deals with both heat transfer and fluid flow where the fluid paths can be complicated and governing equations are non-linear. For most real-life problems associated with fluid dynamics, obtaining an analytical solution is difficult, if not impossible. Experiments that simulate fluid flow using scaled models in air tunnels, water tunnels or towing tanks are used to obtain accurate information, but these experiments can be difficult and expensive to set up. CFD takes the advantage of modern computing power in the application of numerical methods to solve complex fluid dynamic problems. Retrospectively, CFD emphasizes the resolution of the physical processes through the use of digital computers which proceeds by first negotiating the sub-division of the domain into a number of finite, non-overlapping subdomains. This leads to the construction of an overlay mesh of cells covering the whole domain. In general, the set of fundamental mathematical equations are required to be converted into suitable algebraic forms, which are then solved via suitable numerical techniques. The mathematical equations governing the heat transfer and fluid flow of the pipe flow systems within HIFAR and OPAL are those of the conservation of mass, momentum and energy. Nevertheless, the majority of pipe flows in the reactor are turbulent in nature. Direct Numerical Simulation (DNS) is the most accurate approach to turbulence simulation, which directly solves the governing transport equations without undertaking any averaging or approximation other than the numerical approximations performed on them. Through such simulations, all of the fluid motions contained in the flow are considered to be resolved; all significant turbulent structures are required to be adequately captured (i.e., the domain of which the computation is carried out needs to accommodate for the smallest and largest turbulent eddy). Alternatively, Large Eddy Simulation (LES), which essentially resolves the large eddies exactly through the availability of mesh requirement but approximates the small eddies, is still expensive but much less costly than DNS. The results of a DNS or LES simulation contain very detailed information about the flow, producing an accurate realization of the flow while encapsulating the broad range of length and time scales. DNS and LES approaches usually require high usage of computational resources and often cannot be used as a viable design tool in reactor design and analysis because of the enormity of the numerical calculations and the large number of grid nodal points. Meanwhile, a more pragmatic approach is to adopt computational procedures that can still supply adequate information about the turbulent processes but avoid the need to predict all of the effects associated with each and every eddy in the flow. For most engineering purposes, especially in reactor design and analysis, information about the time-averaged properties of the flow (e.g., mean velocities, mean pressures, mean stresses, etc.) are sufficient to satisfy regulatory requirements. In this sense, all details concerning the state of the flow contained in the instantaneous fluctuations can be discarded. This process of only obtaining mean quantities can be achieved by adopting a suitable time-averaging operation on the equations governing the conservation of mass, momentum and energy; these timeaveraged equations are generally known as the Reynolds-Averaged Navier-Stokes (RANS) equations.

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G.H. Yeoh and M.K.M. Ho

Turbulence Modeling The Reynolds-averaged approach to turbulence results in the formulation of the twoequation turbulence model—the standard k-ε model proposed by Launder and Spalding (1974). This model is well established, widely validated and gives rather sensible solutions to most industrially relevant flows. As an alternative to the standard k-ε model, other eddy viscosity models such as RNG k-ε model and reliazable k-ε model proposed by Yakhot et al. (1992) and Shih et al. (1995) are possible recommendations. The improved features of these models have been shown to be aptly applicable to predict important flow cases having flow separation, flow re-attachment and flow recovery. For wall attached boundary layers, turbulent fluctuations are suppressed adjacent to the wall and the viscous effects become prominent in this region known as the viscous sub-layer. The modified turbulent structure of near-wall flow generally precludes the application of the two-equation models such as standard k-ε model, RNG k-ε model and reliazable k-ε model at the near-wall region. One common approach is to adopt the so-called wall-function method; the near-wall region is bridged with logarithmic wall functions to avoid resolving the viscous sub-layer. Nevertheless, it is possible to totally resolve the viscous sub-layer by the application of the model standard k-ω model developed by Wilcox (1998) where ω is a frequency of the large eddies of which the model has also shown to perform splendidly close to walls in boundary layer flows. The standard k-ω model is nevertheless very sensitive to the free-stream conditions and unless great care is exercised, spurious results are obtained in flow regions away from the solid walls. To overcome such problems, the SST (Shear Stress Transport) variation of Menter’s model (1993, 1996) was developed with the aim of combining the favorable features of the standard k-ε model with the standard k-ω model in order that the inner region of the boundary layer is adequately resolved by the latter while the former is employed to obtain numerical solutions in the outer part of the boundary layer. This model works exceptionally well in handling non-equilibrium boundary layer regions such as flow separation.

Grid Generation The arrangement of discrete number of points throughout the flow field, normally called a mesh, is a significant consideration in CFD. For the Molybdenum-99 irradiation facility, grid generation poses the most challenging task because of its inherent intricate geometrical details. In the three case designs considered, the construction of suitable meshes accounts for almost the entire reactor design and analysis. Owing to the complexity of geometry, application of structured and/or unstructured meshes is required. By definition, a structured mesh is a mesh containing cells having either a regular-shape element with four-nodal corner points in two dimensions or a hexahedral-shape element with eight-nodal corner points in three dimensions. Commonly applied in numerous CFD investigations, it basically deals with the straightforward prescription of either an orthogonal mesh or a body-fitted mesh. An unstructured mesh can however be described as a mesh


273

overlaying with cells in the form of either a triangle-shape element in two dimensions or a tetrahedron-shape in three dimensions. For a body-fitted mesh, the mesh construction of the internal region of the physical domain can normally be achieved via two approaches. On one hand, the Cartesian coordinates may be algebraically determined through interpolation from the boundary values. This methodology requires no iterative procedure and it is computationally inexpensive. On the other hand, a system of partial differential equations of the respective Cartesian coordinates may be solved numerically with the set of boundary values as boundary conditions in order to yield a highly smooth mesh in the physical domain. The former is commonly known as the transfinite interpolation method and the latter is typically the elliptic grid generation method (Smith, 1982 and Thompson, 1982). For an unstructured mesh, triangle and tetrahedral meshing are by far the most common forms of unstructured grid generation. In Delaunay meshing, this most commonly adopted grid generation procedure entails the initial set of boundary nodes of the geometry to be triangulated according to the Delaunay triangulation criterion. Here, the most important property of a Delaunay triangulation is that it has the empty circumcircle (circumscribing circle) property (Shewchuk, 2002). All algorithms for computing Delaunay triangulations rely on the fast operations for detecting when a grid point is within a triangle's circumcircle and an efficient data structure for storing triangles and edges. The most straightforward way of computing the Delaunay triangulation is to repeatedly add one vertex at a time, then retriangulating the affected parts thereafter. When a vertex is added, a search is done for all triangles’ circumcircles containing the vertex. Then, those triangles are removed whose circumcircles contain the newly inserted point. All new triangulation is then formed by joining the new point to all boundary vertices of the cavity created by the previous removal of intersected triangles. Delaunay triangulation techniques based on point insertion extend naturally to three dimensions by considering the circumsphere (circumscribing sphere) associated with a tetrahedron. More details on Delaunay triangulation and meshing can be referred in Mavriplis (1997). Besides Delaunay method, other meshing algorithms in unstructured grid generation include the advancing front method (Lo, 1985; Gumbert et al., 1989; Marcum and Weatherill, 1995) and quadtree/octree method (Yerry and Shepard, 1984; Shepard and Georges, 1991). It should be noted that the use of hybrid grids that combine different element types such as triangular and quadrilateral in two dimensions or tetrahedral, hexahedral, prisms and pyramids in three dimensions can provide the maximum flexibility in matching mesh cells with the boundary surfaces or internal solid regions where heat transfer due to conduction needs to be resolved, and allocating cells of various element types in other parts of the complex flow regions. Grid quality can be enhanced through the placement of quadrilateral or hexahedral elements in resolving boundary layers near solid walls or composite materials in solid regions within the pipe flow systems whilst triangular or tetrahedral elements are generated for the rest of the flow domain. This generally leads to both accurate solutions and better convergence for the numerical solution methods.

274


Outline of CFD Model Governing Equations The unsteady RANS equations for the CFD model that reflect the conservation of mass, momentum and energy can be written as: Mass

∂ρ ∂ ( ρ u ) ∂ ( ρv ) ∂ ( ρ w ) + + + =0 ∂t ∂x ∂y ∂z

(1)

x-Momentum

∂ ( ρ u ) ∂ ( ρ u u ) ∂ ( ρv u ) ∂ ( ρ wu ) + + + = ∂t ∂x ∂y ∂z ∂ ⎡ ∂u ⎤ ∂ ⎡ ∂u ⎤ ∂ ⎡ ∂u ⎤ ∂τ xx′ ∂τ xy′ ∂τ xz′ + + + + + + Su μ μ μ ∂x ⎢⎣ ∂x ⎥⎦ ∂y ⎢⎣ ∂y ⎥⎦ ∂z ⎢⎣ ∂z ⎥⎦ ∂x ∂y ∂z

(2)

y-Momentum

∂ ( ρv ) ∂ ( ρ u v ) ∂ ( ρv u ) ∂ ( ρ wv ) + + + = ∂t ∂x ∂y ∂z ∂ ⎡ ∂v ⎤ ∂ ⎡ ∂v ⎤ ∂ ⎡ ∂v ⎤ ∂τ xy′ ∂τ ′yy ∂τ ′yz + + + + + + Sv μ μ μ ∂x ⎢⎣ ∂x ⎥⎦ ∂y ⎢⎣ ∂y ⎥⎦ ∂z ⎢⎣ ∂z ⎥⎦ ∂x ∂y ∂z

(3)

z-Momentum

∂ ( ρ w ) ∂ ( ρ u w ) ∂ ( ρv w ) ∂ ( ρ ww ) + + + = ∂t ∂x ∂y ∂z ∂ ⎡ ∂w ⎤ ∂ ⎡ ∂w ⎤ ∂ ⎡ ∂w ⎤ ∂τ xz′ ∂τ ′yz ∂τ zz′ + + + + + + Sw μ μ μ ∂x ⎣⎢ ∂x ⎥⎦ ∂y ⎢⎣ ∂y ⎥⎦ ∂z ⎣⎢ ∂z ⎦⎥ ∂x ∂y ∂z

(4)

Energy

∂(ρH )

∂ ( ρu H )

∂ ( ρv H )

∂ ( ρ wH )

∂p +Φ+ ∂t ∂x ∂y ∂z ∂t ∂ ⎡ ∂T ⎤ ∂ ⎡ ∂T ⎤ ∂ ⎡ ∂T ⎤ ∂q′x ∂q′y ∂q′z λ λ λ + + + + + x ⎥⎦ ∂y ⎢⎣ z ⎥⎦ ∂z ⎢⎣ z ⎥⎦ ∂x ∂y ∂x ⎢⎣ ∂ ∂ ∂ ∂z qx

+

+

qy

+

qz

=

(5)


275

From the above equations, ( ) indicates time-averaged quantities, t is the time, ρ is the density, u, v and w are the respective velocity components along the Cartesian coordinate directions x, y and z, H is the enthalpy, T is the temperature, μ is the fluid dynamic viscosity and λ is the fluid thermal conductivity. The time-averaged source or sink terms Su , Sv and

S w are given by ∂p′ ∂ ⎡ ∂u ⎤ ∂ ⎡ ∂v ⎤ ∂ ⎡ ∂w ⎤ μ μ μ + + + ∂x ∂x ⎢⎣ ∂x ⎥⎦ ∂y ⎢⎣ ∂x ⎥⎦ ∂z ⎢⎣ ∂x ⎥⎦

(6)

∂p′ ∂ ⎡ ∂u ⎤ ∂ ⎡ ∂v ⎤ ∂ ⎡ ∂w ⎤ + μ + μ + μ − ( ρ − ρ ref ) g ∂y ∂x ⎢⎣ ∂y ⎥⎦ ∂y ⎢⎣ ∂y ⎥⎦ ∂z ⎢⎣ ∂y ⎥⎦

(7)

∂p′ ∂ ⎡ ∂u ⎤ ∂ ⎡ ∂v ⎤ ∂ ⎡ ∂w ⎤ μ μ μ + + + ∂z ∂x ⎢⎣ ∂z ⎥⎦ ∂y ⎢⎣ ∂z ⎥⎦ ∂z ⎢⎣ ∂z ⎥⎦

(8)

Su = −

Sv = −

Sw = −

From above, p′ is the modified averaged pressure defined by p′ = p + 2 3

2 3

ρk +

μ∇ ⋅ V − ρ ref gi xi where p is the dynamics pressure, k is the turbulent kinetic energy,

V ≡ ( u ,v , w ) is the velocity vector, ρ ref is the reference density, gi are the gravitational acceleration components ( 0, − g , 0 ) and xi are the coordinates relative the Cartesian datum

( x, y, z ) and the effects due to the viscous stresses in the energy equation (5) are described by the averaged dissipation function Φ . For weakly compressible flows, it is common practice to transform the energy equation by replacing the heat flux according to the local enthalpy gradient instead of the temperature gradient, viz.,

qx =

μ ∂H Pr ∂x

qy =

μ ∂H Pr ∂y

qz = −

μ H Pr ∂z

(9)

where Pr is the Prandtl number. Also, the pressure work term ∂p ∂t and the averaged dissipation function Φ that represents the source of energy due to work done deforming the fluid element are usually ignored in most practical applications. On the basis of the eddy viscosity model, the Reynolds stresses defined by the stress vector τ′ ≡ τ ij′ where i, j = x, y, z in the momentum equations (2)–(4) is given by

(

− τ′ ≡ −τ ij′ = μt ∇V + ( ∇V )

T

) − 23 μ ∇ ⋅ Vδ − 23 ρ kδ t

(10)

276 where


μt is the turbulent eddy viscosity. The Reynolds flux in the energy equation (5)

defined by qi′ may also be modelled analogously to the eddy viscosity hypothesis as

qi =

μt ∂H

(11)

Prt ∂xi

where Prt is the turbulent Prandtl number. To satisfy dimensional requirements, at least two scaling parameters are required to relate the Reynolds stress to the rate of deformation. In most engineering flow problems, the complexity of turbulence precludes the use of any simple formulae. A feasible choice is the turbulent kinetic energy k and another turbulent quantity which is the rate of dissipation of turbulent energy ε . The local turbulent viscosity

μt can be obtained either from dimensional analysis or from analogy to the laminar viscosity as

μt ∝ ρ vt l . On the latter definition, based on the characteristic velocity vt defined as

and the characteristic length l as k

3/ 2

k

ε , the turbulent viscosity μt can thus be ascertained

according to

μt = C μ ρ

k2

(12)

ε

where Cμ is an empirical constant. In order to evaluate the turbulent viscosity in equation (12), the values of k and ε must be known which are generally obtained through solution of their respective transport equations. After a fair amount of algebra, the final forms of equations for the standard k-ε model developed by Launder and Spalding (1974) for the turbulent kinetic energy k and dissipation of turbulent energy ε can be written as:

∂ (ρk ) ∂t

+

∂ ( ρu k ) ∂x

+

∂ ( ρv k ) ∂y

+

∂ ( ρ wk ) ∂z

=

∂ ⎡ μt ∂k ⎤ ∂ ⎡ μt ∂k ⎤ ∂ ⎡ μt ∂w ⎤ ⎢ ⎥+ ⎢ ⎥+ ⎢ ⎥ + P + G − ρε ∂x ⎣ σ k ∂x ⎦ ∂y ⎣ σ k ∂y ⎦ ∂z ⎣ σ k ∂z ⎦ ∂ ( ρε ) ∂t

+

∂ ( ρu ε ) ∂x

+

∂ ( ρv ε ) ∂y

+

∂ ( ρ wε ) ∂z

=

∂ ⎡ μt ∂ε ⎤ ∂ ⎡ μt ∂ε ⎤ ∂ ⎡ μt ∂ε ⎤ ε ⎢ ⎥+ ⎢ ⎥+ ⎢ ⎥ + ( Cε 1 P + C3 G − Cε 2 ρε ) ∂x ⎣ σ ε ∂x ⎦ ∂y ⎣ σ ε ∂y ⎦ ∂z ⎣ σ ε ∂z ⎦ k where P is the shear production defined by

(13)

(14)


(

P = μ t ∇ V ⋅ ∇V + ( ∇ V )

T

) − 23 ∇ ⋅ V ( ρ k + μ ∇ ⋅ V )

277

(15)

t

and G is the production due to the gravity, which is valid for weakly compressible flows can be written as

μt g ⋅∇ρ ρσ ρ

(16)

where g ≡ ( 0, − g , 0 ) is the gravity vector and C3 and

σ ρ are normally assigned values of

G=−

unity and G in equation (16) is the imposed condition whereby it always remains positive, i.e. max ( G , 0 ) The constants for the standard k-ε model have been arrived through comprehensive data fitting for a wide range of turbulent flows (see Launder and Spalding, 1974): Cμ = 0.09, σk = 1.0, σε = 1.3, Cε 1 = 1.44 and Cε 2 = 1.92. Note that the effect of buoyancy is included in equations (14) and (15) to account for regions in the fluid with very low flow velocities. For the solid regions comprising of various materials, the heat-conduction equation can be expressed as

ρ s C ps

∂Ts ∂ ⎡ ∂Ts ⎤ ∂ ⎡ ∂Ts ⎤ ∂ ⎡ ∂Ts ⎤ λs λs λs = + + +S ∂t ∂x ⎢⎣ ∂x ⎥⎦ ∂y ⎢⎣ ∂y ⎥⎦ ∂z ⎢⎣ ∂z ⎥⎦

(17)

where Ts is the solid phase temperature and S is the internal volumetric heat source accounting for the total heat generated due to the irradiation process. In equation (17), ρ s , C ps and λs denote the thermophysical properties of the solid materials.

Boundary Conditions Based on the mass flow rate, the normal velocity can be calculated and prescribed at the inlet boundary. The temperature of the fluid is also prescribed at this boundary. At the outlet, the static pressure is normally imposed according to fully-developed flow condition. For all variables, the normal derivative at the outlet boundary is equivalent to zero. One possible approach of overcoming the difficulty of modeling the near-wall region of a solid wall is through the prescription of logarithmic wall functions for the standard k-ε model. In order to construct these functions, the region close to the wall can usually be characterized by considering the dimensionless velocity U

+

+

and wall distance y with respect to the local

conditions at the wall. The dimensionless wall distance y

+

is defined as

where very near the wall, y = d, while the dimensionless velocity U

+

ρ uτ ( d − y ) μ

can be expressed in the

278


form as U uτ where U is taken to represent some averaged velocity representing either of the mixture velocity parallel to the wall, uτ is the wall friction phase velocity which is defined with respect to the wall shear stress τ w as

τw ρ .

+

For wall distance of y < 5, the boundary layer is predominantly governed by viscous forces that produce the no-slip condition; this region is subsequently referred to as the viscous sub-layer. By assuming that the shear stress is approximately constant and equivalent to the wall shear stress τ w , a linear relationship between the averaged velocity and the distance from the wall can be obtained yielding

U + = y + for y + < y0+

(18)

+

With increasing wall distance y , turbulent diffusion effects dominate outside the viscous sub-layer. A logarithmic relationship is employed:

U+ =

1

κ

ln ( E y + ) for y + > y0+

(19)

The above relationship is often called the log-law and the layer where the wall distance +

y lies between the range of 30 < y + < 500 is known as the log-law layer. Values of κ (~0.4) and E (~9.8) in equation (19) are universal constants valid for all turbulent flows past +

smooth walls at high Reynolds numbers. The cross-over point y0 can be ascertained by computing the intersection between the viscous sub-layer and the logarithmic region based on the upper root of

y0+ =

1

κ

ln ( E y0+ )

(20)

A similar universal, non-dimensional function can also be constructed to the heat transfer. The enthalpy in the wall layer is assumed to be:

H + = Pry + for y + < yH+ H+ =

PrT

κ

ln ( FH y + ) for y + > yH+

where FH is determined by using the empirical formula of Jayatilleke (1969):

21)

(22)


⎡⎛ Pr ⎞0.75 ⎤ ⎡ ⎛ Pr ⎪⎧ FH = E exp ⎨9.0κ ⎢⎜ ⎟ − 1⎥ ⎢1 + 0.28exp ⎜ −0.007 Prt ⎢⎣⎝ Prt ⎠ ⎥⎦ ⎣ ⎝ ⎪⎩ By definition, the dimensionless enthalpy H

H

+

(H =

w

+

279

⎞ ⎤ ⎪⎫ ⎟⎥ ⎬ ⎠ ⎦ ⎪⎭

(23)

is given by:

− H ) ρ Cμ0.25 k 0.5

(24)

JH

where H w is the value of enthalpy at the wall and the diffusion flux JH is equivalent to the

(

normal gradient of the enthalpy ∂H ∂n

)

wall

perpendicular to the wall. The thickness of the

thermal conduction layer is usually different from the thickness of the viscous sub-layer, and +

changes from fluid to fluid. As demonstrated in equation (20), the cross-over point yH can also be similarly computed through the intersection between the thermal conduction layer and the logarithmic region based on the upper root of

Pr yH+ = PrT

1

κ

ln ( FH yH+ )

(25)

For the rest of the boundary conditions at a solid wall, all wall temperatures are determined using the energy balance. Boundary conditions for the turbulent kinetic energy and the dissipation have its normal derivative at the wall equal to zero and obtained through the relation

ε=

Cμ3 4 k 3 2

κd

(26)

where d is the distance of the nearest grid point from the wall boundary.

Computational Procedure The algebraic forms of the governing equations to be solved by matrix solvers can be formed by integrating the system of equations using the finite volume method over small control volumes. By employing the general variable φ , the generic form of the governing equations can be written initially in the form as

∂ ( ρφ ) + ∇ ⋅ ( ρ Vφ ) = ∇ ⋅ ⎣⎡Γφ ∇φ ⎦⎤ + Sφ ∂t

(27)

280


In order to bring forth the common features, terms that are not shared between the equations are placed into the source term Sφ . By setting the transport variable φ equal to 1,

u , v , w , H , k and ε and selecting appropriate values for the diffusion coefficient Γφ and source term Sφ , special forms of each of the partial differential equations for the continuity, momentum and energy as well as for the turbulent scalars can thus be obtained. The cornerstone of the finite volume method is the control volume integration. In order to numerically solve the approximate forms of equation (27), it is convenient to consider its integral form of this generic transport equation over a finite control volume. Integration of the equation over a three-dimensional control volume ΔV yields:

∂ ( ρφ ) dV + ∫ ∇ ⋅ ( ρ Vφ ) dV = ∫ ∇ ⋅ ⎡⎣Γφ ∇φ ⎤⎦ dV + ∫ Sφ dV ∂t ΔV ΔV ΔV ΔV

∫

(28)

By applying the Gauss’ divergence theorem to the volume integral of the advection and diffusion terms, equation (28) can now be expressed in terms of the elemental dA as

∂ ( ρφ ) dV + ∫ ( ρ Vφ ) ⋅ n dA = ∫ ⎡⎣Γφ ∇φ ⎤⎦ ⋅ n dA + ∫ Sφ dV ∂t ΔV ΔA ΔA ΔV

∫

(29)

Equation (29) needs also to be further augmented with an integration over a finite time step Δt. By changing the order of integration in the time derivative terms, t +Δt ⎛ t +Δt ∂ ( ρφ ) ⎞ ⎛ ⎞ ∫ΔV ⎜⎝ ∫t ∂t dt ⎟⎠ dV + ∫t ⎜⎝ Δ∫A ( ρ Vφ ) ⋅ n dA ⎟⎠ dt = t +Δt t +Δt ⎛ ⎞ ∫t ⎜⎝ Δ∫A ⎡⎣Γφ ∇φ ⎤⎦ ⋅ n dA ⎟⎠ dt + ∫t Δ∫V Sφ dVdt

(30)

In essence, the finite volume method discretises the integral forms of the transport equations directly in the physical space. If the physical domain is considered to be subdivided into a number of finite contiguous control volumes, the resulting statements express the exact conservation of property φk from equation (21) for each of the control volumes. In a control volume, the bounding surface areas of the element are, in general, directly linked to the discretisation of the advection and diffusion terms. The discretised forms of these terms are:

∫ ( ρ Vφ ) ⋅ n dA ≈ ∑ ( ρ V ⋅ nφ ) k

ΔA

f

f

ΔA f

(31)


∫ ⎡⎣Γφ ∇φ ⎤⎦ ⋅ n dA ≈ ∑ ⎡⎣Γφ ∇φ ⋅ n ⎤⎦ f

ΔA

f

ΔA f

281

(32)

where the summation in equations (31) and (32) is over the number of faces of the element and ΔAf is the area of the face of the control volume. The source term can be subsequently approximated by:

∫ Sφ dV ≈ Sφ ΔV

(33)

ΔV

For the time derivative term, the commonly adopted first order accurate approximation entails: t +Δt

∫ t

∂ ( ρφ ) ( ρφ ) − ( ρφ ) dt ≈ ∂t Δt n +1

n

(34)

where Δt is the incremental time step and the superscripts n and n + 1 denote the previous and current time levels respectively. Equation (21) can then be iteratively solved accordingly to the fully implicit procedure by

( ρφ )

n +1

− ( ρφ )

n

Δt

⎛ ⎞ + ⎜ ∑ ( ρ V ⋅ nφ ) f ΔAf ⎟ ⎝ f ⎠

⎛ ⎞ ⎜ ∑ ⎡⎣Γφ ∇φ ⋅ n ⎤⎦ f ΔAf ⎟ ⎝ f ⎠

n +1

= (35)

n +1

+ Sφn +1ΔV

Consider the particular control volume element in question of which point P is taken to represent the centriod of the control volume, which is connected with the respective centroids of other surrounding control volumes. Equation (35) can thus be expressed in terms of the transport quantities at point P and surrounding nodal points with a suitable prescription of normal vectors at each control volume face and dropping the superscript n +1 which by default denotes the current time level as:

aφ

n +1 P P

= ∑a φ

n +1 nb nb

( ρφ ) P ΔVP n

+S

n +1 off

+S

n +1 non

+S

n +1 u

ΔVP +

nb

Δt

(36)

where

aP = ∑ anb + S P ΔVP + ∑ F nb

( ρφ ) P

n +1

n +1 f

+

Δt

ΔVP

(37)

282

G.H. Yeoh and M.K.M. Ho n +1

and the added contribution due to non-orthogonality of the mesh is given by S non , which is required to be ascertained especially for body-fitted and unstructured meshes. Note that the

(

convective flux is given by Ffn +1 = ρ V ⋅ n φ

)

n +1 f

ΔAf For the sake of numerical treatment, the

source term for the control volume in equation (35) has been treated by

Sφn +1ΔV = ( Sun +1 − S PφPn +1 ) ΔVP In equation (36), aP is the diagonal matrix coefficient of

(38)

φPn +1 ,

∑F

n +1 f

are the mass

imbalances over all faces of the control volume and S P is the coefficient that is extracted from the treatment of the source term in order to further increase the diagonal dominance. The k coefficients of any neighboring nodes for any surrounding control volumes anb in equation (36) can be expressed by

anb = D nf +1 + max ( − Ffn +1 , 0 ) n +1

where D f

(39)

is the diffusive flux containing the diffusion coefficient Γφ along with the

geometrical quantities of the particular element within the mesh system. The treatment of the advection term which results in the form presented in equation (39) is known as upwind differencing to guarantee diagonal dominance. In order to reduce the effect of false diffusion caused by upwind differencing, the well-known deferred correction approach is adopted to n +1

treat the off-diagonal contributions Soff

in equation (36) due to higher resolution

differencing schemes. In this study, the coupled solution approach, a more robust alternative to the segregated approach, is adopted to solve the velocity and pressure equations simultaneously. Algebraic Multigrid accelerated Incomplete Lower Upper (ILU) factorization technique is employed to resolve each of the discrete system of linearized algebraic equations in the form of equation (36). The advantages of a coupled treatment over a segregated approach are: robustness, efficiency, generality and simplicity. Nevertheless, the principal drawback is the high storage requirements for all the non-zero matrix entries.

Results and Discussion For the analysis of reactor thermo-hydraulic safety, the central focus often befalls on a ‘bounding case study’ which demarks what can be loosely regarded as the operational envelop of the reactor cooling system. This bounding case study is a simulation of all extreme operational conditions that may occur simultaneously during the life of the reactor. The meaningful operands of these bounding operational conditions are represented by input parameters (such as coolant temperatures, mass flow, etc.) of the system which become the parametric specifications of the computer model.


283

However, before these inputs are specified, a computer model of the irradiation rig must first be generated; complete with flow passages, internal structures and irradiation target(s). Components of the model rig assembly must be built in accordance to engineering blueprints and where possible, minor geometric simplifications are introduced into the model so that simulations remained physically representative without having to waste computational resources or prolong solution times over unnecessarily detailed control volumes. After the model is generated, it is geometrically discretised by the generation of a ‘mesh’ which further segregates each control volumes into sub-volumes. The physical properties of each geometric feature – such as the aluminium in the rig structure are then specified in the corresponding volumes of the model. It is standard practice to conduct a grid-convergence test before the modelled geometry is deemed representative of the physical prototype. To perform this, the amount of control volumes is increased in all three axes until the increase in mesh resolution does not result in any appreciable difference in final key results, such as in the model’s maximum attainable temperature. Finally, the results are analysed and compared to the safety limit. This procedure for this safety analysis can be summarised in five steps: 1. 2. 3. 4.

Determining the bounding case scenario Modelling the geometry, the physics and physical properties Solving the CFD model by numerical approximation techniques Checking the validity of our solution by mesh sensitivity analysis as well as by comparisons with other simpler numerical models such as from one-dimensional simulations reactor nuclear thermo-hydraulic code 5. Analysing and comparing results with safety requirement

‘Rocket-Can’ Design as Used in the HIFAR Reactor The first pipe-cooling irradiation system to be examined is the ‘rocket-can’ design used in HIFAR. This study investigated the maximum temperature of 2.2% 235U enriched UO2 pellets during irradiation. The recently decommissioned HIFAR reactor, a 10 MW nuclear research reactor at the Australian Nuclear Science and Technology Organisation (ANSTO), produced a steady supply of technicium-99 (99Tc) radiopharmaceutical for domestic and international use. Technicium-99 is formed by the radioactive-decay of Molybdenum-99 (99Mo), which itself is a fission product of Uranium-235 (235U). The process of generating technicium-99 started with the loading of seventeen UO2 pellets into a thick aluminium tube shaped like a ‘rocket-can’ (Figure 2). Granulated magnesium oxide (MgO) was packed in with the pellets to assist heat conduction and to control pellet spacing. The rocket-cans were then sealed by welding on a cap and inserted into slots inside a long vertical ‘stringer’ assembly which looked much like a ‘cake-stand’. In turn, the stringer was inserted into a ‘liner’, a hollow tube with flow by-pass inlets at its conical tip. Finally the liner was inserted into the centre of four concentric annular fuel plates which formed the fuel assembly (Figure 1). The purpose of the liner was to separate high speed flow passing through the four outer concentric fuel plates from the slower flow going through the liner cooling the cans. All components: cans, stringer and liner were fabricated

284


using aluminium on account of its unique qualities of neutron transparency, anti-corrosion and good thermal conductivity.

SALVAGE COUPLING PHALANGE FOR ASSEMBLY TO SHIELD PLUG IDENTIFICATION NUMBER

EMERGENCY COOLING WEIR AND SPRAY RING INTERMEDIATE SECTION

THERMOCOUPLE TUBE

COOLANT FLOW

PERFORATED EXTENSION

THERMOCOUPLE TUBE

DOWEL

EMERGENCY COOLING WATER TUBE (NOT USED IN HIFAR)

THERMOCOUPLE COMB

VIEW OF ARROW A

FUEL TUBES

OUTLINE OF LINER LOWER COMB

GUIDE NOSE SPHERICAL SEAT

SKIRT COOLANT FLOW

Figure 1. Schematic drawing of the HIFAR fuel element.

When the HIFAR reactor was operating, 235U in fuel elements underwent fission which absorbed and generated neutrons in a self-sustaining process. The nuclear reaction produced large amounts of heat that was removed by upwardly flowing heavy water (deuterium oxide, D2O) through the fuel element. One and a half percent of coolant flow bypassed into the centre liner for rocket-can cooling. Also, the neutrons produced by the nuclear process in the fuel were absorbed by the UO2 pellets to produce molybdenum that had a short half-life of 2.7 days before beta-decaying into technicium-99. After irradiation, the molybdenum-99 was chemically separated and quickly packaged so that it arrived at clinics in its usable betadecayed form as technicium.


285

Figure 2. Schematic drawing of the inadiation rig and placement of target cans in the liner.

During irradiation, a vast amount of heat was generated in the uranium pellets which must be evacuated. For safety and licensing purposes, it was necessary to demonstrate that the pellet would not melt during irradiation. Thus, neutronic and thermal-hydraulic analyses using numerical methods were used to determine the pellet maximum temperature. This method directly solved the three conservation equations of mass, momentum and energy in the coolant flow domain which was coupled with the conduction physics of the solid domains of the can, magnesium oxide and uranium oxide pellets. The result of this work was critical to the licensing requirements of HIFAR as bounded by operation licence conditions (OLCs) and evaluated safety limits.

Grid Generation of Rocket-Can The complete geometry of can, stringer, liner, fuel plates and flow passages was generated simultaneously and their respective volumes patched as separate logical entities as shown in Figure 3. The simultaneous volume generation meant that separate component surfaces were logically connected and no surface patching was required between components.

286


On the outset, minor geometric details were simplified in regions of little flow-dynamic consequence in-order to accelerate the time to solution convergence. Some work was needed to organise the arrangement of each entity with respects to other parts but the effort expended was outweighed by the benefit of not needing to specify intra-component surface patching. Note in Figure 3 that hexahedral body-fitted volumes were used in the geometry to properly model the thin conduction regions of the liner and fuel plates. The thin solid region of the liner and stringer could only be practically constructed by using a structured mesh because of the high thermal flux across the thin stretch of aluminium. Numerical diffusion over such a thin area of high thermal flux given if we were using unstructured mesh would have been exceedingly pronounced. The additional benefit of using hexahedral control volumes was in their superiority over tetrahedral elements for the modelling of near-wall flows where the correlation-model of wall flow-profiles work best with body-fitted rectangular mesh. This was also represented an economy in the number of rectangular mesh required as compared to tetrahedral mesh to attain the same level of solution accuracy. The pellet-stack geometry consisted of a 119.5 mm vertical cylinder of MgO with a radius of 5 mm. The pellets modelled inside the MgO stack was 3.5mm high with a radius 4.5mm as shown in Figure 4a. In Figure 5a, the rocket-can and stringer are shown simultaneously. Notice the relief windows at the top of the stringer which allowed the coolant flow to pass through as was identical to the real physical prototype. Finally, five volumes of the same model as shown in Figure 3b were stacked on each other to produce the complete stringer assembly such as shown in Figure 9.

(a)

(b)

Figure 3. The modelled ‘block’ of multiple components is shown with the (a) rocket-can and (b) slotted stringer highlighted.


(a)

287

(b)

Figure 4. The modelled stack of (a) UO2 pellets in MgO was built to fit inside (b) the rocket-can’s internal cavity.

(a)

(b)

Figure 5. The model of (a) rocket-can inserted inside stringer and (b) multiple stringers were assembled to produce a complete stringer assembly.

288


Specification of Material Properties The total D2O mass flow through the rig was 16 kg/s with 98.5% passing through the fuel and 1.5% passing through the liner. The material properties of aluminium and heavy water are summarised in Table 1 and the variable conductivities of magnesium oxide and uranium dioxide are shown in Figure 6. Table 1. Material Properties of Aluminium and Heavy Water MATERIAL PROPERTIES -3

Density

kg.m

Specific Heat Capacity

2702

1094.92

-1

903

4.12849

-1

-1

273

0.614259

J.kg .K

W.m .K

Dynamic Viscosity μ

Pa.s

0.000712

Uranium Oxide conductivity vs. Temp.

Magnesium Oxide conductivity vs. Temp. 2.0

Thermal Conductivity [W/(m.K)]

Thermal Conductivity [W/(m.K)]

D2O

-1

Thermal Conductivity

1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 300

Al

8 7 6 5 4 3 2 200

3000

418

1800

3000

Tem p. [K]

Tem p. [K]

(a)

(b)

Touloukian 1970.

Figure 6. Temperature dependent thermal conductivities of (a) magnesium oxide and (b) uranium dioxide.

Computational Geometry The domains of the simulation were specified as follows (the default solid-domain material was aluminium): • • • • • • •

‘Fuel1’, ‘Fuel2’, ‘Fuel3’, ‘Fuel4’– Solid Domain ‘Liner’ – Solid Domain ‘Stringer’ – Solid Domain ‘Can’ – Solid Domain ‘MgO’ – Solid Domain, material: Magnesium Oxide ‘UO2’ – Solid Domain, material: Uranium Oxide ‘Fluid’ – Fluid Domain, fluid: heavy water


289

In summary the CFD problem can be simplified as a list of input and outputs: Inputs • Geometry • Material Properties • Heat Source • Boundary Conditions

Outputs • Temperature Profile T (x,y,z) • Wall Heat Flux q (x,y,z)

UO2 pellets producde heat at a steady rate. The heat was conducted through the MgO into the aluminium rocket can then into the flowing heavy water D2O. The solutions of interest given by CFD included: the temperature profile T (x,y,z) of the whole region and wall heat flux q (x,y,z) over the surface of the rocket can. The subdomain ‘Pellets’ was defined for the entire UO2 domain as an energy source with a volumetric power of 8.4556 x 108 W.m-3. The fuel plates, the structural material (aluminium) and the coolant (heavy water) would also produce heat but the heating from fuel plates was not modeled because it was known that they were cooled efficiently by the faster flowing coolant outside the liner and heating from other aluminium components was negligible. For the boundary conditions, there were two inlets, one at ‘INLET FUEL’ with Normal Speed 2.58816 m.s-1, another at ‘INLET LINER’ with Normal Speed 0.0717642m.s-1 (these were calculated from coolant mass flow value, inlet areas and bypass ratios), both with a static temperature of 318K (45°C) in the 16-pellet simulation. In these simulations, there were two separate flow regions.

Power Density Calculation The power emitted from 2.2% enriched UO2 pellets were calculated by using the Monte Carlo N-Particle (MCNP) transport code for simulating neutronic reactions. For this safety analysis a conservative reactor power of 11MW was assumed. The 16 pellets of a can were modeled discretely (Dia. 9mm × 3.5mm high). Total power per m-3 for 16 pellets = 3204.7 W.m-3 / 3.79 × 10-6 m3 (Volume of 16 Pellets) = 8.4556 × 108 W.m-3. This power density signified the source term for the conduction equation.

Turbulence Solver and Convergence Criteria To account for the turbulent pipe flow, the standard k-ε model was used. Computational predictions were deemed to be converged when the normalized residual mass was less than 1 x 10-4.

Computational Predictions Simulation for the heating of sixteen pellets in a rocket can had been previously modeled by Yeoh and Storr (2000) which was interested in the rocket-can’s maximum surface heat

290


flux to determine the margin for Onset of Nucleate Boiling (ONB). The CFD model of Yeoh & Storr’s simulation had a lesser degree of resolution and modeled the stack of pellets as a single entity instead of discrete pellets. This was partly attributed to the more limited computational resources available at the time. Yeoh & Storr’s CFD result’s were validated to a degree by simple irradiation experiments where metal samples were introduced inside the pellet stack and subsequently examined for evidence of melting in order to estimate maximum temperatures. In this investigation calculated by CFX-10, we were primarily interested in the maximum temperature of the UO2 pellets. Temperature contours in Figure 7a indicate a maximum pellet temperature of 2656°C which remained below the pellet melting temperature of 2847°C. For clarity, the four concentric annular fuel plates, liner, stringer and can in Figure 7 are shown in silhouette. This result was consistent with the maximum temperature of Yeoh & Storr’s large-volume pellet stack at 2277°C. The temperature was understandably lower in Yeoh & Storr’s simulation because the same power was distributed over a larger volumetric space.

(a)

(b)

Figure 7. Section view of temperature contours showing maximum temperatures of (a) UO2 pellets: 2656°C and (b) MgO: 2439°C.

Other maximum temperatures of interest include: the magnesium oxide at 2439°C (mp. 2800°C) in Figure 7b; the aluminium can at 307°C (mp. 660°C) in Figure 8a and the heavy water D2O at 79°C (bp. 120°C at 2 atm) in Figure 8b. The temperatures were highest in the middle of the pellets, decreasing steeply across the width of the pellet and still more steeply across the MgO powder, reflecting its low thermal conductivity. This was in contrast with the low thermal gradient across the aluminium rocket can, due to the high thermal conductivity of aluminium. Figures 9 and 10 show the wall heat flux at the surface of rocket cans, the higher fluxes were colored blue, due to outward flux being defined as negative. There were four high heat flux regions corresponding to the slot opening. Opposite the slot opening in Figure 10, forced convection cooling was restricted by the stringer enclosure and as a consequence caused a lower heat-flux as indicated by the green region of the can surface. The sudden expansion of the open slot at the side of the stringer created a highly turbulent region which assists in


291

forced convection cooling and this resulted in the highest thermal heat flux achieved as indicated in Figure 10 by the dark blue region of the can.

(a)

(b)

Figure 8. Section view of temperature contours showing maximum temperatures of (a) rocket can: 307°C and (b) D2O: 79°C.

Figure 9. Heat-flux contour at the surface of the rocket cans. The view is through the opening in the stringer. The silhouette of the stringer is made out in transparent purple.

Inside the liner, the mean flow accelerated when squeezed into tighter paths between the can and the stringer, as indicated by yellow and orange vectors in Figure 11 and decelerated when entering an expansion, as shown by blue vectors near the tip of the rocket can. The deceleration was basically due to reasons of mass conservation as explained by the Bernoulli equation. The length of an arrow in Figure 11 is proportional to the flow velocity at that point. The long red arrows belonged to the faster flow outside the liner, with velocities of up to 4m.s-1.

292


Figure 10. Heat-flux contour at the surface of the rocket cans. The view is through the enclosed back of the stringer.

Figure 11. Flow velocity vector plot of coolant channels inside outside the liner.

Proposed ‘Annular Can’ Replacement Design for HIFAR Flow visualization and LDV measurements were performed to better understand the fluid flow around the narrow spaces within the X216 irradiation rig, prototypes of annular target cans and liner. A three-dimensional computational fluid dynamics model was used to


293

investigate the hydraulics behavior within a HIFAR fuel element liner model. An interesting feature of the computational model was the use of unstructured meshing, which consisted of triangular elements and tetrahedrons within the flow space (Figure 13), to model the “scaledup” experimental model. This present investigation focused on the evaluation of the CFD model in its capability to predict the complex flow structures inside the liner containing the mock-up X216 rig with two targets. The reliability of the model was validated against experimental observations and measurements.

Description of the Water Tunnel Experimental Apparatus and Methods Figure 12 describes the transparent model of the prototype design of the rig and annular target cans that were placed in the water tunnel facility. In this test section, the liner was sealed so that the flow path was only through the liner inlet holes. An orifice plate was used to measure the flow rates within the facility. Flow visualization of the fluid flow was performed using an Argon Ion laser light sheet, high-resolution digital camera and standard video equipment. The flow was seeded with Iriodin powder concentrations of 0.5 to 1 gram per 3000 liters of water. Particles in the flow were illuminated while they were in the field of the laser light sheet. With the digital images and video footage the flow field was clearly visible.

Figure 12. Points of velocity measurements in X216 rig.

294


Figure 13. Unstructured mesh of the X-216 annular target.

Velocities were measured using a two-dimensional (2D) Dantec LDV system, operated in 1D mode measuring axial velocity components. Data were stored on the computer attached to the LDV hardware. Uncertainty in measurement using the LDV equipment was determined to be a maximum of ± 3.5% for a particular optics configuration but generally less than ±1.8%. Points of velocity measurement are shown in Figure 12. The points measured in the plane shown had been taken with the laser entering at the open side of the mock-up rig and the forward scatter detector viewing through the closed side of the rig. Measurements were also taken using the LDV probe in backscatter mode, and were verified at a number of points by using the forward scatter mode. Backscatter mode was used only for measurements in the grid pattern between the heights of 185 mm and 360 mm, since at the other locations the beams were sufficiently attenuated due to the additional influences of the acrylic interfaces, giving very low data rates.

Computational Details A three-dimensional CFD program ANSYS-CFX5.6 has been employed to simulate the complex thermal-hydraulics behavior in the space within a HIFAR fuel element liner model in the water tunnel. The CFD code solved the conservation equations of mass, momentum and energy. Turbulence of the fluid flow was accounted via a standard k-ε model. Buoyancy


295

effect was included for regions where low velocities were present. This effect was included within the source terms of the momentum equations as well as in the turbulence models. The unstructured meshing was adopted for the construction of the computational model of the mock-up because of the inherent intricate details of the irradiation rig and the placements of the annular targets within the rig. The mesh was prepared in two stages. A surface mesh of triangular mesh elements was initially generated on all the surfaces covering the model components. The volume mesh consisted of tetrahedral elements was then subsequently generated within the fluid flow domain from the surface mesh elements. Figure 13 shows the mesh layout of triangular surface elements around the pin and support tube nose cone of the irradiation rig. For the entire geometrical structure that included the assembly of two annular target cans placed within the prototype rig in the liner and mock-up fuel element, a volume mesh of 605158 tetrahedrons and a surface mesh of 42030 triangular elements were allocated. The governing equations were solved by matrix solution techniques formulated by integrating the system of equations using the finite volume method over small elemental volumes. For each elemental volume, relevant quantity (mass, momentum and turbulence) was conserved in a discrete sense for each control volume. Here, a coupled solver, which solved the hydrodynamic equations (for velocities and pressure) as a single system, was employed. It has been found in the segregated approach that the strategy to first solve the momentum equations using a guessed pressure and an equation for a pressure correction resulted in a large number of iterations to achieve reasonable convergence. By adopting the coupled solver, it has been established that such a coupled treatment significantly outweighed the segregated approach in terms of robustness, efficiency, generality and simplicity. To accelerate convergence for each of the discretised algebraic equations, the Algebraic Multigrid solver was adopted.

Validation Against Water Tunnel Observations and Measurements CFD simulation of the fluid flow through the various components of the fuel element model that included the mock-up rig and annular target cans was performed. Figure 14 illustrates the computed flow distribution inside and outside of the liner nose cone. Based on the experimental flow rate of 1.6965 kgs-1 and a base diameter of 0.3 m of the mock-up fuel element model, there observed a very low flow velocity outside of the liner nose cone in Figure 14(b). Nevertheless, the fluid after being squeezed through the small size bottom and side holes of the liner nose cone caused these interacting merging flows to yield a very highly complicated flow structure consisting of multiple vortices of recirculating flows (see Figure 14(a)). It was also evident that due to the significant acceleration of the flow found near the liner holes, the velocities increased dramatically to a magnitude of 5.0 ms-1 and resulted in large pressure drops. Near the bottom hole of the liner nose cone, the CFD model predicted a normal velocity of approximately of 3 ms-1. This predicted value has been found to be in good agreement with experimental LDV measurement of 2.6 ms-1, which provided confidence to the reliability of the models in the CFD computer program. As the fluid moved vertically upwards, the flow gradually became more uniform.

296


(a)

(b)

Figure 14. (a) Velocity vectors and (b) velocity contours around perforated liner.

(a)

(b)

Figure 15. Flow separation near the pin as examined by (a) CFD and (b) PIV capture.

Another important consideration for the rig and target specification was the incorporation of a pin situated at some distance below the placement of the rig as can be seen in Figure 15. This design feature was implemented in the liner because of safety concerns in the event of a possible accident scenario of the rig falling through to the bottom and impacting on the liner nose cone. The selection of pin size was an important requirement for the rig and target specification. From the flow predictions in Figure 15(a), it could be ascertained that the pin size chosen contributed to only minor flow disturbances in the area between the pin and the bottom surface of the rig nose cone. It was also observed that the majority of the bulk fluid flow was unperturbed and diverged smoothly as it approached the rig. Flow visualization


297

performed during experiments (see Figure 15(b)) projected a similar flow pattern, which further confirmed the reliable predictions of the CFD model.

(a)

(b)

Figure 16. CFD study of flow distribution around annular can.

Figure 16 presents the flow distribution of the fluid traveling between the inner annular can wall and rig outer surface designated for the purpose of illustration as region 1 and the area between the outer annular can wall and the liner inner surface designated as region 2. The fluid flowing within these spaces was found to be rather uniform. These favorable flow structures indicated that axial cooling in the reactor rig along the length of the target could remove the heat effectively for the design where uranium foils are embedded in the sealed annular targets during the irradiation process. An interesting aspect of the model predictions through the velocity contours in Figure 16(b) showed succinctly more fluid moving vertically upwards in region 2 than in region 1. Based on the LDV flow measurements at the discrete locations in regions 1 and 2 in Figure 12, the experiments confirmed the CFD predictions of the different velocities in the two regions. Velocity values of 0.44 ms-1 and 0.502 ms-1 were measured during experiments for regions 1 and 2 respectively. The predicted velocities as depicted by the velocity contours in Figure 16(b) demonstrated the similar trend predicted through the CFD model where region 1 yielded lower velocities compared to the higher velocities in region 2. Figure 17 illustrates the LDV measurements for the unmodified and modified designs of the can flutes affecting the axial velocity distributions for two volumetric flow rates. Changes introduced to the flute designs through the removal of any sharp edges significantly altered the axial velocity distributions. The profiles were flatter thereby resulting in more uniform and lesser wake flow structures. In the same figure, comparison between predicted and measured vertical velocities is also presented. The predicted velocity profiles through the CFD model showed similar encouraging distributions with the measured profiles in the inner

298


and outer channels. Good qualitative agreement was achieved between the predicted and measured velocities. X216 Rig Channel Axial Velocities 0 degree plane 1.61 l/s Inner Channel 1.61 l/s Outer Channel 1.61 l/s IC;RMS 1.61 l/s OC;RMS 1.68 l/s IC 1.68 l/s OC 1.68 l/s IC;RMS 1.68 l/s OC;RMS 1.61 l/s mod IC 1.61 l/s mod OC 1.61 l/s mod IC;RMS 1.61 l/s mod OC;RMS 1.61 l/s IC CFD prediction 1.61 l/s OC CFD prediction

0.7

0.6

Fluid Velocity m/s

0.5

0.4

0.3

0.2

0.1

0 0

50

100

150

200

250

300

350

400

450

500

NOTES: 1) Run @ 1.61 l/s was performed before fin smoothing 2) Run @ 1.61 l/s mod was performed after fin smoothing

Axial Distance From Stem Tip (mm)

Figure 17. Inner and outer channel LDV velocity measurements along X216 Rig vertical axis.

Current ‘Molybdenum-Plate’ Design Used in the New OPAL Reactor The third pipe-cooling irradiation system to be examined is the ‘molybdenum-plate’ design used in the new Open Pool Australian Light-water (OPAL) research reactor. This study investigated the maximum temperature of 20% 235U enriched U-Al compound sealed in aluminium cladding plates. It will also demonstrate the incremental development of the model and the affect this has on the maximum temperature, as more physics and boundary conditions were introduced. The OPAL research reactor is an open-pool design constructed by INVAP, Argentina. The core rests thirteen meters under an open pool of light-water which provides both cooling and radiation protection. Surrounding the core is a Heavy Water Reflector Vessel that is physically isolated from the bulk light water coolant. Its primary purpose is to reflect neutrons back into the core to maintain the critical (nuclear) conditions necessary for steady fission reaction rates during normal reactor operation. The secondary purpose of the reflector vessel is to provide a relatively large volume with high flux for irradiation and neutron beam facilities. A compact reactor core measuring only 0.35 m x 0.35 m x 0.62 m (width, breadth, height) is located 13m below the reactor pool surface. The Reflector Vessel surrounding the Core contains heavy water and accommodates facilities such as irradiation rigs and the cold Neutron Source (see Figure 18). A number of irradiation positions penetrating the Reflector Vessel provide facilitates for the production of radioisotopes. These positions are sealed to


299

prevent the H2O and the D2O from mixing. Moreover, the irradiation positions contain builtin mechanisms for cooling which draws cold H2O from the main pool downwards into the irradiation targets which is then conveyed away through pipes.

Figure 18. OPAL reactor facilities (top view).

Molybdenum-Plate Life Cycle A primary aspect of the OPAL reactor is the production of Technetium-99 (99mTc) for the medical supply of local and international markets. To manufacture Molybdenum-99, a product made of the compound U-Al contained in an aluminium matrix (called meat) is rolled in between two parallel plates of aluminium alloy A96061. The result is a simple, selfcontained plate of aluminium measuring 230 mm x 28 mm x 1.64 mm that contains a sandwich of U-Al ‘meat’ in the centre (see Figure 19). It is of note that the isotopic content of fission uranium is high, at an enrichment of 20% 235U but low enough to abide by international guidelines for LEU (Low Enriched Uranium) fuels and irradiation targets. Assembly of the irradiation rigs are performed in a hot-cell situated at pool level. Fresh, un-irradiated molybdenum plates are loaded—four at a time—into a holder known as the ‘target’ (see Figures 20 and 21). A rig stem is then inserted through the target’s hollow centre to form an assembled irradiation ‘rig’. The rig incorporates two to three irradiation targets along with other flow control devices like nozzles and flow restrictors (see Figure 22). Once assembled, the rigs are transported from the hot-cells into the service pool (underwater) via a hermetically sealed elevator. These rigs could then be remotely lowered to the reflector vessel’s bulk irradiation facilities (see Figure 18).

300


Figure 19. One irradiation target with four molybdenum plates.

Figure 20. Isometric view of inadiation target model.

Figure 21. Cross sectional view of the irradiation target with four molybdenum plates inside.

Before the reactor is brought to critical, all primary cooling and secondary cooling systems are switched on. The rigs cooling flow draws light water from the main pool down


301

into the irradiation rigs before being passed onto radiation decay tanks and heat exchangers after. The fast neutrons radiating from the fuel plates are slowed by the H2O core coolant and are completely thermalised by the D2O between the core and the facility. Having been slowed, these neutrons could then be captured by the Uranium-235 meat resulting in fission. A maximum number of three targets could be loaded in one irradiation rig. This constituted the bounding case to be modeled.

Figure 22. Molybdenum plate target and rig (shown on its side but loaded vertically).

Design Characteristics Compared to the previous method of molybdenum manufacture, the philosophy of this purpose-built system was for high molybdenum yield and low radio-active waste output. The previous design of molybdenum irradiation in the recently decommissioned HIFAR reactor (an old British design of the DIDO class), utilized 2.2% enriched UO2 reactor power fuel pellets packed in bulky aluminium cans. These cans were centrally inserted into concentrically arranged annular fuel plates and differed from the new irradiation rigs that lie outside the core itself. Consequently, the two main changes in molybdenum targets as compared to the previous HIFAR (High Flux Australian Research) reactor are: (1) the geometry, which has changed from a can to a plate, (2) the enrichment, increasing from 2.2% to 20% and (3) the location of the irradiation rig outside the core. This new configuration allows the irradiation of more 235U but also uses a substantially lower mass of aluminium and 238U, the result of which is an increase in yield and a decrease in aluminium waste. Despite these gains in efficiency, the increase in power generated by such a high concentration of enriched uranium poses challenging conditions for heat removal. The thin layer of aluminium gives little impedance to the flow of heat away from the U-Al centre but at the same time, any interruption to the forced convection flow of coolant could cause the cladding to blister or at the very worst melt, as there is very little aluminium mass to which the very large amounts of heat produced can conduct away. Thus, the prediction of flow on and over the molybdenum plates is crucial to ensure no isolated heat spots could develop that would result in the blistering of the clad or in the most extreme case, the melting of clad during irradiation (burn out). Thus, the aim of this numerical study was to show the thermo-hydraulic design of the molybdenum holder was sufficient to remove the heat generated by fission with a sizable margin between bounding operational conditions and extreme (i.e. almost impossible) conditions. After a maximum of 14 days of irradiation, the rig assembly is removed from the reflector vessel and placed in the service pool storage racks to ‘cool’. Loading and unloading

302


of irradiation rigs can be performed whilst the reactor is at full power. The removal of the rig from its irradiation position halts the neutron-irradiation heating, leaving only the nucleardecay heat to be removed by means of natural convection. The decay-heat produced by weaknuclear reactions is much lower than the fission heat—initially at approximately 8% of the power produced during irradiation and decreases exponentially with time.

Methods of Calculations and Model Set-Up The maximum steady state temperatures of the molybdenum-99 rigs as a result of the coupled action of nuclear heating and light-water forced convection cooling were determined. These values could then be used to determine two important margins: (1) the difference between the maximum attainable temperature in the plate and the aluminium cladding blister temperature and (2) the difference between the operational maximum heat flux and the heat flux at the onset of nuclear boiling. For the simulation of heat transfer during irradiation, different parts of the irradiation assembly were separately modeled, meshed and combined by what is termed as ‘patching’. Afterwards, material properties were explicitly specified in each domain, along with the relevant boundary conditions. The built model incorporated nothing more than the necessary areas under examination to optimize the use of computational resources. Other fundamental features such as the setting of appropriate turbulence model, the solution’s mass-residual convergence criteria and the time & space discretisation schemes for the iterative solution process were then set (Tu et al., 2008). Some features of the numerical model, such as the modeling of buoyancy were deemed to have a negligible effect to the overall flow characteristics and were thus not modeled. In order to validate the CFD model, a comparison with a set of independent results was necessary. Available INVAP results using a more simplified one-dimensional code were used as comparative data against the CFD results. In this way, we were able to evaluate the consistency of the numerical models. For those cases which have no corresponding INVAP study, the only method for validation was by means of grid convergence testing and by the comparison of results using different initial values or different convergence criteria. The approach with this numerical study was to start with the most simplified assumptions in the CFD model and to introduce increasing complexities to examine the relative importance of each effect. A series of CFD simulations, utilizing a three-dimensional CFD program ANSYS-CFX10.0, were thus conducted to attain a final validation-case which included all the identified bounding conditions.

Power Distribution of Molybdenum-Plate Targets To undergo irradiation, assembled rigs loaded with two or three targets are inserted vertically inside the reflector vessel. When the reactor is at power, nuclear reactions occur in the molybdenum-plates as thermal neutrons are absorbed by uranium with the effect of producing heat. Since the horizontal neutron flux from the core radiates uniformly in the near vicinity of the circular reflector pool, there is little flux variation in the horizontal plane of the irradiation rig. However, the irradiation rig is very long, so the neutron flux distribution varies


303

appreciably in the vertical direction, resulting in a cosine power distribution along the vertical axis of the molybdenum-rig. Using the Monte Carlo N-Particle (MCNP) transport code for simulating neutronic reactions, it was found that for a three target configuration, the middle position of three targets generated the most power. A separate and independent neutronic calculation conducted by INVAP (MOLY-0100-3OEIN-004, 2003) found that the maximum total power loading was 40.2 kW for one target and 79.6 kW for two targets. Since these results indicate a trend for proportionately lower powers for each additional target, it was deemed that each target could generate no more than 40.2kW by itself during normal reactor operations. However, to increase the safety margin for this analysis, it was specified that each of the three targets would generate a power of 60 kW. The power distribution is defined in terms of the ratio between the maximum and the average power along the vertical position of the plates in the irradiation rig. This can be defined either in terms of the plate surface heat flux or in terms of the volumetric power density, with a ratio known as the Power Peaking Factor (PPF). In ref. Moly-0100, the PPF has been shown to be equal to 1.1 for a single target and 1.3 for a two-target loading. It was thus assumed that a PPF of 1.3 could be used for three targets. This assumption has been validated later by an MCNP model of the three targets loaded for irradiation with a core of fresh fuel. Thus, the bounding case was defined as an irradiation rig loaded with three targets, each containing four plates. The power density was defined as follows:

q = PPF

Q plate V

cos ( B ( z − z0 ) ) = 1.3

15000 cos ( 3.134453 ( z − 0.413) ) (40) 5188.8 10-9

where q PPF Qplate V B z z0

: : : : : : :

Power density, [W.m-3] Power Peaking Factor, [-] Total power per plate, [W] Meat volume, [m3] Adjustable constant, [m-1] Coordinate on the z axis, [m] offset of the centre, [m]

Under nominal conditions, the average flow velocity through most of the rig’s cross section along the plates is 3.8 m.s-1. It was decided to set a more conservative limit with a reduced average flow velocity of 3 m.s-1 for this study.

Boundary Conditions and Domain Restriction Since coolant for the molybdenum-rigs is drawn from the pool, the temperature is essentially the pool water temperature which is regulated at a steady 37°C. To establish a safety margin of 3°C, a value of 40°C was conservatively applied for the inlet coolant boundary condition. It is difficult to ascertain the coolant velocity profile entering the rig

304


because the initial pool-water flow conditions could not be reliably predicted. But seeing as the purpose of this investigation was to examine the heat transfer characteristics of the rig, the exact flow pattern entering the rig was not of primary importance. Furthermore, the reducerexpansion arrangement of the nozzle preceding the three targets had the intended effect of straightening the flow before it passed over the molybdenum-plates. Therefore the flow can be assumed to have a developed turbulent velocity profile before entering the targets’ coolant channels. By these considerations, the total control volume modeled in these simulations only extended from the rig inlet nozzle to the outlet restrictor (see Figure 22) and a fully developed turbulent flow profile was assumed as the inlet condition. To allow the average velocity to develop into a turbulent velocity flow profile, an artificial pipe extension of fifteen hydraulic diameters was modeled before the inlet. Also, lest the flow after the restriction nozzle affect the preceding flow domain, an artificial pipe of 15 hydraulic diameters was also attached to the end of the restrictor. Finally, the outer surface of the rig was conservatively assumed to be adiabatic to attain the highest possible temperature in the coolant, now set to be solely responsible for the removal of heat from the rig.

Figure 23. Cross-sectional view of control volumes representing the irradiation rig.

Figure 24. The control volumes grouped in their respective materials shows each separate entity more clearly.


305

Since it is obvious that the rig geometry is symmetrical in two planes (see Figures 23 and 24), only one-quarter of the irradiation rig needed to be modeled with symmetrical boundaries applied to the ‘cut planes’ to limit the number of mesh elements. In truth, a one-eighth model would have sufficed but it was decided a complete coolant channel would be modeled so that no assumptions need be made of what was subsequently demonstrated as a symmetrical turbulent velocity profile through the pipe. The inlet condition was defined at mass flow rate of 0.6259 kg.s-1, corresponding to the 3 m/s average velocity in the major channel (see Figure 25). The inlet coolant temperature was also conservatively set at a higher temperature of 40°C, providing a 3°C margin above the nominal 37°C pool temperature. At the outlet, a 0 Pa gauge pressure was specified and the cylindrical outer wall was defined as adiabatic. The standard k-ε model was chosen for this highly forced convective problems with automatic adjustments for kinetic and dissipation values as calculated by local velocities.

(a)

(b)

(c)

Figure 25. Quarter-sections of the: (a) nozzle (b) target (c) restrictor; blue indicates flow areas.

Grid Generation For the sake of accuracy and computational efficiency, a body fitted orthogonal mesh was selected to model the irradiation rig. Building quadrilateral surfaces for the body fitted mesh required more time than automatically generated grids but this resulted in the use of less nodes and reduced the amount of mesh-induced solution diffusivity when compared to tetrahedral mesh. The occurrence of this artificial diffusion is a result of interpolation errors arising from highly skewed mesh contact angles of tetrahedral-meshes. In areas where large gradients were expected, the volumes were meshed with relatively high density to capture sharp temperature changes. Similarly, grid optimization was performed in those areas where the temperature and velocity gradients were small by a reduction in mesh density. Within the flow domain, the change of mesh sizes was gradual to prevent numerical instability in the solution. Whilst mesh size transitions in the solid domain were not required to be as stringent because the harmonic-mean interpolation for thermal conduction between solid nodes is not as geometrically sensitive as the solution for convection and diffusion in fluid nodes. As mentioned, highly skewed elements were avoided where possible to minimize numerical diffusion. In Figure 26, the circled area in (a) indicates the mesh is too skewed and has been re-meshed in (b) so that all cell face angles stood between 38° and 142°. This optimization improved the solution maximum temperature result by a difference of 6°C.

306


(a)

(b)

Figure 26. Circled area re-meshed from (a) to (b) to reduce the presence of skewed mesh.

Mesh refinement was checked using three different meshes. Due to limits in computational resources, it was not possible to double the mesh seeding in every direction at the same time. Thus, two new meshes were created, one with a doubling of mesh in the streamwise directions and one with a doubling in the crosswise. In each case, a simulation was run and the maximum temperatures did not vary by more than 1°C. Thus it was shown that grid convergence had been established, with a final node element count of 432,667.

Validation Case This simulation compared the temperatures attained in INVAP’s correlation-based onedimensional code with our CFD model under the same flow and power conditions. The nominal condition study conducted by INVAP evaluated a two target loading scenario with a total power output of 79.4 kW, a cooling flow of 3.6 m.s-1 and an oxide layer thickness of 7 μm as calculated by the Argonne National Lab (ANL) model for oxide layer growth. Since INVAP’s model calculated for two targets whilst the CFD model was built with three, one four-plate target was completely ‘switched off’, so that only two targets produced power as simulated in the INVAP scenario. The inlet mass flow of the CFD model was increased from the safety scenario of 0.6258 kg.s-1 to 0.7510 kg.s-1 to correspond with the average flow velocity increase from 3 m.s-1 (safety study) to 3.6m.s-1 (validation study). INVAP’s input parameters of solid physical properties and oxide layer thickness were also adopted in this validation study. A summary of settings is displayed in Tables 2 and 3. Details of the model can have effects on the criterion maximum temperature. These details include: 1. modeling of the oxide layer 2. modeling of contact conditions between the Plate and the Target Holder 3. power density distribution in the U-Al meat To appreciate differences between model details, a simple CFD model was first built with the assumptions:


307

1. without oxide layer. 2. perfect contact between the plate and the holder 3. a homogeneous power profile distribution across the U – Al meat Table 2. Summary of the settings for the validation case Oxide thickness Number of targets Total Power Inlet flow rate Inlet temperature Outlet gauge pressure

7 μm 2 79.4 kW 0.7510 kg.s-1 37°C 0 Pa

Table 3. Summary of the material properties for the validation case MATERIAL PROPERTIES -1

Molar Mass Density

g.mol kg.m-3

Specific Heat Capacity Thermal Conductivity

J.kg-1.K-1 W.m-1.K-1

Dynamic Viscosity μ

Al

U2Al and Al matrix Oxide

Water

26.98 2702

270 4625.34

18.02 997

903 165

900 148

4181.7 0.6069

-1 -1

kg.m .s

2.25

0.0008899

Effect of Oxide Layer Modelling During irradiation, the layer of aluminium oxide that forms on the molybdenum-plate is relatively thin at 7μm. Physically, this thin layer could not be directly represented in the computer model because the mesh was not fine enough to virtually interpret what was physically there. The oxide layer could not be ignored either because though small, its reduced conductivity had a significant effect on the molybdenum-plate’s maximum temperature.

Figure 27. Step-function thermal conductivity profile.

308


Figure 28. Graphical representation of distance vs. temperature for realistic and modeled thermal conductivity for aluminum and aluminum oxide.

A first attempt was made to simply prescribe a thermal conductivity profile onto the cladding elements as shown in Figure 27. The thermal conductivity of the aluminium varies with respects to cladding depth, with the circled regions of aluminium-oxide possessing a much lower thermal conductivity. However, the result of this simulation was proven a failure as the maximum temperature proved no different from a separate identical simulation run with no oxide-layer. To circumvent this problem without resorting to the use of more computationally expensive mesh, the effect of the oxide layer was mathematically modeled as an integral part of the cladding. This was done by calculating an equivalent conductivity via the formula below which was derived from flux conservation principles (Patankar, 1980):

⎛ 1− f f ⎞ k =⎜ + ⎟ kox ⎠ ⎝ kAl where f =

−1

(41)

δ ox . The oxide layer thickness was assumed to be 7 μm corresponding with δ Al + δ ox

INVAP’s assumption. The equivalent conductivity of the cladding result was thus: ⎧δ ox = 7 μ m ⎪δ + δ = 350 μ m −1 ox 7 ⎪ Al ⎛ 1 − 0.02 0.02 ⎞ −1 −1 f k ⇒ = = 0.02 ⇒ = + ⎨ ⎜ ⎟ = 67.44 W .m .K −1 −1 k W m K 165 . . = 350 165 2.25 ⎝ ⎠ ⎪ Al ⎪k = 2.25W .m −1.K −1 ⎩ ox

This formula modified the aluminium conductivity to attain the correct temperature at the aluminium boundary but at the expense of correct temperatures inside the aluminium itself, as graphically explained in Figure 28. Maximum temperatures from the step-conductivity profile and the average conductivity profile technique are displayed in detail in Table 4. The largest difference in temperature


309

proved to be between the no-oxide and average-conductivity oxide simulations, with a difference of 3.9°C in the U-Al meat. However, the stepwise-conductivity profile technique resulted in only a 0.5°C gain in UAl meat maximum temperature. It is obvious from these results that the step-wise conductivity profile technique was unsuitable. The average conductivity profile technique was adopted as the method for simulating the effect of aluminium oxide on cladding conductivity. Table 4. Comparison of models Max. Temperatures (°C) Reference

Oxide model

Oxide layer

absent

step

average

Contact type

perfect

perfect

perfect

Power distribution

uniform

uniform

uniform

Surface (water side)

59.2

59.2

59.0

Surface (cladding side)

99.1

99.3

100.5

Cladding – Meat interface

100.9

101.3

104.7

Meat – centre line

102.0

102.5

105.9

Effect of Contact Surface Since the thermal conductivity of water was much less than aluminium, a break in the solid conduction path between the molybdenum-plate and the target holder would be detrimental for the conduction of heat away from the plate. Also, as there was a large tolerance between the plate and the recess in which the plate is held, a water gap existed between one side of the plate and the holder. This study examined the maximum temperature difference caused by the presence of this water gap (see Figure 29). During the creation of this geometry, a space between the plate and its holder was made and patched separately to allow different contact conditions to be simulated. Since the inner channel was slightly larger than the outer channel, slight difference in local velocities on either side of the plate would produce a minute pressure difference that forces the plate to one side. To model the reduced thermal-conduction effects of the water gap, solids with the thermal conductivity of water were patched in between the plate and the holder. Effectively, the water gap acted as a low conductivity solid with no coolant advection. This was a reasonable and conservative assumption because the very restricted flow path of the gap would result in very small flow rates that would provide negligible amounts of forced convection cooling. The alternative was to mesh the gap at a higher resolution to adequately solve for the fluid advection which was in practical terms, unnecessary.

310


Figure 29. Detailed view of contact realistic conditions between the plate and its holder.

Table 5. Effect of the contact surface Temperatures (°C) Thermal conduction profile type: Contact type Power distribution Surface (water side) Surface (cladding side)

average perfect uniform 59.0 100.5

average partial uniform 62.1 103.6

Cladding - Meat interface

104.7

107.6

Meat - centre line

105.9

108.8

The difference in contact conditions only proved to be marginal as can be seen in Table 5. The reduction from ‘full contact’ to ‘half contact’ between the plate and the holder only increased the overall maximum temperature by 3°C. This proved to be positive from an engineer’s standpoint because it was undesirable to have heat transfer characteristics that are too sensitive to unpredictable contact conditions. The half-contact condition between plate and holder was thus adopted for future simulations for its more conservative assumption.

Effect of Power Distribution In this study, the effect of a uniform and cosine power distribution on the maximum temperature was examined. A simpler study by INVAP using a one-dimensional heat transfer code provided a simple comparative assessment. According to MCNP calculations conducted by ANSTO (Geoff, 2006), a power density profile distributing a total power of 79.4 kW over eight plates with a power peaking factor (PPF) of 1.3 was calculated as:

79400 8 q = PPF cos ( B ( z − z0 ) ) = 1.3 cos ( 4.97424 ( z − 0.288 ) ) (41) V 5188.8 10-9 Q plate


311

where : Power density, [W.m-3] : Power Peaking Factor, [-] : Total power per plate, [W] : Meat volume, [m3] : Adjustable constant, [m-1] : Coordinate on the z axis, [m] : Offset of the centre, [m]

q PPF Qplate V B z z0

Power Density vs. Vertical Position 2 Targets 2.60E+09 2.40E+09

q [W.m^-3]

2.20E+09 2.00E+09 1.80E+09 1.60E+09 1.40E+09 1.20E+09 1.00E+09 8.00E+08 0

0.2

0.4

0.6

z [m]

Figure 30. Power distribution along the plates (two positions).

Table 6. Power distribution effect on temperatures Temperatures (°C) Validation case

INVAP study

Oxide layer

average

average

—

Contact type

partial

partial

—

uniform

cosine

—


62.1

61.9


103.6

114.9

Oxide - Cladding interface

—

—

109


107.6

120.2

111

Meat - centre line

108.8

121.6

113

Power distribution

105

This definition allowed the use of the same maximum power density (i.e., 2.5 109 W.m-3) as the one used by INVAP to calculate the maximum meat temperature, and also the same

312


total power in two plates. It also respected the PPF of 1.3 for the two position case. Applying the cosine power distribution to the model gave the simulation more realism than if a constant power was applied throughout the plate. Two simulations were run to observe the effect of the difference between a constant power and a cosine distribution power. Using a cosine-shape distribution for the power increased the maximum meat temperature by 12.8°C as shown in Table 5. Since a cosine power distribution (see Figure 30) is more conservatively representative, this model feature was retained for the “bounding case” scenarios.

Conclusive Remarks for the Validation Case Beginning from a simple model and making successive improvements with increasing detailed features, a final validation case was attained which has been proven to be consistent and conservative against INVAP’s one-dimensional heat-transfer study. This course of action has also allowed an appreciation of the sensitivity of temperature changes with respect to varying engineering assumptions. The temperatures on the interface between the cladding and the oxide layer could not be attained in the CFD model, and therefore has not been compared. The CFX commercial software appears to quote the temperature value held at the centre of the control volume on either side of the plate-water interface, so that these results could not be directly compared against INVAP’s study. However, the cladding side surface temperature yields valid and comparable results to INVAP’s plate surface temperature results. Table 7. Summary of the simulations for the validation case. Temperatures (°C) Validation Case

INVAP study

Oxide layer

absent

step

average

average

average

—

Contact type

perfect

perfect

perfect

partial

partial

—

Power distribution

uniform

uniform

uniform

uniform

cosine

—


59.2

59.2

59.0

62.1

61.9


99.1

99.3

100.5

103.6

114.9

Oxide - Cladding interface

—

—

—

—

—

109


100.9

101.3

104.7

107.6

120.2

111

102

102.5

105.9

108.8

121.6

113

Meat - centre line

105

Various hypotheses were tested on the validation case. Several simulations were run which revealed: •

The application of an average thermal conductivity over the whole cladding was more realistic than the definition of a step function to lower the conductivity in the oxide layer.

Pipe Flow Analysis of Uranium Nuclear Heating with Conjugate Heat Transfer • •

313

The assumption of one-side contact between the plate and its holder was more realistic than assuming full aluminium contact on both sides of the plate. The application of a variable power density was more realistic than a simple definition of the average power density.

In conclusion, these results were in good agreement with INVAP’s calculations and as such, this model could now be used to examine even more conservative flow conditions for the purpose of safety analysis. Summary of results are presented in Table 7.

Bounding Case Having demonstrated the consistency of the CFD model against INVAP’s verification data, the CFD model could now be used to investigate the flow conditions of the bounding case scenario. A summary of the material properties are detailed in Table 8. Similar to the validation simulation, a cosine power profile (see Figure 31) for three targets was set for this bounding simulation and a ‘high resolution’ advection scheme with automatic timescale was adopted for the solution process in-order to accelerate solution convergence. A summary of temperature results for oxide thicknesses of 20 μm and 10 μm are displayed in Table 9 below: Table 8. Summary of physical properties MATERIAL PROPERTIES g.mol kg.m-3 J.kg-1.K-1 W.m-1.K-1 kg.m-1.s-1

26.98 2700 903 165

270 4625.34 900 148

4.00E+09 3.50E+09 3.00E+09 2.50E+09 2.00E+09 1.50E+09 0

0.2

0.4

0.6

oxide

2.25

Pow er Density vs. Vertical Position 3 Targets

q [W.m^-3]

Molar Mass Density Specific Heat Capacity Thermal Conductivity Dynamic Viscosity μ

Aluminium U2Al and Al matrix

-1

0.8

z [m ]

Figure 31. Power distribution over 3 plates.

Water 18.02 996.2 4178.8 0.609 0.0008327

314

G.H. Yeoh and M.K.M. Ho Table 9. Calculated temperatures for the bounding case Temperatures (°C)

Oxide layer Surface (water side) Surface (cladding side) Cladding - Meat interface Meat - centre line

20 μm 92.2 183.1 199.5 201.6

10 μm 92.5 181.1 190.9 193

Thus, assuming an average flow velocity of 3 m.s-1 at 40°C, an oxide thickness of 20 μm, and a PPF of 1.3 for a total power of 45 kW over three plates, the main results were obtained: • •

Maximum water temperature: 93°C Maximum meat temperature: 202°C

These results showed a sufficient margin from plate melting and water boiling.

Conclusion The cooling systems for three separate molybdenum irradiation facilities have been solved for maximum mean temperatures using the computational fluid dynamics methodology. In order to accomplish this, the geometry of all three systems were first modeled and geometrically discretised in a variety of structured and unstructured mesh. Secondly, accurate material properties were attributed to corresponding areas of the numeric model. Uranium volumes were then attributed Power densities as calculated by MCNP to simulate the power produced within irradiated targets. The high velocities and thus high Reynolds numbers of all three case studies placed the flow regime within a turbulent setting. As the study was concerned with time-averaged results, the RANS approach for turbulence modeling was most suited and the simple standard k-ε turbulence model was selected over other models for its applicability to fully turbulent flows. Other types of flows, such as large swirling flows or flows with large amounts of laminar-turbulent blending, would have required more sophisticated RANS modeling like the RNG k-ε model, reliazable k-ε model and SST Menter’s model. However, this level of sophistication was unnecessary for the solution of these pipe flow systems and was thus not employed. Computational results of all three case studies have been demonstrated to agree well with independent experimental and numerical studies. The success of these studies further confirms the robustness and versatility of CFD methods in the field of nuclear engineering and will remain a continual feature in the field of thermo-hydraulics.


315

Acknowledgements The authors would like to thank Dr. George Braoudakis (Head of Nuclear Analysis Section) for providing the MCNP power input parameters in these simulations and for proofreading this document; past internship students Mr. Tony Chung (UNSW, Sydney) and Mr. Guillaume Bois (INSA, Lyon) for their assistance in attaining the computational simulations; Mr. David Wassink (Water Tunnel Manager) for his experimental validation data; and, finally, to ANSTO for allowing this work to be published.

References [1] Durance, G. (2006), Private communication, ANSTO. [2] Gumbert, C., Lohner, R., Parikh, P. & Pirzadeh, S. (1989), A package for unstructured grid generation and finite element flow solvers, AIAA Paper 89-2175. [3] Jayatilleke, C. L. V. (1969), The influence of Prandtl number and surface roughness on the resistance of the laminar sublayer to momentum and heat transfer, Prog. Heat Mass Transfer, 1, 193 -321. [4] Launder, B. E. & Spalding, D. B. (1974), The numerical computation of turbulent flows, Comp. Meth. Appl. Mech. Eng., 3, 269-289. [5] Lo, S. H. (1985), A new mesh generation scheme for arbitrary planar domains, Int. J. Numer. Methods Eng., 21, 1403-1426. [6] Marcum, D. L. & Weatherill, N. P. (1995), Unstructured grid generation using iterative point insertion and local reconnection, AIAA Paper 94-1926. [7] Mavriplis, D. J. (1997), Unstructured grid techniques, Ann. Rev. Fluid Mech., 29, 473514. [8] Menter, F. R. (1993), Zonal two equation k-ω turbulence models for aerodynamics flows, AIAA paper 93-2906. [9] Menter, F. R. (1996), A comparison of some recent eddy-viscosity turbulence models, J. Fluids Eng., 118, 514-519. [10] MOLY-0100-3OEIN-004 (2003), Heating calculation of Molybdenum targets, INVAP Report. [11] Patankar, S. V. (1980), Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, New York. [12] Shepard, M. S. & Georges, M. K. (1991), Three-dimensional mesh generation by finite octree technique, Int. J. Numer. Methods Eng., 32, 709-749. [13] Shewchuk, J. S. (2002), Delaunay refinement algorithms for triangular mesh generation, Computational Geometry: Theory and Applications, 22, 21–74. [14] Shih, T.-H., Liou, W. W., Shabbir, A., Yang, Z. & Zhu, J. (1995). A new k-ε eddy viscosity model for high Reynolds number turbulent flows, Comp. Fluids, 24, 227-238. [15] Smith, R. E. (1982), Algebraic grid generation, Numerical Grid Generation, Thompson, J. F. (Ed.), North-Holland, Amsterdam, 137. [16] Thompson, J. F. (1982), General curvilinear coordinate systems, Numerical Grid Generation, Thompson, J. F. (Ed.), North-Holland, Amsterdam, 1-30. [17] Touloukian, Y.S., Powell, R.W., Ho, C.Y., Klemens, P.G. (1970). Thermophysical Properties of Matter, 2.

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[18] Tu, J. Y., Yeoh G. H. & Liu C. Q. (2008), Computational Fluid Dynamics – A Practical approach. Butterworth-Heinemann, Oxford. [19] Wilcox, D. C. (1998), Turbulence Modelling for CFD, DCW Industries, Inc. [20] Yakhot, V., Prszag, S. A., Tangham, S., Gatski, T. B. & Speciale, C. G. (1992), Development of turbulence models for shear flows by a double expansion technique, Phys Fluids A: Fluid Dynamics, 4, 1510-1520. [21] Yeoh, G. H. and Storr, G. J. (2000), A three-dimensional study of heat and mass transfer within the irradiation space of the HIFAR fuel element, Advance Computational Methods in Heat Transfer VI, Sunden, B. and Brebbia, C. A. (Eds.), WIT Press, Southampton, 343-351. [22] Yerry, M. A. & Shepard, M. S. (1984), Automatic three-dimensional mesh generation by the modified-octree technique, Int. J. Numer. Methods Eng., 20, 1965-1990.



Chapter 9

FIRST AND SECOND LAW THERMODYNAMICS ANALYSIS OF PIPE FLOW Ahmet Z. Sahin* Mechanical Engineering Department King Fahd University of Petroleum and Minerals Dhahran 31261, Saudi Arabia

Introduction In a fully developed laminar flow through a pipe, the velocity profile at any cross section is parabolic when there is no heat transfer. But when a considerable heat transfer occurs and the thermo-physical properties of the fluid vary with temperature, the velocity profile is distorted. If the thermo-physical properties of the fluids in a heat exchanger vary substantially with temperature, the velocity and the temperature profiles become interrelated and, thus, the heat transfer is affected. Viscosity of a fluid is one of the properties which are most sensitive to temperature. In the majority of cases, viscosity becomes the only property which may have considerable effect on the heat transfer and temperature variation and dependence of other thermo-physical properties to temperature is often negligible. The viscosity of the liquids decreases with increasing temperature, while the reverse trend is observed in gases (Kreith and Bohn, 1993). Heat transfer and pressure drop characteristics are affected significantly with variations in the fluid viscosity. When the temperature is increased from 20 to 80 oC, the viscosity of engine oil decreases 24 times, the viscosity of water decreases 2.7 times and the viscosity of air increases 1.4 times. Therefore, selection of the type of fluid and the range of operating temperatures are very important in the design and performance calculations of a heat exchanger. In the process of designing a heat exchanger, there are two main considerations. These are the heat transfer rates between the fluids and the pumping power requirement to overcome the fluid friction and move the fluids through the heat exchanger. Although the effect of viscous dissipation is negligible for low-velocity gas flows, it is important for high-velocity *


318

Ahmet Z. Sahin

gas flows and liquids even at very moderate velocities (Shah, 1981; Kays and Crawford, 1993). Heat transfer rates and pumping power requirements can become comparable especially for gas-to-gas heat exchangers in which considerably large surface areas are required when compared with liquid to liquid heat exchangers such as condensers and evaporators. In addition, the mechanical energy spent as pumping power to overcome the fluid friction is worth 4 to 10 times as high as its equivalent heat (Kays and London, 1984). Therefore, viscous friction which is the primary responsible cause for the pressure drop and pumping power requirements is an important consideration in heat exchanger design. On the other hand, efficient utilization of energy is a primary objective in designing a thermodynamic system. This can be achieved by the minimization of entropy generation in processes of the thermodynamic system. The irreversibilities associated with fluid flow through a pipe are usually related to heat transfer and viscous friction. Various mechanisms and design features contribute to the irreversibility terms (Bejan, 1988). There may exist an optimal thermodynamic design which minimizes the amount of entropy generation. For a given system, a set of thermodynamic parameters which optimizes the operating conditions may be obtained. The irreversibility associated with viscous friction is directly proportional to the viscosity of the fluid in laminar flow. Therefore, it is necessary to investigate the effect of a change of viscosity during a heating process for an accurate determination of entropy generation and of the required pumping power. Heat transfer and fluid pumping power in a piping system are strongly dependent upon the type of fluid flowing through the system. It is important to know the fluid properties and their dependence to temperature for a heat exchanger analysis. As a result of heat transfer, the temperature changes in the direction of flow and the fluid properties are affected. The temperature variation across the individual flow passages influences the velocity and temperature profiles, and thereby influences the friction factor and the convective heat transfer coefficient. If the thermo-physical properties of the fluids in a piping system vary substantially with temperature, the velocity and the temperature profiles become interrelated and, thus, the heat transfer is affected. Viscosity of a fluid is one of the properties which are most sensitive to temperature. Therefore, selection of the type of fluid and the range of operating temperatures are very important in the design and performance calculations of a piping system. On the other hand, the exergy losses associated with fluid flow through a pipe are usually related to heat transfer and viscous friction. Dependence of the thermo-physical properties on the temperature affects not only the viscous friction and pressure drop, but also the heat transfer. This implies that the exergy losses associated with the heat transfer and the viscous friction are also affected. The contributions of various mechanisms and design features on the different irreversibility terms often compete with one another. Therefore, an optimal thermodynamic design which minimizes the amount of entropy generation may exist. In other words, a set of thermodynamic parameters which optimize operating conditions could be obtained for a given thermodynamic system. In this chapter, the entropy generation for during fluid flow in a pipe is investigated. The temperature dependence of the viscosity is taken into consideration in the analysis. Laminar and turbulent flow cases are treated separately. Two types of thermal boundary conditions are considered; uniform heat flux and constant wall temperature. In addition, various crosssectional pipe geometries were compared from the point of view of entropy generation and

First and Second Law Thermodynamics Analysis of Pipe Flow

319

pumping power requirement in order to determine the possible optimum pipe geometry which minimizes the exergy losses.

Pipe Subjected to Uniform Wall Heat Flux We consider the smooth pipe with constant cross section shown in Figure 1. A constant and inlet heat flux q′′ is imposed on its surface. An viscous fluid with mass flow rate m temperature T0 enters the pipe of length L . Density

ρ , thermal conductivity k , and

specific heat C p of the fluid are assumed to be constant. Heat transfer to the bulk of the fluid occurs at the inner surface with an average heat-transfer coefficient h , which is a function of temperature dependent viscosity. The effect of viscosity on the average heat transfer coefficient is given by Kays and Crawford (1993)

⎛μ ⎞ h Nu = =⎜ b ⎟ hc. p . Nu c. p . ⎝ μw ⎠

n

(1)

where the exponent n is equal to 0.14 for laminar flow. In the case of turbulent flow the exponent n is equal to 0.11 for heating and 0.25 for cooling. For fully developed laminar flow, constant property heat transfer coefficient hc. p. is given by Incropera and DeWitt (1996).

hc. p. =

k 48 k Nu c. p. = D 11 D

Figure 1. Sketch of constant cross sectional area pipe subjected to uniform wall heat flux.

(2)

320

Ahmet Z. Sahin And for fully developed turbulent flow, constant property heat transfer coefficient hc. p. is

given by

hc. p. =

k k ( f / 8) ( Re − 1000 ) Pr Nu c. p. = D D 1 + 12.7 ( Pr 2 / 3 − 1) f / 8

(3)

where the Reynolds number is

Re =

ρUD . μb

(4)

The average Darcy friction factor, f , for a smooth pipe is also considered to be a function of temperature dependent viscosity and is given by Kays and Crawford (1993) m

f f c. p. f f c. p.

⎛μ ⎞ = ⎜⎜ b ⎟⎟ for liquids ⎝ μw ⎠

(5a)

m

⎛T ⎞ = ⎜⎜ b ⎟⎟ for gases ⎝ Tw ⎠

(5b)

where the exponent m = −0.14 for laminar flow and m = −0.25 for turbulent flow case. The friction factor for constant properties for laminar flow is given by Incropera and DeWitt (1996) as

f c. p . = and for turbulent flow as

64 Re

f c. p . = [0.79 ln(Re) − 1.64]

(6)

−2

To account for the variation of the bulk temperature along the pipe length,

(7)

μb and

therefore Re and Pr, in Eqs. (3) – (7), are related to the bulk fluid temperature halfway between the inlet and outlet of the pipe, as suggested by Kreith and Bohn (1993). Since the temperature variation along the pipe is initially unknown and depends on h , a trial and error procedure is followed to determine both h and f .


321

Total Heat Transfer Rate The rate of heat transfer to the fluid inside the control volume shown in Figure 1 is

p dT = q′′π Ddx δ Q = mC

(8)

= ρUπ D / 4 . In writing Eq. (8), the pipe is assumed to have a circular cross where m 2

section. However, the analysis is not affected by assuming cross-sectional areas other than circular. Integrating Eq. (8), the bulk-temperature variation of the fluid along the pipe is obtained as

T − T0 =

4 q′′ x. ρUDC p

(9)

The temperature variation along the pipe is linear when the viscous dissipation and axial conduction effects are neglected. The temperature gradient for the fluid in this case depends mainly on the magnitude of the heat flux. For a constant heat flux q′′ and average heat transfer coefficient h evaluated at the bulk temperature halfway between inlet and outlet of pipe, the temperature difference between the wall surface and bulk of the fluid is given as

Tw − T =

q′′ . h

(10)

Eq. (9) may be rewritten as

θ =4 where

St x, D

(11)

θ is the dimensionless temperature defined as

θ=

T − T0 q′′ / h

(12)

St =

h . ρUC p

(13)

and the Stanton number is

322

Ahmet Z. Sahin

Total Entropy Generation The entropy generation within the control volume of Figure 1 can be written as (Bejan, 1996)

δ Q − , dS gen = mds Tw

(14)

dT dP − . T ρT

(15)

where the entropy change is

ds = C p Substituting Eq. (8) into Eq. (14),

⎡T − T ⎤ 1 p⎢ w dS gen = mC dT − dP ⎥ . ρ C pT ⎦ ⎣ TTw

(16)

The pressure drop is (Kreith and Bohn, 1993)

f ρU 2 dP = − dx , 2D

(17)

where f is the Darcy friction factor. Integrating Eq. (16) along the pipe length L , using Eqs. (9) and (17), the total entropy generation becomes (Sahin, 1999 and Sahin, 2002)

⎧⎪ ⎡ (1 + 4 Stτλ )(1 + τ ) ⎤ 1 f ρU 3 ⎫⎪ p ⎨ln ⎢ τλ ln 1 4 S gen = mC St + + ( ) ⎬, ⎥ ⎩⎪ ⎣ 1 + τ + 4 Stτλ ⎦ 8 q′′ ⎭⎪

(18)

where the dimensionless wall heat flux is

τ=

q′′ / h T0

(19)

and the dimensionless length of the pipe is

λ=

L . D

(20)


323

Eq. (18) can be written in a dimensionless form as

S

ψ = gen Q / T0

(21)

pT0 ( 4 Stτλ ) Q = mC

(22)

where the total heat transfer is

Thus, the entropy generation per unit heat transfer rate becomes

ψ=

⎫ 1 ⎧ ⎡ (1 + 4 Stτλ )(1 + τ ) ⎤ 1 Ec + f ln(1 + 4 Stτλ )⎬ ⎨ln ⎢ ⎥ 4 Stτλ ⎩ ⎣ 1 + τ + 4 Stτλ ⎦ 8 St ⎭

(23)

where the Eckert number is defined as

Ec =

U2 U2 . = C p (Tw − T ) C p ( q′′ / h )

(24)

Two dimensionless groups arise naturally in Eq. (23), namely,

Π1 = Stλ

(25)

and

Π2 = f

Ec St

(26)

Thus, Eq. (23) becomes in compact form

ψ=

⎫ 1 ⎧ ⎡ (1 + 4τΠ1 )(1 + τ ) ⎤ 1 + Π 2 ln(1 + 4τΠ1 )⎬ ⎨ln ⎢ ⎥ 4τΠ1 ⎩ ⎣ 1 + τ + 4τΠ1 ⎦ 8 ⎭

(27)

1 ⎡ 1+τ ⎤ 1 (1 + 4τΠ1 )1+ 8 Π2 ⎥ . ln ⎢ 4τΠ1 ⎣1 + τ + 4τΠ1 ⎦

(28)

or

ψ=

The first and second terms in Eq. (27) are related to entropy generation due to heat transfer and due to viscous friction respectively. Eq. (27) contains three non-dimensional parameters, namely, τ , Π1 and Π 2 . Among these parameters, τ represents the heat flux imposed on the wall of the pipe q′′ and Π1 represents the pipe length L . Once the type of the fluid and the mass flow rate are fixed, the parameter Π 2 can be calculated on the basis of

324

Ahmet Z. Sahin

temperature analysis. Thus,

τ and Π1 are the two design parameters that can be varied for

determining the effects of pipe length and/or wall heat flux on the entropy generation. For small values of the wall heat flux ( τ << 1 ), Eq. (27) is reduced to

ψ=

1 Π2 ln(1 + 4τΠ1 ) . 32 τΠ1

(29)

For low viscosity, Π 2 / 8 << 1 and Eq. (28) shows that entropy generation becomes

ψ=

⎡ (1 + 4τΠ1 )(1 + τ ) ⎤ 1 ln ⎢ ⎥ 4τΠ1 ⎣ 1 + τ + 4τΠ1 ⎦

(30)

Amount of heat transfer is the primary concern in a heat exchanger design. However, a considerable amount of exergy is destroyed during the heat exchange process. For efficient utilization of energy the entropy generation needs to be minimized. Thus, exergy destruction per unit amount of heat transfer is considered to be a suitable quantity in dealing with the second law analysis of a pipe flow. In the present analysis this ratio is represented by the dimensionless entropy generation, ψ , defined as the entropy generation per unit heat transfer rate for specified pipe inlet temperature. Since Π1 represents the length of the pipe, the dimensionless entropy generation per unit heat transfer rate to the pipe

ψ decreases with the pipe length. The dimensionless entropy

generation ψ approaches a common value for short pipes with a limit

lim ψ =

Π1 →0

τ

1 + Π2 . 1+τ 8

(31)

For the case of constant viscosity, ψ decreases with a slope of

⎡ 1 1 ⎤ ∂ψ = −2τ ⎢1 − + Π2 ⎥ . 2 Π1 →0 ∂Π ⎣ (1 + τ ) 8 ⎦ 1 lim

(32)

ψ decreases and approaches zero asymptotically (and therefore the total heat transfer) as pipe length increases. Since τ represents the amount of heat flux, the dimensionless entropy generation

ψ, defined on the basis of total heat transfer rate to the pipe, goes to infinity when τ = 0 . ψ decreases sharply as τ increases and then starts increasing. A second dimensionless entropy generation may be defined, on the basis of unit heat capacity rate of fluid in the pipe, as


φ=

S gen S = gen , m C p (Q / ΔT )

325

(33)

where ΔT is the increase T ( L) − T0 of the bulk temperature of the fluid in the pipe. Since

p is constant for a fixed mass flow rate, the total entropy generation ( φ ), Q / ΔT = mC which increases with an increase in pipe length, can be given as

φ = 4τΠ1ψ . For the case of constant viscosity,

(34)

φ starts to increase with a rate of

∂φ 1 ⎤ ⎡ τ = 4τ ⎢ + Π2 ⎥ Π1 →0 ∂Π ⎣1 + τ 8 ⎦ 1 lim

and approaches to infinity as pipe length increases. It should be noted that Π 2 is inversely proportional to is constant. Thus the value of

φ is finite for τ = 0 lim φ = τ →0

(35)

τ and therefore the propipe τΠ 2

1 Π1 (τΠ 2 ) 2

(36)

φ decreases with a slope of ∂φ = −Π12 (τΠ 2 ) . τ →0 ∂τ

lim

(37)

Pumping Power to Heat Transfer Rate Ratio The ratio of pumping power to heat transfer rate is

Pr =

π D 2 ΔPU 4

Q

.

(38)

Using Eqs. (8) and (17), the pumping power to heat transfer ratio Pr is

1 Pr = Π 2 . 8

(39)

326

Ahmet Z. Sahin Recalling that the product τΠ 2 is invariable with varying

τ , it can be concluded from Eq. (39) that the pumping power ratio is inversely proportional to τ . Effect of Pipe Cross Section The dimensionless entropy generation given in Eq. (28) can be expressed in terms of Re number as (Sahin, 1998)

⎡ Re ⎢ 1 + τ ψ= ln ⎢ C1τ ⎢1 + τ + C1τ Re ⎣ where C1 = 4 Re Π1 and C2 =

1+C2 Re2

⎛ C1τ ⎞ ⎜1 + ⎟ Re ⎠ ⎝

⎤ ⎥ ⎥ ⎥ ⎦

(40)

Π2 . 8 Re 2

For any given pipe geometry and stream of constant thermophysical property fluid, dimensionless numbers C1 and C2 are constants. Therefore, the dimensionless total entropy generation as given in Eq. (40) is a function of Re and τ only. Values of Nu and ( f Re) for fully developed laminar flow are given in Shah and London (1978) for a variety of pipe geometries. The hydraulic diameter DH for any given pipe geometry can be calculated using the relation:

DH = 4

Ac . p

(41)

Table 1 gives the hydraulic diameters for some common pipe geometries. Similarly, the modified dimensionless entropy generation becomes

φ=

C1τ ψ Re

(42)

and the pumping power to heat transfer ratio is obtained to be

Pr = C2 Re 2

(43)

As the temperature difference between the fluid inlet and the surface of the pipe, τ , is increased, the entropy generation increases, however, the pumping power requirement per unit heat transfer decreases. In the limit when τ approaches zero, the contribution of heat transfer to the entropy generation disappears and


327

limψ = C1C2τ Re

(44)

τ →0

which indicates a linear increase in entropy generation that is solely due to the viscous friction as expected. Table 1. Hydraulic diameters of some common pipe geometries. Pipe Geometry Circular

Hydraulic diameter (DH)

DH =

Square

2

π

Ac

DH = Ac

Triangle

DH =

24 3 Ac 3

Rectangle*

Y DH = 2 X Sinusoidal* **

⎛ ⎜ 1 ⎜ ⎜1+ Y ⎜ X ⎝

⎞ ⎟ ⎟ Ac ⎟ ⎟ ⎠

⎡ ⎞ ⎤ ⎛ ⎛ Y ⎞2 ⎟ ⎥ ⎜ ⎜π ⎟ ⎢ 2 π⎟ ⎥ 2 Y ⎢2 ⎛ Y ⎞ ⎜ ⎝ X⎠ 1 + ⎜ π ⎟ E⎜ , ⎟ + 1⎥ DH = 4 2 π X ⎢π 2⎟ X⎠ ⎜ ⎝ Y ⎛ ⎞ ⎥ ⎢ 1 + ⎜π ⎟ ⎟ ⎥ ⎜ X ⎢⎣ ⎝ ⎠ ⎠ ⎦ ⎝

−1

Ac

* Y/X is the aspect ratio and σ

**

E (k , σ ) = ∫

0

1 − k 2 sin 2 σ dσ

is the elliptic integral of second kind (Tuma, 1979)

Figure 2 shows the variation of ψ with Re for five selected pipe geometries. The crosssection Ac and T0 were constant. The sinusoidal pipe geometry gives the lowest dimensionless entropy generation. As Re is increased, the circular geometry becomes the best choice. The total entropy generation tends to decrease initially and then increases with Re. ψ includes terms form heat transfer and viscous friction. As Re is increased, the heat transfer contribution decreases and that of viscous friction increases. Thus, for low Re number flow, those geometries with higher surface areas are favorable. Circular geometry is the best choice for high Re numbers. Square geometry appears to be good choice after the sinusoidal and circular ones. ψ is found to be considerably high for triangular and rectangular pipes. Therefore, they are inferior choices. Rectangular pipes would yield the lowest

ψ when the

328

Ahmet Z. Sahin

aspect ratio becomes 1 (square geometry). For rectangular pipes with lower aspect ratio, the entropy generation is higher and there may not even exist a minimum. As Re → 0 , the effect of geometry disappears and the dimensionless entropy generation becomes a function of (τ ) , i.e.,

lim ψ = ln (1 + τ )

Re→0

Comparison of pipe geometries using

(45)

φ may be appropriate when the total heat-transfer

rate is important as shown in Figure 3. Favorable pipe geometry appears to be circular but sinusoidal and square geometries yield comparable results. Triangular and rectangular pipe geometries are inferior choices because there is no optimum Re with minimum entropy generation. The pumping power required to overcome viscous friction is shown in Figure 4. The circular geometry is superior to all other geometries. The results are similar to those for the modified dimensionless entropy generation.

Figure 2. Dimensionless entropy generation ψ vs Re for various pipe geometries; and

q′′ = 500 W/m 2 .

Ac = 4 × 10 −6 m 2


Figure 3. Modified dimensionless entropy generation

Ac = 4 × 10 −6 m 2

and

and

vs Re for various pipe geometries;

q′′ = 500 W/m 2 .

Figure 4. Pumping power to heat-transfer ratio

Ac = 4 × 10 −6 m 2

φ

329

q′′ = 500 W/m 2 .

Pr

ve Re for various pipe geometries;

330

Ahmet Z. Sahin

Pipe Subjected to Constant Wall Temperature We consider the smooth pipe with constant cross section shown in Figure 5. The surface and inlet temperature Tw is kept constant. An viscous fluid with mass flow rate m temperature T0 enters the pipe of length L . Density

ρ , thermal conductivity k , and

specific heat C p of the fluid are assumed to be constant. Heat transfer to the bulk of the fluid occurs at the inner surface with an average heat-transfer coefficient h , which is a function of temperature dependent viscosity. The effect of viscosity on the average heat transfer coefficient is given by Shah and London (1978)

⎛μ ⎞ h Nu = =⎜ b ⎟ hc. p . Nu c. p . ⎝ μw ⎠

n

(46)

where the exponent n is equal to 0.14 for laminar flow. In the case of turbulent flow the exponent n is equal to 0.11 for heating and 0.25 for cooling. For fully developed laminar flow, constant property heat transfer coefficient hc. p. is given by Incropera and DeWitt (1996)

hc. p. =

k k Nu c. p. = 3.66 . D D

(47)

Constant property heat transfer coefficient hc. p. for fully developed turbulent flow is given by Eqn. (3). Correlations for the average Darcy friction factor, f , for a smooth pipe are given in Eqns. (5) – (7) for laminar and turbulent flow cases.

Temperature Distribution The rate of heat transfer to the fluid inside the control volume shown in Figure 1 is

δQ = m C p dT = h πD(Tw − T )dx

(48)

where

m = ρUπ D 2 / 4 Integrating Eq. (48), the bulk temperature variation of the fluid along the pipe can be obtained as:

⎡ 4h T = Tw − (Tw − To ) exp ⎢− ⎢⎣ ρU DC p

⎤ x⎥ ⎥⎦

(49)


331

Figure 5. Sketch of constant cross sectional pipe subjected to constant wall temperature.

The temperature variation along the pipe approaches the pipe wall temperature exponentially assuming a uniform heat transfer coefficient evaluated at the bulk temperature halfway between the inlet and outlet of the pipe. Eq. (49) can be re-written as:

⎛ ⎝

θ = exp⎜ − 4 where

St D

⎞ x⎟ ⎠

(50)

θ is the dimensionless temperature defined as,

θ=

T − Tw To − Tw

St =

h . ρUC p

and St is the Stanton number defined as

Total Entropy Generation The total entropy generation within the control volume in Figure 1 can be written as:

δ Q − dS gen = mds Tw where the entropy change is,

(51)

332

Ahmet Z. Sahin

ds = C p

dT dP − T ρT

Substituting Eq. (48) in Eq. (51), the total entropy generation becomes

⎡T − T ⎤ 1 p⎢ w dS gen = mC dT − dP ⎥ ρ C pT ⎦ ⎣ TTw

(52)

The pressure drop in Eq. (52) is given by Bejan (1988)

dP = −

f ρU 2 dx 2D

(53)

Integrating Eq. (52) along the pipe length, L, using Eqs. (49) and (53), the total entropy generation is obtained as (Sahin, 1998 and Sahin 2000)

⎧ ⎡1 − τe −4 Stλ ⎤ 1 Ec ⎡ e 4 Stλ − τ ⎤ ⎫ − 4 Stλ − − + ( 1 ) τ ln S gen = m C p ⎨ln ⎢ e f ⎬ ⎥ 8 St ⎢⎣ 1 − τ ⎥⎦ ⎭ ⎩ ⎣ 1−τ ⎦

(54)

where τ is the dimensionless temperature difference

τ=

Tw − To , Tw

λ is the dimensionless length of pipe

λ=

L , D

and Ec is the Eckert number defined as

Ec =

U2 . C pTw

The total rate of heat transfer to the fluid is obtained by integrating Eq. (48) along the pipe length and can be written as:

Q = m C p (Tw − To )(1 − e −4 Stλ ).

(55)


333

Now defining a dimensionless entropy generation as:

S

ψ = gen Q /(Tw − To ) Eq. (54) can be written as

ψ=

⎧ ⎡1 − τe −4 Stλ ⎤ 1 1 Ec ⎡ e 4 Stλ − τ ⎤ ⎫ −4 Stλ ln − − + ( 1 ) τ ln e f ⎨ ⎬ 1 − e −4 Stλ ⎩ ⎢⎣ 1 − τ ⎥⎦ 8 St ⎢⎣ 1 − τ ⎥⎦ ⎭

(56)

Therefore, two dimensionless groups naturally arise in Eq. (56) as:

Π1 = Stλ

(57)

and

Π2 = f

Ec St

(58)

Thus Eq. (56) can be written in a compact form for the constant viscosity assumption as:

ψ=

1 1 − e −4 Π1

⎧ ⎡1 − τe −4Π1 ⎤ ⎡ e 4 Π1 − τ ⎤ ⎫ 1 −4 Π1 ln − − + Π ( 1 ) τ ln e ⎨ ⎢ 2 ⎥ ⎢ 1 −τ ⎥⎬ 8 ⎣ ⎦⎭ ⎩ ⎣ 1−τ ⎦

which is a function of three non-dimensional parameters, namely these parameters,

(59)

τ , Π1 and Π 2 . Among

τ represents the fluid inlet temperature To and Π1 represents the pipe

length L. Once the type of the fluid and the mass flow rate are fixed, the parameter Π 2 can be calculated based on temperature analysis. Thus,

τ and Π1 are the two design parameters

that can be varied for determining the effects of pipe length and/or inlet fluid temperature on the entropy generation. Since Π1 represents the length of the pipe, the dimensionless entropy generation defined on the basis of total heat transfer rate to the pipe,

ψ , decreases initially and then starts

increasing along the pipe length. The rate of increase in entropy generation approaches a constant value as the total heat transfer rate to the fluid approaches its maximum value of

Q max = m C p (Tw − To )

(60)

For long pipes where e 1 << 1 and e 1 >> τ , it can be shown from Eq. (59) that the entropy generation increases linearly with the slope −4 Π

4Π

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Ahmet Z. Sahin

dψ Π 2 = dΠ 1 2 for the case of constant viscosity. Since τ represents the difference between the temperature of the pipe surface and that of the inlet fluid, the dimensionless entropy generation defined on the basis of total heat transfer rate to the pipe, ψ , increases as τ increases due to the increase in the gap between the bulk and wall temperatures. For small values of τ , the total entropy generation is due to the viscous friction; and in the limit when τ = 0 , the total dimensionless entropy change using Eq. (59) becomes:

limψ = τ →0

1 Π1Π 2 . 2 1 − e −4 Π1

(61)

It should be noted that the dimensionless entropy generation, ψ , in the above analysis is a function of the total heat transfer rate, Q , which in turn depends on the length of the pipe and inlet fluid temperature. However, a modified dimensionless entropy generation can be defined on the basis of unit heat capacity rate of fluid in the pipe as:

φ=

S gen S = gen m C p Q /(ΔT )

(62)

where ΔT is the increase of the bulk temperature of the fluid in the pipe, ΔT = T ( L) − To .

C p is constant for fixed mass flow rate and Noting that Q / ΔT = m

φ = (1 − e −4Π )ψ .

(63)

1

φ , indicating the total entropy generation along the pipe, is expected to increase with the increase in pipe length.

φ and ψ differ only for small values of Π1 . For large values of Π1 , φ = ψ as can be seen from Eq. (63). The modified dimensionless entropy generation defined, based on the unit heat capacity rate through the pipe, φ , shows similar behavior to that of ψ . This was expected, since φ

(

and ψ are related through a constant factor of 1 − e

−4 Π1

) as given in Eq. (63).


335

Pumping Power to Heat Transfer Rate Ratio The pumping power to heat transfer rate ratio is

Pr =

π D 2 ΔPU 4

Q

.

(64)

Using Eqs. (50) and (53), the pumping power to heat transfer rate ratio, Pr is obtained as

Pr =

1 Π1Π 2 . 2 τ (1 − e −4 Π1 )

(65)

Due to an increase in the bulk temperature and a decrease in viscosity, the pumping power to heat transfer rate ratio may decrease initially and then increase, as the total heat transfer rate to the fluid decreases as the bulk temperature approaches the wall temperature as ψ is increased. For large values of ψ and constant viscosity, it can be shown from Eq. (65) that the pumping ratio, Pr, increases linearly with the slope

dPr Π 2 = . dΠ1 2τ For small values of Π1 the effect of the viscosity variation is small and in the limit,

lim Pr =

Π1 →0

Π2 8τ

(66)

which is clearly a function of the fluid viscosity.

Effect of Pipe Cross Section The dimensionless entropy generation as given in Eqn. (59) can be re-written as function of Re as (Sahin, 1998)

ψ=

1 1 − e −C1 / Re

⎧ ⎡1 − τe −C1 / Re ⎤ ⎡ e C1 / Re − τ ⎤ ⎫ −C1 / Re 2 − − + ln τ ( 1 ) Re ln e C ⎨ ⎢ 2 ⎥ ⎢ 1 − τ ⎥⎬ ⎣ ⎦⎭ ⎩ ⎣ 1 −τ ⎦

where C1 = 4 Re Π1 and C2 =

Π2 . 8 Re 2

(67)

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Ahmet Z. Sahin

For any given pipe geometry and stream of constant thermophysical property fluid, dimensionless numbers C1 and C2 are constants. Therefore, the dimensionless total entropy generation as given in Eq. (67) is a function of Re and τ only. Values of Nu and ( f Re) for fully developed laminar flow are given in Shah and London (1978) for a variety of pipe geometries. The hydraulic diameter DH for any given pipe geometry can be calculated using the relation:

DH = 4

Ac . p

Similarly,

φ = (1 − e −C / Re )ψ

(68)

C1C2 Re τ (1 − e −C1 / Re )

(69)

1

and

Pr =

For very low Re number flow, it is interesting to note that the importance of geometry disappears. In the limit as

⎛ 1 ⎞ lim ψ = ln⎜ ⎟ +τ Re→0 ⎝1+τ ⎠

(70)

As the temperature difference between the fluid inlet and the surface of the pipe, τ , is increased, the entropy generation increases, however, the pumping power requirement per unit heat transfer decreases. In the limit when τ approaches zero, the contribution of heat transfer to the entropy generation disappears and

limψ = C1C2 Re τ →0

(71)

which indicates a linear increase in entropy generation that is solely due to the viscous friction as expected. Figure 6 shows the variation of dimensionless entropy generation, ψ , with Re number for five selected pipe geometries; namely circular, square, equilateral triangle, rectangular (aspect ratio of 1/2) and sinusoidal (aspect ratio 3 / 2 ). The cross sectional area of each pipe, Ac, and the inlet fluid temperature, To, are kept constant. For low Re numbers, sinusoidal pipe geometry gives lowest entropy generation of all the pipe geometries considered. However, as the Re number is increased, circular pipe geometry becomes the best choice. In general, the total entropy generation tends to decrease initially and then increases while the Re number is increased.

First and Second Law Thermodynamics Analysis of Pipe Flow The dimensionless total entropy generation,

337

ψ , includes contributions due to heat

transfer and viscous friction. As the Re number is increased, the contribution due to heat transfer decreases and that of viscous friction increases. Therefore, for low Re number flow for which the viscous frictional contribution can be neglected, those pipe geometries with higher surface areas are favorable. This is because of the fact that the total heat transfer is a function of the surface area for fixed cross sectional area and length of pipe. On the other hand, circular pipe geometry is the best choice for high Re numbers where viscous frictional contribution to the entropy generation becomes dominant. Sinusoidal and square pipe geometries appear to be good choices after the circular pipe for the high Re number region. The dimensionless entropy generation is found to be high for triangular and rectangular pipes almost for the entire Re number region. Therefore, they are the worst choices from the point of view of entropy generation. This conclusion is obtained only if the above comparison of entropy generation is made by keeping the flow cross sections of the pipes constant. Comparison of pipe geometries with same equivalent diameter, on the other hand, leads to the results in which triangular and square pipes seem to be better choices than the circular pipe geometry. This may be misleading. Rectangular pipes would yield the best results when the aspect ratio becomes 1, which corresponds to square geometry, and the square geometry is almost nowhere better than the circular one. For lower aspect ratio rectangular pipes there may not even exist a minimum entropy generation. For very low Re number flow, it is interesting to note that the importance of geometry disappears and all the curves approach to the same value as Re number approaches zero. This limit of dimensionless entropy generation is a function of the temperature difference between the inlet fluid stream and surface of the pipe, τ , as

⎛ 1 ⎞ lim ψ = ln⎜ ⎟ +τ ⎝1+τ ⎠

Re→0

The modified dimensionless entropy generation,

(72)

φ , versus Re number is given in Figure

7. The most favorable pipe geometry appears to be the sinusoidal type in this case, for the whole range of Re numbers. The circular geometry is the next favorable geometry as far as the modified dimensionless entropy generation is concerned; however, the differences in entropy generation for the geometries selected are not significant. It should be noted that, there is no optimum Re number to provide a minimum modified entropy generation, since the total heat transfer increases as Re number is increased. Figure 8 shows the pumping power requirement to overcome the viscous friction. Clearly the circular geometry is superior to all other geometries. Sinusoidal geometry appears to be a bad choice from the point of view of pumping power to heat transfer ratio. Triangular pipe is the worst choice in this case.

338

Ahmet Z. Sahin

Figure 6. Dimensionless entropy generation

ψ

versus Re number for various pipe geometries,

Ac = 4 × 10 −7 m 2 and τ = 0.01 .

Figure 7. Modified dimensionless entropy generation geometries Ac = 4 × 10 m and τ = 0.01 .

φ

versus Re number for various pipe


Figure 8. Pumping power to heat transfer ratio

Pr

339

versus Re number for various pipe geometries

Ac = 4 × 10 −7 m 2 and τ = 0.01 .

Appendix: Viscosity Dependence on Temperature Experimentally it is known that the viscosity of liquids vary considerably with temperature. Around room temperature (293 K), for instance, a 1% change in temperature produces a 7% change in viscosity of water and approximately a 26% change in viscosity of glycerol (Sherman, 1990). As a first approximation, the variation of viscosity with temperature can assumed to be linear,

μ (T ) = μref − b(T − Tref ) where b is a fluid dependent dimensional constant and Tref is a reference temperature (= 293 K). This would be a reasonable approximation if the bulk temperature variation is small. However, for highly viscous liquids the variation of viscosity with temperature is exponential and a more accurate empirical correlation of liquid viscosity with the temperature is given by Sherman (1990) as

⎛ T μ (T ) = μref ⎜⎜ ⎝ Tref

n

⎡ ⎛1 ⎞ 1 ⎟⎟ exp ⎢ B ⎜⎜ − ⎢⎣ ⎝ T Tref ⎠

⎞⎤ ⎟⎟ ⎥ ⎠ ⎦⎥

340

Ahmet Z. Sahin

where n and B are fluid dependent constant parameters. In the present work, both the linear and the exponential viscosity models were used. In addition, a constant viscosity model (in which the numerical value of viscosity at a reference bulk temperature is taken as constant) is also included in order to see the significance of the variation of viscosity on the results.

Nomenclature Cp

Specific heat (J/kg K)

D

Diameter (m)

Ec

2 Eckert number, U / ⎡C p q′′ / h ⎤

f

Average Darcy friction factor

h k L m

Average heat transfer coefficient (W/m2 K) Thermal conductivity (W/m K) Length of the pipe (m) Mass flow rate (kg/s)

Nu P Pr Pr q′′ Q

Average Nusselt number, hD / k Pressure (N/m2) Prandtl number Pumping power to heat transfer rate ratio

⎣

(

)⎦

Heat flux (W/m2) Total heat transfer rate (W)

Re Reynolds number, ρUD / μb s Entropy (J/kg K) S gen Entropy generation (W/K)

( ρUC )

St

Stanton number, h /

T T0

Temperature (K) Inlet fluid temperature (K)

p

Tref Reference temperature (=293 K) Tw

Wall temperature of the pipe (K)

U x

Fluid bulk velocity (m/s) Axial distance (m)

Greek Symbols

ΔT Increase of fluid temperature (K)

μ

Viscosity (N s/m2)


μb

341

Viscosity of fluid at bulk temperature (N s/m2)

μref Viscosity of fluid at reference temperature (N s/m2) μw λ

Viscosity of fluid at wall temperature (N s/m2)

Π1

Modified Stanton number, Stλ

Dimensionless axial distance, L / D

Π 2 Dimensionless group, fEc / St

(

)

ψ

Dimensionless entropy generation, S gen / Q / T0

φ ρ

Modified dimensionless entropy generation, S gen / Q / ΔT

τ

Dimensionless wall heat flux, q′′ / h / T0

θ

Dimensionless temperature, ( T − T0 ) / q′′ / h

(

)

3

Density (kg/m )

(

)

(

)

c. p. Constant properties

References [1] Bejan, A., Advanced Engineering Thermodynamics, John Wiley and Sons, New York (1988). [2] Bejan, A., Entropy Generation Minimization, CRC Press, New York (1996). [3] Incropera, F.P. and DeWitt, D.P., Introduction to Heat Transfer, Third Ed., John Wiley, New York (1996). [4] Kays, W. M. and Crawford, M. E., Convective Heat and Mass Transfer, Third Ed. McGraw Hill, New York (1993). [5] Kays, W. M. and London, A. L., Compact Heat Exchangers, 3rd Ed., McGraw Hill, New York (1984). [6] Kreith, F. and Bohn, M.S., Principles of Heat Transfer, fifth ed., West Publ. Co., New York (1993). [7] Shah, R. K., Compact Heat Exchanger Design Procedures, in: Heat Exchangers, Thermal-Hydraulic Fundamentals and Design, Ed: Kakac, S., Bergles, A.E., and Mayinger, F., Mc-Graw Hill, New York (1981). [8] Shah, R.K., and London, A.L., Laminar Flow Forced Convection in Ducts, Academic Press, New York (1978). [9] Sahin, A.Z., Irreversibilities in Various Duct Geometries with Constant Wall Heat Flux and Laminar Flow, Energy - The International Journal, vol. 23, no. 6, pp. 465-473, (1998). [10] Sahin, A.Z., A Second Law Comparison for Optimum Shape of Duct Subjected to Constant Wall Temperature and Laminar Flow, Heat and Mass Transfer, vol. 33, pp. 425-430, 1998.

342

Ahmet Z. Sahin

[11] Sahin, A.Z., Second Law Analysis of Laminar Viscous Flow Through a Duct Subjected to Constant Wall Temperature, ASME Journal of Heat Transfer, vol. 120, no. 1, pp. 7683, (1998). [12] Sahin, A.Z., The Effect of Variable Viscosity on the Entropy Generation and Pumping Power in a Laminar Fluid Flow through a Duct Subjected to Constant Heat Flux, Heat and Mass Transfer, vol. 35, pp. 499-506, (1999). [13] Sahin, A.Z., Entropy Generation in Turbulent Viscous Flow Through a Smooth Duct Subjected to Constant Wall Temperature, International Journal of Heat and Mass Transfer, vol. 43, no. 8, pp. 1469-1478, (2000). [14] Sahin, A.Z., Entropy Generation and Pumping Power in a Turbulent Fluid Flow through a Smooth Pipe Subjected to Constant Heat Flux, Exergy, an International Journal, vol. 2, no. 4, pp. 314-321, (2002). [15] Sherman, F.S., Viscous Flow, McGraw Hill Book Co., New York (1990). [16] Tuma, J.J., Engineering Mathematics Handbook, Second Ed., McGraw Hill Book. Co., New York (1979).



Chapter 10

SINGLE-PHASE INCOMPRESSIBLE FLUID FLOW IN MINI- AND MICRO-CHANNELS Lixin Cheng* School of Engineering, University of Aberdeen, King’s College, Aberdeen, AB24 3UE, Scotland, the UK

Abstract This chapter aims to present a state-of-the-art review on single-phase incompressible fluid flow in mini- and micro-channels. First, classification of mini- and micro-channels is discussed. Then, conventional theories on laminar, laminar to turbulent transition and turbulent fluid flow in macro-channels (conventional channels) are summarized. Next, a brief review of the available studies on single-phase incompressible fluid flow in mini- and microchannels is presented. Some experimental results on single phase laminar, laminar to turbulent transition and turbulent flows are presented. The deviations from the conventional friction factor correlations for single-phase incompressible fluid flow in mini and micro-channels are discussed. The effect factors on mini- and micro-channel single-phase fluid flow are analyzed. Especially, the surface roughness effect is focused on. According to this review, the future research needs have been identified. So far, no systematic agreed knowledge of single-phase fluid flow in mini- and micro-channels has yet been achieved. Therefore, efforts should be made to contribute to systematic theories for microscale fluid flow through very careful experiments.

Keywords: Single-phase flow; Mini-channel; Micro-channel; Incompressible fluid flow; Pressure drop; Friction factor; Surface roughness.

1. Introduction Miniaturization has recently become the key word in many advanced technologies as well as traditional industries. The miniaturized systems are being progressively applied in commercial sectors such as the electronics, chemical, pharmaceutical, and medical industries, *


344

Lixin Cheng

as well as in the military sector, for biological and chemical warfare defense, as an example. With the emergence of micro-scale thermal, fluidic, and chemical systems, the development of ultra-compact heat exchangers, miniature and micro pumps, miniature compressors, microturbines, micro-reactors, micro thermal systems for distributed power production has become an important agenda of many researchers, research institutions, and funding agencies. Miniaturization is essential not simply from the standpoint of producing compact systems, but for the challenge of fluid flow with yet smaller and smaller channel sizes, as is the case for cooling of electronic devices, microfluidic components and sensors. In addition, it has been shown that proper miniaturization by use of mini- and micro-channels can result in higher system efficiency. However, fluid flow phenomena in mini- and micro-channels are quite different from those in conventional size channels. Therefore, it is of great significance to understand these fundamental aspects. Due to the significant differences of fluid flow phenomena in mini- and micro-channels as compared to conventional channels or macro-scale channels, one very important issue should be clarified about the distinction between mini- and micro-scale channels and macroscale channels. However, a universal agreement is not yet clearly established in the literature. Furthermore, although research in mini- and micro-channels has been greatly increasing due to the rapid growth applications which require fluid flows in the tiny channels, the available studies reveal contradictory results in single-phase friction factors, the transition for laminar and turbulent flows and the effect factors. There are still significant discrepancies between the experimental results obtained by different researchers. At first, one very important issue should be the clarification of the distinction between micro-scale channels and macro-scale channels when studying fluid flow in mini- and microchannels. However, a universal agreement is not yet clearly established in the literature. Instead, there are various definitions on this issue [1-6]. Here, just to show one example, based on engineering practice and application areas such as refrigeration industry in the small tonnage units, compact evaporators employed in automotive, aerospace, air separation and cryogenic industries, cooling elements in the field of microelectronics and micro-electromechanical-systems (MEMS), Kandlikar et al. [6] defined the following ranges of hydraulic diameters dh which are attributed to different classifications: Conventional channels: dh > 3 mm. Mini-channels: 200 μ m < dh ≤ 3 mm. Micro-channels: 10 μ m < dh ≤ 200μ m. Transitional micro-channels: 1 μ m < dh ≤ 10μ m. Transitional nano-hannels: 0.1 μ m < dh ≤ 1μ m. Nano-channels: dh ≤ 0.1 μ m. In the case of non-circular channels, it is recommended that the minimum channel dimension, for example, the short side of a rectangular cross-section should be used in place of the diameter d. In the available studies on fluid flow in mini- and micro-channels, some researchers have concluded that the conventional theories work for mini- and micro-channels while others have showed that the conventional theories do not work well. Therefore, these controversies should be clarified. As the channel size becomes smaller, some of the conventional theories for (bulk) fluid, energy and mass transport need to be revisited for validation. There are two

Single-Phase Incompressible Fluid Flo in Mini- and Micro-channels

345

fundamental elements responsible for departure from the conventional theories at mini- and micro-scale. For example, differences in modeling fluid flow in mini- and micro-channels may arise as a result of (i) a change in the fundamental process, such as a deviation from the continuum assumption for fluid flow, or an increase influence of some additional forces, such as electrokinetic forces, etc. (ii) uncertainty regarding the applicability of empirical factors derived from experiments conducted at larger scales, such as entrance and exit loss coefficients for fluid flow in pipes, etc., (iii) uncertainty in measurements at mini- and microscale, including geometrical dimensions and operating parameters. In this chapter, the available studies of single-phase incompressible fluid flow in miniand micro-channels are reviewed to address these relevant important issues.

2. Single-Phase Frictional Pressure Drop Methods in Macro-channels 2.1. Laminar and Turbulent Flow The flow of a fluid in a pipe may be laminar or turbulent flow, or in between transitional flow. Figure 1 shows the x component of the fluid velocity as a function of time t at a point A in the flow for laminar, transitional and turbulent flows in a pipe. For laminar flow, there is only one component of velocity. For turbulent flow, the predominant component of the velocity is also along the pipe, but it is unsteady (random) and accompanied by random components normal to the pipe axis. For transitional flow, both laminar and turbulent features occur.

Q

A

x

VA Turbulent Transitional Laminar t Figure 1. Time dependence of fluid velocity at a point.

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Lixin Cheng

For pipe flow, the most important dimensionless parameter is the Reynolds number Re, the ratio of the inertia to viscous effects in the flow, which is defined as Re =

ρVd μ

(1)

where V is the average velocity in the pipe. The flow in a pipe is laminar, transitional or turbulent provided the Reynolds number Re is small enough, intermediate or large enough. It should be pointed out here that the Reynolds number ranges for which laminar, transitional or turbulent pipe flows are obtained cannot be precisely given. The actual transition from laminar to turbulent pipe flow may take place at various Reynolds numbers, depending on how much the flow is distributed by vibrations of the pipe, roughness of the entrance region, and the like. For general engineering purposes (i.e. without undue precautions to eliminate such disturbances), the following values are appropriate: the flow in a round pipe is laminar if the Reynolds number Re is less than approximately 2100 and the flow in is turbulent if the Reynolds number Re is greater than approximately 4000. For the Reynolds numbers between these two limits, the flow may switch between laminar and turbulent conditions in an apparently random fashion (transitional flow).

2.2. Entrance Region (Developing Flow) and Fully Developed Flow The region of flow near where the fluid enters a pipe is termed the entrance region (developing flow) and is illustrated in Figure 2. The fluid typically enters the pipe with a nearly uniform velocity profile at section 1. As the fluid moves through the pipe, viscous effects cause it to stick to the pipe wall (the no-slip boundary condition). Thus, a boundary layer in which viscous effects are important is produced along the pipe wall such that the initial velocity profile changes with distance along the pipe, x, until the fluid reaches the end of the entrance length, section 2, beyond which the velocity profile does not vary with x. The boundary layer has grown in thickness to completely fill the pipe. Viscous effects are of considerable importance within the boundary layer. For fluid outside the boundary layer (within the inviscid core surrounding the centerline from 1 to 2), viscous effects are negligible. The shape of the velocity profile in the pipe depends on whether the flow is laminar or turbulent, as does the length of the entrance region, le. As with many other properties of pipe flow, the dimensionless entrance length le/d, correlates quite well with the Reynolds number Re. Typical entrance lengths are given by le = 0.06 Re for laminar flow d

(2)

le = 4.4 Re1/ 6 for turbulent flow d

(3)

Calculation of the velocity profile and pressure distribution within the entrance region (developing flow) is quite complex. However, once the fluid reaches the end of the entrance region, section 2 in Figure 2, the flow is simpler to describe because the velocity is a function


347

of only the distance from the pipe centerline, r, and independent of x. The flow after section 2 is termed fully developed flow.

Entrance region flow le Fully developed flow Inviscid core r V

x 1

2 d

Boundary layer

Figure 2. Entrance region, developing flow and fully developed flow in a pipe.

2.3. Single-Phase Friction Pressure Drop Methods The nature of the pipe flow is strongly dependent on weather the flow is laminar or turbulent. This is a direct consequence of the differences in the nature of the shear stress in laminar and turbulent flows. The shear tress in laminar flow is a direct result of momentum transfer among the randomly moving molecules (a microscopic phenomenon). The shear stress in turbulent flow is largely a result of momentum transfer among the randomly moving, finite-sized bundles of fluid particles (a macroscopic phenomenon). The net result is that the physical properties of the shear stress are quite different for laminar than for turbulent flow. The friction pressure drop for both laminar and turbulent flow can be expressed as Δp = f

l ρV 2 d 2

(4)

For fully developed laminar flow in a circular pipe, the friction factor f is simply as f =

64 Re

(5)

ε

. d For turbulent flow, the dependence of the friction factor on the Reynolds number Re is much more complex than that given by Eq. (5) for laminar flow. For fully developed turbulent flow and transition from laminar to turbulent flow. The Moody chart [7] shown in Figure 3 provides the friction factor f, which can be expressed as

and the value of f is independent of the relative roughness

ε⎞ ⎛ f = φ ⎜ Re, ⎟ d⎠ ⎝

(6)

Figure 3. Friction factor as a function of the Reynolds number and relative roughness for circular pipes-the Moody chart [7].


349

For turbulent flow, the friction factor f is a function of the Reynolds number Re and ε relative roughness . The results are obtained from an exhaustive set of experiments and d usually presented in terms of a curve-fitting formulae or the equivalent graphical form. In commercially available pipes, the roughness is not as uniform and well defined as in the artificially roughened pipes. However, it is possible to obtain a measure of the effective relative roughness of typical pipes and thus to obtain the friction factor. Typical roughness values for various pipe surfaces are also given in Figure 3. It should be noted that the values ε do not necessarily correspond to the actual values obtained by a microscopic of d determination of the average height of the roughness of the surface. They do, however, provide the correct correlation for friction factor f. The following characteristics are observed from Figure 3 for laminar flow, friction factor is calculated by Eq. (2), which is independent of relative roughness. For very large Reynolds number Re, the friction factor is a function of ε the relative roughness as d ⎛ε ⎞ f =φ⎜ ⎟ ⎝d ⎠

(7)

which is independent of the Reynolds number Re. For such flows, commonly termed completely turbulent flow (or wholly turbulent flow), the laminar sub-layer is so thin (its thickness decreases with increasing the Reynolds number Re) that the surface roughness completely dominates the character of the flow near the wall. Hence, the pressure drop required is a result of an inertia-dominated turbulent shear stress rather than the viscositydominated laminar shear stress normally found in the viscous sublayer. For flows with moderate values of Re, the friction factor is indeed dependent on both the ε . Flow in the range of 2100< Re <4000 is a Reynolds number Re ad relative roughness d result of the fact that the flow in this transition range may be laminar or turbulent (or an unsteady mix of both) depending on the specific circumstances involved. Note that even for smooth pipes (ε = 0), the friction factor is not zero. That is, there is a head loss in any pipe, no mater how smooth the surface is made. This is a result of the no-slip boundary condition that requires any fluid to stick to any solid surface it flows over. There is always some microscopic surface roughness that produces the no-slip behavior (and thus the friction factor f does not equal 0) on the molecular level, even when the roughness is considerably less that the viscous sublayer thickness. Such pipes are called hydraulically smooth, and for turbulent flow in this flow, the Blasius [8] equation is used for friction factor as f =

where 4000< R e < 100,000.

0.3164 Re0.25

(8)

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Lixin Cheng

The Moody chart shown in Figure 3 is universally valid for all steady fully developed incompressible pipe flows. The following equation from the Colebrook and White [9] is valid for the entire non-laminar range of the Moody chart, as ⎛ε /d 2.51 = −2 log ⎜ + ⎜ 3.7 Re f f ⎝

1

⎞ ⎟⎟ ⎠

(9)

In fact, the Moody chart is a graphical representation of this equation, which is an empirical fit of the pipe flow pressure drop data. A difficulty with its use is that it is implicit ε in the dependence of f. that is, for given conditions (Re and ), it is not possible to solve f d without some sort of iterative scheme. For easy-to-use, Swamee and Jain [10] proposed an explicit Colebrook-White equation as follows: f =

0.25 ⎡ ⎛ ε / d 5.74 ⎢ log ⎜ 3.7 + Re0.9 ⎣ ⎝

⎞⎤ ⎟⎥ ⎠⎦

(10)

2

which matches the Colebrook-White equation within 1% for 10-6 < ε/d < 10-2 and 5000 < Re < 108. Churchill [11] proposed a more complicated expression for all flow regimes and all the relative roughnesses, which agrees well with the Moody diagram: 1/12

−1.5 16 ⎧ ⎫ 12 16 ⎤ ⎡ ⎛ ⎞ 1 37530 ⎪⎛ 8 ⎞ ⎪ ⎛ ⎞ ⎢ ⎥ ⎟ +⎜ f = 8 ⎨⎜ ⎟ + ⎢ 2.457 ln ⎜⎜ ⎟ ⎥ ⎬ 0.9 ⎟ ⎝ Re ⎠ ⎪⎝ Re ⎠ ⎝ ( 7 / Re ) + 2.7ε / d ⎠ ⎣ ⎦ ⎭⎪ ⎩

(11)

For noncircular ducts: the hydraulic diameter dh is used. The hydraulic diameter of a noncircular enclosure is four times the cross sectional area divided by the wetted perimeter. The wetted perimeter is the edge of the cross sectional area that is in direct contact with the flow medium, as dh =

4A Pwetted

(12)

Given the complexities of viscous sublayers, turbulence shear stress and surface roughness, etc., the use of the hydraulic diameter method introduces some limitations. For one, the use of the hydraulic diameter dh is only valid if the ration of the duct’s height to width is less than about 3 or 4 [8]. Use the hydraulic diameter method to estimate head loss in laminar flow can also introduce large errors. This is due to friction from the action of viscosity throughout the whole body of the flow. It is independent of surface roughness and is not associated with the region close to the boundary walls. Although the details of the flows


351

in non-circular conduits depend on the exact cross-sectional shape, many round pipe results can be carried over, with slight modification, to flow in conduits of other shapes. Practical, easy-to-use results can be obtained as follows: regardless of the cross-sectional shape, there are no inertia effects in fully developed laminar pipe flow. Thus, the friction factor can be written as f =

C Re h

(13)

where the constant C is the Poiseuille number which depends on the particular shape of the duct, and Reh is the Reynolds number based on the hydraulic diameter dh. The hydraulic diameter is also used in the definition of the friction factor and the relative roughness. Typical values of C for concentric annulus and rectangle channels are given in Table 1 along with the hydraulic diameter. Note that the value of C is relatively insensitive to the shape of the conduit. Unless the cross section is very “thin” in some sense, the value of C is not too different from its circular pipe value, C = 64 as shown in Eq. (5). Shah and London [12] provide the following correlation for a rectangular channel with short side a and long side b, and a channel aspect ratio ac = a/b. C = f Re = 24 (1 − 1.3553ac + 1.9467 ac2 − 1.7012ac3 + 0.9564ac4 − 0.2537 ac5 )

(14)

Table 1. Friction factors for laminar flow in noncircular ducts [8] Shape

Hydraulic diameter

Concentric annulus d h = d 2 − d1

Rectangle

dh =

a b

2ab a+b

Parameter

C = f Reh

d1/d2 0.0001 0.01 0.1 0.6 1.0

71.8 80.1 89.4 95.6 96

a/b 0 0.05 0.10 0.25 0.50 0.75 1.00

96 89.9 84.7 72.9 62.2 57.9 56.9

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Once the friction factor is obtained, the calculations for noncircular conduits are identical to those for circular pipes. Calculations for fully developed turbulent flow in ducts of noncircular cross section are usually carried out by the Moody chart data for circular pipes with the diameter replaced by the hydraulic diameter and the Reynolds number based on the hydraulic diameter. Such calculations are usually accurate to within about 15%.

3. Single-Phase Frictional Pressure Drop Methods in Mini- and Micro-Channels To explore the fundamental physical mechanisms of fluid flow in mini- and microchannels, many effects, including the size effect, the surface roughness effect, viscous effect, electrostatic force effect in the channel wall, surface geometry and the measurement errors, etc. should be taken into account. The conventional theories for macro-channels are the fundamental to investigation of single-phase fluid flow in mini- and micro-channels. So far, a number of studies have been performed to study the hydrodynamic characteristics of incompressible fluid flow in mini- and micro-channels. However, divergences of conclusions still exist in quite a few fundamental understandings of the microscale fluid flow phenomena. In this section, some recent studies of incompressible fluid flow and in mini- and microchannels are reviewed and the main conclusions from these studies are summarized. There is big contradictory regarding the applicability of the conventional theories to microscale fluid flow phenomena. Several good reviews on fluid flow in mini- and microchannels were presented by Kandliar [6], Celata et al. [13] Guo and Li [14], Sobhan and Garimella [15] and Morini [16]. There are large deviations among the available studies in the literature. Little agreement among the experimental results of single-phase pressure drops in the laminar and turbulent flow regimes by different researchers has been reached. Although some experimental results agree with the conventional theories, most of the experimental results deviate from the conventional theories, with both under- and over-predictions obtained. Especially, the experimental results also show different trends of variation relative to the conventional predictions in the laminar and turbulent flow regimes. In addition, quite different results on the transition from laminar to turbulent regimes have been obtained by different investigators. Some reported that the transition occurred at lower Reynolds numbers while others reported that the transition agreed with the conventional theory. In general, the fundamental aspects of fluid flow in mini- and micro-channels are not well understood so far. Here only some representative studies in the literature are reviewed below to show these differences. Mala and Li [17] conducted experiments of water flowing in micro-channels with diameters from 50 to 254 μm. Their length-to-diameter ratio was from 1200 to 5000 and the materials were fused silica and stainless steel. Their experimental results showed significant higher values than those predicted by the conventional theory. The difference increased as the microtube diameter decreased. The effect is caused by the surface roughness or an early transition from laminar to turbulent flow regime. They found an early transition from laminar to turbulent flow in the Reynolds number from 300 to 900, while the flow changed to fully developed turbulent flow at the Reynolds number larger than 1000 to 1500.


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Yu et al. [18] conducted experiments on nitrogen gas and water flowing in microchannels with diameter 19, 52 and 102 μm. The test Reynolds number ranged from 250 to 20,000. Their experimental results showed that the friction factor was lower than the conventional theory for fully developed laminar flow while the transition from laminar to turbulent flow occurred in the Reynolds number range 2000 to 6000. Xu et al. [19] conducted experiments on water flowing in micro-channels with hydraulic diameter from 50 to 300 um and their test Reynolds number was from 50 to 1500. Their experimental results showed that the transition to from laminar to turbulent flow region did not occur in their test Reynolds number range. They found that the flow characteristics deviated from conventional theory when the channel dimensions were below 100 μm. Their friction factor was smaller than the predicted results by the conventional theory. Later on, Xu et al. [20] reported a similar study of water flow in micro-channels with hydraulic diameter from 30 to 344 and the Reynolds number from 20 to 4000. Their experimental results showed that the experimental friction factors agreed with the conventional theories. Although the channel sizes were similar using the same test fluid, their experimental results are quite different. Judy [21] conducted experiments of water, hexane and isopropanol flowing in fused silica capillary tubes with diameters from 20 to 150 μm. Their experimental results showed that when the tube diameter was lower than 100 μm, the friction factor deviated significantly from the conventional theory. This is in consistence with that of Xu et al. [19]. The deviation is independent of the Reynolds number and depended on the tube diameter. However, later on, they did very accurate experiments on fluid flow in micro-channels with diameters from 15 to 150 μm with the same fluids and the similar test conditions used in their previous study [22]. Their experimental results showed that the friction factors were in good agreement with the conventional theory. The value of the Poiseuille number C was found close to 64, considering the experimental error and it was independent of the Reynolds number for Re < 2000. Gao et al. [23] conducted experiments on water flowing in micro-channels with diameters from 200 to 1923 μm. Their experimental results were in good agreement with the conventional theories. Figure 4 shows their results on the influence of the Reynolds number and channel height on the Poiseuille number. Both the Shah and London [12] and the Balsius [8] correlations suitably predicted their experimental data for laminar and turbulent regimes. No scale effects were found in their experiments for flow hydrodynamics. Similar conclusions were obtained by Caney et al. [24], Grimella and Singal [25], Qi et al. [26], Lee and Lee [27], Warrier et al. [28] and Qu and Mudawar [29] in their single-phase friction pressure drop experiments in microchannels. Celata et al. [30] conducted experiments of R114 liquid flowing in 6 parallel stainless steel microtubes with a diameter of 130 μm and a tube length of 90 mm for both laminar and turbulent flows (Reynolds number from 100 to 8000). The relative roughness of the test tubes was 2.65%. Their experimental results showed that the single-phase pressure drop in laminar flow agreed with the conventional theory. The laminar to turbulent flow transition was in the Reynolds number range of 1880 to 2480. They noted that the high relative roughness played an important role in the laminar to turbulent transition. Figure 5 shows their test results of friction factor in both laminar, transitional and turbulent flows compared to the Blasius

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equation and the Colebrook correlation. The Blasuis correlation underpredicts their data while the Colebrook correlation overpredicted their data.

Figure 4. Influence of Reynolds number and channel height on Poiseuille number: + e = 1 mm, × e = 0.7 mm, ● e = 0.5 mm, e = 0.4 mm, ▲e = 0.3 mm, ◊ e = 0.2 mm, □ e = 0.1 mm, – – – Blasius law [23].

Figure 5. Comparison of experimental friction factors to the Colebrook and the Blasius correlations [30].


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Wibel and Ehrhard [31] conducted experiments on the laminar to turbulent transition of deionized water flowing in rectangular stainless steel microchannels with a hydraulic diameter of about 133 μm. Three aspect ratios of 1:1, 1:2 and 1:5 were studied. Their results showed that the laminar to turbulent transition was in the Reynolds number range of 19002200, which agreed with the conventional theory. Li et al. [32] conducted experiments on single-phase friction pressure drops of deionized water flowing in smooth fused silica microtubes and rough stainless steel microtubes with diameters of 50 to 100 μm and 373 to 1570 μm, respectively. Their test Reynolds number was from 20 to 2400. For the stainless steel tubes, the surface relative roughnesses of 2.4%, 1.4% and 0.95% were tested. Figure 6 shows their experimental friction factors for the fused silica tubes compared to the conventional theory and Figure 7 shows those for stainless steel tubes. Their experimental friction factors were well predicted by the conventional theory for the smooth silica tubes while for the roughness stainless steel tubes, their experimental friction factors were higher than the predictions by the conventional theory and increased with the relative roughness. For deionized water flowing through stainless steel micro-tubes with a relative surface roughness less than about 1.5%, the friction factors also agreed with the conventional theory. When the relative surface roughness was larger than 1.5%, the experimental friction factor of the microtube flow showed an appreciable deviation from the conventional theory.

Figure 6. Comparison of experimental friction factors to the conventional theory for the fused silica tubes [32].

In general, the available experimental results on microscale fluid flow in the literature have demonstrated that there are big disagreements among the studies on the single phase friction pressure drops in terms of the friction factors for laminar and turbulent flows and the laminar to turbulent flow transition. It seems that conventional theory works for relative large channels while it does not work well for relative smaller channels. So far, no systematic agreed theories on microscale single-phase fluid flows have yet been achieved. Quite big

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contradictory exists in the available studies. Some researchers reported that the friction factors for laminar fully developed flow were lower than the conventional theory, some reported that the friction factors for laminar fully developed flow were higher than the conventional values while some reported that the friction factor could be predicted by the conventional theory. Some reported that the Poiseuille number for laminar fully developed flow depends on the Reynolds number while some reported that it did not. In fact, the friction factor depends on the material of the micro-channel walls (metals, semi-conductors and so on) and the test fluid (polar fluid or not), thus evidencing the importance of electro-osmotic phenomena to microscales. Furthermore, the dependence of the friction factor on the relative roughness of the micro-channel wall also exits in laminar flow regime. Regarding the laminar-to-turbulent flow transition in microchannels, no agreement on the transitional Reynolds number has been reached so far.

Figure 7. Comparison of experimental friction factors to the conventional theory for the stainless steel tubes [32].

It should be pointed out that no study on the entrance length (developing flow) in miniand micro-channels is available in the literature so far. This poses a question: whether the correlation for entrance length in macro-channels can be used for mini- and micro-channels? No doubt, it is very important to identify the developing and the fully developed flow correctly in mini- and micro-channels. Simply using the criteria for macrochannels might results in very big errors in the microscale fluid flow research.


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4. Channel Size Effect on Single Phase Incompressible Fluid Flow in Mini- and Micro-channels The effects of the channel size are of significance in fluid flow. These can be the effect of experimental uncertainties, the effect of surface roughness and the like. Furthermore, the effect of surface electrostatic charges may also be significant but beyond the scope of this review. The accurate measurements become more difficult in mini- and micro-channels than in macrochannels. For example, how can we accurately measure the very small flow rate in mini- and micro-channels? For example, Celeta et al. [30] used 6 parallel stainless steel microtubes in their experiments as shown by their test section in Figure 8. This test section provided an easy method to measure the flow rate in the microtubes. However, it poses another question: how could the flow rate be uniformly distributed in each sub-microtube? In fact, the accuracy of the flow rate measurements can greatly affect the pressure drop results. Furthermore, for pressure drop measurements, how could we implement more accurate measurements for the mini- and micro-channels? All these issues should be carefully considered in the experimental facility and test section design [33].

Figure 8. Test section consisting of 6 parallel micro-tubes in the experimental study of Celeta et al. [30].

Careful experimental operation should also be performed for experiments of fluid flow in mini- and micro-channels. As already mentioned in section 3, the studies for the similar test conditions with the same fluids in [19] and [20] showed quite different results. The same case

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is for the studies in [21] and [22]. These two examples have shown the importance of very careful experimental operations in the study of fluid flow in mini- and micro-channels. To perform accurate measurements, very careful test facility design and experimental operation are necessary. Steinke et al. [33] developed an experimental facility for the investigation of single-phase liquid flow in microchannels with a variety of geometries considering the microscale effects. Furthermore, careful experimental uncertainties should be analyzed [13, 34]. The channel geometry evaluation is also very important because the channel dimensions have a major effect on the friction factor calculation. Figure 9 shows the SEM image of the rectangular micro-channel geometry tested by Steinke and Kandlikar [34]. It can be seen how a rectangle channel looks like in microscope, which indicates that the channel shape actually becomes significance for microchannels. Figures 10 and 11 show the SEM images of the circular microchannel geometries measured by Li et al. [32]. Obviously, a very little error in the cross-section area measurement may cause very big errors in the flow rate calculation and thus in the friction factor calculation.

Figure 9. Actual cross section of the tested rectangular microchannel by Steinke and Kandlikar [34]: a = 194 μm, b = 244 μm, L = 10 μm, θ= 85 degrees and dh = 227μm.

The surface roughness plays a very important role in the study of microscale fluid flow [6, 30-32, 35-38]. For fluid flow in rough microchannels, a number of the available studies considering the surface relative roughness effect have shown that the friction factors in microchannels are higher than those in macro-channels, and the surface roughness also leads to the earlier transition from laminar to turbulent. However, some studies have shown that the


dh = 1570

359

μmdh = 624.4μmdh = 373 μm

Figure 10. The SEM images of cross-section and inner surface of three stainless steel test tubes in the experimental study of Li et al. [32].

Cross section [

Surface roughness Figure 11. The SEM images of cross-section and inner surface of a fused silica test tube in the experimental study of Li et al. [32].

relative roughness does not affect the friction factor. Thus, one issue is still open to discuss as what is the relative roughness limitation below or beyond which the channel can be regarded as smooth or rough. For example, to investigate the role of the surface relative roughness effect on pressure drop characteristics in microtubes, Kandlikar et al. [35] conducted experiments of water flowing in two microtubes with diameters of 1067 and 620 μm. For the 1067 μm diameter tube, the effects of the relative roughness from 0.178% to 0.3% on

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pressure drop are insignificant and the tube can be considered smooth. For the 620 μm diameter tube with the same relative roughness values, the pressure drops are increased. In this case, the tube having a relative roughness 0.3% cannot be considered as smooth. Although several studies on the relative surface roughness effects are available [35-38], no general agreed conclusions have been obtained so far. Therefore, it is worthwhile to understand under what condition the surface relative roughness effect can be ignored or cannot be ignored for the fluid flow in mini- and micro-channels.

5. Conclusion The conventional fluid dynamic theories developed for macro-channels are not always applicable to fluid flow in mini- and micro-channels. The available studies on microscale fluid flow in the literature reveal quite a few contradictory and there are still quite big discrepancies among the experimental results by different researchers. Various causes for such deviations are analyzed in this review. Furthermore, the effects of channel size on microscale fluid flow are analyzed. The surface roughness is a very important factor but still not well investigated. So far, no systematic agreed knowledge of microscale fluid flow has yet been achieved. Therefore, efforts should be made to achieve complete theories on fluid flow in mini- and micro-channels.

Nomenclature A a ac b C d e f l Pwetted Re r V x

cross-sectional area of flow channel, m2 length of rectangle, m channel aspect ratio width of rectangle, m constant in Eq. (13) internal tube diameter, m thickness of channel base, m friction factor length, m wetted perimeter, m Reynolds number radical direction mean velocity, m/s x-axis

Greek symbols Δp ε μ

pressure drop, Pa surface roughness, m dynamic viscosity, Ns/m2


361

density, kg/m3 surface tension, N/m

ρ

Subscripts crit e h

critical entrance hydraulic

References [1] S. G. Kandlikar, Fundamental issues related to flow boiling in minichannels and microchannels, Exp. Therm. Fluid Sci., 26 (2002) 389-407. [2] L. Cheng, and D. Mewes, Review of two-phase flow and flow boiling of mixtures in small and mini channels, Int. J. Multiphase Flow, 32 (2006) 183-207. [3] P.A. Kew, and K. Cornwell, Correlations for the prediction of boiling heat transfer in small-diameter channels, Appl. Therm. Eng., 17 (1997) 705-715. [4] S.S. Mehendale, A.M. Jacobi, and R.K. Ahah, Fluid flow and heat transfer at micro- and meso-scales with application to heat exchanger design, ASME Appl. Mech. Rev., 53 (2000) 175-193. [5] L. Cheng, G. Ribatski, and J.R. Thome, Gas-liquid two-phase flow patterns and flow pattern maps: fundamentals and applications, ASME Appl. Mech. Rev., 61 (2008) 050802-1-050802-28. [6] S.G. Kandlikar, S. Garimella, D. Li, S. Colin, and R. King Michael, Heat Transfer and Fluid Flow in Mini-channels and Micro-channels, Elsevier Science & Technology, UK, 2005. [7] N.E. Moody, Friction factors for pipe flow, Trans. ASME, (1944) 671-684. [8] R.M. Olson, Essentials of Engineering Fluid Mechanics, 4th Ed., Harper & Row, New York, 1980. [9] C.F. Colebrook, and C.M. White, Experiments with fluid friction in roughened pipes, Proc Roy Soc. (A), 161 (1937) 367. [10] P.K. Swamee, and A.K. Jain, Explicit equations for pipe-flow problems, J. Hydraulic Div., 102 (1976) 657-664. [11] S.W. Churchill, Friction factor equation spans all fluid flow regimes, Chemical Eng., 7 (1977) 91-92. [12] R.K. Shah, and A.L. London, Laminar Flow Forced Convection in Ducts, Academic Press Inc., New York, 1978. [13] G.P. Celata, Heat Transfer and Fluid Flow in Microchannels, Begell house Inc, New York, 2004. [14] Z.Y. Guo and Z.X. Li, size effect on single-phhase channel flow and heat transfer at microscale, Int. J. Heat Fluid Flow, 24 (2003) 284-298. [15] C.B. Sobhan, and S.V.A. Garimella, Comparative analysis of studies on heat transfer and fluid flow in microchannels, Microscale Thermophysical Eng., 5 (2001), 293-311.

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[16] G.L. Morini, Single-phase convective heat transfer in microchannels: a review of experimental results, Int. J. Therm. Sci. 43 (2004) 631-651. [17] G.M. Mala, and D. Li,, Flow characteristics of water in microtubes, Int. J. Heat fluid Flow, 20 (1999) 142-148. [18] D. Yu, R. Warrington, R. Barron, and T. Ameel, An experimental and theoretical investigation of fluid flow and heat transfer in microtubes, ASME/JSME Thermal Engineering Conference, Vol. 1, ASME, 1995. [19] B. Xu, K.T. Ooi, N.T. Wong, C.Y. Li, and W.K. Choi, Liquid flow in microchannels, Proceedings of the 5the ASME/JSME Joint Thermal Engineering Conference, San Diego, California, March 15-19, 1999. [20] B. Xu, K.T. Ooi, N.T. Wong, and W.K. Choi, Experimental method to calculate the friction factor for liquid flow in microchannels, Int. Comm. Heat Mass Transfer, 27 (2000) 1165-1176. [21] J. Judy, D. Maynes, and B.W. Webb, Liquid flow pressure drop in microtubes, Proceedings of the International Conference on Heat Transfer and Transport Phenomena in Microscale, Banff, Canada, October 15-20, 2000. [22] J. Judy, D. Maynes, and B.W. Webb, Characterization of frictional pressure drop for liquid flows through microchannels, Int. J. Heat Mass Transfer, 45 (2002) 3477-3489. [23] P. Gao, S. L. Person, and M. Favre-Marinet, Scale effects on hydrodynamics and heat transfer in two-dimensional mini and microchannels, Int. J. Therm. Sci., 41 (2002) 10171027. [24] N. Caney, P. Marty, and J. Bigot, Friction losses and heat transfer of ingle-phase flow in a mini-channel, Appl. Therm. Eng., 27 (2007) 1715-1721. [25] S.V. Garimella, and V. Singhal, Single-phase flow and heat transport and pumping considerations in micro heat sinks, Heat Transfer Eng., 25(1) (2004) 15-25. [26] S.L. Qi, P. Zhang, R.Z. Wang, and L.X. Xu, Single-phase pressure drop and heat transfer characteristics of turbulent liquid nitrogen flow in micro-tubes, Int. J. Heat Mass Transfer, 50 (2007) 1993-2001. [27] H.J. Lee, and S.Y. Lee, Heat transfer correlation for boiling flows in small rectangular horizontal channels with low aspect ratios, Int. J. Multiphase Flow, 27 (2001) 20432062. [28] G.R. Warrier, V.K. Dhir, and L.A. Momoda, Heat transfer and pressure drop in narrow rectangular channels, Exp. Therm. Fluid Sci., 26 (2002) 53-64. [29] W. Qu, I. Mudawar, experimental and numerical study of pressure drop and heat transfer in a single-phase micro-channel heat sink, Int. J. Heat Mass Transfer, 45 (2002) 25492565. [30] G.P. Celata, M. Cumo, M. Guglielmi, and G. Zummo, Experimental investigation of hydraulics and single phase heat transfer in 0.130 mm capillary tube, Microscale Thermophysical Eng., 6 (2002) 85-97. [31] W. Wibel, and P. Ehrhard, Experiments on the laminar/turbulent transition of liquid flows in rectangular micochannels, Heat Transfer Eng., 30(1-2) (2009) 70-77. [32] Z. Li, Y.L. He, G.H. Tang and W.Q Tao, Experimental and numerical studies of liquid flow and heat transfer in microtubes, Int. J. Heat Mass Transfer, 50 (2007) 3447-3460. [33] M.E. Steinke, S.G. Kandlikar, J.H. Magerlein, E.G. Colgan, and A.D. Raisanen, Development of an experimental facility for investigating single-phase liquid flow in microchannels, Heat Transfer Eng., 27(4) 2006 41-52.


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[34] M.E. Steinke, and S.G. Kandlikar, Single-phase liquid friction factors in microchannels, Int. J. Therm. Sci., 45 (2006) 1073-1083. [35] S.G. Kandlikar, D. Joshi, and S. Tian, Effect of channel roughness on heat transfer and fluid flow characteristics at low Reynolds numbers in small diameter tubes, in: Proceedings of 35th National Heat Transfer Conference, Anaheim, CA, USA, 2001, Paper: 12134. [36] D. Gloss and H. Herwig, Microchanel roughness effects: a close up view, Heat Transfer Eng., 30(1-2) (2009) 62-69. [37] P. Young, T.P. Brackbill, and S.G. Kandlikar, Comparison of roughness parameters for various microchannels surface in single-phase flow applications, Heat Transfer Eng., 30(1-2) (2009) 78-690. [38] Z.X. Li, D.X. Du, and Z.Y. Guo, Experimental study on flow characteristics of liquid in circular microtubes, Microscale Thermophysical Eng. 7, (2003) 253-265.



Chapter 11

EXPERIMENTAL STUDY OF PULSATING TURBULENT FLOW THROUGH A DIVERGENT TUBE Masaru Sumida* School of Engineering, Kinki University 1 Takaya Umenobe, Higashi−Hiroshima, 739−2116 JAPAN

Abstract An experimental investigation was conducted of pulsating turbulent flow in a conically divergent tube with a total divergence angle of 12°. The experiments were carried out under the conditions of Womersley numbers of α =10∼40, mean Reynolds number of Reta =20000 and oscillatory Reynolds number of Reos =10000 (the flow rate ratio of η = 0.5). Timedependent wall static pressure and axial velocity were measured at several longitudinal stations and the distributions were illustrated for representative phases within one cycle. The rise between the pressures at the inlet and the exit of the divergent tube does not become too large when the flow rate increases, it being moderately high in the decelerative phase. The profiles of the phase-averaged velocity and the turbulence intensity in the cross section are very different from those for steady flow. Also, they show complex changes along the tube axis in both the accelerative and decelerative phases.

Keywords: divergent tube, pulsating flow, turbulent flow, pressure distribution, velocity distribution.

1. Introduction Divergent tubes are an important pipeline component and are widely used as diffusers to convert kinetic energy into pressure energy and as devices to connect two tubes of different diameters in pipework systems and fluid machinery. Therefore, there have been a number of studies, e.g., Singh and Azad [1, 2], Gan and Riffat [3], and Xu et al. [4], devoted to the flow in divergent tubes to date. That is, much attention has been given to the tube geometry, which *

E-mail address: [email protected]. Tel: +81−82−434−7000, Fax: +81−82−434−7011

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affects pressure recovery efficiency and losses; the relationship between performance and the optimum geometry has been investigated. The flow in divergent tubes is highly unstable and then, recently, computational fluid dynamics has been applied to the prediction of mean and turbulent flow characteristics (Gan and Riffat [3], and Xu et al. [4]). However, all these studies concerned steady flow. On the other hand, unsteady flow has become important in connection with problems concerning the starting and stopping or undesirable accidents of pumps and blowers, because fluid machines are becoming better and fluid transport is becoming diversified and faster. Nevertheless, there have been few studies on unsteady flow in divergent tubes. Mizuno and Ohashi [5], and Mochizuki et al. [6] conducted experiments for a two-dimensional diffuser. The former group oscillated one plane, whereas the latter group used periodic flows of a wake generated by a cylinder into the diffuser entrance. Thus their studies were aimed at grasping the flow features, including unsteady separation, and/or establishing a method of controlling the flow. Unfortunately, the periodically volume-cycled, unsteady flow has never been treated, to the author’s knowledge. The purpose of the present study is to treat the problem of volume-cycled, pulsating turbulent flow through a conical divergent tube with a total divergence angle of 2θ =12°. First, the distribution of wall static pressure is measured for the various pulsation frequencies. Subsequently, periodical changes of profiles, such as the phase-averaged velocity and turbulent intensity, are examined under a representative flow condition. Furthermore, knowledge of their characteristics is obtained through comparison with those in the case of steady flow.

2. Experimental Apparatus and Measurement Method 2.1. Experimental Apparatus A schematic diagram of the experimental apparatus is shown in Figure 1. The working fluid is air. The system consists of a pulsating-flow generator, a test tube and measuring devices. Moreover, the pulsating-flow generator is composed of a steady flow and an oscillating flow. The steady flow, i.e., time-mean flow, was supplied, through a surge tank, by a suction blower, which ensured that the flow rate was independent of the superimposed oscillation and the length of the test tube. On the other hand, the volume-cycled oscillating flow was introduced by a piston, with a diameter of 300 mm and a stroke length adjustable from 0 to 1000 mm, controlled by a personal computer. The test tube had a total divergence angle 2θ of 12° and an area ratio m of 6.25. Here the ratio m was (d2/d1)2, d1 (=2a1=80 mm) and d2 (=200 mm) being the diameters at the inlet and the exit, respectively, of the divergent tube. These dimensions are shown in Figure 2, together with the coordinate system. The divergent tube was constructed from 5 accurately machined, transparent acrylic blocks connected via a slip ring. The ring, furthermore, had static pressure holes, 0.8 mm in diameter and spaced 90° apart, and a small hole for inserting a hot-wire probe. Straight transparent glass tubes with lengths of 3700 mm (=46.3d1) and 4200 mm (21d2) were connected to the inlet and the exit of the divergent tube, respectively.

Experimental Study of Pulsating Turbulent Flow through a Divergent Tube

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Figure 1. Schematic diagram of experimental apparatus – 1) upstream tube, 2) divergent tube, 3) downstream tube, 4) oscillating flow generator, 5) surge tank, 6) blower.

Figure 2. Coordinate system and dimensions of test tube.

2.2. Measurement Procedure and Data Acquisition The wall static pressure was measured, using a diffusive-type semiconductor pressure transducer (Toyoda MFG, DD102-0.1F), at 11 stations between z = −22.1d1 in the upstream straight tube and z = 21.9d1 in the downstream one, where z is the length measured along the tube axis from the inlet of the divergent tube. Velocity measurements were performed with a hot-wire anemometer, and the velocity w in the axial direction was obtained at 8 stations including sections of z/d1= -2 and 9.6. The voltage output from the DC amplifier of the pressure transducer and from the anemometer was sampled, with the personal computer, synchronously with a time-marker signal that indicates the position of the piston. The data was processed as follows. For a periodically unsteady turbulent flow, any instantaneous quantity, e.g., the axial velocity w in the pulsation cycle, can be written as w = W + wt .

(1)

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Here, W is the phase-averaged velocity and wt is the fluctuating one. The data at each measuring point was collected, in equal time intervals, for 200 to 500 pulsation cycles. The recorded data were ensemble phase-averaged to obtain W at each phase. Moreover, the turbulence intensity w’ was obtained as the square root of the ensemble phase average of wt2. This procedure was also applied to the phase-averaged value P of the wall static pressure. It was confirmed beforehand that averaging over 200 cycles has no influence on the values of the quantities. The scatter in the results was less than 4 and 6% for the phase-averaged values and the turbulence intensity, respectively. In the hot-wire measurement, it is difficult to determine accurately the direction of flow at the position and time when the ratio of the radial to the axial component of the velocity is rather large. For such a case, errors of a limited extent are not avoidable. Furthermore, it was also checked that the flow properties are symmetric with respect to the tube axis. Additionally, in order to gain insight into the pulsating flow features, the visualization experiment using water was executed for a divergent tube with d1=22 mm. In the experiment, the flow in the horizontal plane including the tube was rendered visible by a solid tracer method using polystyrene particles of about 0.2 mm diameter.

3. Results and Discussion 3.1. Experimental Conditions Pulsating flow in divergent tubes is governed by five nondimensional parameters: the total divergence angle 2θ, the area ratio m, the Womersley number α, the mean Reynolds number Reta and the oscillatory Reynolds number Reos (or the flow rate ratio η). Here, the former two and the remainder are related to the geometry of the divergent tubes and the fluid motion, respectively. The Womersley number is defined as α = a1(ω/ν)1/2, ω being the angular frequency of pulsation and ν the kinematic viscosity of the fluid. The mean Reynolds number is expressed as Reta =d1Wa1,ta /ν and the oscillatory Reynolds number as Reos =d1Wa1,os /ν. Here, Wa1 is the cross-sectional average velocity in the upstream tube and the subscripts ta and os indicate the time-mean and amplitude values, respectively. Finally, the flow rate ratio is given by η =Wa1,os /Wa1,ta (=Reos /Reta). The experiments were performed systematically under the conditions of α =10~40, Reta=20000 and Reos = 10000 (η = 0.5). These conditions were chosen referring to works on pulsating straight-tube flows (for example, Iguch et al. [7]). In order to check the flow entering the divergent tube, preliminary measurements were conducted for the axial velocity at z/d1 = -2 and the pressure gradient in the upstream tube. In the former, integrating the measured velocities over the cross section, a time-dependent flow rate Q was calculated. The result is shown in Figure 3. In the figure, the solid line denotes the first fundamental component developed by the Fourier series, in which the second and higher components are less than 3% compared with the first one and are considerably small. This confirms that the prescribed flow is realized, and also indicates that a sinusoidal flow rate is achieved in the form Q = Qta + Qos sinΘ ,

(2)


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Θ (= ωt) being the phase angle and t the time.

Figure 3. Flow rate variation.

Furthermore, the results obtained for the varying pressure gradient in the upstream tube were in good agreement with the previous ones obtained by Ohmi and Iguchi [8]. For example, the phase difference Φp between the variation of the pressure gradient and the flow rate is shown in Figure 4. Therefore, it is reconfirmed that a fully developed pulsating turbulent flow enters the divergent tube.

Figure 4. Phase difference between axial pressure gradient in the upstream tube and flow rate variation.

Ohmi and Iguchi [8] showed that the pulsating turbulent flow in straight tubes is classified roughly into three regimes depending on a characteristic number of α/Reta3/8. The flow conditions taken up in this study are in the α/Reta3/8 region of 0.2 to 1, which is the intermediate regime in their classification. Thus we can say that the entry flow conditions into the divergent tube in the present experiment are of higher interest in this research field.

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3.2. Wall Static Pressure 3.2.1. Distribution of Wall Static Pressure and Its Variation with Time In divergent tubes, for steady flow, the kinetic energy of the flow converts into the pressure energy and then the static pressure rises in the downstream direction. On the other hand, for pulsating flow with a periodic change of the flow rate, the drop and rise of the pressure in the longitudinal direction are needed to drive the fluid acceleratively and deceleratively, respectively, with time. Hence the amplitude of the pressure variation is predicted to be proportional to the pulsation frequency ω. Consequently, the pressure in the divergent tube shows a complex distribution with time. The wall static pressure P is shown in Figure 5 for the various Womersley numbers. The pressure P is expressed in the form of the pressure coefficient Cp, which is defined as Cp = (PPref) / (ρ Wa1,ta2 /2), Pref being the pressure at the station of z/d1 = -2 in the upstream tube and ρ the density of the fluid. In the figure, the broken lines indicate the time-averaged values and the chain line the result for steady flow at the same Reynolds number as Reta. In the upstream tube, Cp changes in the phase leads of Φp, shown in Figure 4, with flow rate variation. The time-averaged Cp is slightly larger than that for the steady flow at Re=2000. On the other hand, the pressure in the divergent tube rises with an increase of z/d1, except for the part of the period for moderate and high α. That is, the variation of Cp becomes larger with an increase of α. Furthermore, the phase with the largest value of Cp changes from the Θ ≈ 90° (α=10) of the maximum flow rate to the Θ ≈ 150° (α=40) of the middle of the decelerative phase. Conversely, the phase with the minimum value of Cp shifts from the Θ ≈ 270° (α=10) of the smallest flow rate to the Θ ≈ 330° (α=40) of the first half of the accelerative phase. At these phases with the lowest Cp, for the low α (α=10), there is little change of Cp in the longitudinal direction. However, for the moderate and high α (α=20, 40), Cp takes negative values, and the pressure in the divergent tube shows a favorable gradient.

(A) Figure 5. Continued on next page.


371

(B)

(C) Figure 5. Longitudinal distribution of wall static pressure. (○: Θ=0°, ●: Θ= 90°, Δ: Θ= 180°, ▲: Θ=270°, ⎯ - ⎯ : time-averaged, ⎯ ⎯ ⎯ : steady flow at Re=20000). (a) α=10, (b) α=20, (c) α=40

We examine the pressure rise ΔPL in the length L, i.e., between the inlet and the exit of the divergent tube. The representative result is shown in Figure 6, in which ΔPL is nondimensionalized by the dynamic pressure based on Wa1,ta in the upstream tube. In the figure, the waveform of ΔPL is developed with the Fourier series denoted by the solid line. The broken line denotes the result that is theoretically obtained using Bernoulli’s theorem for a quasi-steady flow. It is expressed as

ΔPL = (1 + η sinΘ)2 (1− m-2) .

(3)

Moreover, the symbol ← indicates the pressure rise ΔPL,s for steady flow, at Re=20000, with the same cross-sectional average velocity as Wa1,ta. The pressure rise ΔPL in the pulsating flow changes almost sinusoidally. However, it lags behind the variation of the flow rate. The phase difference ΦΔP between the fundamental waveform of ΔPL and the pulsating flow rate Q becomes large with an increase in the Womersley number. Incidentally, ΦΔP changes from -

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Masaru Sumida

5° to -60° when α increases from 10 to 40. On the other hand, ΔPL is lower than the theoretical value for quasi-steady flow. Furthermore, ΔPL takes approximately zero values for phase in the range of 230 ~ 340° with a small flow rate.

Figure 6. Difference between pressures at the inlet and exit of a divergent tube (← : steady flow at Re=20000).

As has been stated above, for pulsating flow in a divergent tube, the pressure at the exit of the divergent tube rises when the cross-sectional average velocity is large and is in a decelerative phase. That is, ΔPL becomes large from the latter half of the accelerative phase to the middle of the decelerative phase (Θ ≈ 50 ~ 180°), as seen in Figure 6. By contrast, Cp exhibits a small change in the axial direction from the ending of the decelerative phase to the first half of the accelerative phase (Θ ≈ 230 ~ 330°). This is because the kinetic energy to be converted to pressure energy is low and because, to accelerate the fluid in the axial direction, the downstream pressure must be lowed. Therefore, it can be understood that the pressure distribution at the beginning of the accelerative phase shows the favorable larger gradient for higher Womersley number where the fluid is strongly accelerated in the streamwise direction.

3.3. Axial Velocity 3.3.1. Changes in Centerline Velocity with Time and along the Tube Axis In this section, we discuss the axial velocity. To start, we will elucidate the outline of the flow features by focusing on the axial velocity on the tube axis. Figure 7 shows the waveforms of instantaneous axial velocity w, together with those of its phase-averaged one W and its turbulence intensity w’. For comparison with those on the tube axis, the results of the measurement at the radial position r/Rz = 0.75 near the tube wall are also shown in Figure 7. Moreover, the changes in Wc and wc’ along the tube axis are illustrated in Figure 8. The subscript c indicates the values on the tube axis and the broken lines indicate the timeaveraged values.


373

Figure 7. Velocity waveforms. (A) Instantaneous velocity w (B)Phase-averaged velocity W (C) Turbulence intensity w’.

The phase-averaged velocity W exhibits an almost sinusoidal change in time, as displayed in Figure 7(b). However, since the flow extends with increasing z/d1 in both the divergent tube and the downstream straight tube, Wc in these sections is reduced (see Figure 8(a)). In particular, the degree of reduction in the divergent section becomes smaller and larger for the middle, i.e., Θ = 180 and 0°, of the decelerative and the accelerative phase, respectively. Thus, the phase angle at which Wc shows a peak is discernibly delayed compared with the flow rate variation as z/d1 increases. This lag is equivalent to the time at which the fluid flowing into the divergent tube at the maximum flow rate (Θ = 90°) reaches each section. As a result, the time lag at the divergence exit (z/d1 = 7.1) amounts to approximately one-tenth of one cycle. On the other hand, W near the wall (r/Rz = 0.75) in the divergent tube changes almost in synchronous phase with the flow rate. That is, W takes a maximum value at about Θ = 90°. In the case of steady flow, Wc/Wa1 at the Reynolds numbers of 10000 and 30000, corresponding to the minimum and maximum flow rates in pulsating flow, respectively, attenuates axially in the same manner as the flow at Re = 20000. Hence, the Reynolds number has little effect on Wc/Wa1 (the illustration is omitted here). It can be inferred from the above discussion that the differences in the change in Wc along the tube axis in phase are attributed to an unsteady effect. The turbulence intensity w’c becomes large, at any phase, with an increase in z/d1 in the first half of the divergent tube. It shows the peak magnitude in the second half of the decelerative phase, as seen in Figures 7(c) and 8(b). Furthermore, w’c near the divergence exit in the first half of the decelerative phase is about 2.6 times the value at the divergence inlet. Consequently, the variation of w’c over one cycle becomes larger.

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(A)

(B) Figure 8. Change in Wc and w’c along the tube axis (○: Θ=0°, ●: Θ= 90°, Δ: Θ= 180°, ▲: Θ=270°, ⎯ ⎯ ⎯ : time-averaged). (a) Phase-averaged velocity Wc (b) Turbulence intensity w’c.

On the other hand, w’ near the wall (r/Rz=0.75) in the upstream tube is twice as large as w’c, but the phase difference between w’ and Wa is small. In the section immediately behind the divergence inlet, the turbulence intensity is high in the phase with the high flow rate. However, the change of w’ gradually becomes similar to that of waveform w’c with an increase in z/d1.

3.3.2. Distributions of Phase-Averaged Velocity and Turbulence Intensity Figure 9 shows distributions of W and w’ at four representative phases. The broken lines in the figure denote results for steady flow at the Reynolds number of Reta. Moreover, sketches of the stream are displayed in Figure 10, which is based on, with moderate


375

exaggeration, the observation of the water flow. The painted parts enclosed by the broken lines show the main current of the flow.

Figure 9. Distributions of W and w’ in upper and lower figures, respectively (○: Θ=0°, ●: Θ= 90°, Δ: Θ= 180°, ▲: Θ=270°, ⎯ ⎯ ⎯ : steady flow at Re=20000).

In pulsating flow, the phase difference among fluid motions in the divergent tube becomes larger as z/d1 increases. Consequently, the difference between the flow states of the accelerative and decelerative phases becomes considerable. Therefore, the phase-averaged velocity and turbulence intensity exhibit complicated distributions. In the upstream tube (Figure 9; z/d1=−2), the periodic change of the axial velocity leads slightly near the tube wall, whereas it is delayed in the central part of the cross section. Nevertheless, on the whole, the velocity at each phase shows a profile similar to the steady one with a simple 1/7-th-power law. In the divergent tube, the pressure rise in the first half of the accelerative phase is smaller than the theoretical one derived for quasi-steady flow, as shown in Figure 5(b). Moreover, there is little pressure difference in the divergent tube. As a result, the main current in the central part of the cross section reaches the exit plane without extending too much towards the tube wall (Figure 10(b); Θ ≈ 0°). Meanwhile, near the inlet, a local pressure drop occurs, as seen in Figure 5(b). This accelerates the fluid in the neighborhood of the wall, and the phase of the velocity variation is more advanced (z/d1= 1.4). When the flow rate increases and becomes maximum, eddies generated in the shear layer surrounding the main current grow rapidly, accompanying the acceleration of the flow, and become massive vortices. These vortices strongly interact with each other near the divergence exit. Following this, the main stream suddenly begins to meander in the central part of the cross section, resulting in a rapid increase in w’ (Θ = 90°; Figure 10(b)). By the middle of the decelerative phase (Θ = 180°), the pressure at the divergence exit is, as before, about twice as high as that in the case of steady flow at the same flow rate, because the

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pressure there decelerates the fluid flow (Figures 5(b) and 6). Thus the stream becomes unstable. As a result, the massive vortices in the section downstream of the divergence exit collapse and start to decay. In addition, the main current is mixed. At the end of the decelerative phase (Θ = 270°), the turbulence intensity in the section downstream of the divergence exit is considerably attenuated and part of the main current retreats to the first half of the divergent tube (Figure 10(b)). The fluid particle in the vicinity of the wall is carried, locally and momentarily, upstream along the wall. Nevertheless, flow separation and backward flow are indistinct, and the fluid almost always flows downstream.

(A)

(B) Figure 10. Sketches of flow pattern obtained by visualization experiment. (A) Steady flow at Re=20000 (B) Pulsating flow (α=20, Reta=20000, η=0.5).


377

From the above description, in the phase at which the flow rate increases, the degree of acceleration is high in the central part of the cross section. As a result, W near the divergence exit (z/d1 = 5.5) at the maximum flow rate (Θ = 90°) shows a profile with convexity near the tube axis, as seen in Figure 9. In the decelerative phase, on the other hand, the velocity is reduced almost uniformly throughout the cross section because of the stronger action of turbulent mixing in the section further downstream. The distribution of the turbulence intensity differs appreciably from that in a steady flow, as seen in Figure 9. That is, the turbulence intensity is high in the phase from the end of the increase to the first half of the decrease of the flow rate. This is because, during this phase, the shearing layer between the main current and the tube wall rolls up and becomes massive vortices. On the other hand, w’ is low in the first half of the accelerative phase when these vortices are almost decayed. To put it concretely, when the flow rate is large, the turbulence intensity immediately behind the divergence inlet (z/d1 = 0 ∼1.4) takes large values near the wall. Downstream, the region with high w’ extends radially, accompanied by the formation of a shearing layer. Furthermore, the maximum w’/Wa1,ta distribution in this section occurs in the first half of the decelerative phase and becomes about 18% at z/d1=2.7 from the 14% in the upstream straight tube. Downstream, in the middle section of the divergent tube, at z/d1 = 2.7 ~ 5.5, w’ becomes maximum in the vicinity of the inflection point of the W distribution when a fluid flowing at high speed is passed through the section. Subsequently, the location of maximum w’ shifts toward the tube axis. In this phase (Θ ≈ 180°), w’ is about 20 percent higher than that of steady flow with Reta. The maximum value corresponds to 20% of Wa1,ta. In the section downstream of the divergence exit, the distribution of the turbulence intensity is uniformalized in the cross section (z/d1 = 9.6).

5. Conclusion Experimental investigations were performed for pulsating turbulent flow in a divergent tube and the flow field was examined. The principal findings of this study are summarized as follows. (1) The axial distributions of the pressure coefficient Cp in the divergent tube are high in the phase from the latter half of acceleration to the middle of deceleration. In contrast, they are low in other phases, namely, from the ending of the deceleration to the first half of the acceleration. (2) The time-mean value of the pressure rise ΔPL between the inlet and the exit of the divergent tube is larger than that in the steady flow with the Re = Reta. The variation of ΔPL and the phase lag ΦΔP behind the flow rate depend on and increase with α2. (3) The phase-averaged velocity W shows profile swelling in the central part of the cross section when the flow rate increases. The phase at which the value becomes maximum in each plane is delayed from Θ = 90° at the maximum flow rate with an increase in z/d1. However, as the flow rate decreases, the W profile becomes flat downstream of the divergent tube. (4) The turbulence intensity w’ for a high flow rate is maximum at the position on the radius near the inflection point of the W profile. As the phase proceeds, the region

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Masaru Sumida with a large w’ extends towards the wall and the tube axis, and the w’ profile becomes level at the plane of the divergence exit.

Acknowledgements The author would like to thank Mr. J. Morita of Tokuyama Corporation for his assistance.

References [1] [2] [3] [4] [5] [6] [7] [8]

R. K. Singh and R. S. Azad, Exp. Thermal Fluid Sci. 1995, vol. 10, 397-413. R. K. Singh and R. S. Azad, Exp. Thermal Fluid Sci. 1995, vol. 11, 190-203. G. Gan and S. B. Riffat, Appl. Energy 1996, vol. 54-2, 181-195. D. Xu, M. A. Leschziner, B. C. Khoo and C. Shu, Comput. Fluids 1997, vol. 26-4,417423. Mizuno and H. Ohashi, Trans. Jpn. Soc. Mech. Eng. Ser. B 1984, vol. 50-453, 12231230 (in Japanese). O. Mochizuki, M. Kiya, Y. Shima and T. Saito, Trans. Jpn. Soc. Mech. Eng. Ser. B 1997, vol. 63-605, 54-61 (in Japanese). M. Iguchi, M. Ohmi and S. Tanaka, Bull. JSME 1985, vol. 28-246, 2915-2922. M. Ohmi and M. Iguchi, Bull. JSME 1980, vol. 23-186, 2029-2036.


ISBN 978-1-60741-037-9 c 2009 Nova Science Publishers, Inc.

Chapter 12

S OLUTION OF AN A IRFOIL D ESIGN I NVERSE P ROBLEM FOR A V ISCOUS F LOW U SING A C ONTRACTIVE O PERATOR Jan Šimák and Jaroslav Pelant Aeronautical Research and Test Institute (VZLÚ), Prague, Czech Republic

Abstract This chapter deals with a numerical method for a solution of an airfoil design inverse problem. The presented method is intended for a design of an airfoil based on a prescribed pressure distribution along a mean camber line, especially for modifying existing airfoils. The main idea of this method is a coupling of a direct and approximate inverse operator. The goal is to find a pseudo-distribution corresponding to the desired airfoil with respect to the approximate inversion. This is done in an iterative way. The direct operator represents a solution of a flow around an airfoil, described by a system of the Navier-Stokes equations in the case of a laminar flow and by the k − ω model in the case of a turbulent flow. There is a relative freedom of choosing the model describing the flow. The system of PDEs is solved by an implicit finite volume method. The approximate inverse operator is based on a thin airfoil theory for a potential flow, equipped with some corrections according to the model used. The airfoil is constructed using a mean camber line and a thickness function. The so far developed method has several restrictions. It is applicable to a subsonic pressure distribution satisfying a certain condition for the position of a stagnation point. Numerical results are presented.

1.

Introduction

The method described in this chapter is assumed for an airfoil design corresponding to a given pressure distribution. It is an extension of a method for a potential flow and later for an inviscid compressible flow [1], [2] and a laminar viscous flow [3]. It is useful in the cases where a specific distribution is desired. The method is based on the use of an approximate inversion and the solution of the flow around an airfoil. Since the flow is assumed turbulent, a model of turbulence is used to improve the quality of the solution. In the following text the detailed description of the method is given.

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2.

Jan Šimák and Jaroslav Pelant

Description of the Method

The presented method, as most of the others inverse methods, is based on the idea of an approximate inversion. This approximate inverse operator, in the following text denoted by L, is accompanied by a so-called direct operator, denoted by P. This direct operator represents a solution of the flow around and airfoil. The pressure distributions fup and flo on the upper and lower part of the airfoil are prescribed along the mean camber line above the chord line, in the direction from the leading edge to the trailing edge. The chord line is of length b. A new function f on the interval h−b, bi is defined by the following relations f (x) = fup (−x), f (x) = flo (x),

for x ∈ h−b, 0),

for x ∈ h0, bi.

Then the inverse problem can be defined as the following: Find a function u : h−b, bi → R such that P L(u) = f.

(2.1)

The function u will be referred to as a pseudo-distribution. The equation (2.1) is transformed to a contractive operator whose fixed point is searched for. This yields a sequence of pseudo-distributions {uk }∞ k=0 , uk+1 = uk + α (f − P Luk ) .

(2.2)

If this sequence converges, the limit is the solution of (2.1). In order to ensure that, a suitable parameter α ∈ (0, 1i is chosen. The numerical results indicate that the values less then one are usually sufficient. In many examples the value was set to α = 0.6.

3.

Inverse Operator

The approximate inverse operator L, described in this section, is a mapping between a velocity distribution f and a curve ψ representing an airfoil. Since the given distribution is a pressure, it is necessary to transform it to a velocity distribution. The pressure is assumed to be constant across the boundary layer in the normal direction to the airfoil. From that reason, the velocity on the edge of a boundary layer is computed from the pressure on the airfoil. The transformation is derived from relations for the pressure, density, Mach number and speed of sound, γ − 1 2 −γ/(γ−1) p = p0 1 + M , 2 γ − 1 2 −1/(γ−1) ρ = ρ0 1 + M , 2 c2 = γp/ρ.

The derived formula for the velocity related to the velocity in the free stream is 2 + γ − 1 (p /p(x))(γ−1)/γ − 1 v(x) 2 2/M∞ 0 = , v∞ γ−1 (p0 /p(x))(γ−1)/γ

(3.3)

Solution of an Airfoil Design Inverse Problem...

381

where p0 is the pressure at zero velocity, M∞ is the Mach number in the free stream and γ is the Poisson adiabatic constant. The velocity distribution, obtained by the above mentioned transformation (or is simply given), is on an upper and lower side of the airfoil above its chord. This is a straight line connecting the leading and trailing edge of the airfoil. The derivation of the operator will be in two steps. At first, a function describing a mean camber line is derived. This is a line in the middle of an airfoil, its end points are the same as the end points of the chord. The second step is construction of a thickness function. It is the distance between points on a surface and points on a mean camber line. Putting the function describing the mean camber line and the function describing the thickness together we get the operator L. The prescribed distribution has to satisfy a condition, that the stagnation point on the leading edge is located at the beginning of the chord line. Otherwise the method can have troubles near the stagnation point. The required position is ensured by the determination of the appropriate angle of attack.

3.1.

Construction of the Mean Camber Line

The derivation of the function describing the mean camber line is based on the theory of thin airfoils. In this theory, the airfoil thickness is neglected and the airfoil shape is simplified to a line. From this reason we assume that the mean camber line represents an airfoil in the following text. At first, the origin of the coordinate system is set at the beginning of the chord line on the leading edge. The x-axis is set so that the chord line, which length is b, lies on the axis. Let us consider the direction of the flow v∞ parallel to the x-axis or with a small angle of attack α∞ . The main idea is to consider a system of vortices on the airfoil surface. This system determines a circulation along the airfoil. A vortex with an intensity Γ generates a velocity according to the relation 2vπr = Γ, where v is the size of a velocity at some point of the plane and its direction is perpendicular to a line connecting this point and the point of the vortex. The symbol r denotes the distance of these two points. According to this, if a system of vortices is assumed on the interval h0, bi on the x-axis, then there will be a velocity generated by this system in the direction of the y-axis at the point x ∈ h0, bi, given by the formula 1 vˆy (x) = (PV) 2π

Z

b 0

dΓ(ξ) . ξ−x

(3.4)

The symbol (PV) means a principal value of an integral, defined as a limit (PV)

Z

a

b

f (x) dx = lim

ǫ→0

Z

c−ǫ

f (x) dx + a

Z

b

c+ǫ

where c is a point at which the function f (x) has a singularity.

f (x) dx ,

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Let us denote the coordinates of the mean camber line by [x, s(x)], x ∈ h0, bi. Since the both end points of the mean camber line lie on the x-axis, the condition s(0) = s(b) = 0 must be satisfied. Thus, although the formula (3.4) is valid only for points on the x-axis, it can be also used for points on the mean camber line causing only a small error, especially if the camber is small. The vortices on the chord line causing the circulation are unknown but they are determined by the given velocity distribution. The relation between them is quite easy to derive. Assume a closed curve whose image lies over the mean camber line (Figure 1). The cir-

vup

A′

B′ y

dξ A

vlo

B

Figure 1. Derivation of the circulation along the mean camber line. culation along a closed curve is defined as a curvilinear integral of the tangent component of the velocity along this curve. The flow on the upper and lower side has the velocities equal to vup , vlo , the velocities across the mean camber line are zero because it is assumed impermeable. So it holds for the elementary circulation dΓ(ξ) = vup (ξ) − vlo (ξ) dξ, where dξ is the length of the segments AB and A′ B ′ . Next, the velocity of the free stream is denoted by v∞ , its components are x v∞ = v∞ cos α∞ ,

y v∞ = v∞ sin α∞ , where α∞ is the angle of attack. By v(x) = vx (x), vy (x) is denoted the velocity on the mean camber line, which is the free stream velocity disturbed by the vortices. Let α be the angle between the vector v(x) and the x-axis. Then the following must be true:

tan α =

vy (x) vˆy (x) = tan α∞ + x . vx (x) v∞

(3.5)

Since the mean camber line represents an airfoil, the vector of velocity v(x) must be tangential to this line. This yields the relation tan α = s′ .

(3.6)

Introducing a specific circulation dΓ(ξ) dξ and using relations (3.4) and (3.6) in (3.5), the differential equation for the mean camber line is obtained, Z b 1 γ(ξ) (PV) s′ (x) = tan α∞ + dξ. x ∈ h0, bi , (3.7) x 2πv∞ ξ −x 0 γ(ξ) =


383

The equation is completed with boundary conditions s(0) = s(b) = 0. In the following let us assume the velocity v∞ = 1 (this can be achieved using a suitable y x =v normalization). It is also assumed that v∞ ˙ ∞ = 1, that means v∞ is small compared to x v∞ . The equation (3.7) can be rewritten as Z b γ(ξ) ′ s (x) = a − c (PV) dξ, 0 x−ξ where c = 1/(2π) and a = tan α∞ . The angle α∞ is left as a parameter in order to satisfy the boundary conditions. The solution to the differential equation is expressed as Z x Z b γ(ξ) s(x) = ax − c (PV) dξ dt, (PV) 0 0 t−ξ using the boundary condition for x = 0. Changing the order of integration and setting the parameter a to ensure the boundary condition for x = b is satisfied, the final results are obtained. The function s(x) and its derivative s′ (x) are given by Z b Z b b − ξ γ(ξ) 1 1 ′ (PV) γ(ξ) ln (PV) dξ, (3.8) s (x) = dξ − 2πb ξ 2π 0 0 x−ξ Z b Z b b − ξ x − ξ x 1 dξ, γ(ξ) ln s(x) = (PV) γ(ξ) ln (PV) dξ − (3.9) 2πb ξ 2π ξ 0 0 where γ(ξ) = vh (ξ) − vd (ξ).

3.2.

Construction of the Thickness Function

During the derivation of the mean camber line, an airfoil with zero thickness was assumed. By contrast, in this part of the text an airfoil with zero camber is assumed. In the previous part, a system of vortices was assumed in order to derive the mean camber line function s(x). This time, a system of sources is assumed on the chord in order to derive the thickness function. This system generates a velocity potential ϕ(x, y) at some point of the xy-plane. The potential is given by the relation Z b p 1 q(ξ) ln (x − ξ)2 + y 2 dξ, ϕ(x, y) = (3.10) 2π 0

where q(ξ) describes the intensity of the sources. As will be shown in the following text, it is possible to derive a relation between q(x) and the airfoil thickness t(x). From this reason a flow parallel to the x-axis is considered. According to the symmetry of the airfoil it is sufficient to work only with the upper half ˆ = (ˆ of the airfoil. By v vx , vˆy ) is denoted a velocity vector produced by the distribution of sources on the chord. The relation between the thickness and the source intensity is based on the flow rate along the chord (see Figure 2). The difference between velocities at points x and x + dx is dˆ vx and the difference between the thickness at these points is dt. Thus it is necessary (since the flow is considered incompressible) to add or remove some mass using sources in order to achieve balance.

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Jan Šimák and Jaroslav Pelant t + dt

t 0

x + dx

x

v∞ + vˆx

dx

b

v∞ + vˆx + dˆ vx

Figure 2. Illustration for derivation of the thickness function. Mathematically formulated, the balance is 1 (v∞ + vˆx ) t + q dx = (v∞ + vˆx + dˆ vx ) (t + dt) . 2 After some rearrangement and omitting the term dˆ vx dt, the relation is d 1 q(x) = v∞ + vˆx (x) t(x). 2 dx

Under the assumption the airfoil is thin enough, it is true |ˆ vx | ≪ v∞ inside the interval (0, b). Hence, the relation can be simplified by neglecting the term vˆx and the result can be written as q(x) = 2v∞ t′ (x). (3.11) Differentiation of the equation (3.10) yields ∂ϕ(x, 0) 1 vˆx (x) = = (PV) ∂x 2π

Z

b

q(ξ) 0

1 dξ x−ξ

and putting here for q(ξ) results in v∞ vˆx (x) = (PV) π

Z

b

t′ (ξ) 0

dξ . x−ξ

(3.12)

The velocity on the chord is assumed as an average of velocities on the upper and lower side of the airfoil. Thus the velocity increment vˆx can be expressed as vˆx = (vh + vd )/2 − v∞ . Assuming again the free stream velocity v∞ = 1, a new function vp (x) can be defined, vp (x) =

vh (x) + vd (x) − 1. 2

(3.13)

Substituting this into (3.12) yields vp (x) =

1 (PV) π

Z

0

b

t′ (ξ)

dξ . x−ξ

(3.14)

The unknown in this equation is the derivative of the thickness t′ (x) and it is possible to find an analytical solution. This integral equation is a special case of a more general integral equation of the form Z β ϕ(ξ) αϕ(x) + dξ = f (x), x ∈ R, πi L ξ − x


385

R

where L denotes a complex-valued curvilinear integral along a smooth curve L. In our case α = 0, β = −πi and the curve L is an identity on the interval h0, bi. According to [4], the solution of our problem can be written in the form s Z b vp (ξ) ξ(b − ξ) 1 C ′ t (x) = − (PV) dξ + p . π x − ξ x(b − x) x(b − x) 0 After integration with respect to x, the function t(x) is determined by 1 t(x) = − (PV) π

Z

+ C arctan

b 0

Z p vp (ξ) ξ(b − ξ) (PV)

0

x − b/2 p x(b − x)

!

− c,

x

dy p y(b − y)(y − ξ)

!

dξ+

where c and C are two constants determined later. The second integral can be written in the form q q ξ b−x Z x 1 + b−ξ dy 1 x p q (PV) = −p ln q . ξ b−x y(b − y)(y − ξ) ξ(b − ξ) 1 − 0 b−ξ

Finally we obtain the equation describing the airfoil thickness q q ξ b−x Z b 1 + 1 b−ξ x q vp (ξ) ln t(x) = (PV) q dξ + C arctan π ξ 0 1 − b−ξ b−x x

x

x − b/2 p x(b − x)

!

+ c.

(3.15) This equation contains two parameters C and c. If we set x = 0 or x = b, the above mentioned integral equals zero. The second term containing the parameter C is a monotonous function which equals −Cπ/2 in the case of x = 0 or is equal to Cπ/2 in the case of x = b. Since we are interested in an airfoil which is closed on both sides of the chord, we choose C = c = 0. This choice is fully natural because it agrees with the situation where the function vp is set to zero. In this case the velocity on the airfoil equals the free stream velocity and from this reason the corresponding airfoil has zero thickness. Under this condition the mapping between the velocity vp (x) and the thickness function t(x) is unique. It is also obvious that we can obtain an airfoil with prescribed thickness of the trailing edge by a suitable choice of C and c. The resulting equation is q q ξ b−x Z b 1 + 1 b−ξ x q t(x) = (PV) vp (ξ) ln (3.16) q dξ. π ξ 0 1 − b−ξ b−x x At the end, there is necessary to mention one drawback. The airfoil thickness should be positive, of course. But the procedure mentioned above should result in a negative thickness, depending on the input velocity distribution. Since the input is the so-called pseudodistribution, it is necessary to include some control mechanism into the implementation.

386

3.3.


Construction of the Airfoil

y

(−s′ (x), 1)

(1, s′ (x)) t(x) s(x) t(x)

0

x

b

x

Figure 3. Construction of the airfoil. Construction of the airfoil is based on the mean camber line s(x) and the thickness function t(x). Both functions are dependent on the velocity distributions f = {vup , vlo }. The airfoil is expressed as ψ(x) = ψ1 (x), ψ2 (x) , where x ∈ h0, bi. Since the distance between points on the surface and on the mean camber line is given by t(x) (Figure 3), the coordinates of the airfoil are expressed in the form ψ1 (x) = x ∓ t(x) q

s′ (x) 1 + s′ 2 (x)

ψ2 (x) = s(x) ± t(x) q

,

1 1+

s′ 2 (x)

(3.17) .

(3.18)

The upper sign is meant for the upper part of the airfoil and the bottom sign for the lower part. It is easy to see that the following is true, ψ1 (0) = 0,

ψ1 (b) = b,

ψ2 (0) = 0,

ψ2 (b) = 0.

Thus we get a closed airfoil over the chord h0, bi. The derived approximate inverse operator is the mapping L(f ) = L(vup , vlo ) = (ψ1 , ψ2 ). According to the formulation of the inversion, it is necessary to deal with continuous functions. In order to have an airfoil with an acceptable geometry, the functions s(x) and t(x) must be continuous and smooth. This is achieved by assuming a subsonic flow with continuous pseudo-distributions uk for all k. Thus the distributions obtained in each iteration as a solution of the flow problem must be continuous. From the numerical point of view, this restriction is not so strong. The prescribed distribution must be continuous but the intermediate distributions can have discontinuities with small jumps which are smeared by the iterative process and by the integral formulation of the functions s(x) and t(x).


3.4.

387

Numerical Realization

The evaluation of the functions ψ1 (x), ψ2 (x) includes evaluations of integrals, which are done by a suitable numerical quadrature. The integrands are functions in two variables x and y and have singularities for x = y. From this reason it is necessary to avoid this points. The quadrature used in this method is the Chebyschev-Gauss quadrature. The integrals are discretized in the way of b

Z

f (x, y) dy =

0

Z

0

b

p N/2 X p f (x, y) y(b − y) p dy = wk f (x, yk ) yk (b − yk )+RN , (3.19) y(b − y) k=1

where x ∈ h0, bi, N is an even number, wk = 2π/N are quadrature coefficients, p N f (x, η) η(b − η) ∂ 2π RN = N 2 N! ∂y N

is an error xk )/2 are nodes. In this case x ˆk = cos (2k − of the quadrature and yk = (b + bˆ 1)π/N are roots of the Chebyschev polynomial N TN/2 (x) = cos arccos x . 2 Finally, a sequence of points xi is defined, b iπ 1 + cos , i = 0, 1, . . . , N. xi = 2 N In the formula (3.19) is set x = xi , i = 0, 2, 4, . . . , N and the desired quadrature formula Z

0

b

N 2 p 2π X f (xi , y) dy ≈ f (xi , x2k−1 ) x2k−1 (b − x2k−1 ), N

i = 0, 2, 4 . . . , N (3.20)

k=1

is obtained. The distribution of nodes using the Chebyschev polynomial has a favourable property. The density of nodes is higher near the ends of the chord. From this reason the shape of the airfoil is expressed more precisely. The velocity is needed to be known at points xi , i = 1, 3, . . . , N − 1 and the resulting airfoil is evaluated at points xi , i = 0, 2, 4, . . . , N . The number of evaluated points along one side is N/2 + 1 and the total number of different points of the airfoil is 2(N/2 + 1) − 2 = N.

3.5.

Viscous Correction

The approximate inversion described so far can deal with a pressure distribution regardless of viscosity. However, the inverse operator is based on the fact that an airfoil can be

388


represented by streamlines. In the case of an inviscid flow, streamlines are attached to the upper and lower surface and so the airfoil contours can be replaced by them. But this is not true in the case of a viscous flow. Streamlines are influenced by a boundary layer and thus the obtained shape is different from the physical shape. The airfoil is thicker and the camber is smaller. Although the inverse operator has some capability to overcome this fact, in the cases with strong influence of the boundary layer it is not enough. From this reason it is useful to introduce a correction to eliminate this problem. This correction is based on the displacement thickness corresponding to the pressure and the airfoil geometry. Using the mentioned inverse operator, the mean camber line function and the thickness function are computed. Now the correction is applied. First, the thickness is reduced by the displacement thickness. Second, the mean camber line is corrected. Since the geometry of the airfoil is closed and the effective shape is open, it is not possible to simply subtract the displacement thickness form the thickness function. The remedy is to start from the definition of the displacement thickness and derive the correction. The displacement thickness is defined as the following integral, Z δ ρu ∗ δ = 1− dy, (3.21) ρe u e 0 where the subscript e denotes the values on the edge of the boundary layer, u is the velocity across the boundary layer and δ is the boundary layer thickness. Since u/ue ≈ 1 for y ≥ δ (in this case ue means a velocity outside the boundary layer) and the inverse operator assumes an incompressible flow, the formula can be rewritten Z d u dy, for d ≥ δ. δ∗ ≈ d − u e 0 Denoting fx′ (y) = ux (y)/ue (x) (the subscript x denotes the dependency on the location on the airfoil), the relation for the corrected airfoil thickness is tvis (x) = t(x) − δ ∗ (x) = t(x) − d + fx (d), for d ≥ δ. That means tvis = fx (t) for t ≥ δ. Assuming the velocity profile u nonnegative, we can extend this relation for all t nonnegative. Thus the corrected airfoil thickness is determined by tvis = fx (t) (3.22) providing that the velocity profile on the airfoil is known. The velocity profile is obtained by the Pohlhausen’s method [5], which was derived for laminar boundary layers. The approximation of the velocity by the polynomial of fourth order is assumed 3 4 3 ux (y) y y y due (x) δ(x) y =2 −2 + + y 1− . (3.23) ue (x) δ(x) δ(x) δ(x) dx 6ν δ(x) The velocity described by this polynomial satisfies the boundary conditions ux (0) = 0, ux (δ) = ue (x), ∂2u

x (0) ∂y 2

=−

ue (x) due (x) ∂ux (δ) ∂ 2 ux (δ) , = 0, = 0. ν dx ∂y ∂y 2

(3.24)


389

This representation makes sense for a parameter Λ = (δ 2 /ν)(due /dx) from an interval h−12, 12i. If Λ > 12, the velocity profile overshoots the value of the velocity ue (x). If Λ < −12, a separation occurs and the velocity profile is not nonnegative. The boundary layer thickness δ is obtained from the von Kármán’s momentum equation and relations for the boundary layer parameters: due dθ τwall = (2θ + δ ∗ ) ue + u2e , ρ dx dx 2 3 1 due δ δ∗ = δ − , 10 120 dx ν

1 due δ 2 1 37 − − θ=δ 315 945 dx ν 9072 2 µue 1 due δ 2+ . τwall = ρ 6 dx ν

(3.25) (3.26)

due δ 2 dx ν

2 !

,

(3.27) (3.28)

The new thickness tvis is computed using the formula (3.22). After that, the mean camber line is modified to eliminate viscous effects. To do so, a correction function ∆(x) =

∗ − δ∗ δup lo 2

is subtracted from the mean camber line function s(x). The function s(x) is then transformed to satisfy s(0) = s(b) = 0.

3.6.

Finding the Mean Camber Line of a General Airfoil

As was mentioned earlier, the pressure distribution is given along a mean camber line. The inverse operator evaluates the function describing this line, so the mean camber line of the designed airfoil is known. But if the given pressure distribution is based on a known airfoil, it is useful to have a way how to get its mean camber line. Assume the airfoil coordinates are known. The process of the airfoil construction (3.17)–(3.18) can be rewritten as ! 1 s′ (x) ψup x − t(x) p = s(x) + t(x) p , ′2 1 + s (x) 1 + s′2 (x) ! s′ (x) 1 ψlo x + t(x) p = s(x) − t(x) p , x ∈ h0, bi (3.29) ′2 1 + s (x) 1 + s′2 (x)

under the assumptions s(x) ∈ C 1 h0, bi, t(x) ∈ C h0, bi. The symbols ψup (˜ x), ψlo (˜ x) denote the y-coordinates of the upper and lower part of the airfoil (corresponding to xcoordinate x ˜). Linearization of the left hand sides and rearrangement leads to a differential equation for an unknown function s(x), ′ ′ ′ ′ ′ s (x) ψup (x)ψlo (x) + ψup (x)ψlo (x) − s(x) ψup (x) + ψlo (x) = 2s(x) − ψup (x) − ψlo (x),

x ∈ (0, b) .

(3.30)

390


This equation is equipped with boundary conditions s(0) = 0,

s(b) = 0.

′ (x) 6= −1 and The equation was derived under the assumptions that s′ (x)ψup ′ (x) 6= −1. In other words, the tangential vectors of the airfoil shape and the s′ (x)ψlo mean camber line are not perpendicular to each other. This is true with an admissible geometry. The expression enclosed by parentheses in (3.30) can be zero at some points from the interval h0, bi. Thus the derivative s′ (x) at these points is undefined by (3.30). However, the equation then says s(x) = (ψup (x) + ψlo (x))/2, which corresponds to the idea of a mean camber line. If the airfoil is closed and the curves representing upper and lower ′ (x) = ψ ′ (x). parts are smooth, then there exists at least one point x ∈ h0, bi such that ψup lo ′ (x) = −1 or the term in parentheses in (3.30) is zero. From At this point, either s′ (x)ψup that reason it is necessary to be careful when solving the differential equation and utilize both boundary conditions. Generally, the existence and uniqueness of the solution is not guaranteed, but in the common cases the solution is unique.

4.

Direct Operator

This operator represents the solution of the flow problem. Depending on the model of flow used, its formulation can vary. In this case, where the viscous compressible flow is assumed, the model is described by the system of the Navier-Stokes equations.

4.1.

Mathematical Formulation

The Navier-Stokes equations are given by 2

∂ρ X ∂(ρvj ) + = 0, ∂t ∂xj

(4.31)

j=1

2 ∂(ρvi vj + p δij ) X ∂τij = , i = 1, 2, (4.32) ∂xj ∂xj j=1 j=1 2 2 X ∂E X ∂ (E + p)vj ∂ µ µT ∂e + = τj1 v1 + τj2 v2 + + γ . ∂t ∂xj ∂xj Pr PrT ∂xj

∂(ρvi ) + ∂t

2 X

j=1

j=1

(4.33)

Since most of the flow in a real situation is turbulent, the laminar model seems insufficient. To improve the quality of the predicted flow and also the stability of the method, a model of turbulence is included (hence the term with the subscript T in (4.33)). In the case described in this chapter the k − ω turbulence model is used [6], [7]. New variables describing the turbulence properties are introduced, a turbulent kinetic energy k and a specific


391

turbulent dissipation ω. These two variables are linked together by equations 2

2

j=1

j=1

2

2

j=1

j=1

X ∂ ∂ρk X ∂ρkvj + = ∂t ∂xj ∂xj

∂k (µ + σk µT ) + Pk − β ∗ ρωk, ∂xj

(4.34)

X ∂ ∂ρω X ∂ρωvj + = ∂t ∂xj ∂xj

∂ω (µ + σω µT ) + Pω − βρω 2 + CD . ∂xj

(4.35)

For the simplicity, the system can be rewritten into the vector form 2

2

∂w X ∂Fj (w) X ∂Gj (w, ∇w) + = + S (w, ∇w) . ∂t ∂xj ∂xj j=1

(4.36)

j=1

By µT an eddy viscosity coefficient is denoted. This coefficient is given by the formula µT =

ρk . ω

The stress tensor in the N.-S. equations is given by relations 2 ∂v2 4 ∂v1 − − τ11 = (µ + µT ) 3 ∂x1 3 ∂x2 2 ∂v1 4 ∂v2 τ22 = (µ + µT ) − + − 3 ∂x1 3 ∂x2 ∂v2 ∂v1 . + τ12 = τ21 = (µ + µT ) ∂x2 ∂x1

2ρk , 3 2ρk , 3

The production of turbulence Pk on the right hand side of the eq. (4.34) and the production of dissipation Pω in eq. (4.35) are expressed as ∂v1 ∂v2 ∂v1 ∂v2 Pk = τ 11 + τ 12 + , + τ 22 ∂x1 ∂x2 ∂x1 ∂x2 Pk Pω = αω ω , k where τ ij = τij for µ = 0. Finally, the cross-diffusion term CD is given by the relation ρ ∂k ∂ω ∂k ∂ω CD = σD max + ,0 . ω ∂x1 ∂x1 ∂x2 ∂x2 The turbulence model is closed by parameters β ∗ = 0.09, β = 5β ∗ /6, αω = β/β ∗ − √ σω κ2 / β ∗ (where κ = 0.41 is the von Kármán constant), σk = 2/3, σω = 0.5 a σD = 0.5. This choice of parameters resolves the dependence of the k − ω model on the free stream values [7]. In the standard Wilcox model cross diffusion the parameters are √ without ∗ ∗ ∗ 2 ∗ β = 0.09, β = 5β /6, αω = β/β − σω κ / β , σk = 0.5, σω = 0.5 a σD = 0. If the turbulent kinetic energy k is set to zero, the turbulence model has no influence upon the N.-S. equations and the laminar model can be solved.

392

4.2.


Numerical Treatment

In the numerical method, a dimensionless system of equations is solved. The variables p, ρ, v1 , v2 , k and ω are normalized with respect to the critical values of density ρ∗ , velocity c∗ and pressure p∗ and to the characteristic length l∗ . The critical values are the values corresponding to a unit Mach number. The dimensionless system has the same form as (4.31)–(4.35). The relations between the original and normalized variables are the following: vˆ1 =

v1 , c∗

vˆ2 =

k kˆ = 2 , c∗

v2 , c∗ ω ˆ=

ρˆ = ωl∗ , c∗

ρ , ρ∗

p , ρ∗ c2∗ µT µ ˆT = . ρ ∗ c∗ l ∗ pˆ =

(4.37)

The problem is solved by an implicit finite volume method. Details about this method and the numerical solution of a flow in general can be found in many textbooks, for example [8]. Since the coupling between the equations describing the flow and the equations describing the turbulence is only by the viscous terms, it is possible to solve the problem in two parts [6]. When the flow variables p, ρ, v1 , v2 in the time tk+1 are computed using (4.31)–(4.33), the turbulent variables k and ω are assumed to be constant in time with values corresponding to the time tk . On the contrary, when computing the turbulent variables k, ω in a time tk+1 using (4.34) and (4.35), the flow variables p, ρ, v1 , v2 are assumed to be constant in the time tk . That means the solution of the problem consists of two systems, 1. (v1k+1 , v2k+1 , ρk+1 , pk+1 ) = NS(v1k+1 , v2k+1 , ρk+1 , pk+1 , k k , ω k ), 2. (k k+1 , ω k+1 ) = Turb(v1k , v2k , ρk , pk , k k+1 , ω k+1 ). These systems of equations can be solved independently of each other. The solution of the system of equations (4.31)-(4.33) is similar to the way how a laminar problem is solved. The equations are identical with the pure N.-S. equations except the stress tensor τij in viscous terms and the heat flux, which depend on k and ω. Since k and ω are taken as parameters in the system, it is quite easy to modify the existing laminar solver. By wh will be denoted a vector of 6-dimensional blocks wi of the values of an approximate solution on finite volumes Di ∈ Dh . For wh ∈ Rn the vector Φ(wh ) consists of 6-dimensional blocks Φi (wh ) given by ! 2 2 X 1 X X Φi (wh ) = ns Fs,h (wh ; i, j) |Γij | − ns Gs,h (wh ; i, j) |Γij | |Di | j∈S(i)

s=1

− Sh (wh ; i, j) ,

s=1

(4.38)

where |Di | denotes the cell area, |Γij | denotes the length of the edge between Di and Dj , nij = (n1 , n2 ) denotes the outer normal to Di . Functions Fs,h (wh ; i, j), Gs,h (wh ; i, j) and Sh (wh ; i, j) are approximations of F(w), G(w, ∇w) and S(w, ∇w) on the grid Dh . In order to have a higher order method we apply the Van Leer κ-scheme or the Van Albada limiter inside the functions Fs,h . By S(i) is denoted a set of indices of neighbouring elements and by the symbol τ will be denoted the time step. Thus the implicit finite volume


393

scheme in a cell Di can be written as wik+1 = wik − τ k Φi (whk+1 ).

(4.39)

The nonlinear equation above is linearized by the Newton method. The arising system of linear algebraic equations is solved by the GMRES method (using software SPARSKIT2 [9]). The convective terms Fi are evaluated using the Osher-Solomon numerical flux in the case of the flow part and by the Vijayasundaram numerical flux in the turbulent part. The numerical evaluation of a gradient on the edge V2 V3 on a boundary (Fig. 4) is done by the following formulae (the index denotes the value in the corresponding vertex) − (V3,y − V2,y ) (f1 − fwall ) ∂f ≈ , ∂x1 V2 V3 |(V2,x − V1,x )(V3,y − V1,y ) − (V3,x − V1,x )(V2,y − V1,y )| (V3,x − V2,x ) (f1 − fwall ) ∂f ≈ . (4.40) ∂x2 V2 V3 |(V2,x − V1,x )(V3,y − V1,y ) − (V3,x − V1,x )(V2,y − V1,y )| If the edge is inside the domain, another scheme according to Fig. 5 is used,

[V1,x , V1,y ]

[V3,x , V3,y ]

[V2,x , V2,y ]

Figure 4. Scheme for a derivative on a wall. [V4,x , V4,y ]

[V1,x , V1,y ]

[V3,x , V3,y ] [V2,x , V2,y ]

Figure 5. Scheme for a derivative inside the domain. (f3 − f1 )(V4,y − V2,y ) − (f4 − f2 )(V3,y − V1,y ) ∂f = , ∂x1 V1 V3 |(V3,x − V1,x )(V4,y − V2,y ) − (V4,x − V2,x )(V3,y − V1,y )| (f3 − f1 )(V4,x − V2,x ) − (f4 − f2 )(V3,x − V1,x ) ∂f =− . ∂x2 V1 V3 |(V3,x − V1,x )(V4,y − V2,y ) − (V4,x − V2,x )(V3,y − V1,y )|

(4.41)

The points V2 and V4 are centres of corresponding cells, f2 and f4 are values in these cells. The values f1 and f3 are obtained as an arithmetic mean of values of the four neighboring cells.

394

4.3.


Boundary Conditions

In our problem, three types of boundary conditions occur: a condition on a wall, a condition at an inlet boundary and a condition at an outlet boundary. Due to the viscosity, the zero velocity on the wall is prescribed, further the zero turbulent kinetic energy and a static temperature are prescribed. The value of the specific turbulent dissipation ω is obtained by the formula 120µ ωwall = , ρyc2 where yc is the distance between the wall and the centre of a cell in the first row. At the inlet part of the boundary, the velocity vector (v1 , v2 ), the density ρ, turbulent energy k and dissipation ω are prescribed. At the outlet part of the boundary, three variables are prescribed, the static pressure p, turbulent energy k and dissipation ω. The other variables are evaluated from values inside the domain. The values of k and ω on the boundary are values of the free stream and are given in the form of a turbulent intensity I and a viscosity ratio ReT = µT /µ. The turbulent intensity is defined as r 2 k . (4.42) I= 3 v∞ Following this, the relations for k∞ and ω∞ are obtained,

4.4.

2 (v∞ I)2 , 3 ρk µT −1 = . µ µ

k∞ =

(4.43)

ω∞

(4.44)

Mesh Deformation

During the inverse method iterations, the direct operator is applied a number of time. This is of course true, a new airfoil shape is designed in each iteration. Moreover, there is a need to find an appropriate angle of attack, which ensures the required position of the stagnation point on the leading edge. This is done by the rotation of the airfoil. This all results in changes of the computational domain and of the mesh, of course. In order to remove the dependency on the mesh generator used, the actual mesh is deformed to fit the new geometry. The deformation is based on the linear elasticity model which is described in many textbooks. The grid cells are stretched or shrinked and moved to fit the new domain. No grid points are created or deleted, the neighbouring cells are the same in the new grid as in the original grid. The linear elasticity model is described by the equation div σ = f

in Ω,

(4.45)

where σ is a stress tensor and f is some body force. The symbol Ω denotes the domain, its boundary will be denoted by Γ. The stress tensor is expressed using a strain tensor ǫ as σ = λ Tr(ǫ)I + 2µǫ


395

and the components of the strain tensor can be expressed using a displacement function u as ∂uj 1 ∂ui ǫij = + . 2 ∂xj ∂xi The symbols λ and µ are the Lamé constants which describe the physical properties of the solid. They can be expressed using the Young’s modulus E and Poisson’s ratio ν, λ=

νE , (1 + ν)(1 − 2ν)

µ=

E . 2(1 + ν)

Substituting the above mentioned relations into (4.45), the following problem for the displacement is obtained: Find an unknown function u : Ω → R2 such that (λ + µ)∇(div u) + µ∆u = f u = uD

in Ω, on Γ.

(4.46)

The function uD is the displacement on the boundary, which is known. This problem is reformulated into the weak sense. Thus the problem for the mesh deformation can be formulated in the form: Find a function u ∈ H 1 (Ω)2 such that u−u∗ ∈ H01 (Ω)2 , where u∗ represents Dirichlet boundary conditions (that means u∗ ∈ H 1 (Ω)2 , u∗ |Γ = uD ) and the function u satisfies the equation Z Z Z −µ ∇u : ∇ϕT dx − (λ + µ) div u · div ϕ dx = f · ϕ dx (4.47) Ω

for all ϕ ∈

H01 (Ω)2 .

Ω

Ω

The colon operator is defined by the relation A:B =

2 X 2 X i=1 j=1

aij bji ,

A, B ∈ R2×2 .

The weak problem described above can be solved by a finite element method. The domain is discretized by a triangular mesh with nodal points, which are the same as in the mesh for the flow problem. This leads to the fact, that the solution, which represents the movement of the nodal points of the original mesh, is evaluated at the appropriate points. The parameter E is set proportional to the reciprocal values of the cell volumes. This ensures that the most deformation is carried out on large cells instead of the small ones.

5. 5.1.

Numerical Examples Example 1

In this example the given pressure distribution is obtained as a result of a flow around the NACA4412 airfoil. The inlet Mach number is M∞ = 0.6, angle of attack α∞ = 1.57◦ and the Reynolds number Re = 6 · 106 . The flow is described by the use of the turbulence k − ω model. The resulting airfoil is compared to the original one and thus shown the correctness of the method. The results are in Figure 6. The relative error of pressure distribution (measured in L2 -norm) after 40 iterations is 8.32 · 10−4 .

396

Jan Šimák and Jaroslav Pelant 0.0003

1.4 0.0002

0.6

0.0001

||e||

0.8 1.2

0

1

0

0.2

0.4 X 0.6

0.8

1

0

0.2

0.4 X 0.6

0.8

1

P

0.4 0.01

0.8

0.2 p-f

0

Y

0.005 0.6

0 0.4

0

0.2

0.4

X

0.6

0.8

1

-0.005

Figure 6. Example 1. Pressure distribution and resulting airfoil shape, error of the resulting airfoil measured as a norm kψresult − ψN ACA4412 ke , difference between the prescribed and resulting pressure distribution on the chord (values are normalized).

5.2.

Example 2

In this example a laminar flow with low Reynolds number is computed. The starting pressure distribution is computed on the NACA3210 airfoil with parameters Re = 1000, M∞ = 0.6, α∞ = 4.56◦ . At first, the problem is computed without any viscous correction and then the mentioned correction based on the Pohlhausen’s method is used. From the results is evident, that for very low Reynolds numbers the correction is necessary. The comparisons are in Figure. 7.

1.4

0.2 y

0.8

1.2

0

0.6

P

0.4

0

0.2

0.4 X 0.6

0.8

1

0

0.2

0.4 X 0.6

0.8

1

1 0.02 0.2

0.01 p-f

0

Y

0.8 0

-0.01 0.6

0

0.2

0.4

X

0.6

0.8

1

-0.02

Figure 7. Example 2. Pressure distributions and resulting airfoil shapes (solid - correction, dashed - without correction), comparison of the airfoils with the NACA3210 (dotted), difference between the prescribed and resulting pressure distribution along the chord (values are normalized).


6.

397

Conclusion

A numerical method for a solution of an inverse problem of flow around an airfoil was described. The advantage of this method is the weak dependency on the model describing the flow. It is easy to modify the method by assuming a different model. The correction due to viscosity is necessary only with low Reynolds numbers. The presented method has still some drawbacks. These are the dependency of the angle of attack on the prescribed distribution and the applicability on the subsonic regimes only. The method can be improved in the future.

Acknowledgment This work was supported by the Grant MSM 0001066902 of the Ministry of Education, Youth and Sports of the Czech Republic.

References [1] Pelant, J. Inverse Problem for Two-dimensional Flow around a Profile, Report No. Z–69; VZLÚ, Prague, 1998. [2] Šimák J.; Pelant J. A contractive operator solution of an airfoil design inverse problem; PAMM Vol. 7, No. 1 (ICIAM07), pp. 2100023–2100024. [3] Šimák, J.; Pelant, J. Solution of an Airfoil Design Problem With Respect to a Given Pressure Distribution for a Viscous Laminar Flow, Report No. R–4186; VZLÚ, Prague, 2007. [4] Michlin, S. G. Integral Equations; Pergamon Press, Oxford, 1964. [5] Schlichting, H. Boundary-Layer Theory; McGraw-Hill, New York, 1979. [6] Wilcox, D. C. Turbulence Modeling for CFD; DCW Industries Inc., 1998; 2nd ed. [7] Kok, J. C. Resolving the Dependence on Freestream Values for the k − ω Turbulence Model; AIAA Journal 2000, vol. 38, No. 7, pp. 1292–1295. [8] Feistauer, M.; Felcman, J.; Straškraba, I. Mathematical and Computational Methods for Compressible Flow; Clarendon Press, Oxford, 2003. [9] Saad, Y. Iterative Methods for Sparse Linear Systems; SIAM, 2003; 2nd ed.



Chapter 13

S OME F REE B OUNDARY P ROBLEMS IN P OTENTIAL F LOW R EGIME U SING THE L EVEL S ET M ETHOD M. Garzon1 , N. Bobillo-Ares1 and J.A. Sethian2 1 Dept. de Matemáticas, Univ. of Oviedo, Spain 2 Dept. of Mathematics, University of California, Berkeley, and Mathematics Department, Lawrence Berkeley National Laboratory.

Abstract Recent advances in the field of fluid mechanics with moving fronts are linked to the use of Level Set Methods, a versatile mathematical technique to follow free boundaries which undergo topological changes. A challenging class of problems in this context are those related to the solution of a partial differential equation posed on a moving domain, in which the boundary condition for the PDE solver has to be obtained from a partial differential equation defined on the front. This is the case of potential flow models with moving boundaries. Moreover, the fluid front may carry some material substance which diffuses in the front and is advected by the front velocity, as for example the use of surfactants to lower surface tension. We present a Level Set based methodology to embed this partial differential equations defined on the front in a complete Eulerian framework, fully avoiding the tracking of fluid particles and its known limitations. To show the advantages of this approach in the field of Fluid Mechanics we present in this work one particular application: the numerical approximation of a potential flow model to simulate the evolution and breaking of a solitary wave propagating over a slopping bottom and compare the level set based algorithm with previous front tracking models.

1.

Introduction

In this chapter we present a class of problems in the field of fluid mechanics that can be modeled using the potential flow assumptions, that is, inviscid and incompressible fluids moving under an irrotational velocity field. While these are significant assumptions, in the presence of moving boundaries, the resulting equations is a non linear partial differential equation, which adds considerable complexity to the computational problem. In the literature this model is often called the fully non linear potential flow model (FNPFM). Several

400

M. Garzon, N. Bobillo-Ares and J.A. Sethian

interesting and rather complicated phenomenon are described using the FNPFM, as for example, Helle-Shaw flows, jet evolution and drop formation, sprays and electrosprays, wave propagation and breaking mechanisms, etc, see [21], [22], [30], [13]. Level Set Methods (LSM) [31], [33], [34] [37] are widely used in fluid mechanics, as well as other fields such as medical imaging, semiconductor manufacturing, ink jet printing, and seismology. The LSM is a powerful mathematical tool to move interfaces, once the velocity is known. In many physical problems, the interface velocity is obtained by solving the partial differential equations system used to model the fluid/fluids flow. The LSM is based on embedding the moving front as the zero level set of one higher dimensional function. By doing so, the problem can be formulated in a complete Eulerian description and topological changes of the free surface are automatically included. The equation for the motion of the level set function is an initial value hyperbolic partial differential equation, which can be easily approximated using upwind finite differences schemes. Recently, the LSM has been extended to formulate problems involving the transport and diffusion of material quantities, see [3]. In [3] model equations and algorithms are presented together with the corresponding test examples and convergence studies. This led to the realization that the nonlinear boundary conditions in potential flow problems could also be embedded using level set based methods. As a result, the FNPFM can also be formulated with an Eulerian description with the associated computational advantages. Two difficult problems that have been already approximated using this novel algorithm are wave breaking over sloping beaches [16], [17] and the Rayleigh taylor instability of a water jet [20]. Moreover, related to drop formation and wave breaking, it has been recently reported in the literature [46], [45] that the presence of surfactants on the fluid surface affects the flow patterns. The models described in this chapter are the groundwork for solving these complex problems. This chapter is organized as follows: in section 2. we have made an effort to obtain dynamic equations valid for any spatial coordinate system. To do so, we derive the equations using only objects defined in an intrinsic way (i.e., independent of any coordinate system). At the same time, in accordance with the level set perspective, we have avoided as much as possible, the “ material description” (Lagrangian coordinates). Geometric quantities are defined using the level sets and tensor fields in the space. In section 3. a brief description of the Levels Set Method is given using this intrinsic approach. Section 4. is devoted to describe two particular potential flow models, the first one related to drop formation in the presence of surfactants, which combines all the models derived in section 2.. The wave breaking problem is modeled in 2D, code development in 3D is underway. In section 5., we present the numerical approximation and algorithm for the wave breaking problem. Numerical results and accuracy tests are also presented in section 6.. Precise definitions of certain needed geometrical tools, throughout used in this chapter, are shown in Appendix III.

2.

Some Physical Models

Here, we discuss the derivations of fluid problems and their corresponding reformulation using the Level Set Method (LSM) techniques. The brief derivation of known physical laws is used also as a pretext to introduce some preliminary concepts and notation.

Some Free Boundary Problems in Potential Flow Regime...

2.1.

401

Kinematic Relationships

Reference configuration. The configuration of a continuous medium at certain time t is known when the position of each particle is specified. We name Ωt the space region occupied by the continuous medium at that time. Kinematics require the movement description of each particle. To this aim, we must: i. Label the particles. ii. Specify the movement of each particle. The first step is done considering the configuration at an arbitrary instant t0 (reference configuration). Particles are marked by the point P0 ∈ Ωt0 they occupy. Points in Ωt0 are good labels because they are in a 1 to 1 correspondence with the particles (“particles can not penetrate each other”). In what follows we will abbreviate the phrase “particle with label P0 ” by “particle P0 ”. Once all the particles are labeled, it is now possible to undertake the second step. Let P0 ∈ Ωt0 be a particle. Its position P at instant t is given by the function: P = R(P0 , t),

P ∈ Ωt , P0 ∈ Ωt0 .

(1)

According to the reference configuration definition, we have: R(P0 , t0 ) = P0 .

(2)

The mapping Rt , Rt (P0 ) := R(P0 , t) = P , must be invertible: P0 = Rt−1 (P ) ∈ Ωt0 , P ∈ Ωt .

(3)

Lagrangian/Eulerian descriptions. Any tensor field w may be described in two ways, using (1): w = w(P, t) = w(R(P0 , t), t) = w0 (P0 , t). (4) Function w(P, t) corresponds to the so called Eulerian description and w0 (P0 , t) corresponds to the Lagrangian description. As a consequence any tensorial field w admits two time partial derivatives. The “spatial” derivative, corresponding to the Eulerian description: d ∂t w := w(P, t + ǫ) , (5) dǫ ǫ=0

measures the variation rate with time of w from a fixed point in the space. The “convective” derivative, corresponding to the Lagrangian description: d Dt w := w0 (P0 , t + ǫ) , (6) dǫ ǫ=0

gives the variation rate of w following the particle P0 .

402


Velocity field The velocity u = u(P0 , t) of the particle P0 is obtained using the convective derivative (“following the particle”) of the position P = R(P0 , t): u = Dt P,

P = R(P0 , t).

(7)

Obviously, u admits both descriptions: u = u(P, t) = u0 (P0 , t),

P = R(P0 , t).

(8)

Given an arbitrary tensor field w, its spatial and convective derivatives are related using the calculus chain rule and the definition (7): Dt w = ∂t w + ∂u w.

(9)

Here, ∂u w designates the directional derivative of w along u (see Appendix III). The acceleration of particle P0 is obtained by the convective derivative of the velocity field. Using (9), we have: Dt u = ∂t u + u · ∇u. (10) Transport of a vector due to a moving medium. A fluid particle is located at point1 P at time t. After a time ∆t, the same particle is at point R(P, t + ∆t). Clearly, the function R must verify that R(P, t + 0) = P . A nearby particle at same time t is located at P + ǫa, and at t + ∆t is at point R(P + ǫa, t + ∆t). We have again P + ǫa = R(P + ǫa, t + 0). The vector ǫa that connects both particles varies as they move. Denote by Dt ǫa its rate of change with time: 1 [(R(P + ǫa, t + ∆t) − R(P, t + ∆t)) − (R(P + ǫa, t) − R(P, t))] ∆t R(P + ǫa, t + ∆t) − R(P + ǫa, t) R(P, t + ∆t) − R(P, t) = lim − lim . ∆t→0 ∆t→0 ∆t ∆t

Dt ǫa =

lim

∆t→0

The first term of the right hand side of previous equation is, by definition, the particle velocity at P + ǫa, u(P + ǫa, t), and the second term the particle velocity at P , u(P, t). Thus we have: Dt ǫa = u(P + ǫa, t) − u(P, t). Letting ǫ → 0, we obtain the rate of change with time of an infinitesimal vector dragged by the medium: 1 u(P + ǫa, t) − u(P, t) d Dt a := lim Dt ǫa = lim = u(P + ǫa, t) = ∂a u. (11) ǫ→0 ǫ ǫ→0 ǫ dǫ ǫ=0 We denote ∂a the operator that performs the directional derivative along the vector a (see Appendix III). 1

For this calculation we use here the configuration at t as the reference configuration.


403

Fluid volume change as it is transported by the velocity field. Let a, b and c be three small vectors with origin at point P . The volume of the parallelepiped spanned by vectors a, b, c is given by the trilinear alternate form δV = [a, b, c] = a · b × c. The rate of change of this volume, when particles located on its vertices move, is given by Dt δV , and thus we have Dt δV = Dt [a, b, c] = [Dt a, b, c] + [a, Dt b, c] + [a, b, Dt c]. Using now (11) we get Dt δV = [∂a u, b, c] + [a, ∂b u, c] + [a, b, ∂c u], which is also a trilinear alternate form. As in the tridimensional space all these forms are proportional, we can set Dt [a, b, c] = (div u)[a, b, c], (12) which gives us an intrinsic definition of the divergence of the field u. If the continuous medium is incompressible, the volume δV does not change, Dt δV = 0, and we arrive at the incompressibility condition div u = 0. (13)

2.2.

Dynamic Relationships

Conservation of mass. Denote by ρ = ̺(P, t) the volumetric mass density of the continuous medium at point P and at time t. The rate of change of the mass in a small volume δV dragged by the velocity field is, using definition (12), Dt (ρδV ) = (Dt ρ)δV + ρDt δV = (Dt ρ + ρ div u)δV. The mass conservation law is thus Dt ρ + ρ div u = 0.

(14)

Applying general formula (9) to ρ, we have Dt ρ = ∂t ρ + ∂u ρ. In the case of an homogeneous and incompressible medium with uniform initial density ρ0 , using equations (14) and (13), we have Dt ρ = 0 which gives ̺(P, t) = ρ0 . Conservation of the momentum associated with a small piece of continuous medium. From Newton’s second law applied to a fluid volume V we get the relation Z Z Z Dt u ρdV = g ρdV + τ (ds). (15) V

V

∂V

The term in left hand side of this equation is the rate of change with time of the momentum associated with volume V when dragged by the continuous medium. The first term in the right hand side corresponds to the volumetric forces inside V , generated by a vector field

404


per unit mass g, usually the gravitational field. The second term represents the “contact” forces applied by the rest of the medium over the part in V . The Cauchy’s tensor τ is a linear operator field that is obtained from specific relationships which depend on the material, the so called constitutive relations. We are interested in inviscid fluids which verify the Pascal’s law: τ (ds) = −pds, where p is the pressure scalar field. Green’s formula, Z Z −p ds = −∇p dV, ∂V

V

shows that contact forces may be computed as a kind of volume forces with density −∇p. For a small volume δV dragged by the fluid, equation (15) can be written: Dt (u ρδV ) = (g ρ − ∇p)δV.

(16)

Due to the mass conservation law, Dt (ρδV ) = 0, equation (16) leads to the Euler equation: 1 Dt u = ∂t u + ∂u u = g − ∇p. ρ

(17)

If g is a uniform field it comes from the gradient of a potential function: g = −∇U (P ), U (P ) = −g · (P − O), where P − O is the position vector of the point P .

2.3.

Potential Flow

Assuming an irrotational flow regime, curl u = 0, there exists an scalar field φ such that u = ∇φ.

(18)

Outside of the fluid domain, and separated by a free boundary, there is a gas at pressure pa that is assumed to be constant. This means that, within the gas, the time needed to restore the equilibrium is very small compared with the time evolution of the fluid. Therefore, at the fluid free boundary, the boundary condition is just: p = pa .

(19)

Using the vectorial relationship ∇u2 /2 = ∂u u + u × (curl u), and relations (18) and (13) we have 1 2 p ∇ ∂t φ + u + + U = 0. 2 ρ Performing the first integration, 1 p ∂t φ + u2 + + U = C(t), 2 ρ


405

where C(t) is an arbitrary function of time, which can be chosen in such a way that the previous relation can be written: 1 p − pa ∂t φ + u2 + + U = 0. 2 ρ Now using the obvious relation ∂t φ + u2 = ∂t φ + ∂u φ = Dt φ, we finally obtain p − pa 1 + U = 0. Dt φ − u2 + 2 ρ

2.4.

(20)

Advection

On the surface of a continuous medium with a known movement, a certain substance is distributed, which will be named as “charge”. This is adhered to the fluid particles and it is transported by them. In this way a set of particles will always carry the same amount of “charge”. This phenomenon is called advection. The continuous medium surface is implicitly described as the zero level set of a certain scalar function Ψ = ψ(P, t): Γt = {Q|ψ(Q, t) = 0}.

(21)

Vectors a tangent to the surface are characterized by the condition ∂a Ψ = a · ∇Ψ = 0; thus, the tangent vectorial plane at each point of the surface is given by the normal unit vector2 ∇Ψ n= . |∇Ψ|

The function ψ by itself does not specify the particle movement on the surface, just its shape. We need to add the information about how these particles move, e.g., specifying the velocity field on the surface Q ∈ Γt , u = u(Q, t). A small vector a connecting two nearby particles on the surface and dragged by them as they move, has a rate of change given by (11), Dt a = ∂a u.

(22)

Note that a is a tangent vector, n · a = 0. Surface areas. Using the normal vector to the surface, n, a 2–form to calculate surface areas can be constructed:3 ω(a, b) := [n, a, b] = n · a × b, 2

Ψ must increase from the interior to the exterior of the surface to get n outwards. The surface area definition is not made using the Gram determinant of two tangent vectors, because this procedure involves a particular parametrization of the surface. Instead, a 2-form is defined from the volume form in space (“the parallelogram area is the volume of a rectangular prism of unit height”). 3

406


where ω(a, b) is the area spanned by tangent vectors a, b, and [n, a, b] the volume form in the 3D space. As the tangent vectors a, b are transported by the surface movement, the parallelogram area associated to them changes. The rate of change with time is easily obtained: Dt ω(a, b) = (Dt n) · a × b + n · (Dt a) × b + n · a × Dt b. First term of the right hand side of previous equation is zero since a × b is a normal vector and Dt n is tangent: indeed, as n2 = 1, we have Dt n2 = 2n · Dt n = 0. Using (22) we have Dt ω(a, b) = ∂a u · b × n + ∂b u · n × a. This expression is bilinear and alternate with respect the tangent vectors. It must be, at each point on the surface, proportional to the 2–form ω. We denote by Div u, “surface divergence”, the proportionality coefficient: ∂a u · b × n + ∂b u · n × a := (Div u) ω(a, b)

(23)

This definition of Div u does not depend upon the choice of tangent vectors a and b. In Appendix I, the expression for the surface divergence of an arbitrary vector field w using rectangular coordinates is shown. Advection law. Now, let be σ = σ(Q, t), Q ∈ Γt the “charge” surface density. The “charge” δq carried by a small parallelogram, spanned by two small tangent vectors (a, b), of area ω(a, b) is δq = σ ω(a, b). As the “charge” is conserved, the advection law is Dt δq = 0. Now, by definition (23), we have Dt (σω) = (Dt σ)ω + σDt ω = (Dt σ + σ Div u)ω. Hence we arrive to the intrinsic equation for the advection phenomena: Dt σ + σ Div u = 0.

(24)


2.5.

407

Advection-Diffusion

Next, we are going to assume that the “charge” diffuses along particles on the surface according to the Fick’s law:4 j = −α∇σ, where α is the diffusion coefficient, j is the “charge” flux and ∇σ is the “charge” surface density gradient. As σ is only defined on the surface Γt , ∇σ is only defined for tangent vectors: ∇σ · a := ∂a σ, a tangent vector. On the surface Γt let us consider a surface region S, bounded by a curve ∂S. Let be ν the unit vector field tangent to Γt and orthogonal to the curve ∂S at each point. The “charge” that leaves the surface per unit time is the outward flux through the boundary ∂S: Z Z Z j · ν dl = − j · n × dl = n × j · dl. δS

∂S

∂S

Applying now Stokes’ theorem, we have Z Z n × j · dl = A ω(d1 P, d2 P ). ∂S

(25)

S

The 2-form of the surface integral is obtained using the intrinsic formula A ω(a, b) = ∂a (n × j · b) − ∂b (n × j · a).

(26)

We interpret A ω(a, b) as the “charge” per unit time that, by diffusion, leaves the small parallelogram spanned by the tangent vectors (a, b). Now it is straightforward to set the condition for the advection-diffusion mechanism     “charge” rate of change “charge” that leaves within the tangent the parallelogram =− ,     parallelogram (a, b) by diffusion

that is

Dt (σω(a, b)) = −A ω(a, b). In Appendix II the following expression for A is obtained: A = Div j − (Div n) j · n. Hence, using (24) we arrive at the general equation for the advection-diffusion model: Dt σ + σ Div u = − Div j + (Div n) j · n j = −α ∇σ or Dt σ + σ Div u = α Div ∇σ − α(Div n) ∇σ · n (27) 4

Fick’s diffusion law applies when the “charge” particles move randomly without any preferential direction (Brownian movement).

408

M. Garzon, N. Bobillo-Ares and J.A. Sethian The Cartesian expressions5 for Div u, Div ∇σ and Div n are: 1 2 (∇Ψ) ∂ ∂ Ψ − ∂ Ψ∂ Ψ∂ ∂ Ψ , i i j i j i |∇Ψ|3 (∇Ψ)2 ∂i ∂i σ − ∂i Ψ∂j Ψ∂i ∂j σ ,

Div n = (δij − ni nj )∂j ni = Div ∇σ = Div u =

1 |∇Ψ|2 1 (∇Ψ)2 ∂i ui − ∂i Ψ∂j Ψ∂i uj . 2 |∇Ψ|

(28) (29) (30)

Expanding the implicit summands, we obtain the following expressions for the 3D space (i, j = 1, 2, 3): 1 (∂1 Ψ)2 (∂22 Ψ + ∂32 Ψ) + (∂2 Ψ)2 (∂12 Ψ + ∂32 Ψ)+ |∇Ψ|3 +(∂3 Ψ)2 (∂12 Ψ + ∂22 Ψ) − 2∂1 Ψ∂2 Ψ∂1 ∂2 Ψ − − 2∂1 Ψ∂3 Ψ∂1 ∂3 Ψ − 2∂2 Ψ∂3 Ψ∂2 ∂3 Ψ , 1 Div ∇σ = (∂1 Ψ)2 (∂22 σ + ∂32 σ) + (∂2 Ψ)2 (∂12 σ + ∂32 σ)+ |∇Ψ|2 +(∂3 Ψ)2 (∂12 σ + ∂22 σ) − 2∂1 Ψ∂2 Ψ∂1 ∂2 σ − − 2∂1 Ψ∂3 Ψ∂1 ∂3 σ − 2∂2 Ψ∂3 Ψ∂2 ∂3 σ , 1 Div u = (∂1 Ψ)2 (∂2 u2 + ∂3 u3 ) + (∂2 Ψ)2 (∂1 u1 + ∂3 u3 )+ |∇Ψ|2 +(∂3 Ψ)2 (∂1 u1 + ∂2 u2 ) − ∂1 Ψ∂2 Ψ(∂1 u2 + ∂2 u1 ) − − ∂1 Ψ∂3 Ψ(∂1 u3 + ∂3 u1 ) − ∂2 Ψ∂3 Ψ(∂2 u3 + ∂3 u2 ) . Div n =

(31)

(32)

(33)

To obtain the formulas for the plane we assume axial symmetry in the direction 3: ∂3 Ψ = 0,

∂32 Ψ = 0,

∂3 σ = 0,

u3 = 0, ∂3 ui = 0,

|∇Ψ|2 = (∂1 Ψ)2 + (∂2 Ψ)2 . Inserting these values in (31), (32) and (33) we get: Div n = Div ∇σ = Div u =

5

1 (∂1 Ψ)2 (∂22 Ψ) + (∂2 Ψ)2 (∂12 Ψ) − 2∂1 Ψ∂2 Ψ∂1 ∂2 Ψ , 3 |∇Ψ| 1 (∂1 Ψ)2 (∂22 σ) + (∂2 Ψ)2 (∂12 σ) − 2∂1 Ψ∂2 Ψ∂1 ∂2 σ , 2 |∇Ψ| 1 (∂1 Ψ)2 (∂2 u2 ) + (∂2 Ψ)2 (∂1 u1 )− |∇Ψ|2 − 2∂1 Ψ∂2 Ψ(∂1 u2 + ∂2 u1 ) .

(34) (35)

(36)

In the following expressions we use only subscripts because orthonormal bases coincides with their corresponding reciprocal ones. Then, the position of the indices becomes irrelevant.


3.

409

The Level Set Method

The Level Set method is a mathematical tool developed by Osher and Sethian [31] to follow interfaces which move with a given velocity field. The key idea is to view the moving front as the zero level set of one higher dimensional function called the level set function. One main advantage of this approach comes when the moving boundary changes topology, and thus a simple connected domain splits into separated disconnected domains. Let be Γt the set of points lying in the surface boundary at time t. This surface is defined through the zero level set of the scalar field Ψ = ψ(P, t): Γt = {Q|ψ(Q, t) = 0}.

(37)

To identify the fluid particles, the configuration at t0 (reference configuration) is used: Γt0 = {Q0 |ψ(Q0 , t0 ) = 0}.

(38)

The particle movement is specified through the function Q = R(Q0 , t),

(39)

which gives the position Q ∈ Γt of the fluid particle Q0 ∈ Γt0 . The particle Q0 velocity is calculated using the convective derivative Dt (“following the particle”): d u = Dt Q = R(Q0 , t + ǫ) . (40) dǫ ǫ=0 According to definition (37), we have ψ(R(Q0 , t), t) = 0. Deriving with respect to time and applying the chain rule, we obtain ∂t Ψ + u · ∇Ψ = 0.

(41)

which has to be completed with the value of the level set function at time t = 0, usually set to be the signed distance function to the initial front, Ψ(P, 0) = s(P )d(P ), being d(P ) the distance from the point P to the surface at the initial configuration Γ0 , s(P ) = −1 if P ∈ Ω0 and s(P ) = +1 if P ∈ / Ω0 . Now, if we take a fixed 3D domain ΩD that contains the free surface for all times, we can define the initial value problem for the level set function Ψ posed on ΩD : ∂t Ψ + u · ∇Ψ = 0 in ΩD

Ψ(P, 0) = s(P )d(P ) in ΩD

(42) (43)

A graphical interpretation of the level set function evolution is depicted in figure 1 Equation (42) moves all the level set of Ψ, not just the zero level set, and in many physical applications the front velocity is just defined for points lying on the free boundary. Therefore for this equation to be valid on the whole domain we have to extend the velocity u off the front. There exist several extension procedures which will be briefly commented below.

410


t

y (P, 0)

y (P, t)

G0

Gt

Figure 1. Evolution of the level set function.

3.1.

Extension of Functions Defined on the Front

Let us consider a classical result from functional analysis: suppose a domain Ω bounded by a closed surface ∂Ω. If for k ≥ 1 the surface ∂Ω ∈ C k , then for all functions F (x) ∈ ¯ such that Fext |∂Ω = F (x). C k (∂Ω) there exists a function Fext (x) ∈ C k (Ω) In practice, there are several ways to extend any magnitude F defined on the front onto ΩD . As shown in [10] for the numerical stability of the level set equation it is convenient to preserve Ψ as a signed distance function, which is characterized by the property |∇Ψ| = 1. One way is to perform reinitializations of the level set function at chosen times. If this is done periodically, it will smooth the level set function. However, done too often, especially using poor reinitialization techniques, spurious mass loss/gain will occur. Thus, it is important to perform reinitialization both sparingly and accurately. For the potential flow problems presented in this chapter we follow the strategy introduced in [2]. The idea is to extrapolate F given at the front along its gradient. Mathematically the extended variable Fext is the solution of ∇Fext · ∇Ψ = 0. (44) It is straightforward to show that this choice maintains the signed distance function for the level sets of Ψ for all times. For the numerical approximation we proceed as follows: ˆn given a level set function Ψ at time n, namely Ψn , one first obtains a distance function Ψ around the zero level set. Simultaneous with this construction, the extended quantity Fext is obtained satisfying Eq. (44). For a complete explanation of this extension method see [2].

4.

Examples of Potential Flow Models with Moving Boundaries

In this section the governing equations of two interesting physical problems will be formulated using a level set framework. First, drop formation is a complex 3D phenomena driven mainly by capillary forces, which can be modeled using the potential flow assump-


411

tions. It is well-known that the presence of surface surfactants lowers the surface tension affecting drops shape. Secondly, propagation and wave breaking over sloping beaches can also be modeled with the potential flow equations, which are valid until the jet of the wave impinges against the flat water surface. In this case we can formulate the equations in 2D taking a vertical section of the beach, which facilitates the algorithm and code development. The wave numerical simulations will be presented in section 6..

4.1.

Governing Equations for Surface Tension Driven Flows with Material Advection-Diffusion

Let Ωt be the 3D closed fluid domain surrounded by air and Γt the free surface boundary at time t. Suppose that initially a certain amount of surfactants, which are assumed to be insoluble in water, are uniformly distributed on the surface (see Figure 2).

Figure 2. A fluid volume with surface surfactants. For an incompressible and inviscid fluid, the governing equations are the Euler equations (17). On the free boundary the following partial differential equations apply: • The advection-diffusion equation for the surfactant is (27): Dt σ + σDiv u = α(Div∇σ − κ ∇σ · n) on Γt , where σ is the surface density of the surfactant, u is the free boundary velocity, α is the surface diffusion coefficient, κ = Div n = R11 + R12 and R1 , R2 the principal radii of curvature of Γt at each point. • Continuity of the stress tensor between water and air leads to the balance of the surface tension forces, p = pa +γ( R11 + R12 ), where γ is the surface tension coefficient

412

M. Garzon, N. Bobillo-Ares and J.A. Sethian that may depend on the surfactant concentration σ. Thus Eq. (20) becomes 1 γ ∂t φ + (∇φ · ∇φ) + κ + U = 0 on Γt . 2 ρ

• Finally, if Q = R(Q0 , t) is the position of a fluid particle Q0 on the free surface, the definition (40) states Dt Q = u(Q, t), Q ∈ Γt . (45) The complete model equations in 3D are therefore, u = ∇φ in Ωt

∆φ = 0 in Ωt

Dt Q = u on Γt γ 1 Dt φ = −U + (∇φ · ∇φ) − κ on Γt 2 ρ Dt σ = −σDiv u + α(Div∇σ − κ ∇σ · n) on Γt .

(46) (47) (48) (49) (50)

This is the Lagrangian-Eulerian formulation of the model equations. Classical methods to approximate this set of equations are the so-called “front tracking methods”, in which a fixed number of markers are chosen initially and the trajectories of this markers are followed as time evolves. This method suffers difficulties when the free boundary changes topology: these problems are avoided by a level set formulation. Level Set Framework Equation (48) can be directly formulated as the level set Eq. (41). For the velocity field u(Q, t), the trajectory of a fluid particle at initial position Q0 is given by the solution of Dt Q = u(R(Q0 , t), t), R(Q0 , 0) = Q0 .

(51)

Next, let ΩD be a fixed 3D domain that contains the free surface for all times and let G(P, t) and S(P, t) be two functions defined on ΩD such that for every Q ∈ Γt G(Q, t) = φ(Q, t) ,

(52)

S(Q, t) = σ(Q, t) .

(53)

It is important to remark here that G(P, t) and S(P, t) are auxiliary functions defined in ΩD that can be chosen arbitrarily, the only restriction is that they equal φ(Q, t) and σ(Q, t) on Γt respectively. Figure 3 gives an interpretation of this property for a moving curve in 2D. Applying the chain rule in both identities (52) and (53) we obtain γ 1 ∂t G + u · ∇G = −U + (∇φ · ∇φ) − κ, 2 ρ ∂t S + u · ∇S = −σDiv u + α(Div∇σ − κ ∇σ · n).

(54) (55)


413

t G(P, 0)

G(P, t)

f(Q , t)

f(Q , 0)

G0

Q

Gt

Q

Figure 3. Extension of the velocity potential off the front. which holds on Γt . Note that u and the right hand side of Eq. (54) and Eq. (55) are only defined on Γt , and thus, in order to solve these equations over the fixed domain ΩD , these variables must be extended off the front. This strategy has been discussed in Section 3.. Naming 1 γ f = −U + (∇φ · ∇φ) − κ, 2 ρ h = −σDiv u + α(Div∇σ − κ ∇σ · n), the system of equations, written in a complete Eulerian framework, is u = ∇φ in Ωt

(56)

Ψt + uext · ∇Ψ = 0 in ΩD .

(58)

∆φ = 0 in Ωt

Gt + uext · ∇G = fext in ΩD St + uext · ∇S = hext in ΩD

(57) (59) (60)

Here the subscript “ext” denotes the extension of f , h and u onto ΩD .

4.2.

Governing Equations for the Wave Breaking Problem

We now derive our coupled level set/extension potential equations for breaking waves in two dimensions for which a numerical approximation will be also presented. Let Ωt be the 2D fluid domain in the vertical plane (x, z) at time t, with z the vertical upward direction (and z = 0 at the undisturbed free surface), and Γt the free boundary at time t (see Figure 4). We assume also an inviscid and incompressible fluid, and capillary forces are disregarded on the free boundary curve. The model equations in the Lagrangian-Eulerian formulation are therefore:

414


6

z Γt

Q Γ1

6

h

Ωt

?

Γb Figure 4. The domain.

Γb

u = ∇φ in Ωt

∆φ = 0 in Ωt

Dt Q = u on Γt 1 Dt φ = −gz + (∇φ · ∇φ) on Γt 2 φn = 0 on Γb ∪ Γ1 ∪ Γ2 ,

Γ2

x

-

(61) (62) (63) (64) (65)

Let Ω1 be a fixed 2D domain that contains Γt for all times. Following the same embedding procedure for the potential function as in previous section, we obtain the complete 2D Eulerian formulation: u = ∇φ in Ωt

∆φ = 0 in Ωt

Ψt + uext · ∇Ψ = 0 in Ω1 .

Gt + uext · ∇G = fext in Ω1 φn = 0 on Γb ∪ Γ1 ∪ Γ2

(66) (67) (68) (69) (70)

being here f = 12 (∇φ · ∇φ) − gz and fext the extension of f onto Ω1 .

5.

Numerical Approximations and Algorithms

In this section, we provide overviews of the numerical schemes used to approximate the wave model equations. The integral formulation of Eq. (66) is approximated using a liner boundary element method (BEM), which will provide the velocity of the front node representation. More detailed discussions of the various components may be found in the cited references.

5.1.

Initialization

The initial front position Γ0 and initial velocity potential φ(Q, 0), Q ∈ Γ0 , are needed to solve equations (68) and (69) respectively. Given an initial solitary wave amplitude (H0 )


415

and the physical length of the domain (L), Tanaka’s method gives a way of calculating these quantities. Here, we briefly discuss the theoretical basis of this method. Assuming constant depth, the flow field can be reduced to steady state by using a coordinate system that moves horizontally with speed equal to the wave celerity c. The stream function ψ(x, z) is also harmonic and takes constant values at the bottom and at the free surface of the domain. ¿From the definition of stream function and velocity potential we have φx = ψy , φy = −ψx . Under sensible assumptions about the smoothness of φ and ψ, these are just the CauchyRiemann equations which are satisfied by the real and imaginary parts of the function W = φ+iψ, which is called the complex potential and it is a an analytical function of the complex variable Z = x + iz in the domain occupied by the fluid. By interchanging the role of the variables Z and W , we can take φ and ψ as independent variables, since W = φ + iψ provides a one to one correspondence between the physical and complex potential planes. With this transformation, the fluid region is mapped into the strip 0 < ψ < 1, −∞ < φ < ∞ in the W plane with ψ = 1 on the free surface, ψ = 0 on the bottom and φ = 0 at the wave crest. Denote by u, v the horizontal and vertical components of the velocity u, q = |u| and θ the angle between the velocity and the x axis. The complex velocity is defined by dW = φx + iφy = u − iv = qeiθ dZ and it is also analytic in the flow domain. Therefore, the quantity ω = ln(

dW ) = ln q − iθ, dZ

is an analytic function of W , so τ = ln q must be harmonic in the strip 0 < ψ < 1, −∞ < φ < ∞. The Bernoulli condition at the free surface and the bottom condition can be expressed in terms of q and θ as: dq 3 3 = − 2 sin θ on ψ = 1 dφ F θ = 0 on ψ = 0,

(71) (72)

where F is the Froude number defined by F = √cgh . The problem of finding a solitary wave solution can thus be transformed into the problem of finding a complex function ω that is analytic with respect to W within the region of the unit strip 0 < ψ < 1, decays at infinity, and satisfies the boundary conditions (71) and (72). Tanaka’s method provides a way to solve the previous outlined equations in terms of the new variables τ , θ and a full description of the algorithm can be found in [40].

5.2.

The Level Set and Velocity Potential Updating

We use the standard Narrow Band Level Method, introduced by Adalsteinsson and Sethian [2], which limits computation to a thin band around the front of interest. Following the algorithm discussed in [31], we use second order in space upwind differences to

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approximate the gradient in the level set equation, and a first order time scheme to update the solution. For boundary conditions, homogeneous flux boundary conditions are usually chosen, which are implemented by creating an extra layer of ghost cells around the domain whose values are simply direct copies of the Ψ values along the actual boundary. The level set function is built from the initial position of the front by computing the signed distance function. This is done using the Fast Marching Method [36], which is a Dijkstra-like finite difference method for computing the solution to the Eikonal equation in O(N log N ), where N is the total number of points in the computational domain. The velocity and the velocity potential are both initially defined only on the interface. In order to create values throughout the narrow band, which are required to update the fixed grid Eulerian partial differential equations, we use the extension methodology developed by Adalsteinsson and Sethian in [2] to construct appropriate extensions. The idea of building extension velocities was first introduced in [26]; in that approach, the extension velocity at any grid point in the domain was taken as equal to the velocity on the closest point on the front itself. As shown in [7], this is equivalent to solving the equations ∇u · ∇Ψ = 0, ∇v · ∇Ψ = 0 for the velocity components, and in that paper, the equation was solved using a finite difference iteration. In [2], Adalsteinsson and Sethian present a technique for computing this extension velocity using the very efficient Fast Marching methodology. Finally, in [3], this approach was developed to build extension values for arbitrary material quantities whose evolution affects the underlying interface dynamics. The potential equation (69) is a convection equation with a strong non-linear source term, and homogeneous Newmann boundary conditions are imposed on the boundary of Ω1 . To update in time this equation, note that it is similar to (68) except that it has a nonlinear source term, and therefore we use similar schemes. For example a straightforward first order scheme is −x +x n n n Gn+1 i,j = Gi,j − ∆t(max(ui,j , 0)Di,j + min(ui,j , 0)Di,j + −z +z n n n max(vi,j , 0)Di,j + min(vi,j , 0)Di,j ) + ∆tfi,j

where Gni,j − Gni−1,j ∆x n n G i+1,j − Gi,j +x +x n Di,j = Di,j Gi,j = ∆x are the backward and forward finite approximation for the derivative in the x direction (we −z +z have the same expressions for for Di,j and Di,j .) Note that for simplicity we have written u, v, G, f instead of uext , vext , Gext , fext , and we describe a first order explicit scheme with a centered source term. Initial values of G0i,j are obtained by extending φ(x, z, 0)|Γ0 as previously discussed. However, at any time step n it is always possible to perform a new extension of φn (x, z, n∆t) to obtain a better value of Gni,j . A key issue is how one obtains fext in the grid points of Ω1 . There are several ways of doing so. Here we calculate f = 12 (∇φ · ∇φ) − gz on free surface nodes, and use these values together with the condition ∇f · ∇Ψ = 0 to obtain fext . This algorithm for extending quantities defined on the front off the front works very well for the velocity −x −x n Di,j = Di,j Gi,j =


417

field in the case of equation (68), because it maintains the signed distance function for the level sets of Ψ. However, regarding equation (69) for this particular wave problem, and due to the high variations of f along the front together with its topological structure when overturning, the previous method creates strong G and f gradients in Ω1 . This fact limits the grid spacing in Ω1 and the time step needed to maintain accuracy (see the section on numerical experiments).

5.3.

The Boundary Integral Equation and the BEM Approximation

A first order boundary element method is used to approximate equation (66). Boundary integral equations are well suited to moving boundary problems for two principal reasons. First, determining the surface velocity generally requires computing function derivatives on this boundary, which are accurately evaluated within this formulation. Second, remeshing the moving boundary is clearly simpler than remeshing the entire domain. The Laplace equation for the velocity potential (67) is solved by approximating the corresponding boundary integral equation. Boundary conditions are given by (70) and, on the free boundary, at each time step, by the updated potential velocity given by equation (69). The approximation of the integral equation is done by the BEM, which calculates the potential and the potential gradient on the free surface, that is, its velocity u. The boundary integral equation for the potential φ(P ), in a domain Ω(t) having boundary Σ = ∂Ω(t), can be written as Z ∂φ ∂G (73) P(P ) = φ(P ) + lim φ(Q) (PI , Q) − G(PI , Q) (Q) dQ = 0 , PI →P Σ ∂n ∂n where n = n(Q) denotes the unit outward normal on the boundary surface and {PI } are interior points converging to the boundary point P . The Green’s function or fundamental solution (in two dimensions) is G(P, Q) = −

1 log(r) . 4π

(74)

The integral equation is usually written with the ∂G ∂n singular integral evaluated as a Cauchy Principal Value (CPV), resulting in a ‘interior angle’ coefficient c(P ) multiplying the leading φ(P ) term [5, 6]. The reason for employing the seemingly more complicated limit process will become clear in the discussion of gradient evaluation. The exterior limit equation Z ∂φ ∂G lim φ(Q) (PE , Q) − G(PE , Q) (Q) dQ = 0 . (75) PE →P Σ ∂n ∂n yields precisely the same equation: the jump in the CPV integral as one crosses the boundary accounts for the ‘free term’ difference. In this work, a Galerkin (weak form) approximation of Eq. (73) has been employed, and the boundary and boundary functions are interpolated using the simplest approximation, linear shape functions. Thus, the equations that are solved are of the form Z ψk (P )P(P ) dP = 0 , (76) Σ

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where the weight functions ψˆk (P ) are comprised of all shape functions which are nonzero at a particular node Pk [5]. The calculations reported herein employed the simplest approximation, linear shape functions. These approximations reduce the integral equation to a finite system of linear equations, and invoking the boundary conditions allows the solution of the unknown values of potential and flux on the boundary. Details concerning the limit evaluation of the singular integrals can be found in [14]. As noted above, for the wave problem, and moving boundary problems in general, knowledge of the normal flux is not sufficient, the complete gradient of φ is required to compute the surface velocity. The remainder of this section will present the algorithm for computing this gradient. ¿From Eq. (73) a gradient component can be expressed as ∂φ(P ) = lim PI →P ∂Ek

Z Σ

∂G ∂2G ∂φ (PI , Q) dQ . (PI , Q) (Q) − φ(Q) ∂Ek ∂n ∂Ek ∂n

(77)

Once the boundary value problem has been solved, all quantities on the right hand side are known: a direct evaluation of nodal derivatives would therefore be easy were it not for wellknown difficulties with the hypersingular (two derivatives of the Green’s function) integral [28, 29, 27]. As described in [15], a Galerkin approximation of this equation, Z ∂φ(P ) dP = (78) ψˆk (P ) ∂Ek Σ Z Z ∂G ∂φ ∂2G ψˆk (P ) (PI , Q) (Q) − φ(Q) (PI , Q) dQ dP lim PI →P Σ ∂n ∂Ek ∂n Σ ∂Ek allows a treatment of the hypersingular integral using standard continuous elements. Interpolating ∂φ(P )/∂Ek as a linear combination of the shape functions results in a simple system of linear equations for nodal values of the derivative everywhere on Σ; the coefficient matrix is obtained by simply integrating products of two shape functions. However, the complete boundary integrations required to compute the right hand side are quite expensive. The computational cost of this procedure can be significantly reduced by exploiting the exterior limit equation, Eq. (75). It appears to be useless for computing tangential derivatives, since, lacking the free term, the corresponding derivative equation takes the form Z ∂G ∂φ ∂2G 0 = lim (PE , Q) (Q) − φ(Q) (PE , Q) dQ , (79) PE →P Σ ∂Ek ∂n ∂Ek ∂n and the derivatives obviously do not appear. However, subtracting this equation from Eq. (77) yields (with shorthand notation) Z ∂φ(P ) ∂2G ∂G ∂φ = lim − lim (Q) − φ(Q) dQ . (80) PI →P PE →P ∂Ek ∂Ek ∂n Σ ∂Ek ∂n The advantage of this formulation is that now only the terms that are discontinuous crossing boundary contribute to the integral. In particular, all non-singular integrations, by far the most time consuming, drop out. The calculation of the right hand side in Eq. (80) reduces


419

to a few ‘local’ singular integrations, and as these integrations are carried out partially analytically, this produces an accurate algorithm. Further details about the evaluation of Eq. (80) can be found in [15].

5.4.

Regridding of the Free Surface

In a level set formulation the position of the front is only known implicitly through the node values of the level set function Ψ. In order to extract the front, it is possible to construct first order and second order approximations of the interface using local data of Ψ on the mesh (see [9] for example.) Here we use a first order linear approximation of the free surface, which yields a polygonal interface formed by unevenly distributed nodes, which we call LS nodes. As a result of this extraction technique, occasionally one gets front nodes which are very close together, and this can cause difficulties and instabilities for boundary element calculations. To overcome this problem, and also to achieve more front resolution when needed, we employed a front node regridding technique. An initialization point on the front is selected according to a particular criterion, such as maximum value of height, velocity modulus, or front curvature. This point divides the front in two halves and new nodes are chosen so that, lying in the same polygon, they are redistributed by arclength according to the formula: si+1 − si = d0 (1 + si (f0 − 1)) where si denotes the arclength distance from node i to the initialization point (i = 0) and d0 , f0 are user selected parameters. These regridded nodes on the front are used to create the input file for the BEM calculations and are denoted by BEM nodes.

5.5.

The Algorithm

To initialize the position of the front and the velocity potential on the front, we use Tanaka’s method for computing numerical exact solitary waves. The basic algorithm can be summarized as follows: 1. Compute initial front position and velocity potential φ(Q, 0) on Γ0 . 2. Extend φ(Q, 0) onto the grid points of Ω1 to initialize G. 3. Generate Ωt and solve (67), using the Boundary Element Method. This yields the velocity u and source term f on the front nodes. 4. Extend u and f off the front onto Ω1 . 5. Update G using (69) in Ω1 . 6. Move the front with velocity u using (68) in Ω1 7. Interpolate (bi-cubic interpolation) G from grid points of Ω1 to the front nodes to obtain new boundary conditions for (67). Go back to step 3 and repeat forward in time.

420


A more detailed algorithm including regridding is: Initialization: Given Γ0 = Γ0 , φ0 = φ(Q, 0) 1. Calculate Ψ0 and LS nodes. 2. Extend φ0 to obtain G0 . 3. Redistribute LS nodes to obtain BEM nodes. 4. Calculate u0 at BEM nodes. 5. Find u0 and f 0 at LS nodes and extend onto Ω1 . Steps: Given Ψn , φn , un 1. Calculate Ψn+1 and LS nodes. 2. Calculate Gn+1 in Ω1 grid points. 3. Redistribute LS nodes to obtain BEM nodes. 4. Interpolate G on BEM nodes to find φn+1 . 5. Calculate un+1 at BEM nodes. 6. Find un+1 and f n+1 at LS nodes and extend onto Ω1 . Go to step 1 and repeat. 7. If reinitialization (a) Take LS nodes and reinitialize Ψn+1 . (b) Take BEM nodes and extend φn+1 .

5.6.

Numerical Accuracy

The model equations imply that the wave mass and its total energy should be conserved as the wave evolves in time. One way to check the numerical accuracy of the discretized equations is to compute these quantities each time step. The wave mass above z = 0 is given by Z Z Z m(t) =

dΩ =

Ωt

znz ds =

∂Ωt

znz ds

Γt

and the total energy is E(t) = Ep (t) + Ek (t), where Ep (t), Ek (t) denotes the potential and kinetic wave energy respectively. They can be calculated using the expressions Z Z 1 1 Ep (t) = ρg zdΩ = ρg z 2 nz ds, 2 2 Ωt Γt which is the potential energy with respect z = 0, and Z Z Z 1 ∂φ 1 ∂φ 1 Ek (t) = ρ ∇φ · ∇φdΩ = ρ φ ds = ρ φ ds, 2 Ωt 2 ∂Ωt ∂n 2 Γt ∂n


421

where the divergence theorem has been applied to the three formulas and we have used the ∂φ fact ∂n = 0 on Γb , Γ1 , Γ2 for the kinetic energy formula. These integrals are approximated by a composite trapezoidal rule, using the values of the quantities on the free boundary BEM nodes. Note that LS nodes could have been used for m(t) and Ep (t) approximations but we also used BEM nodes for simplicity. The components of the normal vector to the free surface are computed using the level set embedding function to obtain surface geometrical variables. A common procedure to study the accuracy and convergence properties of the discretized equations with respect the mesh sizes and the time step is by means of an analytical solution. A solitary wave propagating over a constant depth is a traveling wave that moves in the x direction with speed equal to the celerity of the wave (c). The velocity potential and the velocity on the front as functions of x are also translated with the same speed c. Therefore, in this case, by calculating initial wave data with Tanaka’s method and translating it, we are able to compute the L2 norms of the errors for the various magnitudes. For the case of a solitary wave shoaling over a sloping bottom, the accuracy can only be checked looking at the mass and energy conservation properties and comparing breaking wave characteristic obtained here with those reported elsewhere, for example in [22].

6.

Numerical Results

The system of equations to be discretized is a non-linear system of strongly coupled partial differential equations. First order in time and second order in space schemes are used for equation (68); first order in time and in space schemes are used for equation (69); and a first order BEM solver is used for the velocity updating. To study the convergence properties of this method and its capability to predict wave breaking characteristics, the numerical results corresponding to the following physical settings are presented: A solitary wave propagating over a constant depth and the shoaling and breaking of a solitary wave propagating over various sloping bottoms.

6.1.

Constant Depth Test

In order to tune the discretization parameters and see how they affect numerical accuracy we performed a series of numerical tests with a solitary wave of H0 = 0.5 m (wave height at the crest) propagating over a constant depth of 1 m. The wave crest is initially located at x = 6.5 m and the domain has L = 15 m of length. In what follows, the units are taken as meters and seconds for length and time, respectively. Let Ω1 = [0, 15] × [−0.3, 1] be the fictitious domain that contains the free boundary for all t ∈ [0, 0.5], ∆x = ∆z the grid size and ∆t the time step. To discretize ∂Ωt , in order to generate the input BEM file, a variable mesh size is used: ∆l = 0.1 for Γ1 and Γ2 , ∆l = 0.2 for Γb , and the regridding parameters for Γt are chosen to be d0 = 0.005, f0 = 10. This gives 193 BEM nodes on the moving front and 98 nodes on the fixed boundaries. The mesh size ∆x = ∆z for Ω1 should be chosen in order to achieve accurate interpolated values of front position and potential on the front. For the time step selection, a first limitation is the CFL condition. While this condition is enough for the stability of the numerical approximation of equations (68) and (69), the accuracy in the numerical solution of

422


equation (69) requires a smaller time step. This is due to the fact that G and the source term f , for this particular wave problem, develops high gradients in Ω1 . Therefore we present the results for the following test cases: • (a) ∆x = 0.1, ∆t = 0.01. • (b) ∆x = 0.1, ∆t = 0.001. • (c) ∆x = 0.01, ∆t = 0.001. • (d) ∆x = 0.01, ∆t = 0.0001. For a given solitary wave parameters (H0 and length L in the x direction) Tanaka’s method gives us the initial wave magnitudes, front location, velocity potential, velocity components at front points and wave celerity c. At any time t, let (xex , zex ), φex , uex , vex be the values of these variables obtained by translating initial values a distance ct along the x direction and spline interpolating in LS nodes. Denote by (xc , zc ), φc , uc , vc the computed values at LS nodes, L2 (z) =k zc − zex kL2 (Γt ) , L2 (φ) =k φc − φex kL2 (Γt ) , L2 (u) =k uc − uex kL2 (Γt ) and L2 (v) =k vc − vex kL2 (Γt ) the L2 norm of the errors. Table 1 shows these errors at the final time t = 0.5 for the various test cases. Table 1. Values of the L2 error norms at t = 0.5 Test (a) (b) (c) (d)

L2 (z) 0.007239 0.009762 0.001476 0.001699

L2 (φ) 0.095254 0.021451 0.011363 0.00424601

L2 (u) 0.025147 0.039635 0.0099744 0.0106674

L2 (v) 0.025856 0.035685 0.009356 0.010188

Figures 5 and 6 show L2 (z), L2 (φ), L2 (u), L2 (v) versus time for cases (c) and (d) respectively. As observed from these results, the L2 error norm in front location and velocity components decreases with mesh size (∆x) but not with the time step. Only the velocity potential gains accuracy when ∆t is reduced accordingly to the above mentioned facts. Regarding wave mass and energy conservation, at each time step we calculate m(t) and E(t) as explained in 5.6. Figures 7 and 8 show the values of |m(t)−m(0)| and |E(t)−E(0)| versus time and same behavior of these quantities with respect discretization parameters is observed. Next, to see if we gain accuracy in the velocity calculations by increasing the number of BEM nodes, we take ∆l = 0.05 on Γ1 and Γ2 , ∆l = 0.1 on Γb , and d0 = 0.001, f0 = 5 on Γt . This gives 1720 BEM nodes on the moving front and 196 nodes for the fixed boundaries. For this discretization of the bEM boundary we run two more cases: • (e) ∆x = 0.01, ∆t = 0.001. • (f) ∆x = 0.01, ∆t = 0.0001.


423

Values of the L2 error norms for case (e) and (f) are almost identical to those obtained for case (c) and (d) respectively, which means that accuracy in velocity is not gained by increasing the number of bEM nodes. However, as is shown in Figure 7, |m(t) − m(0)| has decreased by almost an order of magnitude due to the accuracy in front position and the improvement in the integral approximation to calculate m(t). Figure 9 shows for case (e) the absolute errors in Ep (t), Ek (t), E(t) versus time and, in agreement with the previous discussion, the kinetic energy is much less accurate than the potential energy. From these numerical experiments we conclude that the proposed algorithm converges, but we do not achieve exactly first order convergence with respect discretization parameters. This is due to the strong interdependence of the equations. Note that f depends nonlinearly on u and linearly on z and that the boundary condition imposed on Γt for the bEM solver builds up numerical and round off error as we step forward in time; we note that the level set approach is stable and robust with respect to these small sawtooth instabilities resulting from velocity calculations on very closely spaced nodes, and the use of filtering or smoothing was not required. Case (c) discretization parameters give sufficient accuracy and we show wave profiles, velocity potential and velocity components for various times in Figures 10, 11 and 12 respectively.

6.2.

Sloping Bottom Test

A solitary wave propagating over a sloping bed changes its shape gradually, slightly increasing maximum height and front steepness, till a point where a vertical front tangent is reached. This is usually called the breaking point BP=(tbp , xbp , zbp ), where xbp represents the x coordinate, zbp the height at xbp and tbp the time of occurrence. Beyond the BP, the wave tip develops, with velocities much bigger than the wave celerity, causing the wave overturning and the subsequent falling of the jet toward the flat water surface. Denote this endpoint as EP=(tep , xep , zep ). Total wave mass and total energy should be, theoretically, conserved until EP. However beyond the BP a lost in potential energy and the corresponding gain in kinetic energy is expected, due to the large velocities on the wave jet. Wave breaking characteristics change, mainly according to initial wave amplitude (H0 ) and bottom topography. To study how our numerical method predicts wave breaking we run the following test cases: • (a) H0 = 0.6, L = 25, slope=1 : 22, xc = 6.05, xs = 6 • (b) H0 = 0.6, L = 18, slope=1 : 15, xc = 5.55, xs = 5.4 and compare the results obtained here for case (b) with those reported in [21]. Here xc denotes the x coordinate at the crest for the initial wave and xs the x coordinate where the bottom slope starts. A series of numerical experiments have been made, and optimal discretization parameters found are: ∆x = 0.01, ∆t = 0.0001 and d0 = 0.005, f0 = 10 (approximately 193 BEM nodes) for all cases. Front regridding has been made according to maximum height before the BP and according to maximum velocity modulus beyond BP. Beyond the BP, and due to the complex topography of the wave front, reinitialization of Ψ and new φ(x, z, t) extension has been performed every 1000 time steps.

424


Table 2 shows the breaking characteristics for the test cases. Grilli et all reported in [21] for test (b) values of tbp = 2.41, xbp = 15.64 and zbp = 0.67. The discrepancies can be attributed to the slightly different position of the initial wave (xc = 5.5) and the higher order approximations used in their Lagrangian-Eulerian formulation. Table 2. Breaking characteristics Test (a) (b)

tbp 2.76 2.34

xbp 17.39 15.20

zbp 0.674 0.662

tep 3.36 2.90

xep 20.2 17.8

In Figure 13 we show m(t) versus time for case (a) and (b) and Figures 14 and 15 show the evolution of Ep , Ek and E with time for cases (a) and (b) respectively. Maximum absolute error in wave mass is 0.01 before BP and 0.02 beyond BP and maximum absolute error in total wave energy is 0.02 near the BP. Although this errors could be improved by increasing the number of BEM nodes on the free boundary (as shown in the constant depth cases), it would require considerably more CPU time per run due to the high cost of the BEM solver. Regarding the evolution of the potential and kinetic energy of the wave we observe the expected behavior beyond the BP. Figure 16 shows wave shape for case (a) at t = 0, 1, 2, 2.76, 2.94, 2.14, 3.34 and Figure 17 shows wave shape for case (b) at t = 0, 1, 2, 2.34, 2, 48, 2.68, 2.90. In Figures 18 and 19 we show in more detail the wave profiles from the BP to the EP for cases (a) and (b) respectively. Finally in Figure 20 the front BEM nodes for case (a) and time 3.34 are shown. ¿From these numerical experiments we conclude that the numerical method presented here is capable of reproducing wave shoaling and breaking till the touchdown of the wave jet. Considering that we use only first order approximations of the model equations, a piecewise linear approximation of the free boundary, and a first order linear BEM, the results are quite accurate. The absolute errors in mass and energy seem to be higher than those reported in [21]. This is not surprising due to the fact that in [21] a higher order BEM is used (both higher order elements to define local interpolation between nodes and spline approximation of the free boundary geometry) and time integration for the free boundary conditions is at least second order in time.

6.3.

Sinusoidal Bottom Test

To see how wave shape and breaking characteristics change with bottom topography, we consider two more tests, this time with a sinusoidal shape bottom: • (c) H0 = 0.6, L = 25, xc = 6.05, Ab = 0.5, hmin = 0.5 • (d) H0 = 0.6, L = 25, xc = 6.05, Ab = 0.8, hmin = 0.2 where Ab denote the amplitude of the sinusoidal function that represents the bottom and hmin the minimum depth.


425

As can be seen in Table 3, the breaking characteristics are considerably different for these simulations, and, in particular, case (c) behaves like a spilling breaker rather than the plunging breaker of case (a) and (b). In Figures 21 and 22 we show wave profiles for various Table 3. Breaking characteristics Test (c) (d)

tbp 1.6 1.0

xbp 12.5 10.5

zbp 0.71 0.55

tep 1.96 1.38

xep 14.1 13.6

times corresponding to case (c) and (d) respectively. Measurements for the mass and total energy conservation behave similar to previous cases. In Figure 23 we show the evolution of wave mass for cases (c) and (d). Finally, Figures 24 and 25 show the evolution of Ep , Ek and E corresponding to cases (c) and (d) respectively. These results show that, in response to the bottom topography, wave height follows a sinusoidal curve, as does the potential and kinetic wave energies, with an amplitude related to the sinusoidal bottom amplitude.

7.

Conclusion

To summarize, in this chapter we have derived some physical models related to moving interfaces in an intrinsic way, that is, independent of any coordinate system. Based on these models a complete Eulerian description of potential flow problems for a single fluid, with or without advection-diffusion of material quantities on the front has been stablished. For the case of two-dimensional breaking waves over sloping beaches a coupled level setboundary integral algorithm has been developed. Numerical results and convergence tests show that even first order level set schemes produce quantitative results in a robust and efficient fashion.

Acknowledgements All work was performed at the Lawrence Berkeley National Laboratory, and the Mathematics Dept. of the University of California at Berkeley. First author was partially supported by the Spanish Project MTM2007-65088. Second author was supported by Spanish CGL2006-06401-BTE and CGL2008-03786-BTE projects, both funded by Ministerio de Educación y Ciencia and Fondo Europeo de Desarrollo Regional (FEDER). We want to thank E. Suárez D´ıaz for his help with some of the figures.

Appendix I. The Surface Divergence in Rectilinear Coordinates Given a vector field w, we want to find an expression for the surface divergence Div w using rectilinear coordinates. We start from the definition: ω(a, b) Div w := ∂a w · b × n + ∂b w · n × a,

(81)

426


being a and b arbitrary tangent vectors to the surface, ω(a, b) the area of the corresponding parallelogram and n the unit vector normal to the surface at the same point. To abbreviate the computations we use indices for basis vectors:6 a1 = a,

a2 = b,

a3 = n.

(82)

The reciprocal basis, designed by ai (i = 1, 2, 3), ai · aj = δji ,

i, j = 1, 2, 3,

(83)

is calculated by the formulae: a1 =

a3 × a1 a1 × a2 a2 × a3 , a2 = , a3 = = n. [a1 , a2 , a3 ] [a1 , a2 , a3 ] [a1 , a2 , a3 ]

(84)

According to definition (81), we have for Div w: a2 × a3 a3 × a1 · ∂ a1 w + · ∂ a2 w [a1 , a2 , a3 ] [a1 , a2 , a3 ] = a1 · ∂a1 w + a2 · ∂a2 w = aα · ∂aα w.

Div w =

(85)

In the last expression and below we have used the summation convection: when in a monomial expression we have two repeated indices it must be interpreted as a summation, from 1 to 2 for greek indices and from 1 to 3 for latin indices. Notice that the basis ai is in general different in each surface point. We want now to express Div w using the components and coordinates in a fixed basis (global) ei and the reciprocal one ej , defined7 by the nine equations ej · ei = δij . We set: ai = hji ej , ai = fki ek , w = wj ej ; (fki hkj = δji ). (86) Substituting this expressions in the last term of (85), we have: Div w = fkα ek · ∂hiα ei wj ej = fjα hiα ∂i wj .

(87)

Considering that nj = a3 · ej = fj3 and ni = a3 · ei = hi3 , the coefficient in the previous result becomes: fjα hiα = fjk hik − fj3 hi3 = δji − nj ni . (88) Substituting this result in equation (87), the searched expression is obtained: Div w = δji − nj ni ∂i wj .

(89)

Notice that, as we have anticipated, the final result does not depend on the selected tangent vectors a and b. 6

Latin indices i, j,... go through the values 1, 2 y 3 and greek indices α, β,... go through 1 y 2. The vectors ai (i = 1, 2, 3) must accomplish the condition [a1 , a2 , a3 ] > 0. 7 When ei is an orthonormal basis (Cartesian coordinates) then it coincides with the corresponding reciprocal basis: ei = ei .


427

Appendix II. The Differential Operator A We are going to show that the differential operator A=

1 (∂a (n × j · b) − ∂b (n × j · a)) , ω(a, b)

(90)

appearing in (25), may be written using surface divergences of j and n. To do that, we perform in the previous definition the indicated derivatives, A=

1 (b × n · ∂a j + n × a · ∂b j) + ω(a, b) 1 + (b × ∂a n + ∂b n × a) · j. ω(a, b)

According to definition (81), the first term is Div j. Let be: A = Div j + B.

(91)

In order to identify de second term B, we select the basis ai , following the specified notation in (82). As ∂aα n (α = 1, 2) are tangent vectors to the surface, we can set ∂aα n = Nαβ aβ ,

α = 1, 2.

(92)

Also, for the two terms in B we obtain: b × ∂a n ω(a, b) ∂b n × a j· ω(a, b)

j·

= j·

a2 × N11 a1 = −N11 j · n, ω(a1 , a2 )

= −N22 j · n.

Therefore: B = −Nαα j · n.

(93)

On the other hand, as Nαβ = aβ · ∂aα n, making α = β, summing and using the result (85), we have: Nαα = aα · ∂aα n = Div n. (94) Finally, substituting this result in (93) we arrive to the searched expression: A = Div j − (Div n)j · n.

(95)

Appendix III. Useful Definitions Points and vectors. In our euclidean space we can define two useful operations. Given a point P and a displacement vector a, we define P + a as the point that results translating point P by vector a. Also, given two points A and B, we define A − B as a vector c, so that: A − B := c ⇔ B + c = A. (96)

428


Directional derivative. Given a tensor field w = w(P, t), function of the position P (a point of our Euclidean space) and the time t, we define the directional derivative along the vector a as d w(P + ǫa, t) . (97) ∂a w := dǫ ǫ=0 The result ∂a w is a tensor of the same rank that w. Differentiability of the field w implies that ∂a w is a linear function of the vector a. When w is a vector field, we use, as customary, the special notation: a · ∇w := ∂a w.

(98)

In this case, ∂a w is a linear operator acting on the vector a.

Gradient of a scalar field. Let us consider a scalar field φ = ϕ(P, t). As ∂a φ is a real valued linear function on the argument a, we can define the gradient vector field, ∇φ, a · ∇φ := ∂a φ.

(99)

L2 Errors. (H0=0.5, depth=1) 0.012 front potential u v

0.01

L2 error

0.008

0.006

0.004

0.002

0

0

0.05

0.1

0.15

0.2

0.25 time

0.3

0.35

0.4

0.45

0.5

Figure 5. L2 (z), L2 (φ), L2 (u), L2 (v) vs time for case (c).

Some Free Boundary Problems in Potential Flow Regime... L2 Errors. (H0=0.5, depth=1) 0.012 front potential u v

0.01

L2 error

0.008

0.006

0.004

0.002

0

0

0.05

0.1

0.15

0.2

0.25 time

0.3

0.35

0.4

0.45

0.5

Figure 6. L2 (z), L2 (φ), L2 (u), L2 (v) vs time for case (d).

−3

3

Absolute error in wave mass. (H0=0.5, depth=1)

x 10

(a) (b) (c) (d) (e)

2.5

abs(m(t)− m(0))

2

1.5

1

0.5

0

0

0.05

0.1

0.15

0.2

0.25 time

0.3

0.35

0.4

Figure 7. Absolute error in wave mass.

0.45

0.5

429

430

M. Garzon, N. Bobillo-Ares and J.A. Sethian −3

7

Absolute error in total wave energy. (H0=0.5, depth=1)

x 10

(a) (b) (c) (d)

6

abs(E(t)− E(0))

5

4

3

2

1

0

0

0.05

0.1

0.15

0.2

0.25 time

0.3

0.35

0.4

0.45

0.5

Figure 8. Absolute error in wave total energy. −4

8

Wave Energy. (H0=0.5 depth=1)

x 10

Ep Ek E

abs(E(t)−E(0)

6

4

2

0

0

0.05

0.1

0.15

0.2

0.25 time

0.3

0.35

0.4

0.45

0.5

Figure 9. Absolute error in potential, kinetic and total energy. Case (e). wave shape at several times. (H0=0.5, depth=1) 1

z

0.5

0

−0.5

−1

0

5

10

15

x

Figure 10. Front location at t = 0, 0.1, 0.2, 0.3, 0.4, 0.5. Case (c).

Some Free Boundary Problems in Potential Flow Regime... velocity potential. (H0=0.5, depth=1) 3

2

potential

1

0

−1

−2

−3

0

0.1

0.2

0.3

0.4

0.5 s

0.6

0.7

0.8

0.9

1

Figure 11. Velocity potential at t = 0, 0.25, 0.5. Case (c). u velocity. (H0=0.5, depth=1) 2 1.5

u

1 0.5 0 −0.5

0

0.1

0.2

0.3

0.4

0.5 s

0.6

0.7

0.8

0.9

1

0.7

0.8

0.9

1

v velocity. (H0=0.5, depth=1) 0.6 0.4 0.2 v

0 −0.2 −0.4 −0.6 −0.8

0

0.1

0.2

0.3

0.4

0.5 s

0.6

Figure 12. Velocity components at t = 0, 0.25, 0.5. Case (c). Wave mass 2 slope 1:22 slope 1:15

1.98

mass

1.96 1.94 1.92 1.9 1.88 0

0.5

1

1.5

2

2.5

3

time

Figure 13. Wave mass vs time. Case (a) and (b).

431

432

M. Garzon, N. Bobillo-Ares and J.A. Sethian Wave Energy. (H0=0.6 slope=1:22) 0.8 Ep Ek E

0.75

0.7

0.65

E

0.6

0.55

0.5

0.45

0.4

0.35

0.3

0

0.5

1

1.5

2

2.5

3

3.5

time

Figure 14. Wave energy. Case (a).

Wave Energy. (H0=0.6 slope=1:15) 0.9 Ep Ek E 0.8

0.7

E

0.6

0.5

0.4

0.3

0.2

0

0.5

1

1.5 time

2

Figure 15. Wave energy. Case (b).

2.5

3


433

H0=0.6 slope1:22 4

z

2

0

−2

−4

0

5

10

15

20

25

x

Figure 16. Wave shape at various times. Case (a) H0=0.6 slope1:15 4 3 2

z

1 0 −1 −2 −3 −4

0

2

4

6

8

10

12

14

16

18

x

Figure 17. Wave shape at various times. Case (a). H0=0.6 slope1:22 2

1.5

1

z

0.5

0

−0.5

−1

−1.5

−2 16

16.5

17

17.5

18

18.5 x

19

19.5

20

20.5

Figure 18. Wave shape at various times. Case (a).

21

434

M. Garzon, N. Bobillo-Ares and J.A. Sethian H0=0.6 slope1:15 2

1.5

1

z

0.5

0

−0.5

−1

−1.5

−2 14

14.5

15

15.5

16 x

16.5

17

17.5

18

Figure 19. Wave shape at various times. Case (b). H0=0.6 slope1:22 2

1.5

1

z

0.5

0

−0.5

−1

−1.5

−2 18

18.5

19

19.5

20 x

20.5

21

21.5

Figure 20. Front BEM nodes at t=3.34. Case (a).

22


435

H0=0.6 , sinusoidal bottom 4 3 2

z

1 0 −1 −2 −3 −4

2

4

6

8

10

12

14

16

18

20

x

Figure 21. Wave shape at various times. Case (c).

H0=0.6 , sinusoidal bottom 4 3 2 1 z

0 −1 −2 −3 −4

2

4

6

8

10

12

14

16

18

20

x

Figure 22. Wave shape at various times. Case (d).

Wave mass 2

1.98 (c) (d)

mass

1.96

1.94

1.92

1.9

1.88 0

0.2

0.4

0.6

0.8

1 time

1.2

1.4

1.6

1.8

Figure 23. Wave mass vs time. Case (c) and (d).

2

436

M. Garzon, N. Bobillo-Ares and J.A. Sethian Wave Energy 0.9 Ep Ek E

0.8

0.7

E

0.6

0.5

0.4

0.3

0.2

0

0.2

0.4

0.6

0.8

1 time

1.2

1.4

1.6

1.8

2

Figure 24. Wave energy. Case (c). Wave Energy 0.75 Ep Ek E

0.7

0.65

0.6

E

0.55

0.5

0.45

0.4

0.35

0.3

0.25

0

0.2

0.4

0.6

0.8

1

time

Figure 25. Wave energy. Case (d).

1.2

1.4


437

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Chapter 14

A N EW A PPROACH FOR P OLYDISPERSED T URBULENT T WO -P HASE F LOWS : T HE C ASE OF D EPOSITION IN P IPE -F LOWS S. Chibbaro∗ Dept. of Mechanical Engineering, University of “Tor Vergata”, via del politecnico 1 00133, Rome, Italy

Abstract This article is basically a review of recent works that is aimed at putting forward the main ideas behind a new theoretical approach to turbulent wall-bounded flows, notably pipe-flows, in which complex physics is involved, such as combustion or particle transport. Pipe flows are ubiquitous in industrial applications and have been studied intensively in the last century, both from a theoretical and experimental point of view. The result of such a strong effort is a good comprehension of the physics underlying the dynamics of these flows and the proposition of reliable models for simple turbulent pipe-flows at large Reynolds number Nevertheless, the advancing of engineering frontiers casts a growing demand for models suitable for the study of more complex flows. For instance, the motion and the interaction with walls of aerosol particles, the presence of roughness on walls and the possibility of drag reduction through the introduction of few complex molecules in the flow constitute some interesting examples of pipe-flows with some new complex physics involved. A good modeling approach to these flows is yet to come and, in the commentary, we support the idea that a new angle of attack is needed with respect to present methods. In this article, we analyze which are the fundamental features of complex two-phase flows and we point out that there are two key elements to be taken into account by a suitable theoretical model: 1) These flows exhibit chaotic patterns; 2) The presence of instantaneous coherent structures radically change the flow properties. From a methodological point of view, two main theoretical approaches have been considered so far: the solution of equations based on first principles (for example, the Navier-Stokes equations for a single phase fluid) or Eulerian models based on constitutive relations. In analogy with the language of statistical physics, we consider the former as a microscopic approach and the later as a macroscopic one. We discuss why we consider both approaches unsatisfying with regard to the description of general complex turbulent flows, like two-phase ∗

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S. Chibbaro flows. Hence, we argue that a significant breakthrough can be obtained by choosing a new approach based upon two main ideas: 1) The approach has to be mesoscopic (in the middle between the microscopic and the macroscopic) and statistical; 2) Some geometrical features of turbulence have to be introduced in the statistical model. We present the main characteristics of a stochastic model which respects the conditions expressed by the point 1) and a method to fulfill the point 2). These arguments are backed up with some recent numerical results of deposition onto walls in turbulent pipe-flows. Finally, some perspectives are also given.

1.

Introduction

Turbulent flows are ubiquitous in nature. The boundary layer in the earth’s atmosphere, rivers and canals, the photosphere of the sun, the interstellar medium, most combustion processes, the flow of natural gas and oil in pipelines are just a few examples of turbulent motions. Most turbulent flows are bounded (at least in part) by one or more solid surfaces. Examples include internal flows such as the flow through pipes and ducts; external flows such as the flow around aircraft. Since Reynolds’ experiment in 1883, pipe flow has played an important role in the development of our understanding of turbulent flows. In particular, it is quite simple to measure the drop in pressure over a length of fully developed turbulent pipe flow and hence to determine the skin-friction coefficient. Laminar Poiseuille flow occurs when a fluid in a straight channel, or pipe, is driven by a constant upstream pressure gradient, yielding a symmetric parabolic stream-wise velocity profile. In turbulent states, the mean stream-wise velocity profile remains symmetric, but is flattened in the center because of the increase in velocity fluctuations. A lot of research has been carried out for turbulent wall flows, [1, 3, 4, 5] and, in particular, in the case of pipe flow, experiments for measuring the mean-velocity profile have been successfully performed at moderate to high Reynolds numbers [6, 7]. Thus, we can say that the basic physics of these flows is well-understood, even though the fundamental understanding of how these profiles change as functions of the Reynolds number and of the dissipative mechanisms have yet to be assessed. However, this is not at all such cases where some complex phenomena are added like combustion [35], particle dispersion (two-phase flows) [42], presence of wall-roughness or of complex molecules which cause a drag reduction [8, 9, 10]. In these cases physical understanding remains limited and appears to be scarce compared to that obtained for simpler turbulent flows. The purpose of the present work is to analyse a suitable modeling approach, which has simplified rules compared to the real phenomena, and which is used to simulate the overall and collective behaviour of a complex system. The question is therefore whether the model contains the right physics (thus the need to understand clearly the important phenomena) and then how to reach an acceptable compromise between the simplicity of the model versus its physical realism (thus the need of an appropriate formalism). In this commentary, which tries to propose an overview of recent modeling developments [42, 44, 47], we analyze the case of two-phase flows and, in particular, particle deposition onto walls.

A New Approach for Polydispersed Turbulent Two-Phase Flows

2.

443

A Sketch of the Physics of Turbulent Two-Phase Wall Flows

In this section, we would like to outline the present knowledge about near-wall physics and, more specifically, the physical mechanisms which can be considered as important for particle deposition. Generally speaking, two elements constitute the most significant signatures of those turbulent flows: i) the flow is chaotic ii) quasi-coherent structures are present. It is important to underline that the first point gives information on the statistical nature of these flows, while the second concerns the geometrical one. For the first point, Navier-Stokes equations, which describe accurately a turbulent flow [11], represent a dynamical system with a very large number of degree of freedom [13, 14, 15, 16, 17]. Turbulence is characterized by non-Gaussian velocity fluctuations on a wide range of scales and frequencies. The number of degrees of freedom is of the order of Re9/4 for a Reynolds number Re that is typically 105 ÷ 108 . The existence of such of wide range of scales, and of the acute sensitivity of turbulent flows to small perturbations in initial and boundary conditions (which are never known absolutely) explain the search of a statistical description of such flows. Turbulent structures are identified by flow visualization, by conditional sampling techniques, or by other eduction methodologies; but they are difficult to define precisely. The idea is that they are regions of space and time (significantly larger than the smallest flow or turbulence scales) within which the flow field has a characteristic coherent pattern. Kline and Robinson [18] and Robinson [19] provide a useful categorization of quasi-coherent structures in channel flow and boundary layers. The eight categories identified are the following: 1. Low-speed streaks in the region (0 < y + < 10). 2. Ejections of low-speed fluid outward from the wall. 3. Sweeps of high-speed fluid toward the wall. 4. Vortical structures of several proposed forms. 5. Strong internal shear layers in the wall zone (y + < 80). 6. Near-wall pockets, observed as areas clear of marked fluid in certain types of flow visualizations. 7. Backs: surfaces (of scale S) across which the stream wise velocity changes abruptly. 8. Large-scale motions in the outer layers. The deposition of very small particles is mainly led by diffusion process and Brownian motion. At the same time, it is largely accepted that particle transfer in the wall region and also deposition onto walls are processes which are dominated by near-wall turbulent coherent structures [20, 21, 22]. As seen above, there are many different quasi-coherent structures in wall-flows. Among all these structures, four appear to be determinant for particle deposition: the low-speed streaks in the region 0 < y + ≤ 10, the Ejections of low-speed fluid outward from the wall, the Sweeps of high-speed fluid towards the wall

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and Vortical structures of various size and intensity [23, 24, 25, 26, 27, 28, 29]. Moreover, several experiments have investigated particle transfer in near-wall region and they have found that particles tend to remain trapped along the streaks when in viscous-layer. This migration phenomenon results in overall mean fluxes and it has been referred to as Turbophoresis [30]. This name may be, however, rather misleading since it suggests the existence of some hidden force or mechanism which bring particles towards walls. Instead, a net migration happens because the transfer of particles towards the wall is more efficient than the transfer, due to entrainment, of particles from the wall into outer flow. Furthermore, this net particle flux to the wall is related to quasi-coherent phenomena, which are instantaneous realizations of the Reynolds stress [29, 31]. Sweeps events have been found to be strongly correlated with particle trapping in lowspeed streaks, while ejections events are correlated with particle entrainment in the outer flow [27, 21]. Ejections topology has been found more efficient than sweeps events and this explains particle accumulation in near-wall region. Furthermore, the importance of these mechanisms for particle deposition depends on particle inertia. In particular, light particles follow much more closely sweeps and ejections and their motion towards the wall appears to be very well-correlated with turbulent structures. On the contrary, heavy particles are not so well-correlated with turbulent structures and their motion is less influenced by them in the near-wall region. Given the physical picture, one first important conclusion can be drawn: a model which aims to be appropriate for the description of particle deposition in turbulent two-phase flows has to be (i) a statistical model, to be capable to describe the solutions of the basic equations as being random or stochastic processes. It is necessary to cope with a reduced or contracted description of continuous fields. (ii) To take into account the effect of the most important geometrical structures, we believe that a model which does not consider this step is likely to fail a proper and general description of particle deposition in turbulent flows.

3.

Modeling

Given the framework put forward in the previous section, that is a statistical one which can include some geometrical information, it is necessary to determine which kind of approach can be included in this framework. Let us introduce briefly the basic equations for turbulent two-phase flows [33]. For heavy particles where ρp ≫ ρf , the drag and gravity forces are the dominant forces and the particle equation of motion is reduced to dUp 1 = (Us − Up ) + g. dt τp

(1)

The drag force has been written in this form to bring out the particle relaxation time scale τp =

ρp 4 d p . ρf 3 CD |Ur |

(2)


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In Eq. (1), τp appears as the only scale, and is the time necessary for a particle to adjust to fluid velocities. In the limit when Rep ≪ 1, it is seen from the expression of CD in that range that ρp d2p τp = , (3) ρf 18 νf which is the Stokes value. The drag coefficient is an empirical coefficient that can be estimated through experiments. Various expressions have been put forward, cf. Clift et al. [32], among which an often retained form is   24 1 + 0.15Re0.687 if Rep ≤ 1000, p CD = Rep (4)  0.44 if Rep ≥ 1000. The particle relaxation time scale is a non-linear function of particle properties. In the Stokes regime, it is quadratic in the particle diameter dp . Outside the Stokes regime, the dependence of τp on particle properties and variables, such as Us and Up , is more complicated. Broadly speaking, there are three classes of approaches to compute (two-phase) turbulent flows, which we classify with regard to the level of reduction : 1. Direct numerical simulation (DNS) 2. Large eddy simulations (LES) 3. Probability density function (PDF) 4. Reynolds average Navier Stokes (RANS) methods Using an analogy with the statistical physics language, we can say that the two first approaches can be considered “microscopic”, in the sense that they a have a least degree of modeling. DNS is model-free and can be thought to describe correctly turbulence [12]. LES approach is based upon the idea of modeling only some degree of freedoms, with the purpose of describing accurately the largest part of the degree of freedoms of the problem. The PDF approach is “mesoscopic”. The construction of a reduced state vector can be achieved by a coarse-graining procedure where the system is described on a large enough scale to eliminate some degrees of freedom. Information is therefore lost and this lack of complete knowledge will be reflected by the use of a stochastic description for the remaining degrees of freedom. In this method, the model is not directly written in terms of macroscopic variables but it is introduced at a mesoscopic level: the idea is to build a modeled equation for the pdf. Finally, the last approach is “macroscopic”, it starts by applying some averaging or filtering operator to the exact equations and, hence, we obtain exact but unclosed mean equations in which closure relations are then introduced. Closed mean equations result. Therefore, closure is attempted directly at the macroscopic level, and when it is performed available information is of course strictly limited to the very macroscopic variables that have been explicitly retained in the second step of the procedure, that is a limited number of moments at each point (usually not more than two). It is important to underline here that the PDF approach is different. Here, the exact instantaneous equations are replaced by models

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but still at the instantaneous level. Thus, in the PDF approach, the introduction of a model is made at an upstream level, where far more information is still available (since we model a probability density function) and has not yet been eliminated. From a practical point of view, in PDF approach to two-phase flows mean-field equations (Rij − ǫ ) are used for the fluid, whereas a particle PDF equation is solved by a Monte Carlo method using a trajectory point of view. The PDF model is therefore formulated as a particle stochastic Lagrangian model (a set of Langevin SDEs) [35, 34]. In this formulation, this approach corresponds to a Eulerian/Lagrangian method. There are, finally, some other techniques like Proper orthogonal (POD) [36] or group (SO(3)- SO(2)) [37] decomposition, which have been conceived specifically for the description of turbulent structures and which can be linked to the above methods. These are eduction techniques, which accessing directly to some actual information can rebuild the velocity field and identify the structure components. All these approaches are statistical and, therefore, belong to the suitable framework. Nevertheless, the computational cost is very different among them. More specifically, microscopic approaches are very demanding and, in practice, are both not of great help in engineering applications. Indeed, in the case of a large number of particles and/or of turbulent flows at high Reynolds numbers, the number of degrees of freedom is huge and one has to resort to a contracted probabilistic description. At this level, we can already point out an important drawback which concerns the macroscopic approach. When particle diameters vary considerably from particle to particle or when we are confronted with a situation where particles have completely different histories (highly complicated but local laws), deriving partial differential equations for mean quantities is a thorny issue. The case of complicated source terms happens whenever we want to have particle evaporation or combustion with complex expressions in terms of individual particle properties. When dealing with a distribution of particle diameters, one is faced with the problem of expressing, as a function of mean velocities hUs i, hUp i and the mean particle diameter hdp i, quantities such as h

Us i, τp

h

Up i. τp

(5)

These are complicated functions, due to the complex dependence of τp on particle diameters dp and also on particle and fluid velocities, Eq. (2) and Eq. (4). In theses cases, the Lagrangian PDF (mesoscopic) approach is particularly attractive since it treats these phenomena without approximation while the derivation of closed moment equations is next to impossible unless very crude simplifications are introduced. Concerning the turbulent structure one consideration and two questions will help us to determine which is the best-suited approach, in our opinion. The consideration is that the macroscopic approach is not able to represent properly turbulent structures. Beyond the technical difficulties to achieve physical meaningful closure of two-phase turbulent equations, as said above, this approach is placed at the level of mean equations and the effect of many disparate scales are modeled at the same time through some constitutive relations. In such methods, it appears at least courageous any tentative to add some physical-sound term which take into account of the statistical effect of instantaneous and zero-mean phenomena like quasi-coherent structures. Ad-hoc terms are neither


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general neither under complete control and, therefore, should be considered very carefully as a possible solution of this issue. Therefore, our first main conclusion is that macroscopic RANS approaches should be discarded as appropriate models for polydispersed two-phase turbulent flows and, notably, for particle deposition in wall-bounded flows. Two questions arise about turbulent structures: 1. Which structures are really important? 2. How does one must include these structures? Actually, it is very hard to answer to the first question and it is necessary to analyse it case-by-case. In the example discussed here, DNS simulations and experimental results seem to indicate that sweeps and ejections are particularly relevant. However, much smaller structures, like worms or point-vortex, might be much more effective for explaining internal intermittency in isotropic turbulence [38]. The second question is more probing. In principle, two ways can be explored. The first, the most usual, is to compute directly the turbulent structures, that is to resolve all the scales which are responsible for those geometrical features. Since quasi-coherent structures are fluid-velocity structures, this approach consists in obtaining (in some way) an instantaneous velocity field which contains such information. It is worth emphasising that this approach is based upon the idea of calculating a very accurate instantaneous fluid-velocity field and, therefore, can be pursued only within a microscopic approach. The second possible route is a mesoscopic one. The modeling issue is transferred to the particle phase and, notably, to the problem of building a suitable model for the velocity of the fluid seen. The general form of the Langevin model chosen for the velocity of the fluid seen consists in writing dUs,i = As,i (t, Z) dt + Bs,ij (t, Z) dWj ,

(6)

where the drift vector A and the diffusion matrix B have to be modelled. The complete Langevin equation model can therefore be written dxp,i = Up,i dt,

(7a)

dUp,i = Ap,i dt,

(7b)

dUs,i = As,i (t, Z) dt + Bs,ij (t, Z) dWj ,

(7c)

where the particle acceleration is Ap,i = (Us,i −Up,i )/τp +gi . This formulation is equivalent to a Fokker-Planck equation given in closed form for the corresponding pdf p(t; yp , Vp , Vs ) which is, in sample space. ∂p ∂ ∂ ∂ 1 ∂2 + [Vp,i p ] + [Ap,i p ] + [As,i p ] = (Bs BsT )ij p . (8) ∂t ∂yp,i ∂Vp,i ∂Vs,i 2 ∂Vs,i ∂Vs,j

In this approach, the statistical effect of the most important turbulent structures should be included in eq. (7)c via appropriate terms. Some comments are in order. Although the first way can appear attractive for its accuracy and conceptual simplicity, some major drawbacks undermines its use for engineering

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applications. DNS is certainly a very accurate approach since it is able to reproduce all the scales for both phases and, therefore, together with the experimental studies should be viewed as the preferred method to investigate the fundamentals of turbulent flows [39, 40]. Further, since it is difficult to analyse experimentally many important quantities (frequency and strength of intermittent phenomena, geometrical features of small scales just to make few examples), it appear as an unavoidable searching tool mainly for modeling purposes. Indeed, physical models are based upon such information to guarantee some physical sound basis. However, DNS is a very demanding tool and its practical use in high-Reynolds number and/or geometrical complex flows is at least doubtful. LES is very similar to DNS for wall-bounded flows in terms of computational cost and thus the same remark applies as well [41]. Moreover, LES is a model, even though the model concerns only a part of the energy spectrum. In such sense, it is questionable if this approach is able to reproduce accurately quasi-coherent structures. Of course, it does reproduce some quasi-coherent structures but not all. In particular, small scale structures, which are modeled, are probably not present or, at least, not necessarily well treated. POD and SO(3)-SO(2) techniques are not predictive methods, because they need some information which is provided by previous DNS computations. In particular, SO(3) needs some ”a priori” knowledge of the whole statistical properties of the velocity field at all scales. After getting it, geometry-by-geometry, one may hope for a reduction of the important degrees of freedom by keeping track only of those statistical correlations, isotropic or anisotropic, which are more relevant in the SO(3) decomposition These approaches still aim to compute, even though indirectly, the complete velocity field and, therefore, they join the same category of microscopic approaches. Moreover, they are based upon a projection on a finite (often small for practical purposes) number of chosen eigenstates and, thus, they constitute a contracted model of the complete field. In conclusion, none of these microscopic models appear to be adequate for engineering application and in particular two-phase turbulent pipe flows. On the other hand, mesoscopic approach is radical different. We can say that it is an “active” approach. It is based on “a-priori” choice and not on “a-posteriori” analysis. Indeed, if we know in advance the problem we want to tackle, we can analyse the physical problem on the basis of experimental and DNS results. On this basis, we shall choose which structures appear to be particularly relevant for our own problem. In this way, we avoid the problem which affects LES, POD and SO(3) approaches. Then, we shall try to include the statistical effect of those (and only those) geometrical features which seem more important. Eventually, the model will be written in terms of the known typical statistical signatures (time-frequency, characteristic length scales, life-time...). For this step, DNS simulations will result particularly interesting.

4.

Numerical Results

In this section, we try to illustrate this view with few examples describing particle deposition in a turbulent pipe-flow. The point of depart is the Langevin PDF approach developed


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in the last decade by Minier and his collaborators [42, 34, 43]: dxp,i = Up,i dt 1 dUp,i = (Us,i − Up,i )dt τp ∂hUf,i i 1 ∂hP i dUs,i = − dt + (hUp,j i − hUf,j i) dt ρf ∂xi ∂xj 1 − ∗ (Us,i − hUf,i i) dt TL,i s 2 ˜ ˜ + hǫi C0 bi k/k + (bi k/k − 1) dWi . 3 10

10

(10)

(11)

0

-1

-2

kp/u*

10

(9)

10

10

10

-3

exp.

-4

Standard Stand. + structures

-5

10

-2

10

-1

10

0

+

10

1

10

2

10

3

τp

Figure 1. Deposition rate velocity for the different model used. In all numerical cases the continuous phase is solved via standard k − ǫ model. Experimental results are given for reference (triangle down). The standard results are indicated by the curve labeled with “Standard model” (circles). The results obtained with the new phenomenological model are shown by the curve indicated by “Stand. + structures” (crosses). The results obtained with the phenomenological model derived from DNS are in good agreement with experimental results, in particular small particles deposit only rarely. This model was developed without taking into account quasi-coherent structures and, therefore, it fails to describe correctly the particle deposition, see fig. 1a. Thus, as simple and first step toward considering some of the signatures of coherent structures, a new phenomenological model, built on the basis of DNS results, has been proposed and introduced

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in the numerical simulations [44]. The results obtained with this model are in good agreement with experiments [45] and, hence, show that to take into account some geometrical features of the flow improves significantly the statistical description of particle deposition. Following this suggestion, a more systematic introduction of geometrical features in statistical PDF approach, where coherent structures are introduced as new stochastic terms in the modeled equations, has been recently attempted [47]. The sketch of that model is given in figure 2a, for more details refer to the papers [46, 47]. Standard Langevin Model Core flow y+ = 100 1

0.1

Sweep Ejection

Diffusion

+ p

0.01

k

Outer zone

0.001

0.0001

Possible reentrainment

1e-05

Interface Inner zone

Diffusion

1e-06 0.0001

0.001

0.01

0.1

τp

+

1

10

100

1000

Figure 2. (a) Sketch of the stochastic model of sweeps and ejections structures. (b)Deposition rate for the standard model (down-triangle), the new stochastic model (Full circle) and experimental results (star). The new model is used in a large range of diameters. The deposition rate surges with particle inertia in the range of 1 < τp+ < 70 and slightly decreases for greater inertia. (Courtesy from Physics of fluids). The deposition velocity computed by the new stochastic model for 12 classes of particles is represented in Fig. 2b; it is compared with the experimental data gathered by Papavergos and Hedley [48] and with the results provided by the standard Langevin model. It can be observed that the deposition rate computed by the present model is in fair agreement with the experiments.

5.

Perspectives

There are many complex flows in which geometrical features play a major role and whose modeling is still lacking. A recent stochastic description of geometrical asperities for problems of particle resuspension joins to this category [49]. Nevertheless, in author’s


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opinion, the most intriguing direction to pursue is to put forward a suitable model for bubbly flows. These flows have an enormous importance in a vast spectrum of applications and are very hard to simulate within present methods (microscopic and macroscopic). In these flows the interactions between quasi-coherent structures and particles is particularly relevant since it is well known that particles tend to concentrate within the vortexes [50], at variance with heavy particles, showing a strong preferential-concentration behaviour. The fact that bubbly particles pass most of their time within turbulent structures should make certainly difficult to propose an accurate model without taking them into account. While, for heavy particles, it appears essential to take into account geometrical features in certain situations (like in particle deposition), for bubbly flows it might be true for almost all flows (even for isotropic symmetry). A recent hard effort in DNS simulation of these flows will be of great help for the modeling.

6.

Conclusions

In this commentary, which is based upon recent modeling developments, we have discussed what features should characterize a suitable model for complex turbulent two-phase flows. The most of the attention has been devoted to the case of particle deposition onto walls in turbulent pipe-flows. However, we believe that the rationale can be applied to many other situations and in particular to all those situations in which geometrical features have a not negligible role. We have discussed why a suitable model should be statistical and should be easily linked to geometrical characteristics of the flows. Then, we have analyzed the different statistical approaches available for the description of turbulent two-phase flows and we have tried to explain why the “mesoscopic” PDF approach should be preferred to the others. In particular, we have discussed in some details the issue of modeling quasi-coherent structures and we have emphasized that an active choice of the modeler permits to avoid both too much demanding numerical simulations and a misleading treatment of such structures. Finally, we have shown some recent results obtained through this PDF approach for particle deposition in a turbulent pipe-flow which lend support to this point of view.

Acknowledgments S. Chibbaro’s work is supported by a ERG EU grant. He greatly acknowledges the financial support given also by the consortium SCIRE. More information is available at http://www.consorzio-cometa.it. The author desires to thank in the most particular way Jean-Pierre Minier, since the present author’s point of view is incommensurately related to his highest scientific and pedagogical lessons. Furthermore, I would like to thank him for interesting suggestions for the present manuscript. I thank Dott. Mathieu Guingo for his courtesy in giving me figure 2.

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[45] B. Liu.and K. Agarwal, Aerosol Science, 5, 145 (1974). [46] M. Guingo and J-P. Minier ICMF proceedings, Leipzig (2007). [47] M. Guingo and J-P. Minier Phys. Fluids, 20 053303 (2008). [48] P. G. Papavergos and A. B. Hedley, Chem. Eng. Res. Des. 62, 275 1984. [49] M. Guingo, and J.-P. Minier, J. Aerosol Science 2008, doi: j.jaerosci.2008.06.007.

10.1016/

[50] E. Calzavarini, M. Kerscher, D. Lohse, and F. Toschi. J. Fluid Mech., 607:1324, 2008. [51] R. Volk, E. Calzavarini, G. Verhille, D. Lohse, N. Mordant, J.-F. Pinton, F. Toschi Physica D 237 14-17 (2008) 2084-2089. Reviewed By Prof. Luca Biferale Professor of Theoretical Physics Dept. of Physics and INFN, University of Roma, Tor Vergata Via della Ricerca Scientifica 1, 00133, Roma, Italy ph +39 067259.4595, fax +39 062023507 http://www.fisica.uniroma2.it/ biferale/ skype callto://lucabiferale email [email protected]; [email protected]

INDEX A absorption, 208, 229 academic, xi, 269 accidents, 366 accounting, 127, 141, 277 accuracy, xi, 121, 185, 214, 238, 244, 249, 259, 269, 286, 305, 357, 400, 417, 420, 421, 422, 423, 447 acetylene, 166 actuation, 156 acute, 443 adiabatic, ix, 171, 173, 177, 178, 180, 184, 188, 189, 190, 192, 193, 195, 201, 304, 305, 381 advection-diffusion, 407, 411, 425 AEA, 114, 166 aerosol, xiii, 162, 164, 167, 441 aerosols, viii, 117, 163 aerospace, 344 AFM, 186 Africa, 30, 31, 36, 37 agent, 38 aggregation, 36, 166 agricultural, x, 231 aid, 42, 44 air, 43, 80, 86, 103, 105, 106, 108, 110, 114, 167, 179, 182, 184, 185, 186, 271, 317, 344, 366, 411 air pollution, 167 algorithm, xii, 53, 122, 163, 186, 188, 208, 228, 399, 400, 411, 415, 416, 418, 419, 420, 423, 425 alkali, 38 alternative, 211, 272, 282, 309 alters, 131 aluminium, 283, 284, 286, 288, 289, 290, 298, 299, 301, 302, 307, 308, 309, 313 aluminum oxide, 308 ambiguity, 157 amplitude, 368, 370, 414, 423, 424, 425 Amsterdam, 315, 452 anisotropy, 252 anode, 22 appendix, 212, 213, 222

application, ix, xii, 113, 126, 127, 128, 152, 156, 158, 162, 164, 171, 175, 185, 270, 271, 272, 312, 313, 344, 361, 399, 448 applied mathematics, 163 aqueous solution, 8, 11 aqueous solutions, 8, 11 Arabia, 317 Argentina, 298 argument, 43, 108, 428 arithmetic, 393 aspect ratio, 327, 328, 336, 337, 351, 355, 360, 362 aspiration, 156 assessment, 146, 310 assignment, 207 assumptions, 45, 207, 209, 238, 302, 305, 306, 312, 389, 390, 399, 415 asymptotic, 207, 208, 219 asymptotically, 324 atmosphere, 442 atmospheric pressure, 173 Australia, 41, 117, 269, 270 Australian Research Council, 162 availability, 43, 120, 142, 271 averaging, 45, 46, 99, 238, 271, 368, 445 azimuthal angle, 173

B baths, 167 beaches, 400, 411, 425, 438 beams, 270, 294 behavior, 29, 34, 62, 63, 64, 65, 110, 111, 168, 214, 219, 249, 257, 259, 293, 294, 334, 349, 422, 424 behaviours, 129, 157 benchmark, 53, 75, 111 Bessel, 206, 212, 226 bioreactors, 118 birth, ix, 117, 118, 121, 124 blocks, 185, 366, 392 boiling, 129, 131, 133, 136, 137, 139, 142, 143, 144, 145, 150, 152, 153, 154, 156, 158, 159, 162, 163, 164, 165, 166, 168, 169, 302, 314, 361, 362 Boston, 164

456

Index

bottleneck, 159 boundary conditions, xi, 53, 84, 95, 96, 143, 144, 173, 177, 178, 179, 185, 188, 189, 207, 210, 211, 212, 224, 273, 279, 289, 302, 318, 383, 388, 390, 394, 395, 400, 415, 416, 418, 419, 424, 443 boundary surface, 273, 417 boundary value problem, 211, 418 bounds, 164 Boussinesq, 180 Brazil, 166, 224 breakage rate, 121, 135, 145, 149, 162 bubble, ix, 43, 44, 50, 51, 52, 104, 105, 106, 108, 109, 110, 112, 113, 114, 118, 123, 126, 127, 128, 129, 130, 131, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169 bubbles, ix, 44, 49, 50, 51, 52, 95, 97, 103, 104, 105, 106, 108, 109, 110, 112, 113, 114, 118, 127, 129, 130, 131, 132, 133, 135, 136, 137, 138, 139, 143, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 161, 163, 164, 166, 168, 169, 453 buffer, 98, 101, 103, 105, 244, 246, 261 bun, 406 burn, 301 bypass, 289

C cables, 185 calculus, 402 calibration, 38, 126, 136, 145, 160 Canada, 362 canals, 442 candidates, 28 capacity, 119, 120, 139, 172, 206, 227, 270, 288, 307, 313, 324, 334 capillary, 16, 353, 362, 410, 413 carbonates, 38 carrier, vii, viii, 27, 29, 41, 42, 43, 44, 45, 46, 53, 56, 58, 59, 62, 64, 65, 67, 68, 75, 80, 83, 84, 88, 94, 95, 100, 101, 104, 110, 111 Cartesian coordinates, 172, 179, 273, 426 case study, 282 cast, 186 categorization, 443 cathode, 22 cation, 36, 114 cavities, 131, 140, 152, 185 C-C, 189, 190, 191, 195, 196 cell, 36, 164, 167, 305, 392, 393, 394, 395 CFD, viii, xi, 41, 43, 95, 120, 121, 123, 125, 132, 143, 144, 162, 166, 167, 186, 269, 270, 271, 272, 274, 283, 289, 290, 293, 294, 295, 296, 297, 298, 302, 306, 312, 313, 314, 316, 397

channels, vii, xi, 1, 2, 3, 5, 20, 24, 25, 33, 35, 207, 210, 292, 298, 304, 343, 344, 351, 352, 353, 355, 361, 362 chaos, 173, 174 charge density, 3, 4 chemical composition, 36 chemical reactions, 9 chromatography, 1, 2, 11, 13, 17, 18, 20, 21, 22, 23, 24, 25 circulation, 381, 382 cladding, 298, 301, 302, 308, 309, 310, 311, 312, 314 classes, viii, 41, 42, 44, 53, 54, 56, 58, 59, 62, 63, 64, 111, 125, 126, 143, 144, 149, 153, 154, 155, 156, 160, 163, 445, 450 classical, 259, 410 classification, xi, 114, 343, 369 climate change, 167 clinics, 284 closure, 44, 113, 122, 158, 164, 165, 233, 234, 445, 446 clustering, 97 coagulation, 163, 166 coagulation process, 163 codes, 166 collisions, 46 colon, 395 combustion, xiii, 129, 159, 164, 165, 441, 442, 446 combustion processes, 129 communication, 315 communities, 1 competence, 158 complex systems, 270 complexity, xi, 44, 68, 95, 119, 150, 158, 269, 272, 276, 399 components, 5, 7, 10, 121, 139, 152, 177, 178, 232, 234, 245, 246, 249, 252, 254, 263, 265, 270, 275, 283, 285, 286, 289, 294, 295, 344, 345, 368, 382, 395, 414, 415, 416, 421, 422, 423, 426, 431, 446 composition, 28, 36, 38, 39, 118 comprehension, xiii, 441 computation, 95, 97, 115, 122, 245, 271, 315, 415, 437 Computational Fluid Dynamics (CFD), viii, ix, xi, 41, 117, 118, 129, 185, 186, 269, 292, 314, 316, 366 computational grid, 234, 238 computer technology, 127 computing, 118, 122, 185, 271, 273, 278, 392, 416, 417, 418, 419, 439 concentration, ix, 2, 4, 5, 9, 10, 11, 29, 35, 49, 53, 118, 126, 148, 149, 150, 153, 158, 159, 164, 301, 412 concentric annuli, ix, 171, 173, 174, 175, 229 concrete, 180, 181, 182, 183, 185 condensation, 131, 133, 136, 152, 153, 154, 155, 158, 161 conductance, 5, 7, 185

Index conduction, ix, x, 137, 138, 139, 160, 171, 173, 174, 176, 180, 181, 182, 183, 184, 185, 205, 208, 209, 211, 219, 229, 273, 279, 283, 285, 286, 289, 305, 309, 310, 321 conductivity, x, 3, 7, 172, 182, 205, 206, 208, 209, 217, 224, 275, 284, 288, 290, 307, 308, 309, 312, 319, 330, 340 confidence, 295 configuration, viii, 42, 99, 145, 175, 189, 294, 301, 303, 401, 402, 409 conformity, 64 conservation, 36, 127, 143, 185, 188, 271, 274, 280, 285, 291, 294, 308, 403, 404, 421, 422, 425 constraints, 127 construction, 271, 272, 273, 295, 381, 389, 410, 445 contact time, 52, 136, 162 continuity, 45, 50, 53, 188, 210, 211, 212, 224, 225, 235, 280 control, 53, 136, 145, 186, 208, 228, 236, 279, 280, 281, 282, 283, 286, 295, 299, 304, 312, 321, 322, 330, 331, 385, 447 convection, ix, xi, 118, 138, 160, 165, 171, 173, 174, 175, 176, 177, 180, 181, 182, 184, 185, 189, 201, 205, 207, 209, 229, 269, 290, 291, 301, 302, 305, 309, 416, 426 convective, ix, 53, 137, 171, 172, 173, 176, 179, 206, 282, 305, 318, 362, 393, 401, 402, 409 convergence, 49, 96, 143, 179, 188, 214, 273, 286, 295, 302, 306, 313, 400, 421, 423, 425 convergence criteria, 302 conversion, 8 cooling, xi, 35, 269, 270, 282, 283, 284, 290, 291, 297, 298, 299, 300, 302, 306, 309, 314, 319, 330, 344 copper, 55, 57, 59, 60, 61, 62, 63, 64, 65, 111 correlation, 47, 48, 50, 152, 339, 349, 351, 354, 356, 362 correlations, xi, 42, 152, 156, 343, 353, 354, 448 cosine, 7, 303, 310, 311, 312, 313 costs, 42 couples, 53 coupling, xii, 14, 16, 43, 44, 45, 49, 53, 147, 185, 207, 379, 392 covering, 105, 126, 143, 152, 271, 295 CPU, 424 crack, 438 CRC, 341 critical value, 122, 156, 180, 392 cross-sectional, 137, 227, 251, 321, 351, 360, 368, 371, 372 cryogenic, 344 crystalline, 39 crystallization, vii, 27, 33, 34 crystals, 31, 33 curiosity, xi, 269 curve-fitting, 349 cycles, 368 cyclone, 113 Czech Republic, 379, 397

457

D damping, 239 Darcy, 320, 322, 330, 340 data set, 241 data structure, 273 death, ix, 51, 117, 118, 121, 124, 128, 159 death rate, 51, 121, 128, 159 decay, 301, 302, 376 decomposition, 446, 448 decompression, 29, 34, 35, 37 decoupling, 120 deduction, 110 defects, viii, 27, 36, 186 defense, 344 definition, 14, 17, 20, 68, 138, 184, 272, 276, 279, 311, 312, 313, 351, 388, 401, 402, 403, 405, 406, 409, 412, 415, 425, 426, 427 deformation, 31, 34, 35, 37, 38, 43, 234, 237, 265, 276, 394, 395 degrees of freedom, 443, 445, 446, 448 dehydration, 34, 35 delivery, 42 demand, xiii, 441 Denmark, 163 densitometry, 164 density, viii, 2, 7, 28, 35, 36, 37, 41, 42, 43, 44, 45, 46, 50, 51, 52, 62, 63, 64, 65, 66, 95, 110, 113, 122, 125, 127, 129, 133, 137, 138, 152, 154, 160, 162, 165, 166, 172, 180, 275, 289, 303, 305, 306, 310, 311, 313, 361, 370, 380, 387, 392, 394, 403, 404, 406, 407, 411, 439, 445 dependent variable, 118, 173, 188, 189 deposition, xiii, 42, 442, 443, 444, 447, 448, 449, 450, 451 deposition rate, 450 derivatives, 178, 401, 402, 417, 418, 427 destruction, 35, 43, 324 detachment, 131, 169 detection, 22 deuterium oxide, 284 deviation, 143, 345, 353, 355 diesel, 80, 111 differential equations, 208 differentiation, 142, 225, 226 diffusion, 2, 7, 9, 10, 22, 35, 36, 53, 118, 142, 172, 209, 278, 279, 280, 282, 286, 305, 391, 400, 407, 411, 443, 447 diffusion process, 36, 443 diffusivity, 12, 14, 20, 142, 161, 172, 206, 209, 210, 305 digital images, 293 dilute gas, viii, 41, 44 direct measure, 5, 20 discontinuity, 28 discretization, 421, 422, 423 dislocations, 35

458

Index

dispersion, vii, 1, 2, 3, 10, 13, 14, 15, 16, 17, 18, 19, 22, 23, 25, 44, 52, 105, 112, 113, 114, 118, 133, 136, 146, 160, 166, 442 displacement, 388, 395, 427 distribution, ix, xii, 3, 5, 10, 15, 16, 22, 23, 51, 59, 62, 63, 97, 108, 109, 110, 112, 118, 119, 120, 121, 125, 126, 129, 130, 131, 135, 143, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157, 158, 162, 165, 166, 167, 174, 177, 188, 189, 208, 210, 228, 234, 244, 245, 246, 257, 259, 263, 295, 297, 302, 303, 306, 307, 309, 310, 311, 312, 313, 346, 365, 366, 370, 371, 372, 377, 379, 380, 381, 382, 383, 385, 386, 387, 389, 395, 396, 397, 446 distribution function, 109, 110 divergence, xii, 102, 280, 365, 366, 368, 373, 374, 375, 376, 377, 378, 403, 406, 421, 425 diversity, 44, 118, 270 dominance, 282 Doppler, xi, 56, 254, 270 dust, 42, 164 dynamic viscosity, 275, 360 dynamical system, 443

E earth, 442 eddies, 56, 271, 272, 375 elasticity, 394 electric conductivity, 3 electric current, 2, 7 electric field, vii, 1, 2, 3, 7, 11, 14, 23 electrical power, 176 electrical resistance, 176, 178 electrolyte, 2, 3, 4, 5, 7, 11, 22, 24 electromigration, 11 electroosmosis, 3, 9, 11 electrophoresis, 1, 2, 9, 11, 12, 15, 16, 17, 20, 21, 22, 23, 24, 25 electrostatic force, 352 electrostatic interactions, 2, 11 email, 454 energy, ix, 8, 43, 44, 46, 47, 48, 88, 94, 102, 115, 132, 139, 144, 160, 171, 173, 185, 210, 238, 241, 242, 263, 271, 274, 275, 276, 279, 280, 285, 289, 294, 318, 324, 344, 365, 370, 372, 390, 391, 394, 420, 421, 422, 423, 424, 425, 430, 432, 436, 448 energy transfer, 238 engines, x, 42, 231 England, 437 enlargement, 148 entropy, xi, 8, 318, 322, 323, 324, 325, 326, 327, 328, 329, 331, 332, 333, 334, 335, 336, 337, 338, 341 environment, 28, 38, 43, 86 equilibrium, 10, 35, 56, 68, 104, 164, 182, 404 equipment, ix, 42, 171, 173, 293, 294 erosion, 114 estimating, 142

ethane, 166 ethylene, 166 Euclidean space, 428 Euler equations, 411 Eulerian, xii, xiii, 43, 45, 49, 111, 113, 164, 399, 400, 401, 413, 414, 416, 425, 437, 441, 446 evaporation, 133, 137, 151, 152, 158, 160, 165, 446 evolution, xii, 118, 120, 122, 124, 126, 129, 131, 149, 163, 164, 399, 400, 404, 409, 416, 424, 425 exaggeration, 375 exercise, 111 expansions, 208 experimental condition, 68, 95 extraction, 113, 419

F failure, 308 fauna, 43 fax, 454 feedback, 44 feeding, 112 film, 53, 106, 136, 160, 245 film thickness, 53, 105, 136, 160 financial support, 162, 451 finite differences, 400 finite element method, 395 finite volume, xii, 53, 185, 279, 280, 295, 379, 392 finite volume method, xii, 185, 279, 280, 295, 392 fire, ix, 171, 173 first principles, xiii, 441 fission, 283, 284, 298, 299, 301, 302 flatness, 234, 254, 257, 259, 263, 264, 265 flexibility, 120, 121, 124, 127, 273 flooding, 140, 160 flora, 43 flora and fauna, 43 flow, vii, viii, ix, x, xi, xii, xiii, 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 19, 35, 41, 42, 43, 44, 45, 47, 49, 54, 56, 58, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 80, 82, 83, 84, 85, 86, 87, 94, 95, 96, 97, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 111, 112, 113, 114, 115, 118, 119, 123, 124, 126, 127, 129, 130, 131, 134, 136, 137, 141, 143, 144, 145, 146, 147, 149, 150, 152, 154, 156, 157, 158, 159, 160, 162, 163, 164, 165, 166, 167, 168, 171, 173, 174, 175, 176, 177, 178, 179, 180, 184, 185, 186, 188, 189, 194, 195, 197, 201, 205, 206, 207, 208, 209, 212, 214, 215, 219, 221, 223, 224, 227, 228, 229, 231, 232, 233, 234, 235, 237, 238, 239, 241, 244, 249, 251, 252, 254, 257, 259, 260, 263, 265, 269, 270, 271, 272, 273, 276, 277, 282, 283, 284, 285, 286, 288, 289, 291, 292, 293, 294, 295, 296, 297, 299, 300, 301, 302, 303, 304, 305, 306, 307, 309, 313, 314, 315, 317, 318, 319, 320, 323, 324, 325, 326, 327, 330, 333, 334, 336, 337, 340, 343, 344, 345, 346, 347, 349, 350, 351, 352, 353, 355, 356, 357, 358, 360, 361, 362, 363,

Index 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 379, 380, 381, 382, 383, 386, 388, 390, 392, 393, 395, 396, 397, 399, 400, 404, 410, 411, 415, 425, 437, 439, 441, 442, 443, 444, 450 flow behaviour, 118 flow field, x, 43, 44, 45, 68, 124, 171, 174, 176, 179, 180, 184, 189, 195, 231, 233, 244, 257, 260, 272, 293, 377, 415, 443 flow rate, viii, xii, 42, 96, 99, 101, 103, 104, 105, 106, 107, 108, 112, 185, 206, 227, 277, 293, 295, 297, 305, 307, 309, 319, 323, 325, 330, 333, 334, 340, 357, 358, 365, 366, 368, 369, 370, 371, 372, 373, 374, 375, 377, 383 flow value, 289 fluctuations, 53, 56, 59, 68, 69, 75, 82, 95, 112, 241, 243, 246, 247, 248, 249, 252, 255, 256, 257, 258, 261, 262, 263, 271, 272, 442, 443 fluid, vii, viii, ix, x, xi, xii, xiii, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 20, 22, 24, 27, 28, 29, 32, 34, 35, 36, 37, 41, 43, 54, 56, 58, 65, 68, 80, 95, 97, 102, 106, 112, 115, 120, 129, 132, 134, 142, 143, 159, 162, 163, 164, 171, 173, 174, 175, 176, 177, 185, 186, 201, 205, 207, 208, 209, 210, 211, 214, 217, 218, 219, 221, 224, 232, 269, 271, 275, 277, 279, 288, 292, 293, 294, 295, 296, 297, 305, 309, 314, 317, 318, 319, 320, 321, 323, 324, 325, 326, 330, 332, 333, 334, 335, 336, 337, 339, 340, 341, 343, 344, 345, 346, 347, 349, 352, 353, 355, 356, 357, 358, 360, 361, 362, 363, 365, 366, 368, 370, 372, 373, 375, 376, 377, 399, 400, 402, 403, 404, 405, 409, 411, 412, 413, 415, 425, 437, 441, 442, 443, 445, 446, 447 fluid mechanics, xii, 399, 400 fluid transport, 366 fluidized bed, 112, 118, 129, 164, 165 focusing, 120, 130, 372 foils, 297 Fourier, 368, 371 Fox, 123, 164, 166, 167 fragmentation, 166 France, 164, 168, 231 freedom, xii, 379, 443, 445, 446, 448 freedoms, 445 friction, x, xi, 95, 98, 99, 101, 105, 106, 108, 109, 114, 161, 231, 232, 233, 234, 241, 242, 244, 245, 249, 261, 265, 278, 317, 318, 320, 322, 323, 327, 328, 330, 334, 336, 337, 340, 343, 344, 347, 349, 350, 351, 352, 353, 354, 355, 356, 358, 359, 360, 361, 362, 363 FTIR, vii, 27, 29, 30, 31, 37, 38 fuel, 283, 284, 285, 286, 288, 289, 290, 293, 294, 295, 301, 303, 316 functional analysis, 410 funding, 344

459

G gas, viii, x, 2, 4, 41, 42, 45, 46, 47, 48, 49, 50, 53, 54, 55, 56, 57, 58, 62, 65, 68, 69, 78, 80, 94, 95, 96, 97, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 111, 112, 115, 118, 126, 127, 130, 131, 133, 134, 135, 140, 143, 144, 145, 146, 147, 148, 154, 156, 157, 158, 159, 163, 164, 180, 185, 208, 229, 231, 233, 317, 318, 353, 404, 442 gas phase, 45, 46, 47, 48, 49, 50, 53, 54, 56, 58, 62, 65, 68, 69, 80, 131, 133, 135, 143, 147 gas turbine, x, 42, 231 gases, vii, 43, 179, 317, 320 gasoline, x, 231 gauge, 305, 307 Gaussian, 122, 257, 263 gel, 213 generation, xi, 8, 34, 69, 133, 137, 139, 152, 154, 155, 158, 160, 178, 185, 283, 285, 315, 316, 318, 322, 323, 324, 325, 326, 327, 328, 329, 331, 332, 333, 334, 335, 336, 337, 338, 340, 341 geochemical, 39 geochemistry, 28 geophysical, 28 Germany, 27, 114, 165, 168, 266 glass, 56, 58, 59, 60, 61, 62, 64, 65, 66, 79, 80, 111, 366 glycerol, 339 grain, 35 grains, 29 gravitational constant, 160 gravitational field, 404 gravity, 46, 50, 277, 444 greek, 426 grid generation, xi, 269, 272, 273, 315 grid resolution, 238, 239, 241, 259, 260 grids, 115, 232, 238, 254, 273, 305 groups, 44, 51, 52, 106, 112, 125, 126, 143, 156, 323, 333 growth, 118, 119, 130, 138, 139, 140, 141, 142, 160, 161, 164, 166, 306, 344 growth rate, 142 growth time, 140 guidelines, 299

H half-life, 284 handling, 97, 119, 120, 132, 134, 156, 158, 186, 272 haze, 164 heat, ix, x, xi, 115, 118, 120, 129, 131, 133, 137, 138, 139, 142, 143, 144, 150, 152, 158, 160, 163, 164, 165, 167, 168, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 189, 201, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 218, 219, 220, 221, 222, 223, 224, 227, 228, 229, 231, 232, 233, 234, 269, 270, 271, 273,

460

Index

275, 277, 278, 283, 284, 285, 289, 290, 291, 297, 301, 302, 303, 304, 309, 310, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 330, 331, 332, 333, 334, 335, 336, 337, 339, 340, 341, 344, 361, 362, 363, 392 heat capacity, 139, 206, 227, 334 Heat Exchangers, 228, 341 heat removal, xi, 269, 270, 301 heat transfer, ix, x, xi, 115, 131, 133, 137, 152, 160, 164, 165, 167, 168, 171, 172, 173, 174, 175, 176, 180, 182, 183, 184, 185, 201, 205, 206, 207, 208, 212, 222, 223, 227, 228, 229, 231, 232, 233, 269, 270, 271, 273, 278, 302, 304, 310, 315, 317, 318, 319, 320, 321, 323, 324, 325, 326, 327, 330, 331, 332, 333, 334, 335, 336, 337, 339, 340, 361, 362, 363 heating, 137, 140, 150, 174, 175, 207, 219, 222, 289, 302, 318, 319, 330 heavy particle, 113, 444, 451 heavy water, 284, 288, 289, 298 height, 2, 4, 16, 20, 22, 24, 53, 54, 57, 59, 67, 68, 69, 75, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 111, 143, 298, 349, 350, 353, 354, 405, 419, 421, 423, 425 helicopters, 42 helium, 186 heterogeneity, 31 heterogeneous, 28 hexane, 353 high pressure, 183 high resolution, 313 high temperature, 180 high-speed, 443 histogram, 30, 33 Holland, 204 homogenous, 44, 69, 114, 157, 158 host, 29, 31, 32, 34, 35, 36 human, 42, 43 hybrid, 97, 134, 273 hydration, 34, 37 hydrodynamic, 2, 10, 13, 15, 16, 18, 19, 22, 25, 141, 160, 295, 352 hydrodynamics, 26, 154, 158, 165, 353, 362 hydrogen, 28, 36, 38, 39 hydroxide, 39 hyperbolic, 7, 238, 400 hypothesis, 276

I IAC, 126, 143, 149, 150, 152, 155, 156, 157, 158 identification, 33 identity, 238, 385 images, 293, 358, 359 imaging, 400 imbalances, 282 implementation, ix, 117, 120, 125, 127, 385 inclusion, 36, 37, 167

incompressible, vii, xi, 3, 343, 345, 350, 352, 383, 388, 399, 403, 411, 413, 437, 439 independence, 145 independent variable, 415 indication, 257, 259 indices, 20, 24, 392, 408, 426 industrial, viii, ix, xiii, 42, 44, 114, 117, 118, 120, 121, 122, 171, 186, 441 industrial application, xiii, 441 industry, 344 inert, 43 inertia, 43, 56, 80, 95, 346, 351, 444, 450 infinite, 212, 213, 214, 222 infrared, vii, 38, 39 injection, viii, 22, 42, 43, 95, 96, 99, 100, 101, 102, 103, 104, 106, 107, 108, 110, 112, 131, 163 injections, 42 insertion, 273, 315 insight, 234, 368 instabilities, 419, 423 instability, 122, 157, 173, 174, 175, 305, 400 institutions, 344 instruments, 44 insulation, 143 integration, x, 231, 238, 280, 383, 385, 404, 424 intensity, xii, 102, 147, 149, 249, 259, 265, 365, 366, 368, 372, 373, 374, 375, 376, 377, 381, 383, 394, 444 interaction, viii, xiii, 28, 34, 41, 47, 48, 51, 53, 111, 113, 174, 207, 441 interactions, 2, 11, 43, 110, 115, 131, 156, 158, 186, 451 interdependence, 423 interdisciplinary, 42 interface, 133, 177, 188, 211, 309, 310, 311, 312, 314, 400, 416, 419, 437 interference, 85 international markets, 299 internship, 315 interphase, 42 interpretation, 237, 409, 412 interrelations, 259 interval, 51, 380, 381, 384, 385, 389, 390 intrinsic, vii, 27, 28, 35, 38, 400, 403, 406, 407, 425 inversion, xii, 124, 379, 380, 386, 387 Investigations, x, 231 investigative, 159 ionic, 4, 11 ions, 3, 5, 7, 95, 123, 143, 194, 197, 302, 417, 418, 422, 423 iron, 39 irradiation, xi, 269, 270, 272, 277, 283, 284, 285, 290, 292, 295, 297, 298, 299, 300, 301, 302, 303, 304, 305, 307, 314, 316 isothermal, 45, 49, 113, 129, 131, 133, 143, 144, 145, 156, 158, 163, 177, 189, 191, 192, 193, 194, 195, 201 isotopes, xi, 269, 270 isotropic, 51, 53, 135, 136, 237, 249, 447, 448, 451

Index Italy, 113, 164, 441, 454 iteration, 53, 188, 189, 386, 394, 416 iterative solution, 302

J Jacobian, 122, 172, 187 Japan, 203, 205, 266 Japanese, 378 judge, 179 Jung, 184, 204

K kernel, 168 kinetic energy, 44, 46, 48, 88, 94, 102, 160, 241, 275, 276, 279, 365, 370, 372, 390, 391, 394, 421, 423, 424 King, 317, 343, 361 Kolmogorov, 452

L Lagrangian, 43, 104, 115, 400, 401, 446 Lagrangian approach, 104 lamellar, 31 lamina, vii, x, xi, xii, 45, 162, 163, 166, 174, 179, 185, 207, 208, 214, 223, 228, 229, 231, 232, 233, 235, 276, 315, 317, 318, 319, 320, 326, 330, 336, 343, 344, 345, 346, 347, 349, 350, 351, 352, 353, 355, 356, 358, 362, 379, 388, 390, 391, 392, 396 laminar, vii, x, xi, xii, 45, 162, 163, 166, 174, 179, 185, 207, 208, 214, 223, 228, 229, 231, 232, 233, 235, 276, 315, 317, 318, 319, 320, 326, 330, 336, 341, 342, 343, 344, 345, 346, 347, 349, 350, 351, 352, 353, 355, 356, 358, 361, 362, 379, 388, 390, 391, 392, 396, 397, 442 laminated, 229 land, 183 language, xiii, 441, 445 large-scale, 249 laser, 254, 293, 294 law, x, 97, 98, 180, 205, 208, 214, 224, 233, 324, 354, 375, 403, 404, 406, 407 laws, 400, 446 lead, xi, 7, 42, 68, 69, 94, 111, 195, 255, 269 liberation, viii, 27, 34, 35, 37 licensing, 285 life forms, 43 lifetime, 176, 201 likelihood, 148 limitation, 156, 158, 359, 421 limitations, ix, xii, 44, 118, 120, 123, 350, 399 linear, 124, 139, 175, 186, 213, 244, 249, 278, 321, 327, 336, 339, 340, 393, 394, 399, 404, 417, 418, 419, 424, 428

461

linear function, 428 liquid film, 105, 106 liquid nitrogen, 362 liquid phase, 49, 50, 51, 68, 69, 101, 103, 104, 126, 133, 135 liquids, vii, 113, 163, 228, 317, 318, 320, 339 lithosphere, 39 location, 3, 59, 63, 65, 102, 103, 118, 146, 148, 149, 219, 220, 227, 301, 377, 388, 422, 430 London, 26, 228, 318, 326, 330, 336, 341, 351, 353, 361 longevity, 42 long-term, 28 losses, xi, 43, 318, 319, 362, 366 LSM, 400 lubrication, 133, 146, 149, 157, 160 lying, 409, 419

M machinery, 365 machines, x, 231, 366 magma, vii, 27, 28, 29, 34, 35, 37, 39 magnesium, 28, 37, 143, 283, 285, 288, 290 maintenance, 42 manipulation, 225 mantle, vii, viii, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39 manufacturing, 400 mapping, 380, 385, 386, 401 markets, 299 mass loss, 410 mass transfer, x, 50, 118, 120, 128, 129, 131, 132, 133, 150, 158, 162, 168, 208, 228, 231, 316 mass transfer process, x, 150, 158, 231 matrix, 32, 36, 80, 111, 123, 124, 161, 225, 279, 282, 295, 299, 307, 313, 418, 447 measurement, 148, 152, 163, 294, 295, 352, 358, 368, 372 measures, 401 meat, 299, 301, 306, 307, 309, 311, 312, 314 mechanical energy, 318 media, 185 medications, 42 medicine, 270 melt, vii, 27, 28, 29, 33, 34, 35, 37, 285, 301 melting, 28, 290, 301, 314 melts, vii, 27, 33, 34, 37, 38, 39 MEMS, 344 mesoscopic, xiii, 442, 445, 446, 447, 448, 451 metals, 356 metric, 53, 172, 187 Mexico, 39 Mg2+, 36 microbial, 118 microelectronics, 344 microscope, 358 microscopy, 29

462

Index

microstructures, 38 microtubes, 353, 355, 357, 359, 362, 363 migration, 11, 22, 23, 25, 444 military, 344 mimicking, 69 mineralogy, 39 minerals, 28, 38 miniaturization, 344 Ministry of Education, 397 misleading, 337, 444, 451 MIT, 266 mixing, 56, 58, 80, 95, 113, 114, 186, 233, 299 mobility, 2, 5, 6, 7, 9, 12, 13, 14, 20, 22, 25 modeling, xi, xiii, 47, 115, 122, 164, 167, 186, 233, 234, 269, 270, 277, 302, 306, 314, 345, 439, 441, 442, 445, 447, 448, 450, 451 models, vii, viii, ix, x, xi, xii, xiii, 42, 43, 44, 46, 95, 97, 107, 108, 112, 113, 114, 118, 127, 130, 131, 134, 146, 149, 156, 158, 159, 164, 166, 167, 185, 186, 232, 233, 234, 259, 260, 265, 270, 271, 272, 283, 295, 302, 309, 314, 315, 316, 340, 399, 400, 425, 441, 445, 447, 448 modulation, 46, 58, 62, 63, 64, 65, 103, 108, 118 modulus, 251, 395, 419, 423 molar conduct, 3 molar volume, 36 molecules, xiii, 28, 119, 347, 441, 442 molybdenum, xi, 269, 284, 299, 300, 301, 314 MOM, ix, 117, 121, 122, 124, 126 momentum, 42, 43, 44, 45, 49, 50, 53, 87, 95, 118, 132, 143, 144, 147, 148, 159, 209, 234, 235, 238, 271, 274, 275, 280, 285, 294, 295, 315, 347, 389, 403 monotone, 220 Monte Carlo, 113, 121, 185, 303, 446 Monte Carlo method, 185, 446 Moscow, 27, 37, 39 motion, xiii, 3, 6, 7, 9, 42, 43, 44, 68, 86, 95, 104, 112, 118, 131, 137, 159, 185, 190, 197, 236, 239, 265, 368, 400, 441, 443, 444 motivation, 260 movement, 395, 401, 405, 406, 407, 409 multidimensional, 113, 121 multiphase flow, ix, 113, 117, 118, 119, 120, 122, 125, 127, 128, 134, 164, 167, 168

N nanofabrication, vii, 1 nanometers, 3, 31, 33 nanotubes, 25 NASA, 229 national, 270 natural, vii, ix, 27, 38, 39, 42, 119, 171, 173, 174, 175, 176, 177, 179, 180, 184, 185, 189, 201, 209, 302, 385, 442 natural gas, 442

Navier-Stokes, xii, xiii, 44, 271, 379, 390, 437, 441, 443 Navier-Stokes equation, xii, xiii, 379, 390, 437, 441, 443 neglect, 182 negotiating, 271 net migration, 444 neutrons, 284, 298, 301, 302 New York, 26, 167, 203, 228, 229, 315, 341, 342, 361, 397, 452, 453 Newton, 214, 393, 403 Newtonian, x, 205, 208, 214, 221, 224, 229 nitrogen, 353, 362 nitrogen gas, 353 nodes, 273, 282, 305, 387, 416, 419, 420, 421, 422, 423, 424, 434 nodules, vii, 27, 28, 29 nonlinear, 186, 238, 271, 393, 400, 416, 421, 445 non-Newtonian, x, 205, 208, 214, 224, 229 non-Newtonian fluid, x, 205, 208, 214, 224, 229 non-uniform, vii, 1, 10, 16, 51, 174, 185 non-uniformity, 10, 16 normal, 36, 37, 101, 102, 103, 112, 139, 140, 142, 145, 161, 188, 246, 277, 279, 281, 295, 298, 303, 345, 380, 392, 405, 406, 417, 418, 421, 426 normalization, 383 norms, 421, 422, 423 nuclear, xi, 113, 164, 269, 270, 283, 284, 298, 302, 314 nucleation, 34, 35, 118, 119, 131, 133, 137, 138, 139, 142, 152, 158, 160, 161, 164, 165, 166 nuclei, 118 nucleus, 139 numerical analysis, 185 Nusselt, 133, 152, 174, 175, 182, 185, 206, 211, 213, 214, 220, 221, 222, 228, 340

O observations, viii, xi, 27, 33, 35, 37, 104, 108, 131, 134, 145, 151, 152, 179, 234, 244, 248, 249, 263, 270, 293 oceans, 43 OECD, 168 OH-groups, 28 oil, 75, 80, 111, 317, 442 operator, xii, 225, 379, 380, 381, 386, 387, 388, 389, 390, 394, 395, 397, 402, 404, 427, 428, 445 optical, 29 optical microscopy, 29 optics, 294 optimization, vii, 1, 114, 305 order statistic, 265 organization, 252 orientation, 257, 265 orthogonality, 226 oscillation, 34, 35, 37, 366 oscillations, 249

Index oxide, 143, 283, 285, 288, 290, 306, 307, 308, 309, 312, 313, 314 oxide thickness, 313, 314 oxygen, 28

P pairing, 112 paper, viii, 41, 43, 49, 53, 67, 110, 113, 120, 203, 234, 270, 315, 363, 416 parabolic, 6, 234, 239, 317, 442 parameter, 5, 44, 54, 65, 68, 111, 150, 323, 333, 346, 380, 383, 385, 389, 395 Paris, 113 partial differential equations, xii, 179, 273, 280, 379, 399, 400, 411, 416, 421, 446 particle nucleation, 164 particles, viii, xii, xiii, 41, 42, 43, 44, 45, 53, 54, 55, 56, 57, 58, 59, 62, 63, 64, 65, 66, 67, 68, 69, 79, 80, 83, 85, 86, 87, 88, 94, 95, 111, 113, 114, 117, 118, 119, 121, 122, 125, 128, 159, 162, 164, 347, 368, 399, 401, 402, 403, 405, 407, 409, 441, 443, 444, 446, 449, 450, 451 partition, 133, 137, 139, 142, 152, 158, 168 pathways, 33 Peclet number, 2, 14, 24 pedagogical, 451 performance, 20, 25, 186, 207, 233, 259, 317, 318, 366 periodic, 238, 366, 370, 375 permeability, 96 permittivity, 3, 4 personal, 366, 367 perturbations, 443 petrographic, 29, 33 Petroleum, 317 pharmaceutical, 343 philosophy, 301 physical mechanisms, 352, 443 physical properties, 28, 209, 283, 306, 313, 347, 395 physics, xiii, 42, 110, 112, 167, 185, 283, 285, 298, 441, 442, 443, 445 pipelines, 442 planar, 315 plants, 42 plausibility, 36 play, viii, 42, 450 point defects, 35 Poisson, 4, 5, 381, 395 Poisson equation, 4 Poisson-Boltzmann equation, 5 pollen, 42 pollution, 43, 167 polymer, 42 polymers, 43 polynomial, 387, 388 polystyrene, 368 poor, 112, 410

463

population, vii, viii, ix, 51, 113, 114, 117, 118, 119, 120, 121, 122, 124, 125, 126, 128, 129, 131, 149, 152, 158, 159, 163, 164, 165, 166, 167, 168 porous, 95 Portugal, 203, 266 potential energy, 420, 423 powder, 143, 290, 293 power, viii, x, xi, 41, 42, 43, 118, 143, 176, 178, 180, 181, 182, 183, 184, 185, 205, 208, 214, 224, 269, 270, 271, 289, 290, 301, 302, 303, 306, 307, 310, 311, 312, 313, 314, 315, 317, 318, 319, 325, 326, 328, 329, 335, 336, 337, 339, 340, 344 power plant, 42 power plants, 42 power-law, x, 205, 206, 208, 214, 217, 218, 222, 224, 229 powers, 303 pragmatic, 43, 271 Prandtl, 46, 135, 162, 172, 174, 175, 275, 276, 315, 340 precipitation, viii, 117 prediction, 56, 62, 75, 80, 97, 112, 134, 145, 147, 148, 149, 152, 156, 158, 159, 162, 168, 169, 298, 301, 361, 366 pressure, xii, 2, 4, 6, 11, 14, 16, 23, 27, 28, 33, 34, 35, 37, 45, 49, 50, 53, 56, 80, 94, 95, 96, 141, 143, 148, 160, 165, 172, 173, 183, 236, 270, 275, 277, 282, 295, 305, 307, 309, 317, 318, 322, 332, 346, 347, 349, 350, 352, 353, 355, 357, 359, 360, 362, 365, 366, 367, 368, 369, 370, 371, 372, 375, 376, 377, 379, 380, 381, 387, 388, 389, 392, 394, 395, 396, 404, 442 printing, 400 probability, 121, 263, 446 probability density function, 121, 446 probability distribution, 263 probable cause, 146 probe, 43, 143, 245, 294, 366 production, 47, 48, 51, 102, 128, 160, 270, 276, 277, 298, 299, 344, 391 program, 121, 294, 295, 302 propagation, 400, 411 property, 273, 280, 317, 319, 320, 326, 330, 336, 387, 410, 412 proportionality, 406 proposition, xiii, 441 protection, ix, 171, 173, 298 prototype, 270, 283, 286, 293, 295 PSD, 120, 122, 123, 124, 125, 126, 129, 132 pseudo, 385 pumping, xi, 317, 318, 319, 325, 326, 328, 335, 336, 337, 362 pumps, 344, 366 purification, viii, 27, 36

Q quadtree, 273

464

Index

R radial distribution, ix, 118, 143, 158, 227, 245 radiation, ix, 171, 172, 173, 176, 180, 183, 184, 185, 298, 301 radical, 156, 157, 360, 448 radiopharmaceutical, 270, 283 radiotherapy, xi, 269 radius, vii, 1, 139, 140, 142, 160, 161, 182, 183, 206, 209, 211, 232, 235, 286, 377 random, 52, 136, 157, 229, 345, 346, 444 range, x, 8, 15, 24, 42, 43, 44, 80, 102, 103, 110, 112, 125, 126, 131, 142, 143, 145, 152, 156, 158, 179, 205, 208, 211, 214, 233, 259, 271, 277, 278, 317, 318, 337, 349, 350, 353, 355, 372, 443, 445, 450 Rayleigh, 172, 173, 174, 175, 179, 180, 183, 184, 195, 400 reactants, 36, 42 reaction rate, 298 realism, 312, 442 recovery, 272, 366 recrystallization, 37 rectilinear, 425 recycling, 39 reduction, viii, xiii, 11, 13, 18, 42, 43, 98, 99, 101, 105, 108, 110, 112, 114, 138, 233, 234, 239, 244, 245, 249, 251, 265, 305, 310, 373, 441, 442, 445, 448 refining, 54, 110 reflection, 257 reforms, 137 refrigeration, 344 refrigeration industry, 344 regulatory requirements, 271 relationship, 152, 213, 238, 278, 366, 404 relationships, 35, 404 relaxation, 46, 54, 68, 444, 445 relaxation time, 46, 54, 68, 444, 445 relevance, 121 reliability, xi, 244, 269, 270, 293, 295 research, ix, xi, xii, 1, 42, 119, 131, 152, 159, 171, 173, 185, 269, 270, 283, 298, 343, 344, 356, 369, 442 research and development, 1 researchers, 44, 344, 352, 356, 360 reservoir, 143 resistance, 176, 178, 209, 315 resolution, vii, 1, 2, 20, 24, 98, 126, 156, 158, 246, 260, 271, 282, 283, 290, 309, 313, 419 resources, 119, 126, 185, 271, 283, 290, 302, 306 response time, 80 retardation, 9 retention, 20 Reynolds, viii, x, xii, xiii, 41, 42, 46, 50, 51, 53, 54, 64, 67, 101, 102, 105, 111, 161, 174, 175, 185, 186, 207, 223, 229, 231, 232, 233, 234, 236, 238, 239, 244, 245, 249, 250, 254, 259, 260, 261, 262,

263, 264, 265, 275, 276, 278, 314, 315, 320, 340, 346, 347, 348, 349, 351, 352, 353, 354, 355, 356, 360, 363, 365, 368, 370, 373, 374, 395, 396, 397, 441, 442, 443, 444, 445, 446 Reynolds number, viii, x, xii, xiii, 41, 42, 50, 51, 53, 54, 64, 67, 101, 105, 111, 161, 174, 175, 185, 207, 223, 231, 232, 233, 234, 236, 238, 239, 244, 245, 249, 254, 259, 260, 261, 263, 264, 265, 278, 314, 315, 320, 340, 346, 347, 348, 349, 351, 352, 353, 354, 355, 356, 360, 363, 365, 368, 370, 373, 374, 395, 396, 397, 441, 442, 443, 446 Reynolds stress model, 185, 186 rheology, 208, 221 rivers, 442 robustness, 158, 282, 295, 314 Rome, 441 room temperature, 173, 339 roughness, xi, xiii, 315, 343, 346, 347, 348, 349, 350, 351, 352, 353, 355, 356, 357, 358, 359, 360, 363, 441 rubber, 183 Russia, 27, 37 Russian, 27, 37, 39 Russian Academy of Sciences, 27, 37

S safety, viii, xi, 41, 45, 113, 164, 269, 282, 283, 285, 289, 296, 303, 306, 313 sample, 30, 32, 33, 37, 447 sampling, 443 sand, 42 saturation, 131, 162 Saudi Arabia, 317 scalar, 125, 126, 133, 160, 185, 404, 405, 409, 428 scalar field, 404, 409, 428 scaling, 5, 68, 276 scatter, 62, 294, 368 scientists, 43 SCP, 113, 163, 168 search, 273, 443 searching, 448 seeding, 306 segregation, 31, 35 selecting, 280 selectivity, vii, 1, 2, 20, 21, 23, 24 SEM, 358, 359 semiconductor, 270, 367, 400 sensitivity, 283, 312, 443 sensors, 344 separation, vii, 1, 2, 10, 20, 22, 23, 24, 25, 26, 34, 35, 37, 156, 272, 296, 344, 366, 376, 389 series, 119, 125, 126, 143, 213, 214, 302, 368, 371, 421, 423 shape, x, 14, 31, 43, 136, 149, 159, 231, 239, 251, 346, 351, 358, 381, 387, 388, 390, 394, 396, 405, 411, 417, 418, 423, 424, 430, 433, 434, 435 sharing, 111

Index shear, 50, 68, 69, 94, 95, 106, 111, 112, 115, 134, 141, 142, 153, 159, 160, 161, 163, 164, 214, 232, 234, 249, 250, 262, 263, 265, 276, 278, 316, 347, 349, 350, 375, 443 shear rates, 214 sign, 48, 386 silica, 39, 352, 353, 355, 359 silicate, 35, 38 silicates, 28, 37 silicon, 270 similarity, 19, 64, 214 simulation, ix, x, 11, 42, 43, 44, 45, 80, 99, 104, 106, 110, 112, 114, 123, 126, 163, 166, 167, 169, 171, 174, 175, 185, 186, 201, 231, 233, 237, 244, 245, 257, 259, 270, 271, 282, 288, 289, 290, 295, 302, 306, 308, 312, 313, 445, 451 simulations, viii, ix, x, 41, 49, 95, 96, 99, 106, 108, 117, 121, 123, 126, 127, 129, 130, 143, 144, 145, 150, 156, 159, 177, 231, 233, 234, 238, 241, 244, 249, 252, 254, 271, 283, 289, 302, 304, 309, 310, 312, 315, 411, 425, 438, 445, 447, 448, 450, 451 singular, 417, 418, 419, 437, 438 singularities, 387 sites, 131, 137, 138, 161 skewness, 234, 254, 255, 257, 259, 263, 264, 265 skin, 98, 99, 101, 105, 106, 108, 109, 114, 233 smoothing, 298, 423 smoothness, 415 software, 125, 312, 393 soil, 181, 182, 183, 185 solar, ix, 171, 173 solar energy, ix, 171, 173 solid phase, 277 solubility, 33, 38 solutions, ix, 2, 20, 24, 43, 117, 120, 121, 144, 188, 208, 214, 215, 228, 229, 270, 272, 273, 289, 439, 444 soot, 128, 159, 164, 166 South Africa, 30, 31, 36, 37 Southampton, 316, 438 Spain, 399 spatial, 22, 118, 121, 132, 207, 236, 238, 400, 401, 402 spatial location, 118 species, 2, 3, 9, 10, 20, 22 specific heat, 159, 172, 319, 330 Specific Heat, 288, 307, 313 spectrum, 118, 143, 448, 451 speed, 2, 9, 10, 11, 12, 13, 14, 20, 22, 23, 24, 25, 156, 233, 283, 377, 380, 415, 421, 444 stability, x, 175, 186, 231, 233, 234, 390, 410, 421, 439 stages, 29, 119, 295 stainless steel, 352, 355, 356, 357, 359 standard deviation, 3, 23 standard model, 450 standards, xi, 269 statistical mechanics, 164 statistics, x, 231, 233, 234, 238, 254, 260, 265

465

steady state, 53, 96, 143, 179, 302, 415 steel, 143, 352, 353, 355, 356, 357, 359 stochastic, xiii, 51, 112, 135, 163, 229, 442, 444, 445, 446, 450 stochastic model, xiii, 442, 450 stochastic processes, 444 stoichiometry, 36 storage, 28, 39, 43, 185, 238, 282, 301 strain, 47, 186, 232, 237, 394, 395 strategies, 438 streams, 207, 208, 213, 221, 228 strength, 448 stress, x, 45, 102, 106, 115, 153, 161, 185, 186, 231, 232, 233, 234, 238, 249, 250, 265, 275, 276, 278, 347, 349, 350, 391, 392, 394, 411, 444 stroke, 366 students, 315 subsonic, xii, 379, 386, 397 Sun, 126, 157, 166, 168, 186, 204 superiority, 286 supply, 143, 271, 283, 299 suppression, 167, 233, 239, 265 surface area, 50, 51, 118, 121, 122, 135, 280, 318, 327, 337, 405 surface diffusion, 411 surface region, 407 surface roughness, xi, 315, 343, 349, 350, 352, 355, 357, 358, 360 surface tension, xii, 141, 159, 160, 162, 361, 399, 411 surfactant, 42, 411, 412 surfactants, xii, 43, 399, 400, 411, 439 swarm, 156, 168 swelling, 377 symbolic, 437 symbols, 6, 242, 243, 255, 360, 389, 395 symmetry, 4, 11, 143, 255, 257, 383, 408, 451 synchronous, 373 systems, 42, 52, 53, 118, 119, 120, 121, 122, 128, 136, 159, 165, 166, 167, 169, 270, 271, 273, 300, 314, 315, 343, 344, 365, 392

T Taiwan, 171 talc, 28, 31, 33, 34, 35, 38 tangible, 42 tanks, 118, 271, 301 targets, xi, 269, 293, 295, 297, 299, 301, 302, 303, 304, 306, 307, 313, 314, 315 technology, 165 TEM, vii, 27, 28, 29, 31, 33, 35, 38 temperature, ix, x, xi, 2, 4, 28, 39, 131, 137, 139, 143, 161, 169, 171, 172, 173, 174, 175, 178, 179, 180, 183, 184, 185, 188, 194, 195, 197, 200, 201, 205, 206, 207, 208, 209, 211, 212, 213, 217, 218, 219, 220, 221, 224, 227, 275, 277, 283, 285, 289, 290, 291, 298, 302, 303, 304, 305, 306, 307, 308,

466

Index

309, 310, 311, 312, 313, 314, 317, 318, 319, 320, 321, 324, 325, 326, 330, 331, 332, 333, 334, 335, 336, 337, 339, 340, 341, 394 temperature dependence, xi, 318 temperature gradient, 188, 219, 275, 321 tensor field, 400, 401, 402, 428 Texas, 224 textbooks, 119, 392, 394 theory, xii, 46, 47, 115, 233, 352, 353, 355, 356, 379, 381 Thermal Conductivity, 279, 288, 307, 313 thermal equilibrium, 182 thermal expansion, 172 thermodynamic, 318 thermodynamic parameters, 318 thermo-mechanical, 154, 158 thin film, 105 three-dimensional, ix, 132, 144, 158, 171, 173, 174, 175, 177, 179, 184, 189, 201, 233, 280, 292, 294, 302, 316 threshold, 22 time, x, xi, 2, 3, 9, 22, 23, 34, 37, 43, 44, 46, 52, 54, 63, 68, 85, 96, 105, 114, 121, 126, 132, 136, 138, 140, 142, 143, 161, 185, 231, 238, 269, 271, 273, 275, 280, 281, 286, 290, 299, 301, 302, 305, 306, 345, 368, 369, 370, 372, 373, 383, 392, 394, 400, 401, 402, 403, 404, 405, 406, 407, 409, 410, 411, 412, 413, 416, 417, 418, 419, 420, 421, 422, 423, 424, 428, 429, 430, 431, 432, 435, 436, 443, 444, 445, 446, 451 time consuming, 418 tolerance, 309 topological, xii, 399, 400, 417 topology, 409, 412, 444 total energy, 420, 423, 430 tracers, 80 tracking, xii, 43, 44, 115, 122, 126, 149, 399, 412 trajectory, 123, 412, 446 transducer, 367 transfer, ix, x, xi, 42, 46, 50, 51, 115, 118, 126, 131, 133, 137, 147, 152, 159, 160, 164, 165, 167, 168, 171, 172, 173, 174, 175, 176, 180, 181, 182, 183, 184, 185, 201, 205, 206, 207, 208, 212, 222, 223, 227, 228, 229, 231, 232, 233, 238, 269, 270, 271, 273, 278, 302, 304, 310, 315, 317, 318, 319, 320, 321, 323, 324, 325, 326, 327, 330, 331, 332, 333, 334, 335, 336, 337, 339, 340, 347, 361, 362, 363, 443, 444 transformation, viii, 27, 35, 38, 121, 122, 380, 381, 415 transition, x, xi, 28, 143, 146, 156, 157, 158, 168, 173, 174, 175, 179, 185, 231, 233, 343, 344, 346, 347, 349, 352, 353, 355, 356, 358, 362 transitions, 163, 305 transparency, 284 transparent, xi, 270, 291, 293, 366 transport, vii, xiii, 1, 2, 3, 4, 9, 10, 11, 25, 33, 42, 44, 46, 47, 48, 53, 123, 124, 126, 132, 134, 144, 157,

164, 165, 166, 168, 185, 209, 233, 234, 271, 276, 280, 281, 289, 303, 344, 362, 366, 400, 441 transpose, 225 transverse section, 178, 189, 194, 197, 200 travel, 51, 52, 87, 123, 124, 131 trend, 18, 80, 87, 94, 101, 102, 221, 249, 257, 263, 297, 303, 317 trial, 320 trial and error, 320 triangulation, 273 turbulence, viii, x, xi, xii, xiii, 41, 42, 44, 46, 47, 48, 49, 51, 52, 53, 58, 59, 63, 64, 65, 69, 94, 95, 97, 102, 104, 105, 108, 111, 112, 113, 114, 115, 118, 122, 134, 135, 136, 145, 146, 147, 149, 166, 167, 175, 185, 186, 201, 231, 232, 233, 234, 237, 239, 249, 251, 259, 261, 263, 265, 269, 271, 272, 276, 295, 302, 314, 315, 316, 350, 365, 368, 372, 373, 374, 375, 376, 377, 379, 390, 391, 392, 395, 442, 443, 445, 447 turbulent, vii, viii, x, xi, xii, xiii, 41, 42, 43, 44, 45, 46, 47, 48, 51, 52, 53, 58, 62, 66, 88, 94, 95, 97, 99, 102, 103, 110, 111, 112, 113, 114, 115, 118, 129, 131, 133, 134, 135, 136, 138, 143, 145, 146, 149, 153, 156, 157, 158, 160, 161, 162, 164, 166, 168, 174, 179, 185, 186, 207, 209, 229, 231, 233, 234, 235, 236, 237, 239, 241, 243, 244, 245, 247, 249, 253, 254, 255, 257, 259, 260, 261, 263, 265, 266, 267, 269, 271, 272, 275, 276, 277, 278, 279, 280, 289, 290, 304, 305, 314, 315, 318, 319, 320, 330, 342, 343, 344, 345, 346, 347, 349, 352, 353, 355, 358, 362, 365, 366, 367, 369, 371, 373, 375, 377, 379, 390, 391, 392, 393, 394, 441, 442, 443, 444, 445, 446, 447, 448, 449, 451, 453 turbulent flows, xi, xiii, 110, 179, 186, 207, 260, 277, 278, 314, 315, 343, 344, 345, 347, 353, 355, 441, 442, 443, 444, 445, 446, 447, 448 turbulent mixing, 377 two-dimensional (2D), ix, 25, 126, 130, 171, 173, 174, 175, 179, 180, 181, 184, 185, 188, 294, 362, 366, 425 two-way, 43, 44, 53

U ubiquitous, viii, xiii, 27, 37, 441, 442 Ukraine, 27 uncertainty, 146, 345 uniform, xi, 4, 10, 15, 16, 53, 59, 62, 63, 96, 144, 186, 207, 214, 295, 297, 309, 310, 311, 312, 318, 319, 331, 346, 349, 403, 404 universal gas constant, 4 updating, 421 uranium, xi, 269, 270, 285, 288, 297, 299, 301, 302 uranium oxide, 285 uti, 52, 136

Index

V vacancies, 36 valence, 3 validation, ix, 111, 118, 120, 125, 134, 163, 234, 241, 265, 270, 302, 306, 307, 312, 313, 315, 344 validity, 11, 43, 283 values, 6, 20, 30, 33, 59, 62, 63, 80, 81, 98, 135, 188, 212, 214, 217, 218, 222, 223, 227, 233, 238, 239, 245, 249, 252, 255, 257, 263, 265, 273, 276, 277, 280, 297, 302, 305, 324, 334, 335, 346, 349, 351, 352, 356, 360, 368, 370, 372, 377, 380, 388, 391, 392, 393, 394, 395, 396, 408, 415, 416, 418, 419, 421, 422, 424, 426 vapor, 159, 160, 161, 162, 169 variable, 5, 38, 125, 131, 173, 189, 210, 229, 280, 288, 313, 410, 415, 421 variables, 49, 50, 53, 118, 120, 121, 123, 125, 132, 156, 235, 237, 277, 387, 390, 391, 392, 394, 413, 415, 421, 422, 445 variance, 22, 451 variation, 17, 18, 29, 119, 148, 195, 201, 217, 219, 223, 244, 263, 272, 302, 317, 318, 320, 321, 327, 330, 331, 335, 336, 339, 340, 352, 369, 370, 371, 373, 375, 377, 401 vector, 50, 121, 124, 160, 161, 189, 190, 191, 192, 195, 197, 198, 275, 277, 292, 382, 383, 391, 392, 394, 402, 403, 404, 405, 406, 407, 421, 425, 426, 427, 428, 445, 447 velocity, ix, x, xi, xii, 4, 5, 6, 7, 10, 11, 12, 14, 15, 16, 19, 22, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 67, 68, 79, 80, 84, 88, 94, 95, 96, 97, 98, 100, 101, 102, 103, 104, 105, 106, 107, 108, 111, 114, 118, 121, 123, 124, 127, 129, 131, 136, 138, 142, 143, 144, 147, 154, 156, 157, 158, 161, 166, 168, 172, 177, 178, 187, 188, 189, 190, 195, 197, 205, 206, 208, 209, 210, 214, 217, 224, 227, 231, 232, 233, 234, 235, 236, 237, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 251, 252, 253, 254, 255, 256, 257, 258, 259, 261, 262, 263, 264, 265, 270, 275, 276, 277, 278, 282, 291, 292, 293, 294, 295, 296, 297, 298, 303, 304, 305, 306, 314, 317, 318, 340, 345, 346, 360, 365, 366, 367, 368, 371, 372, 373, 374, 375, 377, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 392, 394,

467

399, 400, 402, 403, 405, 409, 411, 412, 413, 414, 415, 416, 417, 418, 419, 421, 422, 423, 431, 442, 443, 446, 447, 448, 449, 450 ventilation, 176 versatility, 314 Victoria, 117 viscosity, xi, 3, 4, 9, 17, 44, 45, 46, 48, 49, 51, 95, 97, 104, 114, 134, 135, 145, 161, 167, 172, 185, 207, 214, 232, 237, 272, 275, 276, 315, 317, 318, 319, 320, 324, 325, 330, 333, 334, 335, 339, 340, 349, 350, 368, 387, 391, 394, 397 visible, 35, 293, 368 visualization, 108, 292, 293, 296, 368, 376, 443 voids, 32, 33, 35 vortex, 381 vortices, 43, 63, 65, 104, 112, 175, 295, 375, 376, 377, 381, 382, 383

W wall temperature, xi, 173, 177, 185, 189, 201, 214, 229, 279, 318, 331, 334, 335, 341 warfare, 344 water, vii, viii, 27, 28, 29, 33, 34, 35, 37, 38, 39, 65, 102, 103, 112, 142, 143, 166, 183, 271, 290, 293, 294, 298, 300, 303, 309, 310, 311, 312, 314, 317, 339, 352, 353, 355, 359, 362, 368, 375, 400, 411, 423, 437, 440 weakness, 120 welding, 283 wettability, 140 wind, 42 windows, 286 worms, 447 writing, 321, 447

Y yield, 7, 22, 24, 44, 126, 130, 131, 136, 156, 273, 295, 301, 327, 328, 337

Z zeta potential, 3, 4, 5, 16

FLUID MECHANICS AND PIPE FLOW: TURBULENCE, SIMULATION AND DYNAMICS

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