Prof. Michael L. Corradini: Department of Engineering Physics University of Wisconsin, Madison WI 53706

FUNDAMENTALS OF MULTIPHASE FLOW

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Prof. Michael L. Corradini Department of Engineering Physics University of Wisconsin, Madison WI 53706 ll contents © Michael L. Corradini

[email protected] Last Modified: Mon Aug 4 00:56:50 CDT 1997

Contents

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Contents i. Nomenclature ii. List of Figures iii. List of Tables 1. INTRODUCTION AND CONCEPTS 1.1. Definitions 1.2. Flow Patterns 1.3. Pool Boiling 1.4. Flow Boiling 1.5. Condensation 2. FLOW PATTERNS AND FLOW-PATTERN TRANSITIONS 3. MULTIPHASE FORMULATION AND PRESSURE DROP 3.1. Homogeneous Equilibrium Model 3.2. Separated Flow Model 3.3. Two-Fluid Model 3.4. Pressure Drop in a Closed Conduit 3.5. Void Fraction Prediction with the Drift Flux Model 4. SOUND SPEED AND CRITICAL FLOW 4.1. Single Phase Critical Flow 4.2. Homogeneous Equilibrium Model for Critical Flow 4.3. Homogeneous Frozen Flow Model for Critical Flow 4.4. Separated Flow Model for Critical Flow 4.5. Fluids Other Than Water For Separated Flow Model 4.6. The Physical Basis of Choked Two Phase Flow 4.7. Advanced Computational Models 4.8. Observations 5. POOL BOILING 5.1. Bubble Nucleation and Onset of Nucleate Boiling 5.2. Bubble Growth and Nucleate Boiling

Contents

5.3. Pool Boiling Critical Heat Flux 5.4. Film Boiling and the Minimum Film Boiling Point 6. FLOW BOILING HEAT TRANSFER 6.1. Objectives 6.2. Regions of Heat Transfer 6.3. Single-Phase Liquid Heat Transfer 6.4. The Onset of Nucleate Boiling 6.5. Subcooled Boiling 6.6. Saturated Boiling and the Two-Phase Forced Convection Region 7. BURNOUT AND THE CRITICAL HEAT FLUX 7.1. Introduction and Objectives 7.2. Effect of System Parameters on CHF 7.3. Correlation Methods for CHF in Round Tubes With Uniform Heating 7.4. Limits on the Critical Heat Flux 7.5. Mechanisms of Critical Heat Flux 7.6. Prediction of CHF in Annular Flow 7.7. Correlations for CHF With Uniform Heating 7.8. CHF With a Nonuniform Heat Flux 7.9. Correlation of Burnout for Rod Bundles (sub-channel analysis): 7.10. CHF Estimates for Nonaqueous Fluids 7.11. Observations 8. POST-CHF HEAT TRANSFER 8.1. Introduction and Objectives 8.2. Post-CHF Heat Transfer Models and Correlations 8.3. Empirical Correlations 8.4. Non-Equilibrium Empirical Models 8.5. Semi-Theoretical Models 8.6. Transition Boiling 8.7. Transition to Film Boiling 8.8. Observations 9. CONDENSATION

Contents

9.1. Basic Processes of Condensation 9.2. Theoretical Developments of Condensation 9.3. Experimental Investigations 9.4. Separate Effects and Large Scale Tests 9.5. Observations ll contents © Michael L. Corradini


Fundamentals of Multiphase Flow: Nomenclature

contents

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NOMENCLATURE at

Thermal Diffusivity

A

Area

Af

Flow area occupied by liquid

Ag

Flow area occupied by gas

Ai

Internal tube surface or interfacial area

Cp

Specific heat at constant pressure

Cv

Specific heat at constant volume

c

Sound speed

D

Diameter

Dd

Bubble departure diameter

Dh

Hydraulic diameter

Fr

Froude number

f

Friction factor

G

Mass velocity

Gr

Grashof number

g

Gravitational acceleration

h

Convective heat transfer coefficient


i

(or h)

Specific enthalpy

ifg

(or hfg)

Heat of vaporization

j

Volumetric flux (superficial velocity)

k

Thermal conductivity

L

Length Mass flow rate

Nu

Nusselt number

P

Pressure

Pe

Pecklet number

Pr

Prandtl number

Q

Volumetric flow rate

q

Heat transfer rate

Re

Reynolds number

r

Variable radius

Ro

Gas constant

R

Pipe Radius

St

Stanton number

s

Specific entropy

T

Temperature

t

Time

u

specific internal energy


V

Velocity

Vgj

Drift velocity

Vr

Relative velocity

We

Weber number

X

Flow quality

Xeq

Thermodynamic equilibrium quality

Xtt

Martinelli Parameter

u,v,w

Velocity components Specific volume

dP/dz

Pressure gradient

Greek Symbols Void fraction Difference (out-in) of X Volumetric flow quality or thermal expansion coefficient

Film thickness Ratio of specific heats

Interfacial mass flow rate Wavelength Dynamic viscosity


Kinematic viscosity Wetted perimeter

Stress tensor Shear stress Mass density

Surface tension Contact angle

Subscripts d

Dispersed phase

e

External

f

Liquid phase

g

Gaseous phase

i

Internal or interface

m

Mixture

o

Stagnation value (velocity = 0)

r

Relative

w

Wall

cr

Critical value

nb

Nucleate boiling

SP

Single phase


TP

Two Phase

sub

Subcooled

scb

Subcooled boiling

sat

Saturated

ll contents © Michael L. Corradini

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List of Figures

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List of Figures Figure 1.1. Figure 1.2. Figure 1.3. Figure 1.4. Figure 2.1. Figure 2.2. Figure 2.3. Figure 3.1.

Gas-Liquid General Flow Regimes Conceptual Picture of Pool Boiling Diabatic and Adiabatic Flow Conceptual Picture of Condensation Flow Pattern Map for Horizontal Flow (Baker) Flow Pattern Map for Vertical Flow (Hewitt and Roberts) Flow Pattern Map used in TEXAS (Chu) Conceptual Picture of 1-D Two-Phase Flow in a Tube

Figure 3.2. Value of

fo

2

as a function of Pressure and Mass Quality (Martinelli-Nelson)

Figure 3.3. Value of as a function of Pressure and Mass Quality (Martinelli-Nelson) Figure 4.1. Conceptual Picture of Fluid Blowdown Figure 4.2. Conceptual Picture of Single Phase Critical Flow Figure 4.3. Theoretical Values for the Velocity of Sound in Equilibrium, Homogeneous Steam-Water Mixture Figure 4.4. Qualitative Picture of Maximum Mass Velocity Figure 4.5. Comparison of Critical Flow Predictions and Experimental Figure 4.6. Experimental Critical Pressure Ratio Data as a Function of Length Figure 4.7. Predictions of Critical Steam-Water Flow Rates with Slip Equilibrium Model Figure 5.1. Physical Interpretation of Boiling Curve (Farber) Figure 5.2. Qualitative Picture of Pool Boiling Hysteresis (Heat Flux Controlled Boundary Condition) Figure 5.3. Conceptual Surface Roughness and Ideal Cavities Figure 5.4. Pool Boiling Onset of Nucleation Conceptual Model Figure 5.5. Construction of the Pool Boiling Curve Figure 6.1. Conceptual Picture of Forced Convective Boiling with Qualitative Temperature Profile for a Uniform Heat Flux Boundary Condition Figure 6.2.a. Variation of Heat Transfer Coefficient Figure 6.2.b. Variation of Heat Flux Figure 6.3. Conceptual Picture of Forced Convective Boiling with Qualitative Heat Flux Profile for a Constant Temperature Boundary Condition Figure 6.4. Variation in Flow Patterns, Temperature Void Fraction and Quality with Tube Length Figure 6.5. Approximate Values for Void Fraction and Quality for Water in Subcooled Boiling and

List of Figures

Available Models Figure 6.6. Convective Boiling Factor, F Figure 6.7. Nucleate Boiling Suppression Factor, S Figure 7.1. Parametric Effect of CHF with Variation of Initial and Boundary Conditions Figure 7.2. Burnout in Cross Flow over Tubes Figure 7.3. Critical Quality Compared to Proportional Power to Outer Surface Figure 7.4. Comparison of "Local" versus "Integral" Hypotheses for CHF Figure 7.5. Comparison of Uniform to Non-Uniform Heating Effect on CHF Figure 7.6. Bounds of the Critical Heat Flux Figure 7.7. CHF Mechanisms Figure 7.8. CHF Mechanisms as a Function of the Quality and Mass Flux Figure 7.9. Counter Flow Critical Heat Flux Figure 7.10. Conceptual Picture of CHF as Function of Length for Non-Uniform Heating Figure 7.11. Subchannel Concepts Used with CHF Correlations Figure 8.1. Post-CHF Variation of Temperature and Quality as a Function of Axial Length Figure 8.2. Iloeje's Dispersal Flow Model Figure 9.1. Flow of a Laminar Film over an Inclined Surface Figure 9.1.1. Figure 9.2. The Air-Vapor Boundary Layer in Condensation ll contents © Michael L. Corradini


List of Tables

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List of Tables Table 2.1. Flow Pattern Maps for Horizontal Two-Phase Flow Systems Based on Generalized Coordinate Parameters Table 2.2. Flow Pattern Transitions for Horizontal Two-Phase Flow Systems Based on Different Coordinate Parameters Table 2.3. Flow Pattern Maps for Vertical Upward Two-Phase Flow Systems Based on Generalized Coordinate Parameters Table 2.4. Flow Pattern Transitions for Vertical Upward Two-Phase Flow Systems Based on Different Coordinate Parameters Table 3.1. Important Dimensionless Variables in Multiphase Flow Table 3.2. Homogeneous Equilibrium Model Governing Equations Table 3.3. Separated Flow Model Governing Equations Table 3.4. Two Fluid Model Governing Equations Table 3.5. HEM Pressure Gradient Model Table 3.6. Separated Flow Pressure Gradient Table 3.7. Drift Flux Suggested Values Table 7.1. Specific CHF Correlations Table 7.2. Kirby Correlation for Non-Uniformly Heated Round Tubes Table 8.1. Groneveld Empirical Post-Dryout Correlations Table 9.1. Theoretical and Experimental Investigations in Condensation Table 9.2. Summary of Previous Investigations in Condensation ll contents © Michael L. Corradini


INTRODUCTION AND CONCEPTS

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1. INTRODUCTION AND CONCEPTS The importance of fluid flow and heat transfer with a change in phase arises from the fact that many industrial processes rely on these phenomena for materials processing or for energy transfer; e.g. petroleum processing, paper-pulping, power plants. Classical thermodynamics tells us that a phase is a macroscopic state of matter which is homogeneous in chemical composition and physical structure; e.g., a gas, liquid or solid of a pure component. Two-phase flow is the simplest case of multiphase flow in which two phases are present for a pure component. Sometimes the term "multi-component" is used to describe flows in which the phases consist of materials of different chemical substances. For example, a flow of steam and water is a two-phase flow with a single component, while an air-water flow is a two-phase/two component flow. In blood flow the plasma/platelet-corpulses are a two-phase/multi-component flow (liquid/solid). In some applications one can have a single phase of two immiscible liquids (oil-water) flowing and treat this as multi-component flow. There are many common examples of multiphase flow in everyday life, such as rain or snow, a boiling teapot or coffee percolator, steam condensation on walls or a cold glass of beer. The major objective of this 'primer' is to provide the student or practicing engineer with a working knowledge of multi-phase flow and heat transfer fundamentals that can be used in system design and analyses. Specifically, we intend to provide a summary of the current state of knowledge, referenced sources of data and correlations, first principles analysis techniques, and some examples of various industrial applications. This book has been patterned after information in graduate courses that have been offered at the University of Wisconsin and industrial short courses and workshops for over 15 years. Topics include flow patterns, pressure drop and void fraction, critical flow, pool boiling and forced convection heat transfer and condensation. Additional special topics are to be added later. In this introduction we provide an overview into phase change processes by considering every day examples of pool boiling, multi-phase flow boiling and condensation. Also as part of this introduction we introduce some basic definitions.

1.1. Definitions 1.2. Flow Patterns 1.3. Pool Boiling 1.4. Flow Boiling 1.5. Condensation

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INTRODUCTION AND CONCEPTS



Definitions

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1.1. Definitions To model multi-phase flows one is sometimes required to describe properties averaged over the phase both spatially and temporally. A certain familiarity is required with these definitions before we discuss specific phenoma. The multiple phases (and/or components) are usually distinguished by numerical subscripts (1,2...) or for two-phases by subscripts f and g for a liquid-gas system of f and s for a liquid-solid system. (The second phase, component, is usually chosen as the dispersed phase.) For illustration, consider a two-phase air-water flow in a vertical pipe. The total mass flow rate is with volumetric flowrate given by

Every part of the multiphase flow is occupied by one phase or another. One can consider the symbol , as the fraction of an element of volume which is occupied over some time interval by phase i. Obviously, if the volume element and the time interval is chosen to be small enough (infinitesimal) would be 0 and 1 at any instant. However, in actual practice is an average quantity over some macroscopic volume (e.g., channel cross-sectional area) and time interval, and is the "volume fraction" of . For a gas the term void fraction is used. It can also be defined over a cross-sectional area or chord length. Another average flow quantity of interest particularly in boiling or condensation applications is the mass fraction of phase i

where, for liquid-gas flows,

is called the "quality". This quantity should not be confused with the

thermodynamic quality, the ratio of the vapor mass (not mass flow rate) to the total mass. Only if the velocity of the phases are equal do the two definitions become the same; e.g., this is done in the homogeneous equilibrium model. One can also define a mass flux or mass velocity, G, by

and volumetric flux, i, as

With these definitions one can derive a number of useful physical quantities, e.g., the relation between

Definitions

the volume fraction

, and mass fraction,

. For a liquid-gas flow

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Flow Patterns

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1.2. Flow Patterns An important distinction in single phase flow is whether the flow is laminar or turbulent, or whether flow separation or secondary flows exist. This information helps in modeling specific phenomena because one has an indication of the flow character for a particular geometry. Analogously in multiphase flow probably the key toward understanding the phenomena is the ability to identify the internal geometry of the flow; i.e., the relative location of interfaces between the phases, how they are affected by pressure, flow, heat flux and channel geometry, and how transitions between the flow patterns occur. There are two fundamental types of flow patterns (Figure 1.1) one can identify, stratified and dispersed. A stratified flow pattern is one in which the two phases are separated by a continuous interface at a length scale comparable to the external scale of the flow; e.g., a liquid film on a wall with a gas or another immiscible liquid in the center of the channel. The complete separation of the two phases usually occurs due to density differences (horizontal flow) combined with a relatively low mass flowrate of the phase near the wall compared to the other phase in the center of the channel (e.g., vertical annular flow). These separated flow patterns can occur when the phases flow in the same direction (co-current flow) or in opposite directions (counter-current flow). The transition between these two types of stratified flow is governed by the balance between buoyancy and inertial forces. A dispersed flow pattern is one in which one or more phases are uniformly dispersed within a continuum of another phase with a length much smaller than the external scale; e.g., gas bubbles or s olid particles in a liquid or liquid droplets in a gas or another immiscible liquid. In this case the dispersed phase forms into nearly regular shaped particles with their stable size-governed again by a balance of buoyancy, inertial and surface tension forces. The transitional flow regimes between these two fundamental types can take on may geometries. Some of the more common transitional flow patterns are churn-turbulent and slug flow; i.e., dispersed-stratified flows where the discontinuous phase begins to form a continuum near the wall (bubbly-film) or in the center of the channel (wispy-annular). These flow patterns will be discussed in more detail in a subsequent section.

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Pool Boiling

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1.3. Pool Boiling One of the processes associated with a change in phase is evaporation. This is simply the process of conversion of the liquid phase to vapor phase at an interface. This process occurs whenever there is a concentration difference between the liquid phase and its vapor; e.g., water evaporation into the atmosphere of a room where the relative humidity is less than 100 %. Boiling is the process in which a liquid evaporates and forms vapor pockets or regions within the continuous liquid phase. Boiling can take many forms. Consider the common everyday occurrence of a pot of boiling water on top of the stove. In this case a stagnant pool of liquid is heated and boiling occurs in the liquid at the bottom of the bulk liquid pool ( Figure 1.2). This overall process is called pool boiling. To form the vapor phase within the continuous liquid phase one must heat the liquid to a temperature above its saturation temperature, ( is that temperature at which the liquid exerts a vapor pressure equal to the ambient pressure.) If the temperature of the liquid rises far above

(e.g.,

C for

water), the vapor will be formed, "nucleate", as bubbles within the bulk of the liquid causing "volumetric" or "bulk" pool boiling. This type of nucleation is termed "homogeneous nucleation" and is rarely observed in most common boiling situations. Rather, if one were to look at the bottom of the pot of water on the stove one would notice the vapor bubbles nucleating at this heater surface. In this case the temperature of the liquid does not need to be far above Tsat- (e.g., C for water). This nucleation occurs within "preferred-sites: or crevices within the heater surface aided by trapped vapor and gas is termed "heterogeneous nucleation". This type of pool boiling is quite common in many industrial applications.

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Flow Boiling

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1.4. Flow Boiling Flow boiling occurs when all the phases are in bulk flow together in a channel; e.g., vapor and liquid flow in a pipe. The multiphase flow may be classified as adiabatic or diabatic, i.e., without or with heat addition at the channel wall (Figure 1.3). An example of adiabatic flow would be oil/gas flow in a pipeline, or air/water flow. In these cases the flow patterns would change as the inlet mass flow rates of the gas or liquid are altered or as the velocity and void distributions develop along the channel. Boiling would not take place and phase change would only occur if in a one component multiphase flow (e.g., steam-water) the pressure decreases and flashing occurs. Examples of diabatic flow are to be found in the riser tubes of steam generators and boiler tubes in power plants or in the coolant channels between nuclear fuel elements in a boiling water reactor. Boiling occurs on the walls of the channels and the flow patterns change due to vapor production as one observes the flow downstream in the channel due to vapor production. This is an important difference between pool boiling and flow boiling; i.e., that the forced flow of the multiphase system causes flow pattern transitions at a given wall heat flux (or temperature) as the integral power deposited in the fluid increases as it flows along the channel. In all cases of multiphase flow the velocities of the phases are usually not equal. For example in a riser tube of a steam generator the vapor rises faster than the liquid due to buoyancy effects. One may term this velocity inequality as "slip" between the vapor and the liquid. The ratio of these velocities is called the "slip ratio". A better description of the phenomena is to consider it as a relative velocity difference , and experimentally investigate these differences. Flow boiling heat transfer between the phases, can occur under two different boundary conditions, either a specified wall heat flux or a specified wall temperature. The former case is an idealized example of a boiler tube in a fossil fuel boiler and the latter case is an idealized example of a riser tube in a steam generator.

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Condensation

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1.5. Condensation Condensation is the reverse process of evaporation; i.e., the process of conversion of vapor back into liquid. It occurs when a wall (or the vapor near a wall) is cooled to a temperature which is below the saturation temperature corresponding to the vapor pressure. This temperature is commonly called the dew point. A common example of condensation occurs when steam condenses on the walls of the shower in a bathroom (Figure 1.4). Initially the water which condenses nucleates as droplets on the cold wall. As the population of these droplets grow or the rate of condensation increases the droplets coalesce into a film which flows down the wall. The first type of condensation is termed "dropwise" condensation and the second is termed "film" condensation. When the rate of condensation is low (e.g., a noncondensible gas is present) or when the liquid does not "wet" the wall, dropwise condensation occurs. In most engineering components where condensation is a required part of an industrial process film condensation is expected, because of the large mass flux of condensed liquid per unit length of wetted area. Consider the temperature distribution through the vapor and liquid during condensation as shown in Figure 1.4. The temperature decreases in the vapor as one approaches the wall; appreciably if the vapor is superheated or if noncondensibles are present. There is a slight decrease at the liquid-vapor interface due to the difference in pressure driving the mass transfer. Also, there is a temperature drop in the liquid due to its thermal resistance to heat transfer. If one were to plot the heat transfer coefficient as a function of distance down the wall one finds an interesting behavior. Initially the heat transfer coefficient would decrease because more liquid accumulates as a film and acts as a thermal resistance. However, at some liquid flow-rate the film becomes wavy and eventually turbulent. This surface roughness actually reduces the thermal resistance in the flowing liquid film. At this point and beyond the heat transfer increases as the liquid flow increases, with the Prandtl number having a second-order effect. There are other effects which will be discussed later.

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FLOW PATTERNS AND FLOW-PATTERN TRANSITIONS

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2. FLOW PATTERNS AND FLOW-PATTERN TRANSITIONS Multiphase flow is characterized by the existence of interfaces between the phases and discontinuities of associated properties. Single-phase flow can be classified according to the external geometry of the flow channel as well as the 'character' of the flow; i.e., laminar - following streamlines, or turbulent - exhibiting fluctuations and chaotic motions. In contrast multiphase flow is classified according to the internal phase distributions or "flow patterns" or "regimes". For a two-phase mixture of a gas or vapor and a liquid flowing together in a channel, different internal flow geometries or structures can occur depending on the size or orientation of the flow channel, the magnitudes of the gas and liquid flow parameters, the relative magnitudes of these flow parameters, and on the fluid properties of the two phases. A wide variety of multiphase flow patterns has been observed and identified in the literature. For example, Rouhani and Sohel (1983) cited a survey which suggested 84 different flow-pattern labels used in the literature. These variations are due to the subjective nature of flow-pattern definitions and partly to a variety of names given to essentially the same geometric flow patterns. In reality one finds a few basic flow patterns with associated transitions. The rate of exchange of mass, momentum and energy between gas and liquid phases as well as between any multiphase mixture and the external boundaries depends on these internal flow geometries and interfacial area; hence is dependent on flow-pattern. For instance, the relationships for pressure drop and heat transfer are likely to be different for a dispersed flow consisting of bubbles in a liquid (bubbly flow) than for a separated flow consisting of a liquid film on a channel wall with a central gas core (annular flow). This leads to the use of flow-pattern dependent models for mass, momentum and energy transfer, together with appropriate flow-pattern transition criteria. Given the existence of any one pattern, it is possible to model the two-phase flow field and to select a proper set of flow-pattern dependent equations so as to predict the important process design parameters. However, the central task is to predict which flow-pattern will exist under any set of operating conditions as well as to predict the value of characteristic fluid and flow parameters (e.g., bubble or droplet size) at which the transition from one flow-pattern to another will take place. Therefore, in order to accomplish a reliable design of gas-liquid systems such as pipe lines, evaporation and condensers, a prior knowledge of the flow-pattern is required. The common practice in representing flow-pattern data is to classify the flow patterns by visual or other means and to plot the data as a two-dimensional flow-pattern map in terms of particular system parameters. Beginning with the early flow-pattern maps, such as those proposed by Bergelin and Gazley (1949) and Kosterin (1949), there has been an on-going discussion on the selection of appropriate pairs of parameters to represent the flow-pattern transition boundaries. There is a wide variety of flow-pattern maps for two-phase flow in channels. A large number of different parameters have been used to present the experimental data in two-dimensional coordinates based on the superficial velocities, and or and . A good example are the empirical flow regime maps by Baker (1955) for horizontal flow and Hewitt and Roberts (1969) for vertical flow, which are illustrated in Figures 2.1 and 2.2. Note that both of these maps are for small tubes; i.e., on the order of centimeters. Some investigators have attempted to correlate the transition boundaries by a single pair of non-dimensional groups, in the hope that the results would apply to pipe sizes, and fluid and flow parameters other than those of the experimental data used to locate curves. Tables 2.1, 2.2, 2.3 and 2.4 present suggestions of these parameters together with specific references for their applicability in horizontal and vertical pipe flows. The main task is to group together the basic flow structures and define a few basic patterns. This is by no means well defined and indeed many flow patterns exist as individual researchers group the flow patterns somewhat differently depending on their own interpretations. Basically the attitude of most practitioners is to minimize the number of flow pattern groups and to group together the flow structures that has basically the same character pertaining to the distribution of the interfaces. An example of this approach is given in multifluid hydrodynamic models such as TEXAS (Chu, 1986, 1996) where only a bubbly flow or droplet flow is considered for large flow channels (See Figure 2.3.). As noted in the introduction dispersed flow patterns and stratified flow patterns are the two major classifications, with transitional patterns identified between these two groups.




References

J.N. Al-Sheikh, D.E. Saunders and R.S. Brodkey, "Prediction of Flow Patterns in Horizontal Two-Phase Pipe Flow," Can J Chem Engng, 48, pp 21-29, 1970. G.E. Alves, "Cocurrent Liquid-Gas Flow in a Pipeline Contractor," Chem Engng Prog, 50, pp 449-456, 1954. D. Baker, "Simultaneous Flow of Oil and Gas," Oil and Gas J , 53, pp 183-195 1954. O.P. Bergelin, and C. Gazley, Jr., "Cocurrent Gas-Liquid Flow, I. Flow in Horizontal Tubes," Proc Heat Transfer and Fluid Mech. Inst., pp 5-18, Berkeley CA, 1949. C.C. Chu, "TEXAS-A 1-D Model for Fuel-Coolant Interactions," UWRSR-36, University of Wisconsin, 1986. C.C. Chu et al, "TEXAS Computer Code Manual," http://www.engr.wisc.edu, 1996. B.A. Eaton, D.E. Andrews, C.R. Knowles, I.H. Sillberberg and K.E. Broon, "The Prediction of Flow Patterns, Liquid Holdup and Pressure Losses Occurring During Continuous Two-Phase Flow in Horizontal Pipelines," J. Petrol Technol , 19, pp 815-828, 1967. W.C. Galegar, W.B. Stovall and R.L. Huntington, "More Data on Two-Phase Vertical Flow," Petroleum Refiner , 33, pp 208-217, 1954. G.W. Govier, and K. Aziz, "The Flow of Complex Mixtures in Pipes," Van Nostrand Reinhold, New York, 1972. G.F. Hewitt and D.N. Robertson, "Studies of Two-Phase Flow Patterns by Simultaneous X-ray and Flash Photography," Rept AERE-M2159, UKAEA, Harwell, 1969. C.S. Hoogendorn, "Gas-Liquid Flow in Horizontal Pipes," Chem Engng Sci, 9, pp 205-217, 1959. C.J. Hoogendorn, and A.A. Beutelaar, "Effect of Gas Density and Gradual Vaporization on Gas-Liquid Flow in Horizontal Lines," Chem Engng Sci, 16, pp 208-215, 1961. H.A. Johnson, and A. H. Abou-Sabe, "Heat Transfer and Pressure Drop for Turbulent Flow of Air-Water Mixtures in a Horizontal Pipe," Trans ASME , 74, pp 977-987, 1952. C.R. Knowles, The Effects of Flow Patterns of Pressure Loss in Multiphase Horizontal Flow," M.S. Thesis, The University of Texas, Austin, 1965. S.I. Kosterin, "An Investigation of the Influence of Diameter and Inclination of a Tube on the Hydraulic Resistance and Flow Structure of Gas-Liquid Mixtures," Izvest. Akad , Nauk. USSR, 12, pp 1824-1830, 1949. B.K. Kozlov, "Forms of Flow of Gas-Liquid Mixtures and Their Stability Limits in Vertical Tubes," Zhur Tech Fig 24, pp 2285-2288, 1954. L.Y. Krasiakova, "Some Characteristics of Movements of Two-Phase Mixtures in a Horizontal Pipe," Zhur Tech Fig 22, p 656, AERE Lib/Trans 695, 1952. J.M. Mandhane, C.A. Gregory and K. Aziz, "A Flow Pattern Map for Gas-Liquid Flow in Horizontal Pipes," Int J Multiphase Flow, 1(4), pp 537-554, 1974. K. Nishigawa, K. Sekoguchi and T. Fukano, "On the Pulsation Phenomena in Gas-Liquid Two-Phase Flow," Bull. JSME, 12, pp 1410-1416, 1969. T. Oshinowa and M.E. Charles, "Vertical Two-Phase Flow, 1: Flow-Pattern Correlations", Can J Chem Engng, 52, pp 25-35, 1974. S.Z. Rouhani and M.S. Sohel, "Two-Phase Flow Pattern: A Review of Research Result," Progress in Nuclear Energy, 11, pp 217-259, 1983. H.H. Schicht, "Flow Patterns for Adiabatic Two-Phase Flow of Water and Air Within a Horizontal Tube", Verfahrenstecknik , 3, pp 153-172, 1969. D. Scott, "Properties of Co-Current Gas-Liquid Flow," Advances in Chemical Engineering , 4 pp 199-277, Academic Press, New York, 1963. H.C. Simpson, D.H. Rooney, E. Gratton and F. Al-Samarrel, "Two-Phase Flow in Large Diameter Horizontal Lines," Paper H6, European Two-Phase Flow Group Meeting, Grenoble, 1977. P.L. Spedding, and J.J.J. Chen, "A Simplified Method of Determining Flow Pattern Transition of Two-Phase Flow in a Horizontal Pipe," Int J Multiphase Flow , 7(6), pp 729-731, 1981. P.L. Spedding, and V.T. Nguyen, "Regime Maps for Air-Water Two-Phase Flow," Chem Eng Sci, 35, pp 779-793, 1980. T. Ueda, "Studies of the Flow of Air-Water Mixtures," Bull. JSME 1(2), pp 139-145, 1958. J. Weisman, D. Duncan, J. Gibson and T. Crawford, "Effects of Fluid Properties and the Pipe Diameter on Two-Phase Flow Pattern in Horizontal Lines," Int. J. Multiphase Flow, 5, pp 437-452, 1979. J. Weisman and S.Y. Kang, "Flow Pattern Transitions in Vertical and Upwardly Inclined Lines," Int J Multiphase Flow , 7, pp 271-280, 1981. P.D. White, and R.L. Huntington, "Horizontal Co-Current Two-Phase Flow of Fluid in Pipe Lines," The Petroleum Engineer , 27(9), p 40,


1958.

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MULTIPHASE FORMULATION AND PRESSURE DROP

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3. MULTIPHASE FORMULATION AND PRESSURE DROP The focus of this section is to introduce the reader to the modelling of multiphase flow in general, liquid-gas flows in particular and the prediction of pressure drop and void specifically. The main objective is to provide the reader with the basic fundamentals that are needed to formulate the balance equations for multiphase flow as well as the associated constitutive relations and then employ them in the practical prediction of pressure drop in a closed conduit as the first example of their usefulness. As in other sections this is a diverse subject and more detailed discussions of these topics are provided by a number of individuals; e.g., Collier, 1981; Wallis, 1979; Bergles et al., 1981; Hsu and Graham, 1977; and Ishii, 1974. Once the concept of flow patterns for multiphase flow has been introduced, one must develop the governing equations which account for the conservation of mass and energy and the transfer of momentum. Consider the case of steam-water flow in a vertical tube with heat addition at the wall boundary (Figure 3.1). In reality the phases are not exactly in thermodynamic and mechanical equilibrium; i.e., if the channel geometry is variable and the flow rate or heat addition rate is changing rapidly with time or space, then the velocity, temperature and pressure of the two-phases are not necessarily the same at a given spatial position in the channel. For certain conditions one may be able to model the multiphase flow and assume that some or all of these potential variables may be equal between the phases. To cover a broad range of applications a number of models have been developed. We consider three general types of multiphase flow models, the homogeneous flow model, separated flow model and two fluid model. Table 3.1 gives a brief summary of important dimensionless groups that can be used to determine model applicability.

3.1. Homogeneous Equilibrium Model 3.2. Separated Flow Model 3.3. Two-Fluid Model 3.4. Pressure Drop in a Closed Conduit 3.5. Void Fraction Prediction with the Drift Flux Model

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MULTIPHASE FORMULATION AND PRESSURE DROP



Homogeneous Equilibrium Model

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3.1. Homogeneous Equilibrium Model In the homogeneous equilibrium model (HEM) one assumes that the velocity, temperature and pressure between the phases or components are equal. This assumption is based on the belief that differences in these three potential variables (and chemical potential if chemical reactions are considered) will promote momentum, energy, and mass transfer between the phases rapidly enough so that equilibrium is reached. For example, when one phase is finely dispersed in another phase generating large interfacial area, under certain circumstances this assumption can be made; e.g., bubbly flow of air in water or steam in water at high pressures. The resulting equations resemble those for a pseudo-fluid with mixutre properties and an equation of state which links the phases to obtain these mixture thermodynamic properties. Whenever the HEM model is used it is advisable to check the validity of the equilibrium assumptions by using more accurate theoretical models for comparison. For example, rapid acceleration or pressure changes cannot be always accurately modelled with the HEM model; i.e., discharge of flashing vapor-liquid mixtures, or shock wave propagation through a multiphase medium. This is especially true when the pressure change is large when compared to the ambient pressure, or any of the driving potentials are large relative to their reference values. Such a 'rule-of-thumb' is very crude and one must carefully consider the timescales for equilibration of these driving potentials with allowable characteristic times for the problem of interest. The governing equations for the HEM model are presented in Table 3.2 where the geometry has been chosen to be a one-dimensional channel inclined from the horizontal by a known angle, , (Figure 3.1). The microscopic equations have been averaged over the channel cross-sectional area using the techniques first proposed by Ishii (1974), leaving a partial differential equation in time, t , and the axial space dimension, z. The definitions of the mixture thermodynamic properties (e.g., , u, v) consider only two-phases but can be simply extended to more phases or components. The extension to more than one-dimension is not as straight forward and one is referred to the general formulation of Bird, et al. (1960). Note that the inclusion of the mixture thermodynamic properties can be followed from Table 3.2. The multiphase transport properties of viscosity and thermal conductivity (

, k) are another matter,

because it is not clear how one should average their effect in an area-average, mass average or volume-average sense. In many situations such as for pressure drop calculations the mixture transport properties have been arbitrarily averaged on a volume average or mass average basis, e.g.,

However, these averaging schemes are not exact and are usually empirically corrected by fitting coefficients to a set of experimental data. In other situations the effect of multiphases are neglected and the liquid or gas property values for viscosity on the thermal conductivity are used, e.g., when the amount of liquid in the channel is large (low quality or void fraction), the viscosity can be taken to be that of the liquid.

Homogeneous Equilibrium Model

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Separated Flow Model

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3.2. Separated Flow Model In the separated flow model the restriction on equal phase velocities is relaxed and one now models the momentum exchange between the phases and the channel separately with different velocities, e.g., vapor and liquid velocities. The relaxation of equal velocities is most important when the densities between the phases are quite different in the presence of a gravitational potential field or large pressure gradients. Given a density difference, buoyancy effects tend to induce a drift velocity of the lighter phase in the heavier phase. One measure of this density ratio is the Atwood ratio and is defined in Table 3.1. One notices that as this density ratio approaches zero the HEM model would become more valid because the drift velocity would be reduced as the buoyancy of the lighter phase diminishes. The remaining assumptions of equal temperatures and pressures between the phases are usually retained in most applications. This is because it is usually felt the rates of mass and energy exchange are large enough to promote equilibrium. Once again a check with a more detailed model is recommended as the analysis proceeds to verify this assumption. The governing equations for the separated flow model are given in Table 3.3. Similar to Table 3.2 we use the 1-D area averaged formulation for two phases. There are two important differences in the equations that one should notice. First, there are now two momentum equations. In each equation there appears a term which represents the friction force at the phase interface caused by the relative velocity between the phases. If the equations are solved separately then one must develop a constitutive relation model for this momentum transfer term. Second, the properties are not averaged exclusively using the void fraction and density of the phases, but require a separate constitutive relation (Eq. 10 in Table 3.3) that relates the volume fraction to the flowing mass fraction. Traditionally the separated flow model has been used primarily for calculating the pressure drop in a flow channel. In this application the usual method of solution is to add the phase momentum equations and eliminate the need to model the interfacial shear stress, . Then one empirically correlates to obtain a model for the frictional pressure drop for the channel,

, and for the slip ratio,

, or velocity differences

function of volume fraction and properties. The model for momentum equation and the algebraic correlation for

between the phases as a

is substituted back in the combined or

is used as a substitute for the

second balance equation. These types of models are described in more detail when one considers multiphase pressure drop. The drift flux model is a special case of such models, because it is physically based as it predicts void fractions given velocity difference.

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Separated Flow Model



Two-Fluid Model

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3.3. Two-Fluid Model The final type of multiphase model is the multiple fluid model (better known as the two fluid model designating two phases or components). This model treats the general case of modelling each phase or component as a separate fluid with its own set of governing balance equations. In general each phase has its own velocity, temperature and pressure. The velocity difference as in the separated flow is induced by density differences. The temperature differences between the phases is fundamentally induced by the time lag of energy transfer between the phases at the interface as thermal equilibrium is reached. If the multiphase system involves rapidly changing flow conditions due to area changes in steady flow or transient conditions then the time lag for reaching thermal equilibrium between the phases may become significant in comparison to the characteristic time it takes for flow conditions to change. One may estimate this condition by computing a characteristic Fourier number (Table 3.1) for the system under expected flow conditions. Therefore, thermal nonequilibrium becomes important and one must include the possibility of a temperature difference by separate energy balances in a multiphase model for two or more separate fluids. The modelling of pressure nonequilibrium is much more complex (Ishii, 1974). The pressure difference between two phases is caused by three main effects: (1) pressure differences due to surface energy of a curved interface,(2) pressure differences due to mass transfer, (3) pressure differences due to dynamic effects. In the first case the simple existence of an interface (probably curved) requires from overall mechanical equilibrium that some pressure difference exist between the phases. This pressure difference is proportional to the interfacial surface tension and inversely proportional to the radius of curvature and is usually quite small in most applications ( m). The second effect is noticeable when the mass flux due to phase change is large at the interface between the phases; e.g., large evaporation or condensation rates. The final effect is caused by dynamics where one phase has a larger pressure relative to the other phase due to very rapid energy deposition or pressurization effects. A common example of an nduced dynamic pressure difference is the flow of a mixture of air bubbles and water through a converging-diverging nozzle. If the rate of flow is high and the area change dramatic enough the liquid will depressurize quickly as it passes through the nozzle leaving the vapor bubbles at a higher pressure. This dynamic pressure difference will cause the vapor bubbles to grow, overexpand and then oscillate around a new mean pressure. This example takes on the second effect if the situation were steam bubbles in water since mass transfer would also be present. The importance of pressure nonequilibrium between the phases is inversely proportional to the time scale of the rate of phase change or external pressure oscillations. For most applications of two-fluid modelling this pressure nonequilibrium is usually neglected; i.e., only when the rate of phase change and pressure oscillation become of equal time scales does this nonequilibrium effect become important. One way to estimate this is to compare the flow velocity to the speed of sound in the multiphase system (note that computing a mixture sound speed is not a straight forward task): i.e., only when the flow velocity approaches or exceeds the multiphase sound speed would the pressure nonequilibrium may be important.

Two-Fluid Model

The two-fluid model equations are given in Tables 3.4. One should note that when a two-fluid model is used a number of interfacial transport coefficients (

,

,

,

) are defined and require constitutive

relation models to complete the overall model. This approach has an advantage in that the actual transport processes can be rigorously defined, however, the disadvantage is that one is required to model these kinetic processes in detail, which implies a much greater depth of experimental data and insight. The usual method of modelling pressure differences between the fluids is to assume that the pressure is equal in both phases (Tables 3.4, Eq. 6). If, as previously discussed, one finds that pressure nonequilibrium between the phases is important one must introduce a local constitutive relation which accounts for this pressure difference due to dynamic and interfacial effects. For example, in research done with explosive boiling a local momentum equation (e.g., Rayleigh momentum equation) is used to model this difference in the pressure of the two fluids; this allows for dynamic pressure differences as well as the effect of surface tension and mass transfer. The other required constitutive relations for interfacial transfer (

,

,

) are complicated functions of

the fluid velocities and their local properties. These kinetic models are also a strong function of the multiphase flow pattern. The model one would develop for the interfacial shear stress or heat flux is significantly different for a dispersed flow pattern in contrast to a stratified flow pattern. In fact, the interfacial models would be different if one had gas bubbles in a liquid versus liquid droplets in a gas. For example in the former case one would find that the stable characteristic size of gas bubbles at low void fractions might be near the Taylor critical wavelength (dimensionless length-D/A 2 , Table 3.1) where as in the latter case the diameter would be determined from the critical Weber number (We 7-12, Table 3.1) which is only partly related to Taylor instabilities. The final point to make about all the multiphase models is how turbulence is included in the analysis. The first point one should notice is that the multiphase governing equations do not seem to directly include the time-averaging due to local turbulent velocity fluctuations. This is somewhat misleading because when one looks into the complete derivation as performed by Ishii (1974) one finds that and inherently include turbulence effects. The important question then is constitutive relations for how is turbulence modelled in these relations. At the present time turbulence modelling is rather phenomenological when compared to the detailed formulations for single phase flow. The inherent assumption in modelling and is that one can use simple turbulence models (e.g., empirical friction factors, mixing length scales) developed for single phase applications at the local level of the multiphase system and then correct for effects of multiple phases by a combination of phenomenological models averaging techniques for the bulk flow, and using empirical correlations from specific data. The following sections considering pressure drop and critical flow models are good examples of these techniques. More fundamental approaches to include turbulence have begun, but are not discussed in detail here.

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Two-Fluid Model



Pressure Drop in a Closed Conduit

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3.4. Pressure Drop in a Closed Conduit Once the governing equations have been cast the next task involves developing models for the constitutive relations that predict the rate of mass, momentum and energy transfer from one phase to another or from the phases to the surrounding channel. This task is the major endeavor for multiphase flow analysis. If one is concerned with predicting the pressure drop in a channel with multiphase flow then one must develop a model for wall shear stress, , and also a relation for the relative velocity, between the phases if a separated flow or two-fluid model is used. In engineering practice the current method for modelling multiphase pressure drop falls into two categories: (1) Homogeneous models where a relation for and is developed empirically, or (2) Two-fluid or drift flux models where one uses the separated flow model for

and substitutes the empirical relation for

by the complete

solution of each phase momentum equation. Consider the steady-state form of the momentum equation for the HEM model from Table 3.2 (Equ 3)

where in this equation we have assumed the channel flow area is constant and we have substituted the constant mass velocity, G, from the mass balance into the acceleration pressure drop term. Note that this equation is not valid for the case of multiphase flow of a liquid and solid where the solid adheres to the wall. Now the model used for is based on the assumption that we have liquid flowing alone, fo, in the channel and correct for this by using a two-phase multiplier, , (this technique can be extended to more than two phases in a straight forward manner),

where the channel can be represented by the hydraulic diameter, two-phase multiplier,

. By inspection one sees that the

, is equal to

where the first phase is a liquid, f , and the second phase is a gas, g. Now consider that the two-phase friction factor, has the same empirical function form as single phase, and this yields


where one now must use an averaging scheme for the transport property viscosity (e.g., see Eq. 9 for ). The value of the exponent, m, would depend whether one was in a laminar or turbulent flow regime. When these constitutive models are substituted into the original momentum balance the total pressure gradient results (Table 3.5). One should remember that the mass fraction is a thermodynamic property and therefore is a function of two other independent properties, here we have chosen the mixture enthalpy, i, and pressure P, because they are the 'natural' variables calculated for a steady flow. This dependence of on i and P results in partial derivatives with respect to enthalpy and pressure in the acceleration pressure drop term. The enthalpy dependence is found from the energy balance while the pressure effect is directly found by the momentum equation by back substitution. The final observation that one should make concerns the term in the denominator of the total pressure gradient. If one evaluates the quantity in brackets (say for steam-water flow) the result indicates that the value lies between zero and one. Consider the case when the term approaches zero. In this case the change in specific volume (density) as a function of pressure is negligible. This suggests that one can treat the multiphase system as an incompressible fluid. Now as the term in brackets approaches unity the denominator approaches zero and the pressure gradient approaches infinity. Physically when the pressure gradient approaches infinity it implies the existence of shock wave or acoustic wave (weak shock). In the present example the only way this is created is because the variation of the density with pressure is large, this implies the flow , is traveling fast enough that it has equaled the speed of pressure perturbations in the fluid mixture, i.e., the 'sound speed' of that homogeneous mixture . Thus, the HEM model Mach number is approximately given by the term in brackets and has the same physical significance as in single phase flow (see Table 3.1). This is detailed in the following section. For the separated flow model, the original definition for the two-phase friction multiplier is used except now one replaces the analytical model for by an empirical correlation based on data. Probably the most widely used model is that by Martinelli and Nelson ( Figure 3.2 and Figure 3.3) in which pressure gradients for air-water flows at low pressure and steam-water flows at high pressure were correlated as a function of flow regime (e.g., turbulent flow) as well as properties. The resulting correlation for the two-phase friction multiplier, , is given in Figure 3.2 as a function of the dimensionless Martinelli parameter,

, where

Also the void fraction of the gas phase was correlated as a function of the Martinelli parameter (Figure 3.3). In this case one would use this empirical result along with the basic constitutive relation for , and

(Table 3.3, Eq. 10) to find the velocity ratio.

Finally, in a similar manner one would determine the dimensionless Mach member for the separated flow model. Table 3.6 presents the complete pressure gradient term for the separated flow model. Since the


void fraction is a function of the velocity difference as well as the quality, both void and quality appear as function of the enthalpy and pressure at any location in the channel. The quality variation is found by the energy balance and the void fraction must be obtained by a separate model or empirical data (such as Martinelli's data). We consider a phenomenological void model in the next section, that can be used in separate flow models; i.e., drift flux model. As for the two-phase friction multiplier, a number of investigators have developed a data-base and associated correlations (e.g., Thom, 1984, Barozcy, 1966, Chisholm, 1973, and Friedel, 1979). Based on a comparison to a proprietary data bank of HTFS, it was found that : 1. The Homogeneous Equilibrium model is moderately acceptable over a wide-range of data comparisons as an initial estimator. 2. The Martinelli-Nelson correlation is a better predictor and should be used for

and

. 3. The Chisholm correlations should be used for 4. The Friedel correlation should be used for

and

.

.

A recommendation of which model is most suitable depends upon the application. The important consideration that one must remember is the relative density ratios (Atwood ratio) and the influence of the gravitational potential field (Froude number) on causing a drift or allowing a relative velocity to exist between the phases. If these differences are large then one should separate the flow model. For example, for air-water flows at ambient pressure the density ratio is large ( , Atwood Ratio 1) therefore a separated flow model may be dictated. As the density ratio approaches one then a homogeneous model becomes more appropriate for a wide range of applications.

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Void Fraction Prediction with the Drift Flux Model

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3.6. Void Fraction Prediction with the Drift Flux Model As noted in the previous section, when the two phases are considered to have different velocities (e.g., liquid and gas), the relation between void fraction and quality is not analytically calculable, but requires some empirical data which links void and quality. A large number of empirical and semi-empirical methods have been suggested over the last fifty years. The semi-empirical model which seems to have the most physical basis is the drift flux model. It relates the gas-liquid velocity difference to the drift flux (or 'drift velocity') of the vapor relative to the liquid; e.g., due to buoyancy effects. This model has been principally developed by Zuber and Findlay (1965), Wallis (1969) and Ishii (1977), and has been refined since that time by themselves and coworkers. The model is fully developed in Wallis' book and only the essential relationships are presented here. In all two-phase flows, the local velocity and local void fraction vary across the channel dimension, perpendicular to the direction of flow. To help us consider the case of a velocity and void fraction distribution (possibly different) it is convenient to define an average and void fraction weighted mean value of local velocity, v. Let F be parameters, such as any one of these local parameters, and an area average value of F across a channel cross-section would be given as

A void fraction weighted mean value of F may be defined as follows:

Now consider the gas velocity,

, that can vary across the channel. An expression for the weighted mean

gas velocity may be expressed as

Taking now a reference frame moving with the velocity of the center of the volume of the fluid one can define the gas or vapor phase drift velocity by

where j is the volumetric flux of the two-phase flow. This is the velocity of the center of volume of the mixture, and is given by


where

and

are the volumetric flux of the liquid and gas phases, respectively, given by

Using Eqs. 3.10 and 3.13 in Eq. 3.9, the weighted mean velocity of the gas or vapor phase can be expressed as:

where

is the weighted mean drift velocity of the gas phase, and Co is the

distribution parameter defined by

Therefore, Co depends on the form of the velocity and concentration (flow-pattern) profiles. For a given flow pattern, the extensive study of Zuber and Wallis suggest that Co depends on pressure, channel geometry and perhaps the flow rate. The form of Eq. 3.15 suggests a plot of the data in this plane representation. First, when Co and between

plane. There are two important characteristics of

are constant Eq. 3.15 shows that a linear relation exists

and ;SPMlt ; j;SPMgt ;. The slope of such a line gives the value of distribution parameter, Co,

whereas the intercept with the

axis gives the drift velocity,

. Thus, these two parameters can be easily

determined from experiments even when the measured profiles required for the direct calculation of Co are not available. The second important characteristic of the plane pertains to any abrupt measured changes in the value of Co and

; these indicate a change of flow-pattern in the conduct of the

experiment. The average void fraction

at a given location in the flow channel can now be obtained by

rearranging Eq. 3.15 giving:

This equation shows that

can be determined if Co,

and the average gas or vapor volumetric flux

are known for a given flow regime; i.e., bubbly slug churn-turbulent. Noting that

and Co are

flow-pattern dependent quantities then any void fraction predictions based on Eq. 3.17 would reflect the flow-pattern effects on the void fraction. Suggested expressions for Co and

and other flow-pattern dependent void fraction correlations are


presented in Table 3.7 based on the work of these investigators.




<>P


REFERENCES q

C.J. Barozcy, "A Systematic Correlation for Two-Phase Pressure Drop," Chem. Engr. Proj. Symp, Vol 62, No 44, pp 232-249, 1966.

q

A.E. Bergles, et al., Two Phase Flow and Heat Transfer in the Power and Process Industries, Hemisphere, New York, 1981.

q

R. Bird, et al., Transport Phenomena, Wiley, New York, 1960.

q

D. Chisholm, "The Pressure Gradient Due to Friction During the Flow of Boiling Water," Engr & Boiler House Rev, Vol 73, No 8, pp 252-256, 1966.

q

J.G. Collier, Convective Boiling and Condensation, McGraw-Hill, 2nd Edition, New York, 1981.

q

L. Friedel, "Improved Friction Pressure Drop Correlations for Horizontal and Vertical Two-Phase Pipe Flow," Proc European Two-Phase Group Mtg. , JRC-Ispra, 1979.

q

Y.Y. Hsu, H. Graham, Transport Processes in Two-Phase Boiling Systems, Hemisphere, New York, 1977.

q

M. Ishii, Thermo-Fluid Dynamic Theory of Two-Phase Flow, Eyrolles, France, 1974.

q

M. Ishii, "One-Dimensional Drift-Flux Model and Constitutive Equations for Relative Motion Between Phases in Various Two-Phase Flow Regimes," Argonne National Lab Report, ANL 77-47 , October 1977.

q

M. Ishii, N. Zuber, "Drag Coefficient and Relative Velocity in Bubbly, Droplet or Particulate


Flows," AIChE Jul, Vol 25, No. 2, p 843, 1979. q

q q

J.R.S. Thom, "Prediction of Pressure Drop During Forced Circulation of Boiling Water," Intl Jul Heat Mass Transfer, Vol 7 , pp 709-724, 1964. G.B. Wallis, One-Dimensional Two-Phase Flow, McGraw Hill, 2nd Edition, New York, 1979. N. Zuber, J. Findlay, "Average Volumetric Concentration in Two-Phase Systems," Trans ASME Jul Ht Transfer, Vol 87 , p 453, 1969.

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SOUND SPEED AND CRITICAL FLOW

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4. SOUND SPEED AND CRITICAL FLOW Critical flow (sometimes called "choked flow") is important for many industrial applications involving multiphase flow. These applications include the design of throttling valves for refrigeration systems, bypass systems for steam power plants, emergency relief in chemical and nuclear plants and other diverse applications. Most of the research and development in the recent past has been in regard to plant safety analyses. Our discussion of critical flow is based on experimental and model results with this latter subject as the main application, although the models are completely general and applicable to other design issues. One of the observations made in examining multiphase pressure drop was that one could determine if the multiphase flow could be considered compressible or incompressible by examining the acceleration terms in the pressure gradient expression. For instance, in the HEM model for the pressure gradient it was noted that the equation denominator could be considered as where M is a local two-phase flow Mach number; i.e., ratio of mass velocity to a two-phase "acoustic" mass velocity. Analogous to single phase flow, as the Mach number approaches unity the flow must be considered compressible and the overall pressure gradient increases; asymptotically approaching an infinite gradient (shock wave) as the Mach number equals one. Thus, as in single phase flow, there is a relationship in this multiphase flow model between the maximum mass flux through a duct and the local sound speed. This maximum mass flux is identified as the critical flow (or choked flow") of the system, and is the major topic of this section. These notes will begin with a discussion of single phase sound speed and critical flow ("choking"). This logically leads us to consider the homogeneous equilibrium model for sound speed and for the multiphase critical flow. Finally, more sophisticated models considering nonequilibrium effects and the effects of phase velocity differences are discussed. We end the paper with a discussion of these models assumptions and recommendations for usage. In this development, as in the discussion of pressure drop, we assume the flow is approximately one-dimensional. This simplifying assumption allows us to develop analytical expressions and is approximately valid in realistic situations if the change in flow area is not abrupt. Many practical applications can involve at least two-dimensional flow. At worst the one-dimensional assumption affords one a way to estimate behavior in more complex multi-dimensional systems to bound the expected behavior.

4.1. Single Phase Critical Flow 4.2. Homogeneous Equilibrium Model for Critical Flow 4.3. Homogeneous Frozen Flow Model for Critical Flow 4.4. Separated Flow Model for Critical Flow

SOUND SPEED AND CRITICAL FLOW

4.5. Fluids Other Than Water For Separated Flow Model 4.6.The Physical Basis of Choked Two Phase Flow 4.7. Advanced Computational Models 4.8. Observations

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Single Phase Critical Flow

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4.1. Single Phase Critical Flow Consider a large tank containing a stagnant fluid with outlet ( Figure 4.1) to a receiver volume. When a compressible fluid flows through this outlet (pipe, nozzle or orifice), there is a receiver pressure, , compared to the stagnation pressure,

, below which the flow rate (or mass velocity) will not increase.

That is, for any receiver pressure below this critical pressure ratio the discharge flow rate will be constant (Figure 4.2). If one could make simultaneous measurements of the pressure, temperature, and mass flowrate at the outlet of the tank, one would find that the velocity of the mass flow corresponds to the sound speed of the fluid at the outlet exit plane. If one then calculates what the mass velocity will be for an isentropic expansion in a nozzle, the mass velocity is a maximum at just the critical pressure ratio, , at which the fluid attains a velocity equal to its sound speed at these conditions at the outlet exit plane. Under the assumption of isentropic fluid flow (frictionless and adiabatic), this is proved by calculating the pressure ratio when the mass velocity is a maximum, determining the fluid velocity and comparing it to the speed of sound in the fluid [1-2]. Similar calculations can be made for other geometries or expansion processes and it is always found that choking occurs when the local fluid velocity is equal to the local sound speed [3]. Insofar as the flow is isentropic the critical flow velocity like the sound speed is a thermodynamic property. The sound speed, c, is defined as

where the specific entropy, s, is considered to be constant in the process. Let us consider a perfect gas as a representative single phase fluid where

Based on the sound speed definition we find that for a perfect gas

Now we can use this definition of the sound speed and incorporate it into the energy balance with mass


continuity for the fluid to find the mass flowrate as a function of the Mach number.

where the subscript, o, denotes the stagnation condition where the velocity is approximately zero. This shows that, for a given Mach Number, the flow is proportional to the stagnation pressure and inversely proportional to the square root of the stagnation temperature. For this reason, flow test data on compressors and turbines, or indeed on any flow passage which operates over a wide range of pressure and temperature levels, are usually plotted for

as the flow variable. In this way the results of a

given test become applicable to operation at levels of temperature and pressure different from the original test conditions. To find the condition of maximum flow per unit area, we could compute the derivative

and set this derivative equal to zero. From this we would find that the critical pressure

ratio is given by

An equivalent procedure would be to differentiate Eq. 6 with respect to M and set this derivative equal to zero. At this condition, we would find that the maximum flow occurs at M = 1. However, neither of these procedures is necessary inasmuch as we have discussed quite generally that the cross-sectional area for isentropic flow passes through a minimum at Mach Number unity. Therefore, to find

, we need only set M = 1 in Eq. 6. Thus we find

For a given gas, therefore, the maximum flow per unit area depends only on the ratio

. For given

values of the stagnation pressure and stagnation temperature and for a passage with a given minimum area, Eq. 8 shows that the maximum flow which can be passed is relatively large for gases of high molecular weight and relatively small for gases of low molecular weight. Doubling the pressure level doubles the maximum flow, whereas doubling the absolute temperature level reduces the maximum flow by about 30 percent. As Figure 4.2 indicates for a converging nozzle the Mach number reaches unity at the point where the critical pressure ratio,

, is given by equ IX.7 (

- .53 for air,

- 1.4), which also coincides with the

point of minimum area. This is the condition of critical flow or "choking" that was mentioned previously. One should note that the effect of friction and heat transfer can be considered but requires an iterative analytical solution or numerical solution to the one-dimensional problem depending on the complexity of the wall models, since the flowrate directly affects the frictional pressure drop. We consider this later under advanced models.


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Homogeneous Equilibrium Model for Critical Flow

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4.2. Homogeneous Equilibrium Model for Critical Flow In single phase flow the critical flow velocity is identical to the fluid sound speed at "choking" ( M = 1). In general it turns out that the identity of the critical flow velocity and the fluid sound speed is not valid for a multiphase flow. This occurs because the concept of a single sound speed for a mixture of phases loses meaning unless one makes some simplifying assumptions about the fluid phase relationships. There may actually be more than one sound speed; i.e. one for each phase and one for the mixture depending on the flow pattern and geometry. The homogeneous equilibrium model (HEM) for multiphase critical flow is one example where one can make simplifying assumptions and the identity between the critical flow velocity and a mixture sound speed is preserved. In our discussion we consider a two-phase system for illustration. The HEM model is based on two major assumptions: (i) the velocity of the phases are equal, and (ii) the phases are in thermodynamic equilibrium. These assumptions are the same as used for the HEM pressure drop model. In every respect, once these assumptions are made, the single phase gas dynamic relations for critical flow can be translated to multiphase conditions. Therefore, for isentropic flow conditions (frictionless and adiabatic) the pressure drop is due solely to acceleration pressure drop which results in

where is the specific volume of the mixture and the reciprocal of the mixture density for the HEM formulation. We have already evaluated this partial derivative when considering the HEM pressure drop as

where the partial derivatives are at constant enthalpy under frictionless conditions. One can use this expression to find the HEM mixture sound speed by multiplying by the mixture specific volume and taking the square root

Figure 4.3 illustrates how this sound speed varies with gas content for an air-water system at atmospheric pressure. Notice that the mixture sound speed falls dramatically to less than 100 m/s with a small amount of gas present. One must keep in mind that this is an idealized model and that transmission of pressure


waves may occur at higher speeds in the real system but at lower amplitudes, because the actual flow is not completely homogeneous and each phase has a characteristic sound speed which is higher. In using this expression to find the critical mass velocity it is necessary to evaluate the indicated partial derivatives at the outlet conditions (e.g., pressure, quality) where "choking" occurs. One must then work back from the outlet conditions to the stagnation conditions. Usually this involves an iterative procedure even though it is analytical, or numerical integration. A straight forward procedure is presented below for a two-phase liquid-vapor system. To find the critical mass velocity,

, and the associated critical pressure ratio,

which it occurs one could produce the complete curve of

, at and below

(Figure 4.4). Since the flow is

isentropic the entropy at the outlet equals the stagnation entropy. Thus one chooses a downstream outlet pressure, , which determines the saturation entropies, and . The equilibrium thermodynamic quality at the outlet is then

which then defines the enthalpy and specific volume at the outlet

The mass velocity for a particular pressure is found by an energy balance as

One simply reduces the outlet receiver pressure from

until the maximum G is calculated at

. For all

pressures below this critical pressure the flow is choked as is the case for single phase flow. One should note that is not a simple function of the thermodynamic state of the system as for a perfect gas and can vary widely for any particular set of stagnation conditions. The accuracy of the HEM model will be discussed later in comparison to some particular data. In general though, let us simply say that the HEM critical flow model has similar limitations as the HEM pressure drop model when the density ratio is large (low pressure when relative velocity is important) and also at low quality (when the nonequilibrium effect is important). As the stagnation pressure and quality increases the HEM prediction improves. Also as the outlet pipe length to diameter ratio increases the HEM model prediction improves, allowing more time for attainment of equilibrium; however one must then include the effect of pipe friction. One should also remember that the stagnation conditions ought to be representative of a homogeneous geometry (dispersed well-mixed) for the HEM model to successfully estimate the outlet flow. If the initial stagnation condition is better represented by stratified phases rather than co-dispersed phases, one might expect the HEM model not to initially give a reliable estimate until flashing and its induced mixing disperses the phases. More likely depending on the location of the pipe


single phase flow would first occur followed by flashing and then the two-phase flow. For stratified conditions with short tubes nonequilibrium might be considered. A bounding nonequilibrium model is discussed in the next section.

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Homogeneous Frozen Flow Model for Critical Flow

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4.3. Homogeneous Frozen Flow Model for Critical Flow The previous HEM model is applicable to conditions where a liquid and its vapor flow through an outlet in equilibrium. If a homogeneous mixture of a liquid and a gas phase (air-water) flow through an orifice or if nonequilibrium effects are important for a liquid and vapor (due to stratification or a short tube), then mass transfer may be precluded. One might consider this to be a homogeneous flow model where the phases are "frozen" at mass fractions equal to their stagnation conditions. In a sense, given that the phases travel at the same velocity, one may consider this to be another bound on the critical flowrate. This type of model has appeared frequently in the literature (e.g., Ref. 4) as a "homogeneous frozen model" for critical flow, and is based on the following assumptions (i) the velocities of the phases are equal, (ii) there is no heat or mass transfer between the phases, (iii) the gas (or vapor) is modelled as a perfect gas, (iv) the critical flow is defined from gas dynamics and determines the velocity (specific kinetic energy).

These assumptions lead to a familiar expression for the critical pressure ratio

if the following inequality is valid

Under these assumptions one can write down the energy balance

where for the vapor as a perfect gas one can write

and thus the critical mass velocity becomes

Homogeneous Frozen Flow Model for Critical Flow

One can observe the difference between the two models as compared to steam-water data at high and low pressures [5,6] in Figures 4.5 to 4.7. In all cases the critical pressure ratio is underpredicted by this model but the critical mass velocity is better predicted at relatively low stagnation qualities. Both models are in error at very low qualities. The ability to predict the flowrate under these conditions is questionable for most models. As we shall discuss in the next section the final model assumption to consider is that of relative velocity between the phases. This along with nonequilibrium effects combined help explain some of the current discrepancies between model and data.

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Separated Flow Model for Critical Flow

Next: Fluids Other Than Water Up: SOUND SPEED AND CRITICAL Previous: Homogeneous Frozen Flow Model

4.4. Separated Flow Model for Critical Flow One of the key assumptions utilized in the past two models was that the velocity of each phase was equal. This assumption is known to be in error particularly under conditions where the pressure is low and the density ratio can be large. The concept of separated flow which was introduced for pressure drop can also be used here to develop a critical flow model. Let us consider the case where the phases are in thermodynamic equilibrium although there is slip between the phases. As stated previously, phase equilibrium is a good assumption when the phases are initially well-mixed or dispersed, or when the channel is long to allow equilibrium to develop. Once again under isentropic flow conditions we can express the pressure drop as consisting of only acceleration effects (i.e. equ 4.9). In this case one must remember the relationship between density and specific volume contains the velocity ratio

where S is the ratio of

. The expression for acceleration pressure drop (equ. 4.9) is in general

If we eliminate the void fraction, , from this expression using equ. 4.22 we arrive at the general expression for the critical mass velocity

One should note that for equal phase velocities, ( S = 1) we simplify to the HEM expression. The flow quality at this "choking" condition can be found from the energy balance and equ. 4.22 as

Now if one can specify the critical slip ratio at the "choking" condition then one can use equ's 24 and 25 to find X and (2 equs and 2 unknowns). Based on some theoretical arguments and empirical matching of experimental data there are two widely


utilized models [8-10] for the critical velocity slip ratio, . Both models assume that the slip ratio at "choking" conditions is equal to the inverse density ratio to an exponent, m

where Fauske [8,9] assumes

and Moody uses

. The only real justification for either

exponent is comparison to data and matching of the exponent and the critical pressure ratio. Over a wide range of data Fauske's empirical value shows somewhat better agreement. In addition the Fauske model has been solved parametrically and a convenient graphical solution is available (Figure 4.6 and 4.7) with the effect of geometry also empirically included. Therefore, we discuss this model results below, primarily as an example of this class of semi-empirical models. Fauske Model

One interesting note should be made. Because in Fauske's model the exponent is empirically "fit" to data the isentropic flow assumption becomes somewhat of a moot point. One might consider that frictional effects, if important are indirectly accounted for in the empirical part of the model. However, this is only D ratios. If one uses the model outside of its range of data then additional frictional correct for similar L / effects may have to be included. As one will see in the graphical results this would be in the range of pipe length to diameter ratio of 15 or greater. Let us discuss the model results in general and then examine the effect of geometry. Figure 4.6 is a plot of the choked mass velocity as a function of stagnation enthalpy at various critical pressures for steam-water mixtures. For any enthalpy and choked pressure one can immediately locate the choked mass velocity. In order to calculate the critical pressure at the choked plane, it is necessary to refer to Figure 4.6 Here D of the test section [9]. The choked flow is the choked pressure ratio is shown as a function of the L / calculated using several different equations depending on the length to diameter ratio of the outlet pipe section. D - O - Sharp edged orifice. There is no choked flow. The following equation should be used 1. L / just as it is shown below. This is essentially the orifice equation for single phase incompressible flow as an estimate for an orifice.

where

is the density of the initial mixture.

D < 3 - Nozzles and short tubes. Use Equation 4.27 with 2. O < L / from 4.6 in Region I replacing . There are substantial departures from equilibrium for these conditions. Perhaps the jet breaks

free of the tube but the space between the tube and the jet is filled with vapor at a pressure higher than the back pressure. D < 12 - Middle length tubes. For this geometry no convenient prediction scheme exists as 3. 3 < L / departures from equilibrium, which are neither negligible nor governing, exist. for these conditions replacing by in Region II from 4.6 overestimates the flow rate to some extent. Apparently,


the jet is breaking up as equilibrium is being established. D < 40. The flow rate is almost independent of the tube length so the 4. 12 < L / from 4.6 in Region III can be used to determine the critical flow rate from 4.7. Almost the whole pressure drop is due to momentum changes and wall friction is negligible. D < 40. Wall friction forces become increasingly important, but very gradually so, and the 5. L / critical flow rate drops slowly. Here it is suggested that one can begin by guessing a critical mass velocity and going to Figure 4.7 to obtain the corresponding critical pressure. Figure 4.6 can be used to estimate the pressure 40 L / D's upstream from the discharge. From that point to the entrance, the appropriate single or two phase pressure drop can be estimated with existing correlations. If the calculated entrance pressure is the right one, the initial mass velocity assumption is correct. If not, try another assumption. Small changes in the assumed mass velocity will change the pressure drop appreciably and the calculation will coverage rapidly.

The experiments justifying the calculation method just described are performed on an apparatus such as illustrated in Figure 4.11. A typical pressure-length curve is shown. Data comparisons reported in References [8] and [12], among others, are shown in Figures 4.12 and 4.13 for low and high quality cases. The comparison between the Fauske prediction methods and the data in this quality range is good. As can be seen, the homogeneous model predicts poorly in the moderately low quality region but gives better predictions as the quality increases. Similar experiments have been run on a long capillary with Freon 12 in and 18-foot tube .042 inches in D is over 1000 in the two phase region II. The pressure and temperature trace is shown diameter. The L / in Figure 4.14. The steeper pressure gradient in the two phase region, beyond 12.5 feet, is evident on this figure. As mentioned in Reference [9], a number of other choked flow theories exist for separated flow, which do about as well with the data of these Figures (4.12-13) and as that of Fauske (e.g., Ref. 10). The details of the assumptions somewhat differ but the result is about the same. In fact one s hould realize that the agreement may be the result of compensating errors.

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Fluids Other Than Water For Separated Flow Model

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4.5. Fluids Other Than Water For Separated Flow Model The bulk of critical two-phase flow data is for water. Calculations have been reported for liquid metals in Reference [12] and cryogens in Reference [13]. No predictions for other fluids have been independently worked out so what can be recommended for other fluids is based on judgement. The recommendation is first the critical pressure ratio curves of Figure 4.7. may be used for other fluids. It is then recommended that the homogeneous equilibrium model, for the fluid in question be used to calculate a critical flow mass velocity. Now, one might assume that the ratio of the actual flow to the HEM flow is the same for water and the fluid in question when the density ratio and quality is the s ame. This procedure is sparsely tested against experimental data. One should carefully compare the fluids to water based on similar critical property ratios, density ratios, latent heats and other thermodynamic properties. ll contents © Michael L. Corradini


The Physical Basis of Choked Two Phase Flow

Next: Advanced Computational Models Up: Fluids Other Than Water Previous: Fluids Other Than Water

4.6. The Physical Basis of Choked Two Phase Flow For a single phase flow choking occurs where the velocity of the gas is equal to the sonic velocity. This occurs at a particular section in any given apparatus and, as far as all the experiments show, a true choke occurs. The flow rate is absolutely independent of the discharge pressure as long as this pressure is below the choked pressure. We have no such foundation for predicting a "choking" condition in a two phase flow [17]. Experiments reported in Reference [14] show that the velocity of sound in a two-phase mixture is a function of flow regime, frequency and whether the pressure wave is a rarefaction or condensation wave. The sound velocity in a two phase mixture is, by no means, a thermodynamic property. For a two phase flow then the choking velocity and the sound velocity are not equal as they are for a single phase system. If we now confine our attention to critical two-phase flows alone the problem is still not simple. None of the models mentioned in the References show the existence of a choking condition without making assumptions beyond those which can be justified experimentally. In this section we would like to examine the characteristics of the critical flow model. These characteristics are as follows: 1. When a choke occurs, downstream pressure no longer matters, 2. Downstream geometry no longer matters, 3. Upstream geometry does not matter so long as the enthalpy and pressure at the choked plane is unaltered, 4. The velocity ratio is as given by Equation 4.26, and 5. Thermodynamic equilibrium exists.

It becomes evident by examining the details of test data closely that assumption (1) is not valid and the downstream pressure does, in some small way, affect the flow rate. There are no experiments that show conclusively this is not true. As far as using this concept for engineering design is concerned, these departures from a true choked flow are of no consequence but they do cast doubt on the idea of the "choking" condition as a real physical occurrence for two-phase flow. Experiments reported in Reference 16 show that downstream geometry does matter. Figure 4.6 shows several test section geometries while Figure 4.7 shows the corresponding critical mass velocity data. What this data shows is there is no plane at which choking occurs. Apparently there is only a very small increase in flow with a decrease in discharge pressure but this changes the fluid mechanics of the flow occurring over the entire tube. Upstream geometry clearly matters or it would not have been necessary to use Figure 4.10 to predict choked flow. For the same choke plane pressure and enthalpy, the choked flow mass velocity clearly depends on the upstream geometry. This is never true of a single phase flow.

The Physical Basis of Choked Two Phase Flow

Measurements of void made by Fauske and reported in Reference 8 show that the velocity ratio is very different from that calculated from Equation 4.26. If comparison is made with other models, they still do not show the behavior indicated by the data of Figure 4.17. Clearly, no simple velocity ratio expression has the needed properties to fit the range of data. Thermodynamic equilibrium cannot exist for the data of Figure 4.17, for, if it did, the pressure drop D's less would not be large enough to pass the flow. Thermodynamic equilibrium does not exist for L / than (12) either, as it would not be necessary to distinguish between the L / D's and the longer ones. Therefore, one can say that none of the assumptions listed earlier are really true under all the conditions of interest.

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Advanced Computational Models

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4.7. Advanced Computational Models The more general case for critical flow involves consideration of heat addition, friction along the pipe wall and non-equilibrium effects in velocity and temperature. Examples of such models are given in References 20-22. The complete set of steady-state conservation equations can be written as (see Reocreux, Ref. 23).

where is the wall shear stress and q' is the linear heat addition rate, both externally imposed. In addition one also has the equation of state for the fluid

As an example for an ideal gas one can substitute the algebraic expressions for pressure, p, and enthalpy, i, into the balance equations and we write eqns 28-30 in matrix form

with

, and

Notice that the frictional, heat addition and area change terms appear as external sources in the expression on the right-hand side. Reocreux [23] used this example to indicate that the momentum and heat transfer relations are algebraic and not included in the derivative terms. The solution to this matrix is found by inverting the matrix B. This can only be done if the determinant of B is non-zero. Mathematically, when the determinant is zero this is a condition when the pressure gradient is infinite and we have reached the critical flow condition, and solve for the velocity which it achieves. For the case of an ideal gas we recover the same results as discussed previously (eqns 4.5 to 7). A, This suggests the difficulties that arise when two phases occupy the channel. In single phase flow dA / and q' were imposed externally upon the fluid. In two phase flow consider how these quantities will be


partitioned between the two phases; i.e., how will the flow area be occupied by the two phases and how will the shear forces be distributed among the two phases? This last consideration will determine the void fraction. Also how energy will be distributed between the two phases; a thermal equilibrium consideration? Thus we can sense that mechanical and thermal non-equilibrium considerations will be important in relation to two-phase critical flow. Giot and Meunier [24] have shown that the Moody and Fauske models outlined above are equivalent to requiring that the determinant of the three mixture conservation equations, augmented by a fourth condition involving a linear combination of the space derivaties of S, p, X , and vanishes. Thus they find again the relationship between critical flows and the vanishing determinant of the conservation equations for single-phase compressible flows. In non-equilibrium models, one needs a mechanistic description of the mass and energy exchanges between the two phases. For example, one can use the theory of heat transfer limited bubble growth to arrive at a vapor generation law needed to close the system of equations in this case [21, 22]. Many models addressing the various aspects briefly touched upon above have appeared in the literature over the years. There is yet no model that gives perfect agreement with all the data and there will probably be none in the future either. Indeed, certain non-equilibrium aspects, such as nucleation on the walls and flow-pattern dependent velocity ratios depend on secondary variables (such as wall roughness and gas content of the liquid in the case of nucleation) and are difficult or impossible to quantify in general. Non-equilibrium models can be either empirical, mechanistic (i.e. based on a phenomenological description of the vapor generation mechanisms), or based on two-fluid models incorporating automatically the interfacial heat and mass exchange mechanisms. In empirical models non-equilibrium vapor generation is handled by introducing, for example, a coefficient that allows only a fraction of the equilibrium vapor generation to occur. Relaxation models for phase change have also been proposed by several authors; in this case the actual vapor generation is proportional to the difference between the actual quality and the equilibrium quality. Mechanistic models attempt to model the mechanisms of nucleation and interfacial phase change. There are inherent difficulties, however, in all critical flow models that become evident when this approach is used. For example, the rate of nucleation depends on the roughness and other properties of the walls as well as on the amount of dissolved air in the system; such effects are very difficult to quantify, as already mentioned. Introducing a grid in the flow area promoted nucleation and lowered the superheating of the liquid with significant impact on the critical flow characteristics during the Moby-Dick experiments [25]. Often the realism introduced into the model by modeling of the physical phenomena is shadowed by totally unreasonable values of certain parameters that must be used to fit the experimental data. The six-equation formulation provides automatically the features necessary for describing departures from mechanical and thermal non equilibrium. The methodology that can be used to detect the presence of critical flow conditions when the six equations are being integrated is well understood [20, 24, 26-28]. In a way quite analogous to the one shown previously for single-phase gas flows, one finds that critical flow occurs when the determinant of the coefficients of the solution variables becomes zero at a certain location. Regarding modeling of the physics, however, there are difficulties with this approach also since we do now know yet the correct form of certain terms that enter into the conservation equations and influence the results significantly; the virtual mass terms have such an effect as well as the mass exchange rate [22].


In spite of the fact that the two-fluid 6-equation models are capable of predicting the occurrence of a critical flow condition, for several practical reasons (computational speed, singularity of the solutions near the critical point, not knowing a priori the location of the choking point, fine nodalization requirements, the numerical schemes used, etc.) a separate choking model is necessary. This separate critical flow criterion is derived from an analysis of the characteristics of the governing; if the real part of all the characteristics is non-negative, no information can propagate upstream, and the exit pressure cannot affect any more the flow upstream from the break. The critical velocity is thus calculated from an algebraic equation obtained by requiring that the largest characteristic velocity be equal to zero. For system calculations, the critical flow condition derived in this fashion is applied at the choking point whose location is known a priori. The NRC computer codes (RELPA5/MOD2, TRAC/BF1 and TRAC-PFI/MOD1) use a simplified two-phase choking criterion that is based on the assumption of thermal equilibrium between the phases, at variance with the physics reality. The so-called "RELAP5 two-phase choking criterion" (used also in TRAC-BF1) [22] is derived from such a formulation (4-equation, thermal equilibrium) with additional simplifications for obtaining an explicit expression for the characteristic roots; choking occurs when a hybrid mixture velocity becomes equal to the homogeneous equilibrium (HE) sound velocity. In this model, the arbitrary simplifications needed to arrive at the final mixture velocity have no justification other than the fact that they yield good results. A different approach is used in CATHARE where the critical condition is derived from the basic thermal-hydraulic model. CATHARE [30] uses a full 6-equation model and the critical condition is found by solving a sixth-degree polynomial that retains all the terms (velocity and temperature difference between the phases are both considered). The code has also the capability, however, of integrating the conservation equations all the way through a critical flow condition. One should note, however, that CATHARE uses a fully implicit numerical scheme that allows use of very fine meshes near the break without consequence on the time step.

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Observations

Next: POOL BOILING Up: SOUND SPEED AND CRITICAL Previous: Advanced Computational Models

4.8. Observations and Recommendations Based on the previous discussion, the following conclusions can be drawn: 1. The Homogeneous-Equilibrium Model (HEM) underpredicts the critical flow rates for short pipes and near liquid saturation or subcooled upstream conditions. 2. The equilibrium slip models of Fauske, Moody and others, although successful for long tubes, underpredict the critical flow rates for short pipes. This is particularly true if the upstream condition is subcooled or near saturation. 3. Effects of thermal nonequilibrium must be taken into account for short pipes. However, it is not D, or both are important in clear whether the pipe length, L, or the pipe length-to-diameter ratio, L / determining the effects of thermal nonequilibrium. 4. More mechanistic models do exist for critical flow that can account for the nonequilibrium effects of velocity differences and temperature differences between the phases; but require iterative or numerical solution. 5. At present, there is no general model or correlation for critical flow which is valid for a broad range of pipe lengths, pipe diameters, and upstream conditions including subcooled liquid. The more sophisticated the model used for a more precise design, the more prudent it is to consider some data benchmarking under similar conditions. References

1. J. Keenan, Thermodynamics, J. Wiley and Sons, Inc., New York (1941). 2. E. Obert, R. Gaggioli, Thermodynamics, McGraw Hill, New York (1960). 3. A.H. Shapiro, The Dynamics and Thermodynamics of Compressible Fluid Flow, Vol. 1, Chap. 4, Ronald Press, New York (1953). 4. R.E. Henry, H.K. Fauske, "The Two-Phase Critical Flow of One-Component Mixtures in Nozzles, Orifices, and Short Tubes," Jnl. Heat Transfer, p 179 (May 1971). 5. D.J. Maneely, "A Study of the Expansion Process of Low Quality Steam Through a deLaval Nozzle," LICRL-6/30, Univ of CA (1962). 6. K.F. Neusen, "Optimizing the Flow Parameters for the Expansion of Low Quality Steam," UCRL-6152, University of CA (1962). 7. M.E. Deich et al., "Investigation of Flow of Wet Steam in deLaval Nozzles," High Temperature, V7, p 294 (1966). 8. H.K. Fauske "Contribution to the Theory of Two-Phase One-Component Critical Flow," ANL-6633, Argonne Nat Lab (1961). 9. H.K. Fauske, "The Discharge of Saturated Water Through Tubes," Chem Engr Prog Symp Ser, V61, p 210 (1965).

Observations

10. F.J. Moody, "Maximum Flow Rate of a Single Component Two-Phase Mixture," Jnl Heat Transfer, Tran ASME Series C, V87, No 1, p 134 (Feb 1965). 11. M.M. Bolstad, R.C. Jordan, "Theory and Use of the Capillary Tube Expansion Device," Jnl. of ASRE, Vol 6, p 518 (1948). 12. H.K. Fauske, "Two Phase Critical Flow with Application to Liquid-Metal Systems (Mercury, Cesium, Rubidium, Potassium, Sodium, and Lithium, ANL 6779 (October 1963). 13. R.V. Smith, "Choking Two-Phase Flow Literature Summary and Idealized Design Solutions for Hydrogen, Nitrogen, Oxygen and Refrigerants 12 and 11," NBS Tech. Note 179, Supt. of Documents, Washington, D.C., U.S. Government Printing Office (1967). 14. R.E. Henry, M.A. Grolmes, H.K. Fauske, "Pressure-Pulse Propagation in Two-Phase One and Two-Component Mixtures," ANL 7792 (March 1971). 15. F.R. Zaloudek, "The Low Pressure Critical Discharge of Steam-Water Mixtures from Pipes," Report HW-68934, Rev. General Electric Co. (1961). 16. R.E. Henry, "A Study of One and Two-Component Two-Phase Critical Flows at Low Qualities," ANL 7430 (March 1968). 17. P. Griffith, "MIT Critical Two-Phase Notes," Cambridge, MA (1976). 18. P. Saha, "A Review of Two-Phase Steam-Water Critical Flow Models with Emphasis on Thermal Non-Equilibrium," NUREQ/CR-0417, Brookhaven Nat Lab (Sept 1978). 19. R.R. Schultz, et al., "Marvihen Critical Flow Data: A Summary of Results and Code Assessment Applications," Nuclear Safety, V25, No 6 (Dec 1984). 20. J.A. Boure, et al., "Highlights of Two-Phase Critical Flow," Int'l Jnl Multiphase Flow, V3 pp 1-22 (1978). 21. M.N. Hutcherson, Numerical Evaluation of the Henry-Fauske Critical Flow Model, MDC-N9654-100, McDonnell Douglas Co, (July 1980). 22. V.K. Ransom, et al., RELAP5/MOD2 Code Manual, NUREG/CR-5273, EGG-2555 (1987). 23. M. Reocreux, "Contribution a l'etude des debits critiques en ecculement disphasique eau-vapeur," doctoral thesis, L'Univ. Scientifique et Medicale de Grenoble (1974) . Also: Contribution to the Study of Critical Flow Rates in Two-Phase Water Vapor Flow," NUREG-Tr-0002 (1977). 24. M. Giot and A. Fritte, "Modeling in Critical Flow," Proc. of the NATO Advanced Research Workshop on the Advances in Two-Phase Flow and Heat Transfer, Spitzingsee, FRG, 31 Aug.-3 Sept. (1982). M. Giot and D. Meunier "Methodes de Determination du Debit Critique en Ecoulements Monophasiques et Diphasiques a un Consituant," Energie Primaire, 4, No. 102, 23 (1968). 25. M. Reocreux, "Experimental Study of Steam-Water Choked Flow," in Transient Two-Phase Flow - Proceedings of the CSNI Specialists Meeting, August 1976, Atomic Energy of Canada, 2, 637-6669. (1977) 26. J. Boure, A.A. Fritte, M. Giot and M.L. Rocreux, "Highlights of Two-Phase Critical Flow: On the Links Between Maximum Flow Rates, Sonic Velocities, Propagation and Transfer Phenomena in Single and Two-Phase Flows," Int. J. Multiphase Flow, 3, 1-22. 27. C.W. Hirt and N.c. Romero, "Application of a Drift-Flux Model to Flashing in Straights Pipes," 1st CSNI Spec. Meet. on Transient Two-Phase Flow, Toronto, (1976).

Observations

28. D. Durack and B. Wendroff, "Relaxation and Choked Two-Phase Flow," 2nd CSNI Spec. Meeting on Transient Two-Phase Flow, Paris, France, (1978). 29. G. Yadigaroglu and M. Andreani, "Two-Fluid Modeling of Thermal-Hydraulic Phenomena for Best-Estimate LWR Safety Analysis," pp. 980-996 in NURETH-4, Proceedings Fourth Int. Topical Meeting on Nuclear Reactor Thermal-Hydraulics, Karlsruhe, 10-13 Oct. 1989, U. Mueller, K. Rehme and K. Rust (editors), G. Braun, Karlsruhe. 30. A. Forge, R. Pochard, A. Porracchia, J. Miro, H.G. Sonnenburg, F. Steinhoss and V. Teschendorff, Comparison of Thermal-Hydraulic Safety Codes for PWR Systems, Graham and Trotman eds, (1988). ll contents © Michael L. Corradini


POOL BOILING

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5. POOL BOILING When a liquid is in contact with a surface maintained at a temperature above the saturation temperature of the liquid, boiling will eventually occur at that liquid-solid interface. Conventionally, based on the relative bulk motion of the body of a liquid to the heating surface, the boiling is divided into two categories; pool boiling and convective boiling. Pool boiling is the process in which the heating surface is submerged in a large body of stagnant liquid. The relative motion of the vapor produced and the surrounding liquid near the heating surface is due primarily to the buoyancy effect of the vapor. Nevertheless, the body of the liquid as a whole is essentially at rest. Though the study on the boiling process can be traced back to as early as the eighteen century (the observation of the vapor film in the boiling of liquid over the heating surface by Leiden in 1756), the extensive study on the effect of the very large difference in the temperature of the heating surface and the liquid, , was first done by Nukiyama (1934). However, it was the experiment by Farber and Scorah (1948) that gave the complete picture of the heat transfer rate in the pool boiling process as a function of . Applying the Newton's law of cooling, , the heat transfer coefficient, h, was used to characterize the pool boiling process over a range of as illustrated by the boiling curve in Figure 5.1.

by Farber and Scorah

Farber and Scorah conducted their experiments by heating the water at various pressures with a heated cylindrical wire submerged horizontally under the water level. From the results, they divided the boiling curve into 6 regions based on the observable patterns of vapor production. Region I, is so small that the vapor is produced by the evaporation of the liquid into gas nuclei on the exposed surface of the liquid. Region II, is large enough that additional small bubbles are produced along the heating surface but later condense in the region above the superheated liquid. Region III, is enough to sustain "nucleate boiling", with the creation of the bubbles such that they depart and rise through the liquid regardless of the condensation rate. Region IV, an unstable film of vapor was formed over the heating surface, and oscillates due to the variable presence of the film. In this region, the heat transfer rate decreases due to the increased presence of the vapor film. Region V, the film becomes stable and the heat transfer rate reaches a minimum point. In Region VI, the is very large, and "film boiling" is stable such that the radiation through the film becomes significant and thus increases the heat transfer rate with the increasing . This behavior as described above occurred when the temperature of the wire was the controlled parameter, . If the power is the controlled variable then the increase in the power (or heat flux, q") in Region III results in a jump in the wire surface temperature to a point in Region VI, (Figure 5.2). This point of transition is known as the critical heat flux and occurs due to hydrodynamic fluid instabilities as discussed later. This results in the stable vapor film being formed, and the wire surface temperature increases as the heat transfer resistance increases for a fixed input power. If the power is now decreased, the vapor film remains stable in Region VI and the decreases to the

POOL BOILING

minimum point for film boiling within Region V. At this point the vapor film becomes unstable and it collapses, with "nucleate boiling" becoming the mode of energy transfer. Thus, one passes quickly through Region IV and III to a lower wire surface temperature. This "hysteresis" behavior is always seen when the power (or heat flux) is the controlled parameter. On the effect of the pressure, Farber and Scorah suggested that increasing the pressure, for the same temperature difference, would result in the decreasing of the size of the bubbles. At the same time, the film becomes thinner and less circulation would be observed. This effect is counter balanced by the increased density of the vapor and the attendant increase in its enthalpy. Thus, the increase in pressure initially increased the heat transfer rate in pool boiling. The objective of this section is to present an overall picture of the pool boiling process with an emphasis on the practical models used (1) to identify the transition between natural convection and nucleate boiling, as well as nucleate and film boiling, and (2) to estimate the heat flux during nucleate and film boiling given the difference between the heater surface temperature and the bulk liquid. First, we consider the process of bubble nucleation and then begin to "construct" the conceptual picture of the pool boiling curve with suggested quantitative models.

5.1. Bubble Nucleation and Onset of Nucleate Boiling 5.2. Bubble Growth and Nucleate Boiling 5.3. Pool Boiling Critical Heat Flux 5.4. Film Boiling and the Minimum Film Boiling Point

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Bubble Nucleation and Onset of Nucleate Boiling

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5.1. Bubble Nucleation and Onset of Nucleate Boiling Vapor may form from a liquid (a) at a vapor-liquid interface away from surfaces, (b) in the bulk of the liquid due to density fluctuations, or (c) at a solid surface with pre-existing vapor or gas pockets. In each situation one can observe the departure from a stable or a metastable state of equilibrium. The first physical situation can occur at a planar interface when the liquid temperature is fractionally increased above the saturation temperature of the vapor at the vapor pressure in the gas or vapor region. Thus, the liquid "evaporates" into the vapor because its temperature is maintained at a temperature minimally higher than its vapor "saturation" temperature at the vapor system pressure. Evaporation is the term commonly used to describe such a situation which can also be described on a microscopic level as the imbalance between molecular fluxes at these two distinctly different temperatures. We consider this conceptual picture again, when condensation is later considered in Section 9. When considering the other two situations of vapor formation, a vapor bubble or "nucleus" must be formed and be mechanically and thermally stable. Consider the simplest case of a spherical vapor bubble of pressure, , with a saturation temperature of in its liquid with pressure, , which corresponds to its saturation temperature

. Mechanical equilibrium requires that

where is the interfacial surface tension and r is the bubble radius of curvature. If the liquid is also in thermal equilibrium with the vapor, , which then implies . If one uses simple thermodynamics, combining equation 5.1 with the equality of local temperatures, one finds the needed liquid superheat for the vapor bubble to exist

One can now relate the superheat required within the bulk of the liquid or at a solid-gas-liquid interface to the size of nuclei. In a bulk liquid, thermal fluctuations always exist as a small but finite cluster of molecules can take on higher than macroscopic average energies (i.e., temperature). As the bulk liquid increases in its superheat these molecule clusters can take on "vapor-like" energies with increasing probability, and possibly form a stable "vapor nucleus." This process of vapor bubble nucleation is referred to as "homogeneous nucleation." One can use a thermodynamics approach to estimate the degree of liquid superheat necessary to form a stable vapor nucleus; e.g., for atmospheric pressures. However, a


statistical mechanics approach (Blander 1975, Skripov, 1970) provides a more complete picture of the nucleation rate

where

as

is the collision frequency

k is the Boltzman's constant, h is Planck's constant and

is liquid molecular density

.

is

the free energy of formation for the vapor nucleus of radius, r , given by

where

is evaluated at

and

is the saturation pressure at

increases, the surface tension decreases and

. As the liquid superheat

increases. Thus, at a particular

the nucleation rate

increases markedly and this corresponds to the "homogeneous nucleation" temperature,

; e.g.,

for water at atmospheric pressure, which corresponds to vapor nuclei radii of ; SPMlt ;1 micron. Such a superheat value for the onset of vapor nucleation is far above experimental observations for water, under commercial applications, thus it is not the primary mode of vapor nucleation, under normal circumstances. Nevertheless, it must be considered as operating conditions change (e.g., pressure) especially for organic liquids. Finally, consider the situation where a vapor/gas pocket exists near a solid surface in a liquid ( Figure 5.3). Container surfaces can provide sites for vapor formation. This third method of vapor generation from pre-existing vapor nuclei is called "heterogeneous nucleation." Examples of such pre-existing nuclei include noncondensible gas bubbles held in an emulsion in the liquid pool or gas/vapor filled cracks or cavities on container surfaces (Figure 5.3). The latter example is probably the most common circumstance for vapor bubble nucleation. In fact, one could derive the required liquid superheat necessary for the case of an ideal cavity of known radius. One finds it is substantially lower than that needed for homogeneous nucleation, because the cavity radius is much larger. Thus, the bubble requires less superheat and associated pressure difference for thermal and mechanical stability. It is hypothesized that the maximum superheat occurs at the throat of any cavity where the radius is smallest for most aqueous fluids with large contact angles. (Note: the contact angle is the angle through the liquid between the solid-liquid interface and the liquid-vapor interface, and depends on liquid-surface chemistry; e.g., water and commercial steel ). Only a small fraction of all cavities become effective sites for vapor nucleation, because one must consider the balance between the required superheat for a cavity of radius, , and the temperature gradient from the wall, , to the bulk liquid at saturation, ; as depicted by Hsu (1962) Figure 5.4 gives a conceptual picture of the model. As the heat flux at the wall is increased, the wall temperature,


which is probably representative of the vapor bubble and local liquid temperature, exceeds the saturation temperature. The liquid will locally vaporize and the vapor nuclei in the cavity will grow toward the cavity throat at the heater surface. If one assumes that the liquid temperature gradient from the wall to the bulk is approximately linear, then the requirement for mechanical and thermal stability of a vapor nucleus at the cavity exit is that the whole bubble should be in a liquid region of boundary layer size, where the temperature is at least above a value of equation 5.2, for

which satisfies the equilibrium condition of

. If there is a sufficiently large array of cavity sizes this "onset of nucleation" will

first occur when the liquid temperature profile is tangent to the line of thermal and mechanical equilibrium (Figure 5.4). One can algebraically eliminate the cavity radius, , from the two equations by equality of temperature and slope and find the relation between the heat flux,

, at

which the "onset of nucleate boiling", ONB, occurs

where all properties are evaluated at

. Now if there are no cavities at this size the heat flux must

increase so that the superheat temperature increases to a point where a cavity first exists and the temperature profile intersects the equilibrium curve in Figure 5.4. One should notice that this model only provides a stability line where the "onset of nucleate boiling" may first occur. To find the particular heat flux and superheat pair one must look for the intersection of this stability line with natural convection mode of heat transfer that would exist prior to boiling ( Figure 5.5)

and Gr is the liquid Grashof number for

and Pr is the liquid Prandtl number. For water at

atmospheric pressure this model predicts an "onset of nucleate boiling" for a superheat less than C, which corresponds to a cavity size of about 50 microns. In practice the superheat may be as high as C for very smooth, clean metallic surfaces, which indicates larger cavities were not available on the surface. As a very rough guide (Brown, 1967) aqueous fluids seem to have active sites ; SPMlt ; 10 microns, organic ; SPMlt ; 5 and cryogens ; SPMlt ; 1.5 microns.

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Bubble Growth and Nucleate Boiling

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5.2. Bubble Growth and Nucleate Boiling Once the vapor bubble on the heater surface is large enough at a given temperature that nucleation is assured, it will begin to grow in size. Remember, Equation 5.2 indicates that as the vapor bubble radius increases the required for stability dramatically decreases. Now the actual continues to increase with the associated pressure difference, , causing the bubble to grow. The first phase of growth is controlled by the large DP that initially exists balanced by the inertia of the surrounding liquid; i.e., inertial growth. As the bubble expands the is maintained by vaporization of the surrounding liquid, caused by energy transfer from the superheated liquid. The second phase of growth is controlled by the rate of energy transfer from the liquid to the vapor-liquid interface to produce vapor and maintain the pressure; i.e., thermal growth. The inviscid spherical momentum equation, sometimes called the Rayleigh equation, can be used under the assumption of constant to predict the bubble growth for the first phase of inertial growth

where R is the bubble radius and is found by the initial superheat . If heat transfer to the vapor-liquid interface controls the rate of vaporization then the bubble is small and growth is controlled by the rate of heat transfer through the liquid across the liquid superheat and the second stage of growth is estimated by

where

,

and

are the latent heat, thermal conductivity and thermal diffusivity of the liquid. Mikic

et al., (1971) developed a complete spherical bubble growth model giving


where Ja is the Jacob

and b is a constant that depends if the spherical bubble is near a

surface or an infinite medium (2/3). This model asymptotically reverts to the inertial or thermal models for very small or very large times. At some diameter the buoyancy of the vapor bubble at the surface overcomes the restraining effects of the surface tension attaching the bubble to the heater surface. A force balance on the bubble ( Figure 5.3) results in the following form to estimate this departure diameter

The proportionality constant depends on the bubble contact angle and stochastic processes; i.e., in actuality the observed bubble departure diameter has a statistical distribution around some mean given by Equ 5.11. Also the frequency of these bubbles departing the surface is important to know to estimate the energy carried away by bubble nucleation. This has been observed by a number of researchers to be governed by the growth rate (like Equ 5.10) and the waiting time for the vapor to reemerge from a surface cavity, i.e., the "waiting time." In fact for aqueous fluids, the waiting time is considered to be about equal to the bubble growth time at its departure diameter; i.e., from Equ 5.10. A number of investigators have correlated this time (or inverse frequency) to be given by the bubble diameter divided by the bubble rise velocity in a liquid

In either case the actual frequency is also statistically distributed about some correlated average; e.g., 20-50 for aqueous liquids. With this background one can estimate the nucleate boiling heat flux as being the energy carried away by vaporizing bubbles on the heater surface

where (n / a) is the number of active cavity sites per unit area; i.e., a function of heater surface-liquid combination. Rohsenow (1962) suggested the following correlation based on this concept

where

is the empirical constant linked to the fluid-surface combination (

commercial steel - Vachon et al., 1965 and the exponent, m, on the Prandtl number, and 1.7 otherwise. Also note that

for water and , is 1.0 for water

is the superheat between the heater wall and the liquid saturation


temperature, with properties evaluated at saturation conditions. This model for nucleate pool boiling is also shown in Figure 5.5. Notice that the heat flux increases with the cube of the liquid superheat, therefore

.

This correlation is one of many (e.g., Forster, Zuber, 1955; Lienhard, 1976) which shows similar behavior of the heat flux being a strong function of the liquid superheat; , m = 2 - 3. Rohsenow suggested that these correlations all need to be empirically "normalized" by test data for any particular liquid-heater combination for a set of initial conditions, although the functional dependencies are quite general. As Figure 5.5 indicates the nucleate boiling curve needs to be joined to the natural convection region. In practice, the presence of mixed heat transfer modes (i.e., natural convection and nucleate boiling) is real and the most straight forward way is to interpolate between these regions; .

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Pool Boiling Critical Heat Flux

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5.3. Pool Boiling Critical Heat Flux Critical heat flux (CHF) in pool boiling is an interesting phenomenon. As Figure 5.2 indicates, if one controls the input heat flux, there comes a point where as the heat flux is increased further the heater surface temperature undergoes a drastic increase. This increase originally was not well understood. Kutateladze (1951) offered the analogy that this large abrupt temperature increase was caused by a change in the surface geometry of the two phases. In fact, Kutateladze first empirically correlated this phenomenon as analogous to a gas blowing up through a heated porous plate cooled by water above it. At a certain gas volumetric flowrate (or superficial velocity, ) the liquid ceases to contact the heated surface and the gas forms a continuous barrier. Kutateladze concluded this by measuring the increasing electrical resistance between the plate and water as a function of the increase gas flowrate. Thus, pool boiling CHF may be thought of as the point where nucleate boiling goes through a flow regime transition to film boiling with a continuous vapor film separating the heater and the liquid. More generally, one may say CHF is the condition where the vapor generated by nucleate boiling becomes so large that it prevents the liquid from reaching and rewetting the surface. Consider this final physical picture of the critical heat flux,

, where the liquid is prevented from

reaching the heater surface by the flow of vapor generated by boiling,

where

is that critical superficial velocity preventing the liquid flow. A simple force balance on the

liquid as droplets,

where

, is given by

is assumed to determined by the characteristic Taylor wavelength (Equ 5.11), which results in

a velocity of

Combining these relations one obtains a general expression for CHF in pool boiling

where the constant, Co, is found to be in the range of 0.12 to 0.18; e.g., Zuber (1958) theoretically

Pool Boiling Critical Heat Flux

, Kutateladze (1951) correlated data for Co = 0.13, and Lienhard (1976) correlated

estimated data for Co = 0.15.

All of these previous discussions focused on the case where the liquid pool was at its saturation temperature. If the stagnant pool is maintained at a temperature below saturation, subcooled, the vapor bubbles can condense before they get very far from the heater surface. Thus, the heater power can go into directly heating the liquid and actual vapor superficial velocity is decreased; thus increasing the allowable heat flux before CHF occurs. Ivey and Morris (1962) correlated this subcooling effect as a multiplicative correlation to ,

where

is the degree of subcooling in the liquid.

The final point to emphasize is the location of the CHF point on the pool boiling curve of Figure 5.5. Critical heat flux appears as a horizontal line of the pool boiling curve and its intersection with the nucleate boiling curve indicates the temperature at which CHF occurs.

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Film Boiling and the Minimum Film Boiling P oint

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5.4. Film Boiling and the Minimum Film Boiling Point Once the critical heat flux is exceeded the heater surface is blanketed by a continuous vapor film; i.e., film boiling. Under this condition one must find the heat transfer resistance of this vapor film as well as consider the additional effect of radiation heat transfer at very high heater surface temperatures through this vapor film ( C). Bromley (1950) used the approach first developed by Nusselt for film condensation to predict the film boiling heat transfer coefficient for a horizontal tube

where is the difference in temperature between the heater wall and the saturation temperature. Bromley also considered the film boiling heat transfer coefficient for a vertical wall. This simplest of solutions is obtained assuming the vapor film is laminar and that the temperature distribution through the film is linear. For a vertical flat surface various boundary conditions may be imposed: (1) zero interfacial shear stress ( (2) zero interfacial velocity ( (3) zero wall shear stress (

= 0)

= 0) = 0)

For the first of these boundary conditions, the local heat transfer coefficient h( z) at a distance z up the surface from the start of film boiling is given by

The average coefficient h( z) over the region up to a distance z is given by:

If one defines

Film Boiling and the Minimum Film Boiling P oint

Note the analogy with natural convection. The values of C for the first of the above boundary conditions is 0.943 and for the second is 0.667. This thin vapor film over a horizontal surface is unstable and large bubbles form and break away. The characteristic spacing of these bubbles ( ) is determined by a balance of surface tension and gravitational forces and is given by:

where one substitutes this length scale, (

), for the tube diameter or wall length.

The final point to discuss is what happens if one decreases the heat flux while in film boiling. Once the continuous vapor film is formed it is hydrodynamically stable even if the heat flux decreases below the critical heat flux. In fact, this hystersis effect persists until the heat flux decreases to a point where the superficial velocity of the vapor formed at the heater surface is too low to "levitate" the liquid above the "continuous" film; in fact the film oscillates substantially and this physical picture may only be qualitatively correct. Zuber (1958) developed the expression for the minimum film boiling heat flux based on the concept of minimum superficial vapor velocity in a saturated pool and resulted in

Notice that this expression is similar in form to the CHF model except the density in the velocity term is the liquid density and comes about from a force balance on vapor bubbles leaving the interface. This minimum film boiling heat flux can be combined with the heat transfer coefficient to predict the minimum film boiling for a saturated liquid pool. Note that once again it is the intersection of a horizontal stability line and the film boiling regime. Wu (1989) considered the more complicated effects of subcooling and curved heater surfaces on the and . It was found a simple model was not found that was universally acceptable, thus one should look into this theoretical model. Dhir and Purohit (1978) developed a purely empirical fit to data for in water for steel or copper spheres. This is a widely used correlation but has limited applicability. References q

M. Blander, J.L. Koltz, "Bubble Nucleation in Liquids," AIChE Journal, Vol 21, No. 5, p 833, 1975.

q

B.P. Breen, J.W. Westwater, "Effect of Diameter of Horizontal Tubes on Film Boiling Heat Transfer," Proc Nat'l Heat Transfer Conf , Houston TX, 1962.

q

L.A. Bromley, "Heat Transfer in Stable Film Boiling," Chem Engr. Progress, Vol 46 , p 221, 1950.

Film Boiling and the Minimum Film Boiling P oint q

W.T. Brown, "A Study of Flow Surface Boiling," PhD Thesis, MIT Mechanical Engr Dept, 1967.

q

J.G. Collier, Convective Boiling and Condensation , McGraw Hill Book Company, 1981.

q

F.A. Farber, R.L. Scorah, "Heat Transfer to Water Boiling Under Pressure," Transaction ASME, Vol 79, p 369, 1948.

q

H. Forster, N. Zuber, "Bubble Dynamics and Boiling Heat Transfer," AIChE Journal , Vol 1, p 532, 1955.

q

C.Y. Han, P. Griffith, "The Mechanism of Heat Transfer in Nucleate Pool Boiling P + I: Bubble Imitation, Growth, Departure," Int'l Jul Heat Mass Transfer, Vol 8 , p 887, 1965.

q

Y.Y. Hsu, "On the Size of Range of Active Nucleation Cavities on a Heating Surface," Trans ASME Journal Heat Transfer, Vol 84 , p 207, 1962.

q

H.J. Ivey, D.J. Morris, "On the Relevance of the Vapor Liquid Exchange Mechanism for Subcooled Boiling Heat Transfer at High Pressure, AEEW-R Report 137 , 1962.

q

S.S. Kutateladze, "A Hydrodynamic Theory of Changes in the Boiling Process Under Free Convection," Akod. Nauk SSSR Tech Nauk, Vol 4 , p 529, 1951.

q

J.H. Lienhard, "Correlation of the Limiting Liquid Superheat," Chem Engr Science, Vol 31, p 847, 1976.

q

J.J. Lorentz, B. Mikic, W. Rohsenow, "The Effect of Surface Conditions on Boiling Characteristic," Proc Int'l Heat Transfer Conf , Tokyo, JP, 1974.

q

B. Mikic, W. Rohenow, P. Griffith, "On Bubble Growth Rates," Int'l Jol Heat Mass Transfer, Vol 13, p 657, 1970.

q

S. Nukiyama, "The Maximum and Minimum Values of the Heat "Q" Transmitted from Metal to Boiling Under Atmospheric Pressure," Inst Jul Heat Mass Transfer, Vol 9 , p 1419, 1994.

q

W. Rohsenow, "A Method of Correlating Heat Transfer Data for Surface Boiling of Liquids," Trans ASME, Vol 74, p 969, 1952.

q

V.P. Skirpov, P.A. Pavlov, "Teplo. Vsyo. Temp," Vol 8, No. 4, p 833, 1970.

q

R. Vachon, "Evaluation of Constants for the Rohsenow Pool Boiling Correlation," Nat'l Heat Transfer Conf , Seattle, August 1967.

q

L.K. Wu, "A Theoretical Model for the Minimum Film Boiling Point," PhD Thesis, University of Wisconsin-Madison, 1987.

q

N. Zuber, "On the Stability of Boiling Heat Transfer," Trans ASME, Vol 80, P 711, 1958.

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FLOW BOILING HEAT TRANSFER

Next: Objectives Up: No Title Previous: Film Boiling and the

6. FLOW BOILING HEAT TRANSFER 6.1. Objectives 6.2. Regions of Heat Transfer 6.3. Single-Phase Liquid Heat Transfer 6.4. The Onset of Nucleate Boiling 6.5. Subcooled Boiling 6.6. Saturated Boiling and the Two-Phase Forced Convection Region ll contents © Michael L. Corradini


Objectives

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6.1. Objectives One of the many applications of multiphase heat-transfer is to be able to predict the temperature of the wall of a boiling surface for a given heat flux or the variation of wall heat flux for a known wall temperature distribution. In this section we focus on the methodology to estimate the wall temperature or the wall heat flux depending on the appropriate boundary condition. We focus on describing the regions of heat transfer, locating the onset of nucleate boiling and finally estimating the wall condition. ll contents © Michael L. Corradini


Regions of Heat Transfer

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6.2. Regions of Heat Transfer Consider the case of a vertical tube heated uniformly over its length with a known heat flux and with subcooled liquid entering at its base. This prescribed heat flux boundary condition for a one dimensional situation is the simplest case for our consideration. More complex situations can subsequently be explained. At some mass flowrate the liquid may be totally evaporated over the length of the tube. Figure 6.1 shows a conceptual picture of the various flow patterns encountered over the tube length, the qualitative temperature profile, together with the corresponding heat transfer regions. As the liquid is being heated to its saturation temperature and the wall temperature remains below the condition necessary for nucleation, heat transfer is single-phase convective heat transfer to the liquid (Region A). At some point along the tube, the condition adjacent to the wall allows the stable formation of vapor from wall nucleation sites. Initially, vapor formation takes place in the presence of subcooled liquid (Region B) and this heat transfer region is known as subcooled nucleate boiling. In the subcooled nucleate boiling region, (B), the wall temperature remains essentially constant a few degrees above the saturation temperature, while the mean bulk fluid temperature is increasing to the saturation temperature. The amount that the wall temperature exceeds the saturation temperature is known as the degree of superheat, , and the difference between the saturation and the bulk fluid temperature is known as the degree of subcooling,

.

The transition between regions B and C, the subcooled nucleate boiling region and the saturated nucleate boiling region, is clearly defined from the thermodynamic viewpoint. It is the point at which the liquid reaches the saturation temperature (thermodynamic equilibrium quality, = 0) found on the basis of a simple energy balance calculation. Vapor generated in the subcooled region is present at the transition between regions B and C ( = 0); thus, some of the liquid must be subcooled to ensure that the liquid bulk (mixing cup) enthalpy equals that of saturated liquid. This effect occurs as a result of the transverse temperature profile in the liquid and the subcooled liquid flowing in the center of the channel will only reach the saturation temperature at some distance downstream of the point, = 0. In the regions C to G, a variable characterizing the heat transfer mechanism is the thermodynamic mass "quality" (

) of the

fluid. The "quality" of the liquid-vapor mixture at a distance, z, is given on a thermodynamic basis as:

As the quality increases in the saturated nucleate boiling region a point may be reached where transition in the heat transfer mechanism occurs. The transition occurs between the process of "boiling" and the process of "evaporation". This transition is preceded by a change in the flow pattern from bubbly or slug


flow to annular flow (regions E and F). In the latter regions the thickness of the thin liquid film on the heating surface is often such that the effective thermal conductivity is sufficient to prevent the liquid in contact with the wall being superheated to a temperature which would allow bubble nucleation. Heat is carried away from the wall by forced convection in the film to the liquid film-vapor core interface, where evaporation occurs. Since nucleation may be completely suppressed, the heat transfer process may no longer be called "boiling". The region beyond the transition has been referred to as the two-phase forced convective region of heat transfer (Regions E and F). For the example described above the critical heat flux condition occurs at some critical value of the quality, where the complete evaporation of the liquid film occurs. This particular CHF transition is known as "dryout" and is accompanied by a rise in the wall temperature for channels operating with a controlled surface heat flux. The area between the dryout point and the transition to saturated vapor (Region H) has been termed the liquid deficient region (corresponding to the drop flow pattern) (Region G), where post-CHF heat transfer occurs. This condition of "dryout" often puts an effective limit on the amount of evaporation which can be allowed to take place in a channel at a particular heat flux. It is also useful to qualitatively describe the progressive variation of the local wall surface temperature (or local heat transfer coefficient) along the length of the tube for this prescribed heat flux boundary condition. The local heat transfer coefficient can be established by dividing the surface heat flux (constant over the tube length) by the difference between the wall temperature and the bulk fluid temperature. Typical variations of these two temperatures with length along the tube are shown in Figure 6.1. The variation of heat transfer coefficient with length along the tube for the conditions represented in Figure 6.1, is given in Figure 6.2 (curve(i), solid line). In the single-phase convective heat transfer region, the wall temperature is displaced above the bulk fluid temperature by a relatively constant amount, (the heat transfer coefficient is approximately constant) and is modified only slightly by the influence of temperature on the liquid physical properties. In the subcooled nucleate boiling region the temperature difference between the wall and the bulk fluid decreases linearly with length up to the point where x = 0. The heat transfer coefficient, therefore, increases linearly with length in this region. In the saturated nucleate boiling region the temperature difference and, therefore, the heat transfer coefficient, remains constant. Because of the reduced thickness of the liquid film in the two-phase forced convective region the difference in temperature between the surface and the saturation temperature decreases and the heat transfer coefficient increases with increasing length or mass quality. At the CHF point the heat transfer coefficient decreases dramatically from a very high value in the forced convective region in a value near to that expected for heat transfer by forced convection in saturated steam. As the quality increases through the liquid deficient region, so the vapor velocity increases and the difference in temperature between the surface and the saturation value decreases with a corresponding rise in the heat transfer coefficient. Finally, in the single-phase vapor region ( ) the wall temperature is once again displaced by a constant amount above the bulk fluid temperature and the heat transfer coefficient levels out to that corresponding to convective heat transfer of a single-phase vapor flow. Now consider the other boundary condition of a vertical tube heated by a uniformly constant temperature along its length with an inlet subcooled liquid flow. The situation once again will cause the entering liquid to 1) nucleate on the heated wall, 2) reach saturation at some downstream axial location, and 3) begin bulk boiling. The difference in this case is that the heat flux will vary along the channel, although the overall multiphase flow regimes would the same (Figure 6.3). The realistic situation may become more complex for a horizontal flow orientation or for the heating surface becoming non-uniformly heated


or with rod bundles. Nevertheless, the general methodology does not change. Throughout our discussion here we focus on an approximately one-dimensional flow orientation. The above comments have been restricted to the case of a relatively low uniform heat flux supplied to the tube wall. The effect of progressively increasing the surface heat flux while keeping the inlet flow-rate constant, can now be considered with reference to Figures 6.2a and 6.2b. Figures 6.2a shows the heat transfer coefficient plotted against mass quality with increasing heat flux as a parameter (curves (i,ii)). 6.2b shows the various regions of two-phase heat transfer in forced convective boiling on a three-dimensional diagram with heat flux, mass quality and temperature as co-ordinates-"the boiling surface," Curve (i) relates to the conditions shown in Figure 6.1 for a low heat flux being supplied to the walls of a tube. The temperature pattern shown in Figure 6.1 will be recognized as the projection (temperature-quality co-ordinates) of curve (i). Curve (ii) shows the influence of increasing the heat flux. Subcooled boiling is initiated sooner, the heat transfer coefficient in the nucleate boiling region is higher but is unaffected in the two-phase forced convective region. Dryout CHF occurs at a lower mass quality. Curve (iii) shows the influence of a further increase in the heat flux. Subcooled boiling is initiated earlier and the heat transfer is again higher in the nucleate boiling region. As mass quality increases, before the two-phase forced convective region is initiated, and while bubble nucleation is still occurring, a large reduction in the cooling process takes place. This transition is essentially similar to the critical heat flux phenomenon in saturated pool boiling and has been termed "departure from nucleate boiling" (DNB).

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Single-Phase Liquid Heat Transfer

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6.3. Single-Phase Liquid Heat Transfer Figure 6.4 shows an idealized form of the flow patterns and the variation of the surface and liquid temperatures in the regions designated by A, B, and C for the case of a uniform wall heat flux. Under steady state one-dimensional conditions the tube surface temperature in region A (convective heat transfer to single-phase liquid), is given by:

and where q;SPMquot ; is the heat flux, Per is the heated perimeter, G is the mass velocity, A is the flow area and is the liquid specific heat. Also is the temperature difference between the wall surface and the mean bulk liquid temperature at a given length z from the tube inlet, h is the heat transfer coefficient to single-phase liquid under forced convection. The liquid in the channel may be in laminar or turbulent flow, in either case the laws governing the heat transfer are well established; for example, heat transfer in turbulent flow in a circular tube can be estimated by the well-known Dittus-Boelter equation.

D > 50 and Re > 10,000. This relation is valid for heating in fully developed vertical upflow in z /

where

is the hydraulic diameter,

is the liquid viscosity and

is the liquid thermal conductivity.

For the case of a given constant wall temperature, the temperature difference will decrease, as well as the heat flux. From an energy balance this is represented by a logarithmic decrease in the temperature difference. ll contents © Michael L. Corradini

[email protected]

Single-Phase Liquid Heat Transfer

Last Modified: Tue Sep 2 15:06:55 CDT 1997

The Onset of Nucleate Boiling

Next: Subcooled Boiling Up: FLOW BOILING HEAT TRANSFER Previous: Single-Phase Liquid Heat Transfer

6.4. The Onset of Nucleate Boiling In a similar fashion to pool boiling heat transfer if the wall temperature rises sufficiently above the local saturation temperature pre-existing vapor in wall sites can nucleate and grow. This temperature, , marks the onset of nucleate boiling for this flow boiling situation. From the standpoint of an energy balance this occurs at a particular axial location along the tube length, . Once again for a uniform flux condition, Equ (2) becomes

We can arrange this energy balance to emphasize the necessary superheat above saturation for the onset of nucleate boiling

Now that we have a relation between

and

we must provide a stability model for the onset of

nucleate boiling. In a manner similar to the pool boiling situation one can formulate this model based on the metastable condition of the vapor nuclei ready to grow into the world. There are a number of correlation models for this stability line of . Using this approach, Bergles and Rohsenow (1964) obtained an equation for the wall superheat required for the onset of subcooled boiling. Their equation is valid for water only, given by

where q;SPMquot ; is the surface heat flux in

and p is the system pressure in bar. An alternative

expression by Davis and Anderson (1966) valid for all fluids is:

where is the liquid-vapor surface tension, Equations (10) and (11) are in good agreement with each other for the case of high pressure water flows. Frost and Dzakowic (1967) have extended this treatment to cover other liquids. In their study nucleation was assumed to occur when the liquid temperature, T ( y)

The Onset of Nucleate Boiling

was matched to the temperature for bubble equilibrium, than

, at a distance

where

rather

used in past work. Thus Eq. (11) becomes:

We would recommend this final general expression to be used with the energy balance to determine and for a specified uniform heat flux. In the case of a known wall temperature one can find the heat flux at the onset of nucleate boiling as well as its corresponding location.

Next: Subcooled Boiling Up: FLOW BOILING HEAT TRANSFER Previous: Single-Phase Liquid Heat Transfer ll contents © Michael L. Corradini


Subcooled Boiling

Next: Saturated Boiling and the Up: FLOW BOILING HEAT TRANSFER Previous: The Onset of Nucleate

6.5. Subcooled Boiling The onset of nucleate boiling indicates the location where the vapor can first exist in a stable state on the heater surface without condensing or vapor collapse. As more energy is input into the liquid (i.e., downstream axially) these vapor bubbles can grow and eventually detach from the heater surface and enter the liquid. In our past discussion we have suggested that the onset of nucleate boiling occurs at an axial location (Point A on Figure 6.4) before the bulk liquid is saturated (Point C on Figure 6.4). Likewise the point where the vapor bubbles could detach from the heater surface would also occur at an axial location (Point B on Figure 6.4) before the bulk liquid is saturated. Now this axial length over which boiling occurs when the bulk liquid is subcooled is called the "subcooled boiling" length. This region may be large or small in actual size depending on the fluid properties, mass flow rate, pressures and heat flux. It is a region of inherent nonequilibrium where the flowing mass quality and vapor void fraction are non-zero and positive even though the thermodynamic equilibrium quality and volume fraction would be zero (Figure 6.4 and X ( z)); since the bulk temperature is below saturation. There are two ways to handle this region. Consider the boundary condition of a uniform heat flux on the heater surface. First, for conditions where this is a small actual length relative to the total heater wall axial length, one may neglect this length and its effect on heat transfer and pressure drop. If the subcooled boiling length is appreciable relative to the total heater wall length, then we need to locate the point where the vapor bubbles detach and boiling bubbles are stable in the flow as well as the associated vapor mass fraction and void fraction in this subcooled region. Figure 6.5 depicts the order of magnitude of the vapor void fraction and mass fraction for a steam-water system, as well as lists the various models available for estimating values for this region. The typical approach is to determine the amount of vapor superheat necessary to support the vapor bubble departure from the heater wall surface (Point B on Figure 6.4). Once this is determined one can find the axial location where this would occur. From this point on one can then calculate the vapor void fraction and mass fraction in the channel. As Figure 6.5 indicates there are only a few complete models developed for the subcooled region. In these notes we focus on the models developed by Levy (1967) and Zuber (1968) based on relatively good agreement with published data, and the relative ease in applying the subcooled boiling model. These models, although developed and checked with primarily steam-water data, can be applied to other fluids. The first objective is to determine the amount of superheat necessary

to allow vapor bubble

departure and then the axial location where this would occur. A force balance was used by Levy (1967) and Staub (1967) to estimate the degree of superheat necessary for bubble departure.

Subcooled Boiling

where is the radius of vapor bubble at departure, and where the first term represents the buoyancy force (destabilizing force), the second term represents the drag force (destabilizing force), and final force term is due to surface tension (stabilizing force); where , and are proportionality constants. In this conceptual model the bubble radius , is assumed to be proportional to the distance to the tip of the vapor bubble, , away from the heated wall. One can then calculate this distance

where the constant is based on empirical data and the buoyancy force was found to be negligible. Levy's fundamental assumption was that the temperature at the tip of bubble, with size , should be at least the saturation temperature,

. This superheat temperature,

universal temperature profile relation between

and

, was then found by using the

.

Now using the local energy balance one can relate the local bulk temperature,

, to the superheat

temperature difference at location B,

Then the energy balance along the length of the channel can be used to find the axial location of bubble departure, ,

The final term in the Levy model for

and

to determine is tw given by

Subcooled Boiling

where the single phase friction factor is used for this estimate. Zuber proposed an empirical relation to determine given by

Once the location of vapor bubble departure is determined, one then knows the subcooled boiling region and the total subcooled boiling length; onset of nucleate boiling ( Point A, Figure 6.4), vapor bubble departure, (

, Point B) and saturated boiling point (

, Point C). The next step is to model the

axial variation in quality and void fraction. Levy used an semi-empirical approach to estimate the non-equilibrium variation in void and quality. It was assumed that the "true" mass fraction, x( z), is related to the thermodynamic equilibrium quality, , by the relationship

where

is the thermodynamic equilibrium quality at the point of bubble departure (Point B of

Fig. 6.4), given by Eqn (6.1). This simple empirical relationship satisfies the following physical boundary conditions: 1. as 2.

then X' = 0 as well as

;

at point B is also 0:

3.

for

.

The void fraction is found by the drift flux model given by

where

is the one-dimensional average inlet velocity of the liquid, Co is the drift flux distribution

parameter (

1.1 chosen by Levy), and

is the drift velocity given by

One should note that since the void in subcooled boiling is near the wall, the common concept of Co is questionable for this application. Zuber took a similar approach in which an axial liquid temperature distribution was arbitrarily chosen

Subcooled Boiling

between the point of zero void fraction, negligible,

, and the point where the non-equilibrium effects become

(Point D). Based on the logic presented previously one could take

(Point A) and

(Point B). The "true" quality is given by

Zuber also used the drift flux model to predict the void fraction and used a constant of 1.41 instead of 1.18 in expression for drift velocity. Either model gives a complete estimate of the subcooled boiling region for location, quality and void. With this description one can then calculate the two-phase pressure drop in the subcooled region and modify the single phase heat transfer coefficient in the energy balance (Eqns. 6.2 - 6.4) to determine the wall temperature given the heat flux. In the following section we discuss how to do this with the two-phase heat transfer coefficient. The treatments of Levy and Zuber fit the experimental data reasonably, and both can be applied to other fluids than water.

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Saturated Boiling and the Two-Phase Forced Convection Region

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6.5. Saturated Boiling and the Two-Phase Forced Convection Region Once the bulk of the fluid has heated up to its saturation temperature, the boiling regime enters saturated nucleate boiling and eventually two-phase forced convection ( Figure 6.1). We once again want to be able to find the wall condition for this situation; e.g., wall temperature for a given heat flux. This section references various models for heat transfer within this regime. The major recommendation is to use the Chen correlation (1966), because it provides a good fit to data, and is well-behaved in the asymptotic limits. One should note that the heat transfer coefficient is so large that the temperature difference between the wall and the bulk fluid is small allowing for large errors in the prediction, without serious consequences. The saturated nucleate boiling and two-phase forced convection regions may be associated with an annular flow pattern. Heat is transferred by conduction or convection through the liquid film and vapor and is generated continuously at the liquid film/vapor core interface as well as possibly at the heat surface. Extremely high heat transfer coefficients are possible in this region; values can be so high as to make accurate assessment difficult. Typical figures for water of up to 200

have been reported.

Following the suggestion of Martinelli, many workers have correlated their experimental results for heat transfer rates in the two-phase forced convection region in the form:

where

(and

) is the value of the single-phase liquid heat transfer coefficient based on the total (or

liquid component) flow and

is the Martinelli parameter for turbulent-turbulent flow. A number of

relationships of the form of this equation have been proposed and have been extended to cover the saturated nucleate boiling region. These correlations do, however, have a high mean error ( 30 Chen (1966) has proposed a correlation which has been generally accepted as one of the best available. The correlation covers both the saturated nucleate boiling region and the two-phase forced convection region. It is assumed that both nucleation and convective mechanisms occur to some degree over the entire range of the correlation and that the contributions made by the two mechanisms are additive:

where

is the contribution from nucleate boiling and

the contribution from convection through the

liquid film. An earlier, very similar approach which has been used successfully for design in the process


industries over a number of years was given by Fair (1960). In the Chen correlation,

, the convective contribution is given by a modified Dittus Boelter form,

The parameter F is a function of the Martinelli parameter,

(Figure 6.6). The equation of Forster and

Zuber (1974), was taken as the basis for the evaluation of the "nucleate boiling" component,

. Their

pool boiling analysis was modified to account for the thinner boundary layer in forced convective boiling and the lower effective superheat that the growing vapor bubble sees. The modified Forster-Zuber equation becomes:

where S is a suppression factor defined as the ratio of the mean superheat seen by the growing bubble to the wall superheat is represented as a function of the local two-phase Reynolds number (Figure 6.7). Curve fits to the functions shown in these figures and F and S respectively are:

This correlation fits the available experimental data remarkably well (with a standard deviation of 11 To calculate the heat transfer coefficient , for a known heat flux ( q), mass velocity and quality the proposed steps are as follows: 1. calculate

which is given by:

2. evaluate F from Figure 6.6 or Equ 34. 3. calculate

from Eq. 32.

4. calculate

from

and

5. evaluate S from Figure 6.7 using the calculated value of


6. calculate 7. calculate 8. plot

for a range of values of from Eq. (31) for the range of for the

range against

values and interpolate

, at q;SPMquot ;.

The correlation developed by Chen is the best available for the saturated forced convective boiling region in vertical ducts and is recommended for use with all single component non-metallic fluids. The final point to emphasize with the Chen correlation is that it can be used over the whole region of saturated nucleate boiling and two-phase forced convection. In fact because it has the proper asymptotic limits as X approaches zero or one, one can extend its use into the subcooled boiling region with X ' substituted for . In this way the original energy balance for the one-dimensional flow in the channel can be consistently used from subcooled single phase heat transfer to saturated nucleate boiling and two-phase forced convection. Consistency in the predictive methodology is a key benefit of such an approach. References q

q

A.E. Bergles and W.M.Rohsenow, "The Determination of Forces Convection Surface Boiling Heat Transfer," Trans. ASME, J. of Heat Transfer , 86c, 365 , 1964. E.J. Davis and G.H. Anderson, "The Incipience of Nucleate Boiling in Forced Convection Flow," AIChE Journal , 12 (4), 774-780, 1966.

q

W. Frost and G.S. Dzakowic, "An Extension of the Method of Predicting Incipient Boiling on Commercially Finished Surfaces." Paper presented at ASME AIChE Heat Transfer Conf. Paper 67-HT-61, Seattle, 1967.

q

W.M. Rohsenow, "A Method of Correlating Heat Transfer Data for Surface Boiling of Liquids," Trans. ASME , 74,969, 1952.

q

W.M. Rohsenow and J.A. Clark, "Heat Transfer and Pressure Drop Data for High Heat Flux Densities to Water at High Sub-critical Pressure," Heat Transfer and Fluid Mechanics Institute , Stanford University Press, Stanford, California, 1951.

q

F.Kreith and M. Summerfield, "Heat Transfer to Water and High Flux Densities With and Without Surface Boiling." Trans. ASME , 71(7), 805-815, 1949.

q

E.L Piret and H.S. Isbin, "Two-Phase Heat Transfer in Natural Circulation Evaporators," A.I.Ch.E. Heat Transfer Symposium , St. Louis, Chem, Engng, Prog. Symp. Series, 50(6), 305, 1953.

q

W.M. Rohsenow, "Heat Transfer With Evaporation," Heat Transfer. A symposium held at the University of Michigan during the summer of 1952. Published by University of Michigan Press, 101-150, 1952.

q

R.W. Bowring, "Physical Model Based on Bubble Detachment and Calculation of Steam Voidage in the Subcooled Region of a Heated Channel," OECD Halden Reactor Project Report . HPR-10, 1962.

q

J.G. Collier, "Convective Boiling and Condensation." Published by McGraw Hill Book Co. (UK) Ltd, 1972.

q

q

J.C. Chen, "Correlation for Boiling Heat Transfer to Saturated Liquids in Convective Flow." Int. Eng. Chem. Process Design and Development , 5,322, 1966. J.R. Fair, "What You Need to Design Thermosyphon Reboilers," Petroleum Refiner . 39(2), 105-124, 1960.

Saturated Boiling and the Two-Phase Forced Convection Region q

P. Foster and N. Zuber, "Point of Vapor Generation and Vapor Void Fraction." 5th Int. Heat Transfer Conf , 1974.

q

P. Griffith, J.A. Clark and W.M. Rohsenow, "Void Volumes in Subcooled Boiling Systems'. Paper 58-HT-19", ASME-AIChE Heat Transfer Conference , Chicago, August 1958; also Technical Report No. 12 (MIT).

q

R.W. Bowring, "Physical Model Based on Bubble Detachment and Calculation of Steam Voidage in the Subcooled Region of a Heated Channel." OECD Halden Reactor Project Report HPR-10 , 1962.

q

F.W. Staub, "The Void Fraction in Subcooled Boiling-Prediction of the Initial Point of Net Vapour Generation." Paper presented at the 9th National Heat Transfer Conference , Seattle, preprint, ASME 67-HT-36, Aug. 1967.

q

S. Levy, "Forced Convection Subcooled Boiling Prediction of Vapour Volumetric Fraction." Int. J. Heat Mass Transfer , 10, 951-965, 1967.

q

P.G. Kroeger and N. Zuber, "An Analysis of the Effects of Various Parameters on the Average Void Fractions in Subcooled Boiling," Int. J. Heat Mass Transfer , 11, 211-233, 1968.

q

J.R.S. Thom, W.W. Walker, T.A. Fallon and G.F.S. Reising, "Boiling in Subcooled Water During Flow up Heated Tubes or Annuli." Paper 6 presented at the Symposium on Boiling Heat Transfer in Steam Generating Units and Heat Exchangers held by Inst. Mech. Engrs. Manchester (England) 15-16, Sept. 1965.

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BURNOUT AND THE CRITICAL HEAT FLUX

Next: Introduction and Objectives Up: No Title Previous: Saturated Boiling and the

7. BURNOUT AND THE CRITICAL HEAT FLUX 7.1. Introduction and Objectives 7.2. Effect of System Parameters on CHF 7.3. Correlation Methods for CHF in Round Tubes With Uniform Heating 7.4. Limits on the Critical Heat Flux 7.5. Mechanisms of Critical Heat Flux 7.6. Prediction of CHF in Annular Flow 7.7. Correlations for CHF With Uniform Heating 7.8. CHF With a Nonuniform Heat Flux 7.9. Correlation of Burnout for Rod Bundles (sub-channel analysis): 7.10. CHF Estimates for Nonaqueous Fluids 7.11. Observations ll contents © Michael L. Corradini


Introduction and Objectives

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7.1. Introduction and Objectives The critical heat flux condition was originally discussed in the context of pool boiling. In this situation it represents a condition in which the flow boiling regime dramatically changes near the heater wall. This is the most important flow regime transition when considering the performance of heat exchange equipment with evaporation and boiling. The critical heat flux condition is characterized by a large decrease in the heat transfer coefficient from the heater wall to the bulk fluid. The reason for the reduction is due to the flow regime transition when continuous liquid is replaced by continuous vapor at the heater wall, whether locally or over a larger area. Let us consider the critical heat flux for two different boundary conditions. For the purpose of this section, CHF is defined as follows: 1. For a surface with a controlled heat flux (e.g., with electrical heating, radiant heating or nuclear heating), CHF is defined as that condition under which a small increase in the surface heat flux leads to a large increase in the wall temperature. 2. For a surface whose wall temperature is controlled (e.g., one heated by a condensing vapor), CHF is defined as that condition in which a small increase in wall temperature leads to a large decrease in heat flux. The term "burnout' is sometimes taken as being synonymous with the terms "critical heat flux" (CHF), "departure from nucleate boiling" (DNB), "boiling crisis" and "dryout." In this discussion the implication of "burnout" is that the rise in surface temperature is sufficient to cause physical damage to the heat surface. ll contents © Michael L. Corradini


Effect of System Parameters on CHF

Next: Correlation Methods for CHF Up: BURNOUT AND THE CRITICAL Previous: Introduction and Objectives

7.2. Effect of System Parameters on CHF In a typical CHF experiment with a uniform heat flux on a heater tube, CHF first occurs at the end of the channel. Figure 7.1 shows conceptually the effect of the various system parameters. One notes the following behavior: 1. For a given pressure ( P), fixed mass flux ( G), tube length ( L) and tube inside diameter (

) the

critical heat flux (CHF) increases approximately linearly with the inlet subcooling (i.e., the difference in enthalpy, , between saturated liquid and inlet liquid). This relationship occurs over fairly wide ranges, but has no fundamental significance, except to indicate more energy goes into saturating the fluid. If a very wide range of inlet subcooling is used, then departures from linearity are observed. 2. For fixed P,

, L and

, CHF rises approximately linearly with G at low values of G but

then rises much less rapidly for high G values; this is discussed later. 3. For fixed P,

, G, and

, critical heat flux decreases with increasing tube length L.

However, the power input required for burnout,

, increases at first rapidly, and then less

rapidly, also as shown in the figure. For very long tubes, the critical power may appear to asymptote to a constant value independent of tube length in some cases. Again, this only applies over a limited range of length. 4. For fixed P, G,

and L, CHF increases with tube diameter,

, the rate of increase

decreasing as the diameter increases. Finally, the effect of system pressure on CHF is similar to that encountered in pool boiling. The parametric effects illustrated in Figure 7.1 are typical of those encountered for upflow. Experimental data for downflow shows surprisingly little difference to upflow particularly for large mass velocities. Critical heat flux in cross flow over a tube bundle was investigated by Lienhard and Eichorn (1976) and their results are illustrated in Figure 7.2. 7.2. For low cross flow velocities, the vapor departure from the heater surface follows the characteristic 3-dimensional jet form also occurring in pool boiling. The critical heat flux was, in this region, close to the value of pool boiling as shown in Figure 7.2. 7.2. At high liquid velocities, the pattern of vapor departure changes to a two-dimensional form as shown and the critical heat flux begins to increase with increasing liquid velocity. A wide variety of data has been obtained for burnout in annuli and rod bundles. The two main reasons for the interest in this geometry are: (1) In the nuclear industry, the main interest is CHF in rod bundle geometries. The annulus might be regarded as a "single rod bundle" or "subchannel."

Effect of System Parameters on CHF

(2) By making the outer channel wall transparent, it is sometimes possible to view the processes occurring on the inner (heated) surface, illustrating the CHF mechanism.

Most of the data obtained have been for the case where the inner surface is heated. However, more recently, data has appeared where both surfaces are heated and the fraction of the power input to, say, the outer surface is varied. Typical of this latter data is that of Jensen and Mannov (1974), some of which is illustrated in Figure 7.3. 7.3. For a fixed inlet subcooling, the critical quality (quality at burnout) initially increases as the fraction of power on the outer surface is increased. In this region, CHF occurs first on the inner surface. As the fraction of power on the outer surface is further increased, a maximum burnout quality is reached, and beyond this point burnout begins to occur first on the outer surface and the critical quality decreases with increasing fractional power on that surface.

Next: Correlation Methods for CHF Up: BURNOUT AND THE CRITICAL Previous: Introduction and Objectives ll contents © Michael L. Corradini


Correlation Methods for CHF in Round Tubes With Uniform Heating

Next: Limits on the Critical Up: BURNOUT AND THE CRITICAL Previous: Effect of System Parameters

7.3. Correlation Methods for CHF in Round Tubes With Uniform Heating For a given mass flux, fluid physical properties (i.e., pressure for a given fluid), tube diameter and for uniform heat flux, it is found that the data for a range of the tube length and inlet subcooling can be represented approximately by a single curve of critical heat flux against mass quality at the CHF location as sketched in Figure 7.4(a). The initial implication of the relationship between CHF and quality at this location is that the local quality conditions govern the magnitude of the burnout heat flux at that locality; this is termed the "local conditions hypothesis." The same data can also be plotted in terms of the quality of CHF, CHF point,

, and "boiling length" at the

. The boiling length is defined as the distance from the point in the channel at which

bulk saturation (i.e., zero thermodynamic quality) condition are attained. This type of plot is illustrated in Figure 7.4(b). This plot can be regarded as indicating a relationship between the fraction of liquid evaporated at CHF ( ) as a function of the boiling length to burnout, indicating the possibility of some "integral" rather than "local" phenomenon. In fact, it is easy to transform the relationship into the

relationship; the boiling length is given from a one-dimensional

energy balance as

which is equivalent to

where

is the hydraulic diameter, G the mass flux,

the latent heat of evaporation,

the local

thermodynamic quality and q" the heat flux. If the critical heat flux is related to the quality of CHF by the expression:

then it follows that:

Correlation Methods for CHF in Round Tubes With Uniform Heating

for the given hydraulic diameter and mass flux, it follows, therefore, that:

The vast majority of correlations for critical heat flux fall into either of these two categories. Although these correlations are equivalent for uniformly heated channels, they give quite different results when the heat flux is nonuniform. The question obviously arises as to which of the two forms is best suited to the prediction of CHF with nonuniform heating, the case which occurs most often in practical applications. Figure 7.5 shows a comparison between data for uniform heating and a nonuniform axial (cosine distribution) heating where local values of CHF and CHF quality are given by Alekseev (1964). As will be seen later, there is a considerable difference in critical heat flux at a given local quality for the uniform and nonuniform heated tubes respectively. Note that, with the nonuniform heating, CHF could occur first upstream of the end of the channel. Remember that this is only an example of nonuniform heating in the axial direction. One may also have circumferential variation of the flux or variable flux within a rod bundle. These must also be handled based on empirical data.

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Limits on the Critical Heat Flux

Next: Mechanisms of Critical Heat Up: BURNOUT AND THE CRITICAL Previous: Correlation Methods for CHF

7.4. Limits on the Critical Heat Flux To understand the limits on the critical heat flux it is useful to compute its upper and lower bounds. The simplest situation to consider is the case of a uniform heat flux on the heater wall. For a variable heat flux these limits apply but one must numerically integrate the equations. The lower bound on CHF would be that heat flux which first causes the heater wall to rise to the fluid saturation temperature, . By a one-dimensional energy balance on the channel one finds

where h is the single phase heat transfer coefficient. If we set the wall temperature equal to the fluid saturation pressure we find the minimum critical heat flux for any axial location in the uniformly heated channel, given by

The upper bound on the critical heat flux is that uniform heat flux which would cause the fluid to completely evaporate . Once again we can use the one-dimensional energy balance for the fluid and set the thermodynamic equilibrium quality to one and solve for the maximum CHF for a given axial location

One can conceptually visualize these two bounds on the critical heat flux by looking at their variation as a function of the axial location in the channel and the inlet subcooling when one computes the critical heat flux from correlation for uniform heating it should be within these bounds ( Figure 7.6). ll contents © Michael L. Corradini


Mechanisms of Critical Heat Flux

Next: Prediction of CHF in Up: BURNOUT AND THE CRITICAL Previous: Limits on the Critical

7.5. Mechanisms of Critical Heat Flux An understanding of the CHF mechanisms is useful for the user of empirical correlations and prediction methods and for devising means to avoid the occurrence of the phenomenon. Let us consider CHF mechanisms and their regions of operation for concurrent flow conditions. A large number of alternative mechanisms for CHF have been proposed but four concepts which appear to have been reasonably well established experimentally are illustrated in Figure 7 and areas follows: 1. Formation of hot spot under growing bubble ( Figure 7.7 (a)). Here, when a bubble grows at the heated wall, a dry patch forms underneath the bubble as the micro-layer of liquid under the bubble evaporates. In this dry zone, the wall temperature rises due to the deterioration in heat transfer. When the bubble departs, the dry patch may be rewetted and the process repeats itself. However, if the temperature of the dry patch becomes too high, then rewetting does not take place and gross local over heating (hence CHF) occurs. This mechanism was proposed, for instance, by Kirby (1966, 1967). 2. Near-wall bubble crowding and inhibition of vapor release ( Figure 7.7 (b)). Here, a "bubble boundary layer" builds up on the surface and vapor generated by boiling at the surface must escape through this boundary layer. When the boundary layer becomes too crowded with bubbles, vapor escape is impossible and the surface becomes dry and overheat giving rise to burnout. This mechanism is discussed, for instance by Tong et al. (1972). 3. Dryout under a slug or vapor clot. In plug or slug flow, the thin film surrounding the large bubble may dry out giving rise to localized overheating and hence burnout. Alternatively, a stationary vapor slug may be formed on the wall with a thin film of liquid separating it from the wall; in this case, localized drying out of this film given rise to overheating and burnout. This mechanism has been investigated, for instance, by Fiori and Bergles (1968) (Figure 7.7 (c)). 4. Film dryout in annular flow (Figure 7.7 (d)). Here, in annular flow, the liquid film dries out due to evaporation and due to the partial entrainment to the liquid in the form of droplets in the vapor core. This mechanism is discussed in more detail below. One should note that each of these mechanisms has a direct relation to the two-phase flow regime in which CHF occurs; e.g., bubble crowding in subcooled nucleate boiling, vapor clotting in slug flow, or film dryout in annular flow. Thus CHF is fundamentally a condition where liquid cannot rewet the heater wall because of the rate of vapor production impeding the liquid flow back to the hot surface. As the flow regime changes (e.g., bubbly, slug, annular) due to variations in mass velocity, pressure or geometry the particular mechanism which prevents the liquid to rewet the heater surface changes, but the basic principle remains the same. One might actually again use quality as a method to describe this. Figure 7.8 shows a conceptual picture of these four mechanisms as a function of G and . Once again the quality contains the effect of many of these mentioned variables. One different situation which might occur is the case of counter current flow and the CHF mechanism associated with it. In this case liquid drains

Mechanisms of Critical Heat Flux

down the channel wall due to gravity as vapor, produced along its length, flows upward ( Figure 7.9). In this situation one holds up the liquid flow into the whole channel due to the production of vapor along the whole channel length. This liquid holdup can be likened to a flooding phenomenon at the entrance of the channel causing film dryout lower in the channel. Oscillatory behavior of the CHF dryout may also occur in this situation.

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Prediction of CHF in Annular Flow

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7.6. Prediction of CHF in Annular Flow Critical heat flux in annular flow is one of the most important mechanisms from a practical point of view since, for channels of reasonable length, the first occurrence of CHF may likely occur in this region. Film flow measurements have been used to investigate CHF mechanisms in two different ways: 1. Measurement of the film flow rate at the end of a heated channel as a function of power input to the channel. The film flow rate at the end of the channel decreases with increasing power and the onset of CHF corresponds quite closely to the point at which the film flow rate becomes zero. This confirms the film dryout (as distinct from a film boiling, for instance) mechanism for critical heat flux. 2. Although measurements of film flow rate at the end of the test section are useful in demonstrating the dryout mechanism, they do not demonstrate the conditions along the test section which have led to the occurrence of CHF. To achieve this, measurements of film flow rate may be made along the test section for a heat flux corresponding to the critical heat flux and then, for constant inlet conditions, and for the known CHF value making measurements of the film flow rate at the end of channels of various heated lengths, less than the length of the channel for which CHF is known. The results of such measurements indicate that CHF occurs when the processes of droplet entrainment, droplet deposition and evaporation lead to the condition in which the film flow rate becomes zero. The rate of evaporation can be calculated from the local heat flux, provided the latent heat is known; for adiabatic flows, the rates of entrainment and deposition can be calculated using the methods described for any annular flow. The important question which arises is whether the rate of entrainment and/or deposition is affected by the presence of a heat flux normal to the surface. The effects which could be caused by a heat flux include: 1. The effect of nucleate boiling in the film giving rise to additional entrainment due to the bursting of bubbles through the film surface and; 2. The effect of a vapor flux away from the surface changing the hydrodynamic boundary condition and possibly inhibiting drop deposition onto the surface in the presence of evaporation. Over a range of conditions, the entrainment/deposition processes are relatively unaffected by the presence of a heat flux, thus, the annular flow prediction methods can be applied to CHF prediction in the annular flow regime. The procedure is to write a mass balance equation for the liquid film and to integrate this equation along the channel from known (or assumed) boundary conditions, and determine the point at which the film flow rate goes to zero.

Next: Correlations for CHF With Up: BURNOUT AND THE CRITICAL Previous: Mechanisms of Critical Heat

Prediction of CHF in Annular Flow



Correlations for CHF With Uniform Heating

Next: CHF With a Nonuniform Up: BURNOUT AND THE CRITICAL Previous: Prediction of CHF in

7.7. Correlations for CHF With Uniform Heating The importance of the critical heat flux phenomenon has led to the development of a very large number of CHF correlations. It is beyond the scope of this section to present a comprehensive description of these many correlations. In fact, the most accurate correlations are not publicly available but rather proprietary. Thus, Table 7.1 is provided to give examples of the various types of correlations adopted for CHF in tubes, annuli and rod bundles when the applied heat flux is spatially uniform. One should first note that the thermodynamic quality is the prime independent variable in most of these correlations. This parameter inherently contains through the energy balance the effect of mass velocity, G, pressure, P, inlet subcooling, , and geometry, and L. In addition some of these correlations have included second order effects for these same variables and as well as others. Finally, one should be cautious not to apply these correlations outside of their range of applicability based on the test data gathered. The reason for this final point is that since these CHF correlations are quite empirical, it is prudent not to extrapolate the empirical correlation fit. The sample correlations provided give a good range of data to apply toward a correlation.

Correlations for CHF With Uniform Heating



CHF With a Nonuniform Heat Flux

Next: Correlation of Burnout for Up: BURNOUT AND THE CRITICAL Previous: Correlations for CHF With

7.8. CHF With a Nonuniform Heat Flux In the introduction to the CHF concept correlation we illustrated the effect of uniform heating of a channel versus nonuniform heating. In general there are two ways to handle nonuniform heating: (1) Develop a correction for a particular type of nonuniform heating pattern. (2) Develop specific CHF correlations for nonuniform heating.

Let us consider each approach below. The differences in

for nonuniform heating led Tong, et al., (1966) to propose the so-called

"F-factor" method for the prediction of critical heat flux in nonuniform axially heated channels. Their method is illustrated schematically in Figure 7.10, the three curves shown in this figure are respectively: 1. The flux profile q"( Z ). 2. The critical heat flux

predicted for the local quality condition using a standard

correlation for uniform heating. 3. The critical heat flux,

for the nonuniform heating situation calculated by multiplying

by the F-Factor defined as:

where

is the distance from the channel inlet to the point at which the burnout flux is being

predicted and

where G is in

is given by the empirical expression:

.

Nonuniform heating experiments have been investigated in the past, primarily for axial variation of the heat flux. Each of these studies have resulted in an empirical correlation for CHF with nonuniform heating. The basic assumption in directly correlating the CHF data from such tests is that one correlates the CHF values for the actual local conditions (e.g., G, , P). Kirby (1966) used some of the past data and more recent data from Harwell to develop such a correlation ( Table 7.2). This is suggested for the


conditions listed for nonuniform axial heating situations as an alternative for the F-factor. Table 7.2. Kirby Correlation for Non-Uniformly Heated Round Tubes CorrelationEquation




Correlation of Burnout for Rod Bundles (sub-channel analysis):

Next: CHF Estimates for Nonaqueous Up: BURNOUT AND THE CRITICAL Previous: CHF With a Nonuniform

7.9. Correlation of Burnout for Rod Bundles (sub-channel analysis): It is possible to correlate data for CHF in rod bundle geometries by using overall, mixed flow, types of correlations and these are moderately successful. However, much more precise predictions can be made by using correlations based on "sub-channel analysis." In this form of analysis, the rod bundle is sub-divided into flow zones or sub-channels and conditions are calculated for these individual sub-channels, taking account of cross-mixing between adjacent channels and of cross-flows generated by pressure differences between the sub-channels. Two types of sub-channel have them employed as illustrated in Figure 7.11: 1. "Coolant" sub-channels which are bounded by lines joining the center of individual rods as illustrated. A discussion of the application of this form of sub-channel analysis to the prediction of burnout i rod bundles is given by, for instance, Bowring (1972). 2. "Rod centered" sub-channels in which the flow zone is bounded by lines of symmetry as illustrated in Figure 7.11 (b). The application of this form of sub-channel analysis is discussed, for instance, by Gaspari et al, (1974). In the "coolant" subchannel method, the usual practice is to devise specific CHF correlations for the sub-channel; in the application of the "rod-centered" sub-channel method, Gaspari et al., were able to employ a standard round-tube burnout correlation to the sub-channels (on an equivalent diameter basis) with reasonable success. ll contents © Michael L. Corradini


CHF Estimates for Nonaqueous Fluids

Next: Observations Up: BURNOUT AND THE CRITICAL Previous: Correlation of Burnout for

7.10. CHF Estimates for Nonaqueous Fluids Although the general qualitative trends are similar to nonaqueous fluids (freons, liquid metals, cryogens) and water, there has been no completely successful correlation for both water and other fluids. Most experiments have been limited in scope, although the work of Stevens et al., (1964, 1966) has been fairly comprehensive. The best recommendation, if data is available for a particular fluid, is to correlate the data in a manner similar to that previously discussed in Section 7.3. If data is not available or quite limited then some scaling laws might be used. One set of scaling laws which is suggested is given by the dimensionless groups of

where

From such a set of groupings comes a suggested dimensionless form for the critical heat flux

One should be cautious in applying these estimates particularly if data is limited to check the predictions under particular conditions. ll contents © Michael L. Corradini


Observations

Next: POST-CHF HEAT TRANSFER Up: BURNOUT AND THE CRITICAL Previous: CHF Estimates for Nonaqueous

7.11. Observations We have presented a summary of CHF with an emphasis on the fundamental mechanisms of CHF. It is important to remember that one must use empirical correlations only within their range of applicability, and if correlations do not exist to use available data in a manner suggested to develop the engineering correlations. One area which has little data to support development is transient critical heat flux (e.g., Moxon and Edwards (1967)). The general observation from this limited data is that under transient conditions tends to be larger than CHF under similar steady-state conditions. It is therefore recommended that consideration of CHF being exceeded under transients (flow or power) be based on local conditions and employing appropriate steady state correlations. References q

G.V. Alekseev, "Burnout Heat Fluxes Under Forced Flow," Proc. of Third Int'l Conference on Peaceful Uses of Atomic Energy, Geneva, August 1964.

q

F. Biancone, et al., "Forced Convection Burnout and Hydrodynamic Instability Experiments for Water at High Pressure Pt. I." EUR 2490e , 1965.

q

R.W. Bowring, "A Simple But Accurate Round Tube, Uniform Heat Flux, Dryout Correlation Over the Pressure Range 0.7-17 MN/M2." AEEW-R789, 1972.

q

q q

J.E.Casterline, "Burnout in Long Vertical Tubes With Uniform and Casine Heating Using Water at 1000 Psia," Topical Rep No. 1. Task XVI Columbia University 1964 (TID 2103) also Topical Rep No. 5 , "Second Experimental Study of Dryout in a Long Vertical Tube With a Cosine Heat Flux." J.G. Collier, "Convective Boiling and Condensation," McGraw Hill Publishers, NY, 1972. M.P. Fiori and A.E. Bergles, "Model of Critical Heat Flux in Sub-Cooled Flow Boiling." MIT Report - DSR70281-56 , 1968.

q

G.P. Gaspari, A. Hassid and F. Lucchini, "A Rod Centered Sub-channel Analysis With Turbulent (enthalpy) Mixing for Critical Heat Flux Prediction in Rod Clusters Cooled by Boiling Water." Proc. 5th Int Heat Transfer Conf. , 4, 295-299, 1974.

q

A.Jensen and G. Mannov, "Measurements of Burnout, Film Flow Film," 1974.

q

G.J.Kirby, "A New Correlation of Non-Uniformly Heated Round Tube Burnout Data." AEEW-R500, 1966.

q

J.F. Kirby, R. Staniforth and L.H. Kinneir, "A Visual Study of Forced Convective Boiling." Part II: Flow Patterns and Burnout for a Round Test Section," AEEW-R506 , 1967.

q

D.H. Lee and J.D. Obertelli, "An Experimental Investigation of Forced Convection Burnout in High Pressure Water - Pt. II." AEEW-R309, 1963.

q

D.H. Lee, "An Experimental Investigation of Forced Convection Burnout in High Pressure Water Pt. III," AEEW-R355 , 1965.

Observations q

D.H.Lee, "An Experimental Investigation of Forced Convection Burnout in High Pressure Water Pt. IV." AEEW-R479, 1966.

q

J.H.Lienhard and R. Eichhorn, "Peak Boiling Heat Flux on Cylinders in a Cross Flow." Int J Heat Mass Transfer , 19, No. 20, 1135-1141, 1976.

q

D. Moxon and P.A.Edwards, "Dry Out During Flow and Power Transients." AEEW-R553, 1967.

q

G.F. Stevens and G.J. Kirby, "A Quantitative Comparison Between Burnout Data for Water at 1000 lb/ and Freon 12 at 155 lb/ (abs) and by Water at 1000 lb/ in Vertical Upflow." AEEW-R327 , 1964.

q

G.F. Stevens and R.W. Wood, "A Comparison Between Burnout Data for 19 Rod-Cluster Test Sections Cooled by Freon 12 at 155 lb/ (abs) and by Water at 1000 lb/ in Vertical Upflow." AEEW-R468, 1966.

q

H.S. Swenson, R.J. Carver and C.R.Kakarala, "The Influence of Axial Heat Flux Distribution on the Departure From Nucleate Boiling in a Water-Cooled Tube." Paper presented to the Winter Annual Meeting of the ASME , Nov 1962, New York. Paper 62-WA-297, 1962.

q

H.S. Swenson, et al., "Non-Uniform Heat Generation Experimental Programme." QPR Nos. 4, 5, 6, 7 and 8, BAW-3238 , 1964.

q

N.E Todreas and W.M. Rohsenow, "The Effect of Non-Uniform Axial Heat Flux Distribution," M.I.T.-9843-37 ; see also paper to 1966 Int. Heat Transfer Conf., Chicago, paper 89, 1965.

q

L.S. Tong and G.F. Hewitt, "Overall View Point of Film Boiling CHF Mechanisms," ASME Paper No. 72-HT-54 , 1972.

q

L.S. Tong, H.B. Currin and T.S. Larsen, "Influences of Axially Non-Uniform Heat Flux on DNB." WCAP-2767 . Published in CEP Symp Ser, 62 (64), 1966.

Next: POST-CHF HEAT TRANSFER Up: BURNOUT AND THE CRITICAL Previous: CHF Estimates for Nonaqueous ll contents © Michael L. Corradini


POST-CHF HEAT TRANSFER

Next: Introduction and Objectives Up: No Title Previous: Observations

8. POST-CHF HEAT TRANSFER 8.1. Introduction and Objectives 8.2. Post-CHF Heat Transfer Models and Correlations 8.3. Empirical Correlations 8.4. Non-Equilibrium Empirical Models 8.5. Semi-Theoretical Models 8.6. Transition Boiling 8.7. Transition to Film Boiling 8.8. Observations ll contents © Michael L. Corradini


Introduction and Objectives

Next: Post-CHF Heat Transfer Models Up: POST-CHF HEAT TRANSFER Previous: POST-CHF HEAT TRANSFER

8.1. Introduction and Objectives In some situations, heat exchanger equipment is operated in a two-phase region beyond the critical heat flux point; e.g., fossil fueled boilers or steam generators, under steady conditions and nuclear reactor cores under transient conditions. Because of these and other similar situations there exists a continuing need for more accurate information about heat transfer coefficients in post-CHF flow regimes. Significant improvements improvements in understanding have been made over the past few years. In particular, the development of experimental techniques has allowed the determination of the complete forced convection boiling curve including the "transition boiling" region for particular values of local vapor quality and mass velocity. The objective of this section is to review the subject of Post-CHF heat transfer and indicate the "state-of-the-art." "state-of-the-art." Although much of this work is specifically relevant to aqueous fluids the techniques used are beneficial in the design of other heat exchange equipment in the power and process industries. ll contents © Michael L. Corradini


Post-CHF Heat Transfer Models and Correlations

Next: Empirical Correlations Up: POST-CHF HEAT TRANSFER Previous: Introduction and Objectives

8.2. Post-CHF Heat Transfer Models and Correlations Three types of modelling approaches have been attempted: 1. Correlations of an empirical nature which made no assumption whatever about the mechanism involved in post-CHF heat transfer, but solely attempts a functional relationship between the heat transfer coefficient and the independent variables. This assumes that the vapor and liquid are at the saturation temperature and in thermodynamic equilibrium. 2. Correlations which recognize that departure from thermodynamic equilibrium condition can occur and attempt to calculate the "true" vapor quality and vapor temperature, different from .A conventional single-phase heat transfer correlation for the vapor is then used to calculate the heated wall temperature. 3. Semi-theoretical models where attempts have been made to examine and derive equations for the various individual hydrodynamic and heat transfer processes occurring in the heated channel and relate these to the wall temperature (or heat flux depending on the boundary conditions). Groeneveld (1973) has compiled a bank of carefully selected data drawn from a variety of experimental post-dryout studies in tubular, annular and rod bundle geometries for steam/water flows. ll contents © Michael L. Corradini


Empirical Correlations

Next: Non-Equilibrium Empirical Models Up: POST-CHF HEAT TRANSFER Previous: Post-CHF Heat Transfer Models

8.3. Empirical Correlations After one has determined the axial location,

, in the channel for the first occurrence of critical heat

flux using correlations (Section 7.0), the next step is to determine the heat transfer coefficient in the post-CHF region. This heat transfer coefficient is to be used for the determination of the wall temperature given the wall heat flux or vice versa for a known wall temperature. A considerable number of empirical equations have been presented by various investigators for the estimation of heat transfer rates in the post-dryout region. Almost all of these equations are modifications of the well-known Dittus-Boelter type relationship for single-phase flow and take no account of the non-equilibrium effects discussed above. Rather thermodynamic equilibrium is assumed between the vapor and liquid. Various definitions of the "two-phase velocity" and physical properties are used in these empirically modified forms and a number of correlations result. Each of these correlations is based on only a limited amount of experimental data and Groeneveld, therefore, proposed a new correlation for each geometry optimized using his bank of selected data. This is the recommended correlation if one is to use the simplified equilibrium approach to a post-CHF analysis. The Groeneveld corrections for tubes and annuli have the form:

and the coefficients a, b, c and d are given in Table 8.1, together with the ranges of independent variables on which the correlations are based.

Empirical Correlations



Non-Equilibrium Empirical Models

Next: Semi-Theoretical Models Up: POST-CHF HEAT TRANSFER Previous: Empirical Correlations

8.4. Non-Equilibrium Empirical Models As mentioned previously our goal is to predict the heat transfer coefficient after the critical heat flux has been exceeded. If one used the empirical correlations of the past section, it is clear these are applicable between the axial location of CHF, (Figure 8.1), and the location where the equilibrium quality, equals one (

1,

). Remember, these past correlations are based on the assumptions that

thermodynamic equilibrium exists between the vapor and liquid, and one temperature, quality,

, with

, determines the thermodynamic state of the fluid. However, if one considers the more

realistic situation where the liquid droplets are at a different temperature, must again determine the "true" quality variation in difference to

, than the vapor

, one

to then determine the wall

temperature given the heat flux. This situation is analogous to our discussion in subcooled boiling when = 0 but actually X '( z) > 0 now in the post-CHF region, but X ' ( z < 1) (Figure 8.1). Therefore, our first objective is to determine the axial variation of this "true" quality, and the point where 1 (i.e., ). Consider the case of a constant heat flux. Now in this non-equilibrium approach we assume that the heat flux can be divided into two portions; one directly heating the vapor, q" g , and one directly into the liquid, q"f , causing it to evaporate.

where we define e as

and it is assumed for simplicity that e = f ( Z ). If we realistically consider the temperature difference of and

, then one must realize that e is actually a function of position; for a constant e we get

Now from a one-dimensional energy balance on the channel the thermodynamic quality is given by

where for

= 1 we can find the location of

Non-Equilibrium Empirical Models

Now we can use the same energy balance method to find X '( z)

where once again we only use the portion of the total heat flux which goes into evaporation. The location where X '( Z ) = 1 is defined as and is given by

so that we can solve for this fraction of the total flux, e, as

where we can correlate e based on measurements of these quantities. Once we have a correlation for e we can derive an expression for the heat transfer coefficient from it definition. Note that the heat transfer coefficient would be different between the wall-vapor and wall-liquid. Groeneveld has suggested such a correlation for e given by

This approach is not necessarily recommended compared to others, because although it includes some degree of realism it still remains quite empirical for the key variable.

Next: Semi-Theoretical Models Up: POST-CHF HEAT TRANSFER Previous: Empirical Correlations ll contents © Michael L. Corradini


Semi-Theoretical Models

Next: Transition Boiling Up: POST-CHF HEAT TRANSFER Previous: Non-Equilibrium Empirical Models

8.5. Semi-Theoretical Models A comprehensive theoretical model of heat transfer in the post-CHF region must take into account the various paths by which it transferred from the surface to the bulk vapor phase. Six separate mechanisms may be identified: 1. heat transfer from surface to liquid droplets which impact with the wall ("wet" collisions) 2. heat transfer form the surface to liquid droplets which enter the thermal boundary layer but which do not "wet" the surface ("dry" collisions). 3. convection heat transfer from the surface to the bulk vapor 4. convective heat transfer from the bulk vapor to suspended droplets in the vapor core 5. radiation heat transfer from the surface to the liquid droplets 6. radiation heat transfer from the surface to the bulk vapor. One of the first semi-theoretical models proposed was that of Bennett et al., (1964) which is a one-dimensional model starting from known equilibrium conditions at the point of CHF. It was originally assumed that there is negligible pressure drop along the channel by Groeneveld (1974) revised the equations to allow for pressure gradient and flashing effects. It was also assumed that droplets could no longer approach the surface. Therefore, mechanisms (a) and (b) were not considered. More recently Iloeje et al, (1974) have proposed a three step model taking into account mechanisms (a), (b) and (c). The physical picture postulated by Iloeje is shown in Figure 8.2. Liquid droplets of varying sizes are entrained in the vapor cor and have a random motion due to interactions with eddies. Some droplets arrive at the edge of the boundary layer with sufficient momentum to contact the wall even allowing for the fact that, as the droplet approaches the wall, differential evaporation coupled with the physical presence of the wall leads to a resultant force trying to repel the droplet (Gardner, 1974). When the droplet touches the wall a contact boundary temperature is set up which depends on the initial droplet and wall temperature and on

If this temperature is less than some limiting superheat

for the liquid then heat will be transferred, first by conduction until a temperature boundary layer is built up sufficiently to satisfy the conditions necessary for bubble nucleation. Bubbles will grow within the droplet Rao (1974) ejecting part of the liquid back into the vapor stream. The remaining liquid is insufficient in thickness to support nucleation and therefore, remains until it is totally evaporated. The surface heat flux transfered by this mechanism can be arrived at by estimating the product of the heat transferred to a single drop and the number of droplets per unit time and per unit area which strike the wall. Iloeje et al., attempted to quantify the various mechanisms identified above and finally arrive at a somewhat complex expression for the droplet-wall contact heat flux ( ). The droplet mass flux to the wall can be estimated from one of a number of turbulent deposition models

Semi-Theoretical Models

Hutchinson (1971). It is important to appreciate that at very low values of

the heat flux (

) predicted from the Iloeje model must coincide with the droplet deposition flux contribution of the Hewitt film flow model of critical heat flux. The trends with vapor quality and mass velocity appear qualitatively correct but a rigorous quantitative comparison with, for example, the prediction of Whalley et al., (1973) is needed for a range of working fluids. Two basic approaches have been taken to estimate the heat flux (

) to droplets entering the thermal

boundary layer but which do not touch the wall. This heat flux can be estimated as the product of the heat flux that would occur across a vapor film separating the droplet from the heating surface and the fractional area covered by such droplets. This approach has been adopted with slight modifications by Iloeje (1974), Groenveld (1974), Plummer (1974) and Course (1974). All of these approaches follow the original treatment by Bennett. Thus, one should begin here for such a theoretical approach.

Next: Transition Boiling Up: POST-CHF HEAT TRANSFER Previous: Non-Equilibrium Empirical Models ll contents © Michael L. Corradini


Transition Boiling

Next: Transition to Film Boiling Up: POST-CHF HEAT TRANSFER Previous: Semi-Theoretical Models

8.6. Transition Boiling Various attempts have been made to produce correlations for the "transition" film boiling region corresponding to the left of the minimum in the boiling curve ( Figure 5.1). These correlations have usually been combined with expressions for the stable film boiling region. The difficulty with this approach is that in flow boiling the physical meaning of this minimum is difficult to understand. Groeneveld and Fung have tabulated the various correlations available for forced convection transition boiling of water. The earliest experimental study was that of McDonough et al., who measured transition film boiling heat transfer coefficients for water boiling over the pressure range 800-2000 psia inside a 0.15" ID tube heated by NaK. The correlation they offered was:

(British Engr. Units) Tong (1974) suggested the following equation for combined transition and s table film boiling at 2000 psia with wall temperatures less than

.

This, in turn, was revised to cover both transition and film boiling regions

This equation was derived from 1442 data covering a pressure of 1000 psia, G from 0.28x10 6 to 3.86x106 lb/hr.ft2 and = proposed for low pressure situations.

from 65 to 985°F. A similar but separate equation was



Transition to Film Boiling

Next: Observations Up: POST-CHF HEAT TRANSFER Previous: Transition Boiling

8.7. Transition to Film Boiling Consider the case when the mass velocity decreases to very low value. As G decreases the flow situation becomes more like pool boiling where the post-CHF regime becomes film boiling. Because the liquid is displaced from the heating surface by a vapor film and the uncertainties associated with bubble nucleation are removed, film boiling is very amenable to analytical solution. This is similar to condensation where stratified flow occur with a liquid film. In general, the problem is treated as an analogue of film-wise condensation and solutions are available for horizontal and vertical flat surfaces, and also inside tubes under both laminar and turbulent conditions with and without interfacial shear. It is not, however, proposed to review all these solutions here and the reader is referred to the comprehensive reviews of Clements and Colver (1970), of Hsu (1972) and of Bressler (1972). ll contents © Michael L. Corradini


Observations

Next: CONDENSATION Up: POST-CHF HEAT TRANSFER Previous: Transition to Film Boiling

8.8. Observations The post-CHF heat transfer regime has been summarized with three different approaches to estimate the heat transfer coefficient. We would recommend the empirical equilibrium or non-equilibrium approach for an estimate of post-CHF heat transfer rates. However, if the wall temperature and degree of superheat (or heat flux for the other boundary condition) are of crucial importance of semi-theoretical approach might be considered for detail analysis. This latter recommendation will require new data for specific geometries to determine the droplet-gas and droplet-wall heat transfer rates accurately. References q

H. Amm and G. Ulrych, "Comparison of Measured Heat Transfer Coefficients During Reflooding a 340-rod Bundle and Those Calculated From Current Heat Transfer Correlations." Paper E7, European Two-Phase Flow Meeting , Harwell, 1974.

q

A.W. Bennett, "Heat Transfer to Mixtures of High Pressure Steam and Water in an Annulus." AERE-R4352, 1964.

q

R.G. Bressler, "A Review of Physical Models and Heat Transfer Correlations for Free Convection Film Boiling." Adv Cryogenic Engr , 17, 382-406, 1972.

q

L.D. Clements and C.P. Colver, "Natural Convection Film Boiling Heat Transfer," Indstr. Engr Chem 62(9), 26-46, 1970.

q

A.F. Course and H.A. Roberts, "Progress With Heat Transfer to a Steam Film in the Presence of Water Drops - a First Evaluation of Winfrith SGHWR Cluster Loop Data," AEEW-M1212. Paper presented at European Two-Phase Flow Group Meeting , Harwell, 1974.

q

G.C. Gardner, "Evaporation and Thermophertic Motion of Water Drops Containing Salt in a High Pressure Steam Environment." Paper presented at European Two-Phase Flow Conference , Harwell, England, 1974.

q

E.M. Greitzer and F.H. Abernathy, "Film Boiling on Vertical Surfaces." Int J Heat Mass Transfer , 15, 475-491, 1972.

q

D.C. Groeneveld, "Effect of a Heat Flux Spike on the Downstream Dryout Behaviour," J of Heat Transfer , 121-125, 1974.

q

D.C. Groenveld, "Post-Dryout Heat Transfer at Reactor Operating Conditions." AECL-4513. Paper presented at the Nat Topical Meeting on Water Reactor Safety, ANS, Salt Lake City UT, 1973.

q

Y.Y. Hsu, "Review of Film Boiling," Adv in Cryogenic Engr 17, 361-381, 1972.

q

P. Hutchinson, G.F Hewitt and A.E. Dukler, Chem Eng Sci 26 , 419, 1971.

q

O.C. Iloje, D.N Plummer, W.M. Rohsenow, P. Griffith, "A Study of Wall Rewet and Heat Transfer in Dispersed Vertical Flow." MIT Dept of Mech Engr Report , 72718-92,1974.

q

D.N. Plummer, O.C. Iloeje, W.M. Rohsenow, P. Griffith and E. Ganic, "Post Critical Heat

Observations

Transfer to Flowing Liquid in a Vertical Tube." MIT Dept of Mech Engr Report 72718-91, 1974. q

P.S. Rao and P.K. Sarma, "Partial Evaporation Rates of Liquid Particles on a Hot Plate Under Film Boiling Conditions," Con J of Chem Engr , 52 (3), 415-419, 1974.

q

L.S. Tong, "Film Boiling Heat Transfer at Low Quality of Subcooled Region." Proceedings 2nd Joint USAEC-EURATOM Two-Phase Flow Meeting , Germantown. Report CONF-640507, p 63, 1974.

q

G.B. Wallis and J.G. Collier "Two-Phase Flow and Heat Transfer." Notes for a Summer Course, July 15-26, Thayer School of Engineering, Dartmouth College, Hanover, NH, USA. III, 33-46, 1968.

q

P.B Whalley, P. Hutchinson and G.F. Hewitt, "The Calculation of Critical Heat Flux in Forced Convection Boiling," AERE-R7520, 1973.

Next: CONDENSATION Up: POST-CHF HEAT TRANSFER Previous: Transition to Film Boiling ll contents © Michael L. Corradini


Condensation

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Next: Basic Processes of Condensation Up: CONDENSATION Previous: Observations

9. CONDENSATION Condensation phenomena occur in many industrial applications. In this section the focus is on the determination of the condensation heat transfer coefficient and the overall energy balance is left to the reader. There are two idealized models of condensation (i.e., filmwise and dropwise). The former occurs on a cooled surface which is easily wetted. The vapor condenses in drops which grow by further condensation and coalesce to form a film over the surface, if the surface-fluid combination is wettable; if the surface is non-wetting rivulets of liquid flow away and new drops then begin to form. This review and discussion will mainly deal with filmwise condensation. The phenomena of dropwise condensation results in local heat transfer coefficients which are often an order of magnitude greater than those for filmwise condensation. Even though condensation phenomena can be classified into these categories of dropwise and film condensation the initial period of condensation would evolve into a film and probably would not affect the overall pressure-temperature response unless drop condensation is promoted (Slaughterbeck, 1970). Rates of heat transfer for film condensation can be predicted as a function of bulk and surface temperatures, total bulk pressure, surface and liquid film characteristics, bulk velocity and the presence of noncondensible gases. Even though film condensation has been investigated extensively, the majority of these studies were devoted to laminar film condensation (laminar bulk flow and laminar film). Since the vapor flow in heat exchange equipment may be turbulent, models and recent data are also reviewed for the condensation flux with a turbulent mixture flow. A simple engineering correlation or model is preferred many times for use in engineering design studies and with existing computer system analyses (Schmitt, et al., 1970; Tagami, 1965). Previous theoretical and experimental investigations are reviewed, in particular, the effects of the presence of noncondensable gases and of the vapor velocity. These effects along with the effects of geometry and scale are of major interest at this time. Because the flow regime for the condensation heat transfer is well defined (stratified flow), the reader will find a much greater propensity for detailed mathematical analyses for simple geometries. Condensation on a vertical or horizontal flat plate, which can be extended to any arbitrary geometry, is the main focus of this discussion, because of its generality. The detailed review for film condensation outside a tube can be found in the work of Lee (1982). The usual modification is to replace the length scale, L, by tube diameter, D, with a slight change in the proportionality constant. Table 1 is provided as a summary of the work on condensation at the present time.

9.1. Basic Processes of Condensation 9.2. Theoretical Developments of Condensation 9.3. Experimental Investigations 9.4. Separate Effects and Large Scale Tests

Condensation

9.5. Observations next

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Basic Processes of Condensation

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9.1. Basic Processes of Condensation By analogy with the process of evaporation, liquid may form in one of three ways corresponding to the existence of an unstable, metastable or stable equilibrium state. Let us briefly look at each one of these to understand the condensation process. In practical engineering design of heat exchange equipment the stable condensation situation needs to exist. Consider a liquid drop of radius, r *, in equilibrium with its surrounding vapor at a system temperature, , and pressure, . The vapor pressure, under equilibrium conditions, is higher than the vapor pressure,

, for a planar surface, and this difference is given by

where R is the vapor gas constant and is the surface tension between the liquid and vapor. With the local condition of mechanical equilibrium for the liquid droplet and its pressure, ,

one can find the liquid pressure in the droplet as

One can also use the Clausius-Clapeyron relation to calculate for this simple situation the saturation temperature, , of the droplet above the vapor temperature for maintaining this equilibrium

Analogous to boiling, the rate of nucleation of these liquid droplets depends on whether one considers homogeneous nucleation or heterogeneous nucleation processes. For homogeneous nucleation the rate expression, dn / dt , is quite similar to that for boiling,

where r * can be found either from Eq.2 or 3. The term, volume and

is a collision frequency

, is the number of vapor molecules per unit

for vapor collisions given by


where m is the mass of one vapor molecule. One should note that in a similar fashion to boiling this nucleation rate is altered if it occurs on solid surfaces since the work required to form a critical size nuclei (r *) is reduced due to wetting of the solid surface. Now in reference to the more stable situations of a vapor condensing on a planar surface covered by its own liquid one must consider the local mass transfer situation. Consider a pure saturated vapor at a pressure, , and a temperature, , condensing on its own liquid phase whose surface temperature is . The phenomenon of such an interface mass transfer can be viewed from the standpoint of kinetic theory as a difference between two quantities; a rate of arrival of molecules from the vapor space towards the interface and a rate of departure of molecules from the surface of the liquid into the vapor space. When condensation takes place the arrival rate exceeds the departure rate. During evaporation the reverse occurs, and during an equilibrium the two rates are equal and there is no net mass transfer. From kinetic theory it can be shown that, in a stationary container of molecules, the mass rate of flow (of molecules) passing in either direction (to right or left) through an imagined plane is given by

where

= flux of molecules (mass per unit time per unit area) M =molecular weight R = universal gas

constant P and T = pressure and temperature related by the saturation line. Equation 7 is the starting point for many theories of interfacial phase change. In general it can be stated that the net molecular flux through an interface is the difference between these fluxes in the directions from gas to liquid and vice-versa,

Since the condition close to the surface is not one of static thermal equilibrium, for any significant rate of evaporation or condensation, it is really not meaningful to make use of the thermostatic pressure and temperature on each side of the interface. Rather there is a concentration and therefore, a temperature difference, , across this interface which drives the mass transfer. Strictly speaking one should solve the Boltzman transport equation with appropriate boundary conditions and asymptotes which are conditions of thermal equilibrium at several mean free path distances from the interface. However, some considerable success for engineering purposes has been achieved by using simplified kinetic theory techniques and applying correction factors to the resulting predictions of this mass transfer and the associated temperature difference. In most practical situations the energy removal rate from this interface controls the condensation rate. Only in the presence of noncondensable gases (continuum) or at low pressure (non-continuum) is this temperature difference, , important to consider. We will investigate the case of the presence of noncondensable gases during condensation; one can get a physical feeling of the magnitude of this temperature difference. next

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Theoretical Developments of Condensation

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9.2. Theoretical Developments of Condensation Stationary Pure Vapor

For filmwise condensation of a "stationary" saturated vapor, Nusselt (1916) presented the first analytical solution for heat transfer on a plane surface (Fig.9.1 ) with the following assumptions (Collier, 1981): 1. the flow of condensate in the film is laminar, 2. the fluid properties are constant, 3. subcooling of the condensate may be neglected, 4. momentum convective changes through the film are negligible, 5. the vapor is stationary and exerts no drag on the downward motion of the condensate 6. heat transfer through the film is by conduction only.

The mean value of the heat transfer coefficient over the whole surface was given by

One should note here that for a tube, L is replaced by the tube diameter, D, and 0.943 becomes 0.725. This model had been extended to include the effects of Nusselt's assumptions. In particular, Bromley (1952) considered the effects of subcooling within the liquid film and Rohsenow (1956) also allowed for a non-linear distribution of temperature through the film due to energy convection. The results indicated that the latent heat of vaporization, , in Eq. (9) should be replaced by

However, it should be noted that in most engineering applications, the value of

is small (typically

less than 0.001) and can be neglected. Sparrow and Gregg (1959) removed assumption (4) and included inertia forces and a convection term within the condensate film by using a boundary layer treatment for the condensate film. For common fluids with Prandtl numbers around and greater than unity, inertia effects are negligible for values of less than 2.0 For liquid metals with very low Prandtl numbers, however, the heat transfer coefficient falls below the Nusselt prediction with increasing

when

is greater than 0.001.


Poots and Miles (1967) have looked at the effect of variable physical properties (assumption 2) on vertical plates. More recently Koh et al. (1961) and Chen (1961) included the influence of the drag exerted by the vapor on the liquid film. Both results show that the interfacial shear stress can reduce heat transfer due to the effect of "hold up" of the condensate film for low values of Pr , but this effect is small and steadily decreases with increasing Pr for Prandtl number greater than unity. As a conclusion for pure steam-water condensation ( Pr ;SPMgt ; 1), Nusselt's assumptions can be accepted for a stationary vapor without noncondensable gas in practical engineering situations. Moving Pure Vapor

The effects of vapor velocity and its associated drag on the condensate film have been found to be significant in many practical problems. For the case of vapor flow parallel to a horizontal flat plate, Cess (1960) presented uniform property boundary layer solutions, obtained by means of similarity transformations by neglecting the inertia and energy convection effects within the condensate film and assuming that the interfacial velocity was negligible in comparison with the free stream vapor velocity. Shekriladze and Gomelauri (1966) simplified the problem and also considered the case of an isothermal vertical plate with similar assumptions (1973). Mayhew et al. (1966, 1987) attempted to expand Nusselt's simple approach to take account of vapor friction as well as momentum drag. South and Denny (1972) proposed an interpolation formula for the interfacial shear stress in a simplified manner as Mayhew. However, such an interpolation formula only led to a small difference in the heat transfer rate. Jacobs (1966) used an integral method to solve the boundary layer by matching the mass flux, shear stress, temperature and velocity at the interface. The inertia and convection terms in the boundary layer equations of the liquid film were neglected. The variation of the physical properties and the thermal resistance at the vapor-liquid interface were also neglected. Since Jacobs used an incorrect boundary condition for the vapor boundary layer, Fujii and Uehara (1972) solved the same problem with the correct boundary condition. In addition, the velocity profile in the vapor layer was taken as a quadratic. They presented the numerical results and their approximate expressions for the cases of free convection, forced convection, and mixed convection. The results show good agreement with numerical calculations and with Cess' approximate solution (1960). The current recommendation in this area is the model developed by this latter work as a best estimate. One should be cautious as the Nusselt number increases because this implies a higher vapor and film flow with accompanying film turbulence, not accounted for in these models. For design purposes the recommendation is to use Nusselt laminar film model (Equ. 9.9), since it will predict a slightly lower heat transfer coefficient, with a thicker condensate film. Stationary Vapor with a Noncondensable Gas

A noncondensable gas can exist in a condensing environment and leads to a significant reduction in heat transfer during condensation. A gas-vapor boundary layer (e.g., air-steam) forms next to the condensate layer and the partial pressures of gas and vapor vary through the boundary layer as shown in Fig. 9.2. The buildup of noncondensable gas near the condensate film inhibits the diffusion of the vapor from the bulk mixture to the liquid film and reduces the rate of mass and energy transfer. Therefore, it is necessary to solve simultaneously the conservation equations of mass, momentum and energy for both the condensate film and the vapor-gas boundary layer together with the conservation of specied for the vapor-gas layer. At the interface, a continuity condition of mass, momentum and energy has to be


satisfied. For a stagnant vapor-gas mixture, Sparrow and Eckert (1961) and Sparrow and Lin (1964) solved the mass, momentum and energy equations for laminar film condensation on an isothermal vertical plate by using a similarity transformation. Sparrow and Eckert (1961) considered the notion of the vapor-gas mixture from the downward motion of the condensate film, whereas Sparrow and Lin (1964) included free convection arising from density differences associated with composition differences. These analyses indicated that the condensing rate is dependent on the bulk gas mass fraction, the vapor-gas mixture Schmidt number,

and

. The numerical calculations show that the effect of the

noncondensable gas increases with increasing Schmidt number and increasing value of

. Since

free convection arises from both the temperature and the concentration difference in a boundary layer, it is important when the vapor and the noncondensing gas have significantly different relative molecular weights and that its importance increases with increasing bulk gas mass fraction and increasing values of . Minkowycz and Sparrow (1965, 1966) also included free convection arising from temperature differences. In addition, the effect of interfacial resistance, superheating, thermal diffusion and property variation in the condensate film and in the vapor-gas mixture were considered and concluded to be less important except for superheating. To reduce the computation time, Rose (1969) presented an approximate integral boundary layer solution assuming uniform properties except for density in the buoyancy term. Plausible velocity and concentration profiles for the vapor-gas boundary layer were used and it was assumed that these two layers had equal thickness. The results showed quite good agreement with those of Minkowycz and Sparrow and this is recommended for use. Moving Vapor with a Noncondensable Gas

For a laminar vapor-gas mixture case, Sparrow, et al. (1967) solved the conservation equations for the liquid film and the vapor-gas boundary layer neglecting inertia and convection in the liquid layer and assuming the stream-wise velocity component at the interface to be zero in the computation of the velocity field in the vapor-gas boundary layer. Also a reference temperature was used for the evaluation of properties. The results showed that the effect of noncondensable gas for the moving vapor-gas mixture case is much less than for the corresponding stationary vapor-gas mixture. A moving vapor-gas mixture is considered to have a "sweeping" effect, thereby resulting in a lower gas concentration at the interface (compared to the corresponding stationary vapor-gas mixture case). Also, the ratio of the heat flux with a noncondensable gas to that without a noncondensable gas was calculated to be independent on the bulk velocity. The computed results reveal that interfacial resistance has a negligible effect on the heat transfer and that superheating has much less of an effect than in the corresponding free convection case. Koh (1962) and Fujii et al. (1977) solved this problem without the simplifying assumptions used by Rose (1969) except for uniform properties and showed good agreement with the approximate analysis. Instead of solving a complete set of the conservation equations, Rose (1980) used the experimental heat transfer result for flow over the flat plate with suction (1979). Denny et al. (1971, 1972) also considered the case of downward vapor-gas mixture flow parallel to a vertical flat plate. They presented a numerical solution


of similar mass, momentum and energy equations for a vapor-gas mixture by means of a forward marching technique. Interfacial boundary conditions conditions at each step were extracted from a locally valid Nusselt type analysis of the condensate film. Local variable properties in the condensate film were evaluated by means of the reference temperature concept, while those in the vapor-gas layer were treated exactly. Asano et al. (1978) treated the condensate film as in the Nusselt analysis but ass umed the interfacial shear stress was the same as that for single-phase flow over an impermeable plate. The analytical model described above was solved using only a laminar vapor-gas (or pure vapor) boundary layer except for Mayhew (1966). Whitley (1976) proposed a simple model, which uses the analogy between heat and mass transfer for forced convection condensation of a turbulent mixture boundary layer by neglecting the interfacial velocity and treating the surface of the condensate film to be smooth. Kim (1990) improved on Whitley's approach for forced convection and natural convection applications by extending it to a wavy/turbulent film. By using well accepted correlations for a flat plate geometry, the solution procedure is simplified to computing the condensate film thickness and the local Reynolds and Sherwood numbers in the downstream direction. This leads to a computationally efficient efficient solution, which can be easily expanded to include more detailed models of the condensate film. Total heat flow is controlled by the gas phase heat transfer and the heat flow through the condensate film. Therefore, the total condensation heat transfer coefficient can be written as:

Gas phase heat transfer consists of convection heat transfer and the latent heat released as a result of mass transfer. Radiation heat transfer can be neglected in the temperature range of interest (30C). Hence hgas is given by

where

is defined as:

The analogy between momentum-heat-mass transfer is used in order to find the heat and mass transfer coefficients along the plate. The friction factor and Stanton number are correlated by

For a smooth surface, the local skin friction factor can be correlated with the local Reynolds number, and a result like Whitley is obtained

By substituting equation 15 into equation 14, the local Nusselt number is obtained,


Utilizing Reynold's analogy between heat and mass transfer, equation 16 is modified to obtain the local Sherwood number:

The turbulent Prandtl and Schmidt numbers in these equations can be replaced with the equations derived by Jischa and Rieke (Kim, 1990). They derived the following results from transport equations of turbulent kinetic energy, heat flux and mass flux, as

where coefficients

and

were fitted to experimental data. The recommended values are given in the

following table. Finally, the local Nusselt and Sherwood numbers for a smooth surface can be obtained by substitution of equations 18 and 19 in equations 16 and 17,

Heat and mass transfer coefficients can now be solved from 20 and 21, respectively. The condensation heat transfer coefficient hcond can then be obtained by substituting the following definition into equation 13,

where G is the mass transfer coefficient and W is the mass fraction. The boundary layer thickness is reduced due to the apparent suction effect of the condensation process. This leads to larger temperature and concentration gradients close to the interface that, consequently, increase heat and mass transfer rates. The following correction factors were implemented to account for the suction effect,

where

is defined as:


where

is defined as:

where

is defined as:

The droplets or waves that form on the condensate interface can increase the shear stress and lead to enhanced turbulent mixing at the interface. The effective surface area of the interface is also increased due to droplets and waves. The case where waves and droplets are present was modelled as a rough surface (Kim, 1990). Kim integrated the non-dimensional temperature profile and expressed the resulting Stanton number as a function of the turbulent Prandtl number, friction factor and a roughness parameter,

where the roughness parameter

where

is based on experimental correlation,

is the shear stress at the wall.

The Nusselt number can be solved from equation 20,

The Sherwood number can be obtained utilizing the Reynold's analogy and equations 31 and 32,


In the aforementioned equations, the effect of the condensate interface structure is included in the surface roughness parameter . In Kim's original model the interfacial waves were considered to influence the roughness parameter. A condensate film Reynolds number of 100 was used as a critical threshold value for wave formation. Kim used Wallis' correlation to account for waviness of the interface by linking it to the condensate film thickness . The current recommendation in this area would be to use this simple engineering correlation of Kim as an estimate for most situations. If more exact estimates are necessary then other more detailed fluid mechanics analyses could be used for the bulk gas flow. next

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Experimental Investigations

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9.3. Experimental Investigations Stationary Pure Vapor

A number of earlier experimental results (before 1950) show some difference with the predictions of the Nusselt theory (McAdams, 1954). The differences can be attributed to one or more of the following reasons: 1) significant forced-convection effects; 2) presence of noncondensable gas; 3) waviness and turbulence within the condensate film; 4) presence of dropwise condensation. More recently, Mills and Seban (1967) condensed steam on a copper vertical flat plate and Slegers and Seban (1969) conducted some experiments with n-butyl alcohol. These tests support the Nusselt theory for pure stationary vapor condensation. Moving Pure Vapor

Mayhew and Aggarwal (1973) experimented with pure steam condensing on a flat surface. To avoid air in-leakage, the experiments were carried out at pressures slightly above atmospheric. Good agreement is obtained between the experimental results and the calculated values by their own theory. It is very interesting to note that the results for the counter-current flow cases are always appreciably higher than those predicted by the author's own model and indeed always higher than the corresponding co-current velocity vapor values. The reason was investigated and explained as follows in the original paper; An obvious explanation was provided by dye-injection tests which showed that, with counterflow, no laminar film flow could be achieved. The film was torn off the plate (i.e. flooding occurred at quite moderate values of vapor velocity. Similar observations with parallel flow confirmed that the film was always both laminar and smooth. From work with noncondensing films it was expected that rippled flow would be encountered over part of the surface at the higher velocities used. In fact remarkable smooth films were observed suggesting that mass transfer, and possibly also surface tension effects on the non-isothermal film, must have had a stabilizing effect. More recently Asano et al. (1978) reported their data for the condensation of pure saturated vapors on a vertical flat copper plate and showed good agreement with the authors' own model. Stationary Vapor with a Noncondensable Gas

Perhaps the earliest definitive experiment of the effect noncondensable gas was done by Othmer (1929), who introduced air mole fractions of up to 11 The experimental heat transfer coefficient data of Hampson (1951) and Akers et al. (1960) were 20 Al-Diwany and Rose (1973) reported heat transfer measurements for steam condensing in the presence at air, argon, neon and helium. The vapor-gas mixture was passed into the steam chamber via flow straighteners which provided uniform flow of the mixture towards the condensing surface so as to preclude forced convection effects. The experimental data for steam-air,

Experimental Investigations

steam-argon and steam-neon showed satisfactory agreement with the predicted theoretical values of Sparrow but for steam-helium showed a lower value than the theoretical values. Recently, DeVuono and Christensen (1984) reported their experiment of natural convection of a steam-air mixture at pressures above atmospheric to 0.7 MPa to investigate the effect of pressure. The experiments were performed on a horizontal copper tube with 7.94 cm O.D. by 1.22 m of active condensation length. The tube was mounted in a cylindrical pressure vessel 1.52 m O.D. by 3.35 m long. Saturated steam was supplied by an external source and allowed to diffuse to the tube resulting in steady-state, natural convection conditions. An expression, which is a function of , percent noncondensable gas by volume (Y

where

MPa

0.0 < Y < 14.0 C. Even though this experiment was done over a large range of pressure for a containment analysis and showed a significant effect of pressure, the pipe geometry and length scale make it questionable to apply this correlation to a large scale system. Unfortunately, the experiment results were not compared with any other theoretical model. Moving Vapor with a Noncondensable Gas

Rauscher, Mills and Denny (1974) performed experiments of filmwise condensation from steam-air mixtures undergoing forced flow over 0.74 in. O.D. horizontal tube. The heat transfer coefficient at the stagnation point was reported for bulk air mass fractions 0 - 7 next

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Separate Effects and Large Scale Tests

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9.4. Separate Effects and Large Scale Tests The previous experiments were separate effect tests in which model development was an integral part of the research. There have been other data collected in which direct empirical correlations have resulted or in which analysis is not completed. These relevant experiments are summarized in Table 9.2. We consider both separate effects data and large scale tests.

Separate Effects Experiments Typically, the heat transfer coefficients have been observed to decrease significantly with increased noncondensable gas mass fraction under various conditions and test geometries. The degradation of heat transfer is caused by the accumulation of a noncondensable gas layer near the cold wall through which the vapor must diffuse. Buoyancy Forces in a Stagnant Gas Mixture

Cho and Stein (1988) investigated condensation of steam in the presence of air and helium on a small horizontal plate facing down with stagnant flow conditions. The results with air were successfully modelled by taking into account the buoyancy forces caused by different molecular weights of participating gases. Since helium is a lighter gas than steam, a suppression of natural convection was expected. However, the tests with a moderate helium content showed higher heat transfer rates than predicted by a diffusion analysis. A convective heat transfer mechanism caused by fog and mist formation was hypothesized. Fog that formed near the cold surface was observed to form localized swirls and generally move in downward direction. These visual observations seemed to confirm the presence of hypothesized natural circulation. A similar geometrical arrangement was also used by Kroger and Rohsenow (1968). Potassium vapor was condensed in the presence of argon and helium. The diffusion theory successfully predicted the experimental data with helium. In the case of argon, experimental results indicated a superimposed natural circulation flow. Vapor phase instabilities and secondary flow cells were also reported by Spencer, Chang and Moy (1970). They investigated the condensation of Freon-113 in the presence of helium, nitrogen and carbon-dioxide on a vertical surface under stagnant conditions. Both visual observations and heat transfer measurements were performed. The results, indicate a modest effect of noncondensable gas molecular weight. Dehbi (1991) studied the influence of an air/helium noncondensable mixture on the condensation heat transfer under stagnant conditions. The condensing surface consisted of a 3500 mm long vertical copper tube. Helium mass fraction was varied from 1.7 to 8.3 weight percent. Dehbi reported that the heat transfer rates decreased with a increased helium mass fraction. When the helium mass fractions were relatively high, sharp stratification patterns were observed as helium migrated to the top of the test vessel and air/stream mixture stayed at the bottom. The natural convection patterns in all these data suggest that scale dependence must be strongly considered. Forced Flow

As mentioned previously Dallmeyer (1970) studied condensation of

and

on a vertical plate in the presence of air. The results

showed that heat transfer rates increased with the Reynolds number and the vapor concentration. Dallmeyer performed detailed measurements of the velocity, temperature and concentration profiles near the wall with laminar and turbulent flow. Measured profiles illustrated the apparent suction effect of the condensation that increases the gradients near the wall and thus leads to higher heat and mass transfer rates in the laminar flow region. Condensation process and, in particular, high condensate mass fluxes were observed to dampen the turbulence level in the turbulent region. Barry (1987) performed condensation experiments with the mixture of steam and air. His apparatus consisted of a horizontal plate facing upwards. The velocity and mass ratio range was chosen so that it covered the conditions that are likely to exist in a containment during an accident in a developing parallel flow situation. Barry's results show expected the effects of velocity and the mass ratio as mentioned previously. Kutsuna, Inoue and Nakanishi (1987) studied condensation of steam on a horizontal plate (facing up) in the presence of air. They also reported increased heat transfer rates due to forced convection. Their results, indicate the expected effects of noncondensable gas concentration and velocity on the heat transfer coefficients. Tests were performed with higher steam content than the tests by Barry, and consequently, the heat transfer coefficient were also significantly higher. When the tests are performed with a high steam content, the heat transfer results become very sensitive to the air content. This may be the reason why the data scatter is markedly higher than in Barry's experiment. Unfortunately, the experimental uncertainties were not discussed. Pressure

Several workers have investigated the effect of the system pressure with stagnant flow conditions (no forced convection). The heat transfer rates are reported to increase with system pressure (e.g., Gerstmann, 1964), because the densities of gas components increase with pressure. Cho and Stein (1988) reported that an increase in the system pressure (0.31 MPa to 1.24 MPa) also influenced the mode


of condensation on a downward facing surface with helium as the noncondensable gas. Higher pressures led to mixed mode of condensation (filmwise and dropwise condensation coexisting) with a downward facing polished surface. Nuclear reactor safety evaluations have prompted studies of the effect of pressure on condensation heat transfer under transient conditions (large concentrations of noncondensable gases). Robinson and Windebank (1988) studied the effect of pressure in the range of 0.27-0.62 MPa with an air/stream mixture. The noncondensable gas mass fraction was varied from 24 to 88 percent. The heat transfer rates were measured with a cooled disk that was placed inside a pressure vessel. The results show that heat transfer rates increase with pressure and decrease with the mass ratio of noncondensable gas. Robinson and Windebank noted that the velocity field due to the steam injection might have had an effect on their results. The magnitude of the induced velocities within the vessel were stated to be below 2 , although no detailed measurements were performed. Similar tests were also conducted by Dehbi. Heat transfer rates were measured at three different pressures (0.15, 0.275 and 0.45 MPa). The noncondensible gas mass fraction in the tests ranged from 25 to 90 percent. The experimental apparatus consisted of a three meter and one half long cooled tube (0.038 m Dia) in a pressure vessel. The motivation behind using a relatively large vertical dimension was to simulate the length scale of internal containment structures. Surprisingly narrow pressure vessel was used (L/D = 10). This led to difficulties to establish homogeneous test conditions in the vessel. In the tests, the mass ratio of air was 8-33 percent greater in the upper part of the vessel than in the lower part. Secondly, the flow field created by the natural convection may have been affected by the sidewalls. Therefore, the results by Dehbi have some unspecified uncertainty. He confirmed the observations of Robinson that heat transfer rate increases with system pressure. Condensate Film Structure

Several studies have been done to address the effect of condensate film characteristics on the heat transfer rates. The condensate film characteristics depend on its flow field and the nature of the condensing surface, e.g. roughness, wetting and orientation. Forced flow induces interfacial instabilities that increase the heat transfer rates by reducing the thickness of gas phase laminar sublayer and enhancing the mixing of both the liquid (condensate film) and gas phase. Barry (1987) studied the effects of interfacial structure caused by shear. Since the condensation length was relatively short, a film injection system was used to produce a condensate film that was sufficiently thick for measurements. The qualitative results suggested that enhanced mixing, which is caused by the interfacial film structure, somewhat compensated for the effect of the noncondensable gas. The surface finish has a major effect on the mode of condensation for a downward facing surface and it is the wetting characteristics of the surface that ultimately determine this. Dropwise condensation is likely to exist on non-wetting surfaces and filmwise condensation is likely on wetting surfaces. In dropwise condensation mode with polished metal surfaces, the heat transfer characteristics are likely to change due to oxidation of the surface or tarnishing. Thus, one cannot precisely know the wetting characteristics as surface aging occurs. Gerstmann and Griffith (1967) studied the condensation of pure, stagnant Freon-113 and water vapor at atmospheric pressure. Heat transfer measurements and visual observations of the interfacial behavior were made. Gerstmann and Griffith observed several distinct flow regimes in the condensate film depending on the angle of inclination. Unstable condensate film with pendant drops and lengthwise ridges existed at the horizontal position. The characteristic length scale of these formations was on the order of the Taylor wavelength. The ridge formation was associated with the presence of a noncondensable gas. When the surface was tilted, the condensate waves developed into "roll waves." The waves were fully developed at about 20 degrees of inclination. The influence of the condensate film on the heat transfer rates was successfully analyzed using an assumption of quasi-steady state with force and energy balance equations. Generally, heat transfer rates were found to decrease with increasing inclination angle. The presence of lengthwise ridge waves induced by noncondensable gas was also reported by Spencer et al. (1970). However, no discussion of the effect of the ridge waves on the heat transfer rates was given.

Integral and Large Scale Experiments In addition to the heat transfer data, integral and large experiments provide valuable background information about physical conditions such as gas concentrations, temperatures, prevailing flow fields and system pressures in a particular circumstance. The data from the experiments can be used as integral benchmarks for models. In the subject area of condensation these data usually involve the study of condensation in large containment structures for nuclear reactor safety. The first integral experiments were performed by Jubb and Kolflat (1960). The results of these integral tests were correlated with the experimental parameters. However, some of the parameters that Jubb and Kolflat used were uniquely dependent on the experimental apparatus. Therefore, the resulting correlations were not generally applicable to anything other geometry. Uchida et al. (1965) and Tagami (1965) performed experiments using a model of a containment structure (3.4 meters dia and 6.4 meters in height). Three water cooled cylinders inside the containment structure were used as test surfaces. Uchida measured the post-blowdown heat transfer rates using a vertical surface with subcooling of C. The dimensions of the test surface were 0.14 by 0.3 meters (width and height). The pressure in the tests varied from 0.1 to 0.3 MPa. The heat transfer rates were reported to decrease with an


increasing concentration of noncondensable gas. Contrary to the findings reported in the separate effects section, Uchida et al., reported that the heat transfer rates depend only on the mass ratio and not on the molecular weight of the participating gases, local velocities or pressure. Uchida's and Tagami's results can be correlated in metric units as, Uchida:

Tagami:

where W is defined as the mass fraction of noncondensable gas. The geometrical aspects and the effect of velocity field were ignored. Therefore, caution should be used to extrapolate results from the correlations for the long sections of structural walls. Unfortunately, it is quoted and used in safety analyses. The CVTR test series was conducted using a full scale structure of a decommissioned nuclear power plant (Schmitt, 1970). The steam was injected through a diffuser (0.25 meters dia and 3 meters in height) located three meters above the operating floor. Three tests were conducted. The heat transfer rates into the wall were computed from the measured temperature profiles in the wall using the inverse conduction method. The velocity field was measured by ultrasonic anemometers. The experimental data from the CVTR test were used by Kim (1990) as one data set to benchmark his condensation model. The major conclusion from the analysis of these tests was that local gas velocities were needed to accurately predict the data. Historically, integral tests have been used to find simple correlations that would predict the heat transfer rates. These correlations have gained wide acceptance and are regularly used in safety analyses. In this light, it is surprising to find out that until recently, most of the data from these integral tests have been based on a very limited number of measurements of the prevailing conditions. Therefore, these correlations generally have a very limited value in making accurate predictions of heat transfer rates through different geometries.


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Observations

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9.5. Observations Condensation phenomena can be classified by the presence of noncondensable gas, the gas mixture velocity, the flow characterization (laminar or turbulent) of the gas mixture and the condensate film and the interface condition as shown in Table 9.1, which presented the summary of the theoretical and experimental investigations discussed. For all cases with a simple geometry except the turbulent gas mixture boundary layer and the wavy interface of both the pure vapor and the vapor-air mixture case, it is seen that numerical solutions of the conservation equations and more approximate analytical solutions of the conservation equations agree well with the corresponding experimental work. As the geometry of the condensing surface becomes more complex more prototypic experiments must be performed. Examples of these cases are provided in the previous section for integral containment tests. The presence of a turbulent gas mixture (natural or forced convection) or a wavy/turbulent film interface complicates the analysis even for simple geometries. Examples of separate effect tests and correlations under a variety of conditions were also presented in the previous section. In these situations theoretical analysis of this turbulent condition is still needed as is consideration of the effect of geometric scale. This may require multi-dimensional, multi-fluid modelling of the condensation process both near the wall and gas boundary layers as well as in the bulk gas mixture. If this approach is taken then one must address the appropriate scaling of these calculations to produce scaling of these calculations to produce useful condensation heat transfer design correlations or procedures.

References q

W.W. Akers, S.H. Davis, Jr and J.E. Crawford, "Condensation of a Vapor in the Presence of a Noncondensing Gas," Chemical Engineering Progress Symposium Series, No 30, Vol 56 , pp 139-144, 1960.

q

H.K. Al-Diwany and J.W. Rose, "Free Convection Film Condensation of Steam in the Presence of Noncondensing Gases," Int J Heat Mass Transfer, Vol 16 , pp 1359-1369 1973.

q

K. Almenas and U. Lee, "A Statistical Evaluation of the Heat Transfer Data Obtained in the HDR Containment Tests," University of Maryland, 1984.

Observations q

K. Asano and Y. Nakano, "Forced Convection Film Condensation of Vapors in the Presence of Noncondensable Gas on a Small Vertical Flat Plate," J of Chem Engr of Japan , 1978.

q

J.J. Barry, "Effects of Interfacial Structure on Film Condensation," PhD Thesis, University of Wisconsin, 1987.

q

L.A. Bromley, "Effect of Heat Capacity of Condensate," Ind. Eng. Chem., Vol 44, pp. 2966-2969, 1952.

q

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Prof. Michael L. Corradini: Department of Engineering Physics University of Wisconsin, Madison WI 53706

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