Heat Transfer Module Users Guide with comsol.

Heat Tr ansfer Modul Module e User´s Guide

VERSION 4.3b

Heat Transfer Module User’s Guide © 1998–2013 COMSOL Protected by U.S. Patents 7,519,518; 7,596,474; and 7,623,991. Patents pending. This Documentation Documentation and the Programs Prog rams described herein are furnished under the COMSOL Software License Agreement ( www.comsol.com/sla www.comsol.com/sla)) and may be used or copied only under the terms of the license agreement. COMSOL, COMSOL Multiphysics, Capture the Concept, COMSOL Desktop, and LiveLink are either registered trademarks or trademarks of COMSOL AB. All other trademarks are the property of their respective owners, and COMSOL AB AB and its subsidiaries and products products are not affiliated with, endorsed by, sponsored by, or supported supp orted by those trademark owners. For a li st of such trademark owners, see www.comsol.com/tm www .comsol.com/tm.. Version: May 2013 COMSOL 4.3b

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Part No. CM020801

C o n t e n t s Chapter 1: Introduction About the Heat Transfer Transfer Module

2

Why Heat Transfer is Important to Modeling . . . . . . . . . . . . How the Heat Transfer Module Improves Your Modeling . . . . . . . .

2 2

The Heat Transfer Module Physics Guide. . . . . . . . . . . . . . 3 Where Do I Access the Documentation and Model Library? . . . . . . 11 Overview of the User’s User’s Guide

14

C h a p t e r 2 : H e a t Tr a n s f e r T h e o r y Theory for the Heat Transfer Transfer User Interfaces

What is Heat Transfer? . . . . . The Heat Equation . . . . . . . A Note on Heat Flux . . . . . . Heat Flux and Heat Source Variables Variables

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18

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18 19 21 23

About the Boundary C onditions for the Heat Transfer User Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . 34 Radiative Heat Transfer in Transparent Media . . . . . . . . . . . . 36 Consistent and Inconsistent Stabilization Methods for the Heat Transfer User Interfaces . . . . . . Moist Air Theory. . . . . . . . . . About Heat Transfer with Phase Change . Theory for the Thermal Contact Feature. About the Heat Transfer Transfer Coefficients

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38 40 46 48 53

Heat Transfer Coefficient Theory . . . . . . . . . . . . . . . . 54 Nature of the Flow—the Grashof Number . . . . . . . . . . . . . 55 Heat Transfer Coefficients — External Natural Convection. Convection . . . . . . . 56 Heat Transfer Coefficients — Internal Natural Convection . . . . . . . 58 Heat Transfer Coefficients — External Forced Convection . . . . . . . 59

CONTENTS

|i

Heat Transfer Coefficients — Internal Forced Convection . . . . . . . 59 About Highly Conductive Layers

61

Theory of Out-of-Plane Heat Transfer

63

Equation Formulation . . . . . . . . . . . . . . . . . . . . . 64 Activating Out-of-Plane Heat Transfer and Thickness . . . . . . . . . 64 Theory for the Bioheat Transfer Transfer User Interface

65

Theory for the Heat Transfer in Porous Media User Interface

66

About Handling Frames in Heat Transfer

68

Frame Physics Feature Nodes and Definitions . . . . . . . . . . . . 68 Conversion Between Material and Spatial Frames . . . . . . . . . . 72 References for the Heat Transfer Transfer User Interfaces

75

Chapter 3: The Heat Transfer Branch About the Heat Transfer Transfer Interfaces

78

The Heat Transfer Interface

81

Domain, Boundary, Edge, Point, and Pair Nodes for the Heat Transfer User Interfaces Interf aces Heat Transfer in Solids . . Translational Motion . . Heat Transfer in Fluids . .

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Initial Values. . . . . . . . . Heat Source. . . . . . . . . Heat Transfer with Phase Change Thermal Insulation . . . . . . Temperature . . . . . . . .

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93 94 96 99 . . . 99

Outflow . . . . . . . . . . . . . . . . . . . . . . . . . Symmetry . . . . . . . . . . . . . . . . . . . . . . . .

ii | C O N T E N T S

84 86 88 89

100 101

Heat Flux. . . . . . . . . . . . . . . . . . . . . . . . . Surface-to-Ambient Surface-to-Ambient Radiation . . . . . . . . . . . . . . . . .

101 103

Periodic Heat Condition . . . Boundary Heat Source. . . . Continuity . . . . . . . . Thin Thermally Resistive Layer.

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104 104 106 106

Thermal Contact . . . . . . . . . . . . . . . . . . . . . . Line Heat Source . . . . . . . . . . . . . . . . . . . . . .

108 111

Point Heat Source Pressure Work . Viscous Heating . Inflow Heat Flux . Open Boundary .

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112 112 113 114 115

Convective Heat Flux . . . . . . . . . . . . . . . . . . . .

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Highly Conductive Layer Nodes

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Highly Conductive Layer . . . . . . . . . . . . . . . . . . .

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Layer Heat Source Edge Heat Flux . Point Heat Flux . Temperature . .

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120 121 122 123

Point Temperature emperat ure . . . . . . . . . . . . . . . . . . . . . Edge Surface-to-Ambient Radiation . . . . . . . . . . . . . . . Point Surface-to-Ambient Radiation . . . . . . . . . . . . . . .

124 125 125

Out-of-Plane Heat Transfer Nodes

127

Out-of-Plane Convective Heat Flux . . . . . . . . . . . . . . . Out-of-Plane Radiation . . . . . . . . . . . . . . . . . . . Out-of-Plane Heat Flux . . . . . . . . . . . . . . . . . . .

127 129 130

Change Thickness . . . . . . . . . . . . . . . . . . . . .

130

The Bioheat Transfer Interface

132

Biological Tissue . . . . . . . . . . . . . . . . . . . . . . Bioheat . . . . . . . . . . . . . . . . . . . . . . . . .

133 134

The Heat Transfer in Porous Media Interface

136

Domain, Boundary, Edge, Point, and Pair Nodes for the Heat Transfer in Porous Media User Interface . . . . . . . . . . . .

137

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CONTENTS

| iii

Heat Transfer in Porous Media . . . . . . . . . . . . . . . . . Thermal Dispersion . . . . . . . . . . . . . . . . . . . . .

137 143

Chapter 4: Heat Transfer in Thin Shells The Heat Transfer Transfer in Thin Shells User Interface

146

Boundary, Edge, Point, and Pair Nodes for the Heat Transfer Transfer in Thin Shells User Interface . Heat Flux. . . . . . . . Thin Conductive Conduct ive Layer. . . Heat Source. . . . . . .

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148 149 150 151

Initial Values. . . . . . . . Change Thickness . . . . . Surface-to-Ambient Surface-to-Ambient Radiation . Insulation/Continuity Insulation/Continuity . . . . Change Effective Thickness . .

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152 152 153 153 154

Edge Heat Source . . . . . . . . . . . . . . . . . . . . . Point Heat Source . . . . . . . . . . . . . . . . . . . . .

154 155

Theory for the Heat Transfer in Thin Shells User Interface 156

About Heat Transfer in Thin Shells . . . . . . . . . . . . . . . Heat Transfer Equation in Thin Conductive Shell . . . . . . . . . . Thermal Conductivity Tensor Components . . . . . . . . . . . .

156 156 157

Chapter 5: Radiation Heat Transfer The Radiation Branch Versions of the Heat Transfer User Interface

The Heat Transfer with Surface-to-Surface Radiation User Interface . . The Heat Transfer Transfer with Radiation in Participating Media User Interface . . . . . . . . . . . . . . . . . . . . . . . . Domain, Boundary, Edge, Point, and Pair Nodes for the Radiation Branch Versions of the Heat Transfer User Interface . . . . . . . .

iv | C O N T E N T S

160

160 161 161

The Surface-To-Surface Radiation User Interface

164

Domain, Boundary, Edge, Point, and Pair Nodes for the Surface-to-Surface Radiation User Interface . . . Surface-to-Surface Radiation (Boundary Condition) . Opaque . . . . . . . . . . . . . . . . . Diffuse Mirror . . . . . . . . . . . . . . .

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167 168 172 173

Prescribed Radiosity . . . . . . . . . . . . . . . . . . . . Radiation Group . . . . . . . . . . . . . . . . . . . . . .

174 177

External Radiation Source . . . . . . . . . . . . . . . . . .

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Theory for the Surface-to-Surface Radiation User Interface

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Wavelength Dependence of Surface Emissivity and Absorptivity . . . . The Radiosity Method for Diffuse-Gray Surfaces . . . . . . . . . .

182 188

The Radiosity Method for Diffuse-Spectral Surfaces. . . . . View Factor Evaluation . . . . . . . . . . . . . . . About Surface-to-Surface Radiation . . . . . . . . . . . Guidelines for Solving Surface-to-Surface Radiation Problems .

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190 192 194 196

Radiation Group Boundaries . . . . . . . . . . . . . . . . .

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The Radiation in Participating Media User Interface

199

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Domain, Boundary, Edge, Point, and Pair Nodes for the Radiation in Participating Media User Interface . Radiation in Participating Media . . . Opaque Surface . . . . . . . . . Incident Intensity . . . . . . . . .

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201 202 203 205

Continuity on Interior Boundary . . . . . . . . . . . . . . . .

206

Theory for the Radiation in Participating Media User Interface

Radiation and Participating Media Interactions . Radiative Transfer Equation . . . . . . . . Boundary Condition for the Transfer Equation. Heat Transfer Equation in Participating Media . Discrete Ordinates Method . . . . . . . .

207

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207 208 209 210 211

Discrete Ordinates Method Implementation in 2D . . . . . . . . .

212

CONTENTS

|v

References for the Radiation User Interfaces

214

Chapter 6: The Single-Phase Flow Branch The Laminar Flow and Turbulent Flow User Interfaces

216

The Laminar Flow User Interface. . . . . . . . . . . . . . . . The Turbulent Flow, k-  User Interface . . . . . . . . . . . . .

216 219

The Turbulent Flow, Low Re k-  User Interface . . . . . . Domain, Boundary, Pair, and Point Nodes for Single-Phase Flow Fluid Properties . . . . . . . . . . . . . . . . . . Volume Force . . . . . . . . . . . . . . . . . . . Initial Values. . . . . . . . . . . . . . . . . . . .

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221 223 224 226 227

Wall . . Inlet. . . Outlet . . Symmetry

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228 231 233 236

Open Boundary . . . . Boundary Stress . . . . Periodic Flow Condition . Fan . . . . . . . . .

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237 237 239 240

Interior Fan . . Interior Wall . Grille . . . . Flow Continuity

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242 244 245 246

Pressure Point Constraint . . . . . . . . . . . . . . . . . . More Boundary Condition Settings for the Turbulent Flow User Interfaces . . . . . . . . . . . . . . . . . . . . . . . .

246 247

Theory for the Laminar Flow User Interface

250

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Theory for the Inlet Boundary Condition . . . . . Additional Theory for the Outlet Boundary Condition Theory for the Fan Defined on an Interior Boundary . Theory for the Fan and Grille Boundary Conditions .

vi | C O N T E N T S

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250 251 253 254

Non-Newtonian Flow: The Power Law and the Carreau Model . . . .

257

Theory for the Turbulent Flow User Interfaces

260

Turbulence Modeling . . . . . . . . . . . . . . . . . . . .

260

The k- Turbulence Model . . . . . . . . . . . . . . . . The Low Reynolds Number k-  Turbulence Model . . . . . . . Inlet Values for the Turbulence Length Scale and Turbulent Intensity Theory for the Pressure, No Viscous Stress Boundary Condition .

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264 270 273 274

Solvers for Turbulent Flow . . . . . . . . . . . . . . . . . . Pseudo Time Stepping for Turbulent Flow Models . . . . . . . . .

274 275

References for the Single-Phase Flow, User Interfaces

276

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Chapter 7: The Conjugate Heat Transfer Branch About the Conjugate Heat Transfer User Interfaces

280

Selecting the Right User Interface . . . . . . . . . . . . . . . The Non-Isothermal Flow Options . . . . . . . . . . . . . . .

280 282

Conjugate Heat Transfer Options . . . . . . . . . . . . . . .

283

The Non-Isothermal Flow and Conjugate Heat Transfer, Laminar Flow and Turbulent Flow User Interfaces

285

The Non-Isothermal Flow, Laminar Flow User Interface . . . . . . . The Conjugate Heat Transfer, Laminar Flow User Interface . . . . . . The Turbulent Flow, k-  and Turbulent Flow Low Re k-  User Interfaces . . . . . . . . . . . . . . . . . . . . . . . . Domain, Boundary, Edge, Point, and Pair Nodes Settings for the

285 289 289

NITF User Interfaces. Fluid . . . . . . . Wall. . . . . . . . Interior Wall . . . .

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292 293 300 302

Initial Values. . Open Boundary Pressure Work Viscous Heating Symmetry, Heat

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303 303 304 305 305

Symmetry, Flow . . . . . . . . . . . . . . . . . . . . . .

306

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CONTENTS

| vii

Theory for the Non-Isothermal Flow and Conjugate Heat Transfer User Interfaces

Turbulent Non-Isothermal Flow Theory . . . . . . . . . . . . .

308

310

References for the Non-Isothermal Flow and Conjugate Heat Transfer User Interfaces

315

Chapter 8: Glossary Glossary of Terms

viii | C O N T E N T S

318

1

Introduction This guide describes the Heat Transfer Module, an optional package that extends the COMSOL Multiphysics® modeling environment with customized physics interfaces for the analysis of heat transfer.

This chapter introduces you to the capabilities of this module. A summary of the physics interfaces and where you can find documentation and model examples is also included. The last section is a brief over view with links to each chapter in this guide. • •

About the Heat Transfer Module Overview of the User’s Guide

1

About the Heat Transfer Module In this section: • • • •

Why Heat Transfer is Important to Modeling How the Heat Transfer Module Improves Your Modeling The Heat Transfer Module Physics Guide Where Do I Access the Documentation and Model Library? Overview of the Physics and Building a COMSOL Model in the COMSOL Multiphysics Reference Manual

Why Heat Transfer is Important to Modeling

The Heat Transfer Module is an optional package that extends the COMSOL Multiphysics modeling environment with customized user interfaces and functionality optimized for the analysis of heat transfer. It is developed for a wide audience including researchers, developers, teachers, and students. To assist users at all levels of expertise, this module comes with a library of ready-to-run example models that appear in the companion Heat Transfer Module Model Library. Heat transfer is involved in almost every kind of physical process, and can in fact be the limiting factor for many processes. Therefore, its study is of vital impor tance, and the need for powerful heat transfer analysis tools is virtually universal. Furthermore, heat transfer often appears together with, or as a result of, other physical phenomena. The modeling of heat transfer effects has become increasingly important in product design including areas such as electronics, automotive, and medical industries. Computer simulation has allowed engineers and researchers to optimize process efficiency and explore new designs, while at the same time reducing costly experimental trials. How the Heat Transfer Module Improves Your Modeling

The Heat Transfer Module has been developed to greatly expand upon the base capabilities available in COMSOL Multiphysics. The module supports all fundamental

2 |

CHAPTER 1: INTRODUCTION

mechanisms including conductive, convective, and radiative heat transfer. Using the physics interfaces in this module along with the inherent multiphysics capabilities of COMSOL Multiphysics, you can model a temperature field in parallel with other features—a versatile combination increasing the accuracy and predicting power of your models. This book introduces the basic modeling process. The different physics interfaces are described and the modeling strategy for various cases is discussed. These sections cover different combinations of conductive, convective, and radiative heat transfer. This guide also reviews special modeling techniques for highly conductive layers, thin conductive shells, participating media, and out-of-plane heat transfer. Throughout the guide the topics and examples increase in complexity by combining several heat transfer mechanisms and also by coupling these to physics interfaces describing fluid flow—conjugate heat transfer. Another source of information is the Heat Transfer Module Model Library, a set of fully-documented models that is divided into broadly defined application areas w here heat transfer plays an important role—electronics and power systems, processing and manufacturing, and medical technology—and includes tutorial and verification models. Most of the models involve multiple heat transfer mechanisms and are often coupled to other physical phenomena, for example, fluid dynamics or electromagnetics. The authors developed several state-of-the art examples by reproducing models that have appeared in international scientific journals. See Where Do I Access the Documentation and Model Library?. The Heat Transfer Module Physics Guide

The table below lists all the interfaces available specifically with this module. Having this module also enhances these COMSOL Multiphysics basic interfaces: Heat Transfer in Fluids, Heat Transfer in Solids, Joule Heating, and the Single-Phase Flow, Laminar interface.

ABOUT THE HEAT TRANSFER MODULE

| 3

If you have an Subsurface Flow Module combined with the Heat Transfer Module, this also enhances the Heat Transfer in Porous Media interface. The Non-Isothermal Flow, Laminar Flow (nitf) and Non-Isothermal Flow, Turbulent Flow (nitf) interfaces found under the Fluid Flow>Non-Isothermal Flow branch are identical to the Conjugate Heat Transfer interfaces (Laminar Flow and Turbulent Flow) found under the Heat Transfer>Conjugate Heat Transfer branch. The difference is that Fluid is the default domain node for the Non-Isothermal Flow interfaces. In the COMSOL Multiphysics Reference Manual: • • •

Studies and the Study Nodes The Physics User Interfaces For a list of all the interfaces included with the COMSOL Multiphysics basic license, see Physics Guide.

PHYSICS USER INTERFACE

ICON

TAG

SPACE DIMENSION

AVAILABLE PRESET STUDY TYPE

spf

3D, 2D, 2D axisymmetric

stationary; time dependent

Turbulent Flow, k-

spf



Turbulent Flow, Low Re k-

spf


stationary with initialization; transient with initialization

nitf



Fluid Flow Single-Phase Flow

Laminar Flow* Turbulent Flow

Non-Isothermal Flow

Laminar Flow

4 |



ICON

TAG

SPACE DIMENSION



nitf




nitf



Heat Transfer in Solids*

ht

all dimensions


Heat Transfer in Fluids*

ht

all dimensions


Heat Transfer in Porous Media

ht

all dimensions


Bioheat Transfer

ht

all dimensions


Heat Transfer in Thin Shells

htsh

3D


nitf




nitf




nitf



ht

all dimensions


Turbulent Flow

Heat Transfer

Conjugate Heat Transfer

Laminar Flow Turbulent Flow

Radiation

Heat Transfer with Surface-to-Surface Radiation


| 5


ICON

TAG

SPACE DIMENSION


Heat Transfer with Radiation in Participating Media

ht

3D, 2D


Surface-to-Surface Radiation

rad

all dimensions


Radiation in Participating Media

rpm

3D, 2D


jh

all dimensions


Electromagnetic Heating

Joule Heating*

* This is an enhanced interface, which is included with the base COMSOL package but has added functionality for this module. THE HEAT TRANSFER MODULE STUDY CAPABILITIES

Table 1-1 lists the Preset Studies available for the interfaces most relevant to this module.

Studies and Solvers in the COMSOL Multiphysics Reference Manual

6 |


TABLE 1-1: HEAT TRANSFER MODULE DEPENDENT VARIABLES AND PRESET STUDY AVAILABILITY PHYSICS INTERFACE

TAG

DEPENDENT VARIABLES

PRESET STUDIES*

Y R A N O I T A T S

T N E D N E P E D E M I T

 

 

N O I T A Z I L A I T I N I H T I W Y R A N O I T A T S

N O I T A Z I L A I T I N I H T I W T N E I S N A R T













FLUID FLOW>SINGLE-PHASE FLOW

Laminar Flow

spf

u, p

Turbulent Flow, k- 

spf

u, p, k, ep


spf

u, p, k, ep, G

FLUID FLOW>NON-ISOTHERMAL FLOW

Laminar Flow

nitf

u, p, T

Turbulent Flow, k- 

nitf

u, p, k, ep, T


nitf

u, p, k, ep, G, T

Heat Transfer in Solids**

ht

T

Heat Transfer in Fluids**

ht

T

Heat Transfer in Porous Media**

ht

T

Bioheat Transfer**

ht

T


htsh

T

 

 

HEAT TRANSFER

  

  

 

 

 

 

HEAT TRANSFER>CONJUGATE HEAT TRANSFER

Laminar Flow**

nitf

u, p, T

Turbulent Flow, k- **

nitf u, p, k, ep, T

Turbulent Flow, Low Re k-**

nitf

u, p, k, ep, G, T

HEAT TRANSFER>RADIATION

Heat Transfer with ht Surface-to-Surface Radiation**

T, J






| 7

TABLE 1-1: HEAT TRANSFER MODULE DEPENDENT VARIABLES AND PRESET STUDY AVAILABILITY PHYSICS INTERFACE

TAG

DEPENDENT VARIABLES

T, I ( radiative

PRESET STUDIES*

Y R A N O I T A T S

T N E D N E P E D E M I T



  



Heat Transfer with Radiation in Participating Media**

ht


rad

J


rpm

I (radiative intensity)

 

T, V



N O I T A Z I L A I T I N I H T I W Y R A N O I T A T S

N O I T A Z I L A I T I N I H T I W T N E I S N A R T

intensity)

HEAT TRANSFER>ELECTROMAGNETIC HEATING

Joule Heating**

jh

* Custom studies are also available based on the interface. ** For these interfaces, it is possible to enable surface to surface radiation and/or radiation in participating media. In these cases, J and I are dependent variables. SHOW MORE PHYSICS OPTIONS

There are several general options available for the physics user interfaces and for individual nodes. This section is a short overview of these options, and includes links to additional information when available. The links to the features described in the COMSOL Multiphysics Reference Manual (or any external guide) do not work in the PDF, only from within the online help. To locate and search all the documentation for this information, in COMSOL Multiphysics, select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree.

8 |


To display additional options for the physics interfaces and other parts of the model tree, click the Show button ( ) on the Model Builder and then select the applicable option. After clicking the Show button ( ), additional sections get displayed on the settings window when a node is clicked and additional nodes are available from the context menu when a node is right-clicked. For each, the additional sections that can be displayed include Equation, Advanced Settings, Discretization, Consistent Stabilization, and Inconsistent Stabilization. You can also click the Expand Sections button ( ) in the Model Builder to always show some sections or click the Show button ( ) and select Reset to Default to reset to display only the Equation and Override and Contribution sections. For most nodes, both the Equation and Override and Contribution sections are always available. Click the Show button ( ) and then select Equation View to display the Equation View node under all nodes in the Model Builder . Availability of each node, and whether it is described for a particular node, is based on the individual selected. For example, the Discretization, Advanced Settings, Consistent Stabilization, and Inconsistent Stabilization sections are often described individually throughout the documentation as there are unique settings. SECTION

CROSS REFERENCE

Advanced Physics Sections The Model Wizard and Model Builder Show Discretization Discretization Discretization (Node) Discretization—Splitting of Compile Equations Show More Options and Expand Sections

complex variables

Show Stabilization Numerical Stabilization Weak Constraints and Constraint Settings Constraint Settings Override and Contribution Physics Exclusive and Contributing Node Types Consistent and Inconsistent Stabilization

OTHER COMMON SETTINGS

At the main level, some of the common settings found (in addition to the Show options) are the Interface Identifier, Domain, Boundary, or Edge Selection, and Dependent Variables.


| 9

At the nodes’ level, some of the common settings found (in addition to the Show options) are Domain, Boundary, Edge, or Point Selection, Material Type, Coordinate System Selection, and Model Inputs. Other sections are common based on application area and are not included here. SECTION

CROSS REFERENCE

Coordinate System Selection

Coordinate Systems

Domain, Boundary, Edge, About Geometric Entities and Point Selection About Selecting Geometric Entities Interface Identifier

Material Type Model Inputs

Pair Selection

Predefined Physics Variables Variable Naming Convention and Scope Viewing Node Names, Identifiers, Types, and Tags Materials About Materials and Material Properties Selecting Physics Adding Multiphysics Couplings Identity and Contact Pairs Continuity on Interior Boundaries

THE LIQUIDS AND GASES MATERIALS DATABASE

The Heat Transfer Module includes an additional Liquids and Gases material database with temperature-dependent fluid dynamic and thermal properties. For detailed information about materials and the Liquids and Gases Material Database, see Materials in the COMSOL Multiphysics Reference Manual .

10 |


Where Do I Access the Documentation and Model Library?

A number of Internet resources provide more information about COMSOL, including licensing and technical information. The electronic documentation, context help, and the Model Library are all accessed through the COMSOL Desktop. If you are reading the documentation as a PDF file on your computer, the blue links do not work to open a model or content referenced in a different guide. However, if you are using the online help in COMSOL Multiphysics, these links work to other modules, model examples, and documentation sets. THE DOCUMENTATION

The COMSOL Multiphysics Reference Manual describes all user interfaces and functionality included with the basic COMSOL Multiphysics license. This book also has instructions about how to use COMSOL and how to access the documentation electronically through the COMSOL Help Desk. To locate and search all the documentation, in COMSOL Multiphysics: • • •

Press F1 or select Help>Help ( ) from the main menu for context help. Press Ctrl+F1 or select Help>Documentation ( ) from the main menu for opening the main documentation window with access to all COMSOL documentation. Click the corresponding buttons ( or ) on the main toolbar. and then either enter a search term or look under a specific module in the documentation tree. If you have added a node to a model you are working on, click the Help button ( ) in the node’s settings window or press F1 to learn more about it. Under More results in the Help window there is a link with a search string for the node’s name. Click the link to find all occur rences of the node’s name in the documentation, including model documentation and the external COMSOL website. This can help you find more information about the use of the node’s functionality as well as model examples where the node is used.

A B O U T T H E H E A T TR A N S F E R M O D U L E

| 11

THE MODEL LIBRARY

Each model comes with documentation that includes a theoretical background and step-by-step instructions to create the model. The models are available in COMSOL as MPH-files that you can open for further investigation. You can use the step-by-step instructions and the actual models as a template for your own modeling and applications. In most models, SI units are used to describe the relevant properties, parameters, and dimensions in most examples, but other unit systems are available. To open the Model Library, select View>Model Library ( ) from the main menu, and then search by model name or browse under a module folder name. Click to highlight any model of interest, and select Open Model and PDF to open both the model and the documentation explaining how to build the model. Alternatively, click the Help button ( ) or select Help>Documentation in COMSOL to search by name or browse by module. The model libraries are updated on a regular basis by COMSOL in order to add new models and to improve existing models. Choose View>Model Library Update ( ) to update your model library to include the latest versions of the model examples. If you have any feedback or suggestions for additional models for the librar y (including those developed by you), feel free to contact us at [email protected] CONTACTING COMSOL BY EMAIL

For general product information, contact COMSOL at [email protected] To receive technical support from COMSOL for the COMSOL products, please contact your local COMSOL representative or send your questions to [email protected] An automatic notification and case number is sent to you by email.

12 |


COMSOL WEBSITES

COMSOL website Contact COMSOL Support Center Download COMSOL Support Knowledge Base Product Updates COMSOL Community

www.comsol.com www.comsol.com/contact www.comsol.com/support www.comsol.com/support/download www.comsol.com/support/knowledgebase www.comsol.com/support/updates www.comsol.com/community

A B O U T T H E H E A T TR A N S F E R M O D U L E

| 13

Overview of the User’s Guide The Heat Transfer Module User’s Guide gets you started with modeling using COMSOL Multiphysics®. The information in this guide is specific to the Chemical Reaction Engineering Module. Instructions how to use COMSOL in general are included with the COMSOL Multiphysics Reference Manual . As detailed in the section Where Do I Access the Documentation and Model Library? this information can also be searched from the COMSOL Multiphysics software Help menu. TABLE OF CONTENTS, GLOSSARY, AND INDEX

To help you navigate through this guide, see the Contents, Glossary , and Index. HEAT TRANSFER THEORY

The Heat Transfer Theory chapter starts with the general theory underlying the heat transfer interfaces used in this module. It then discusses theory about heat transfer coefficients, highly conductive layers, and out-of-plane heat transfer. The last three sections briefly describe the underlying theory for the Bioheat Transfer, Heat Transfer in Thin Shells, and Heat Transfer in Porous Media interfaces. THE HEAT TRANSFER USER INTERFACES

The module includes interfaces for the simulation of heat transfer. As with all other physical descriptions simulated by COMSOL Multiphysics, any description of h eat transfer can be directly coupled to any other physical process. This is particularly relevant for systems based on fluid-flow, as well as mass transfer. General Heat Transfer

The Heat Transfer Branch chapter details the variety of Heat Transfer interfaces that form the fundamental interfaces in this module. It covers all the types of heat transfer— conduction, convection, and radiation—for heat transfer in solids and fluids. About the Heat Transfer Interfaces provides a quick summary of each interface, and the rest of the chapter describes these interfaces in details. This includes the highly conductive layer and out-of-plane heat transfer physics features and the Heat Transfer in Porous Media interface. The Heat Transfer with Participating Media (ht) interface is also described as it is a Heat Transfer interface where surface-to-surface radiation is active by default.

14 |


Bioheat Transfer

The Bioheat Transfer Interface section discusses modeling heat transfer within biological tissue using the Bioheat Transfer interface. Heat Transfer in Thin Shells

The Heat Transfer in Thin Shells chapter describes the interface, which is suitable for solving thermal-conduction problems in thin structures. Radiative Heat Transfer

Radiation Heat Transfer chapter describes the Surface-to-Surface Radiation, the Heat Transfer with Surface-to-Surface Radiation, and the Radiation in Par ticipating Media interfaces. THE CONJUGATE HEAT TRANSFER USER INTERFACES

The Conjugate Heat Transfer Branch chapter describes the Non-Isothermal Flow Laminar Flow (nitf) and Turbulent Flow (nitf) interfaces found un der the Fluid Flow branch, which are identical to the Conjugate Heat Transfer interfaces. Each section describes the applicable interfaces in detail and concludes with the underlying theory for the interfaces. THE FLUID FLOW USER INTERFACES

The Single-Phase Flow Branch chapter describe the single-phase laminar and turbulent flow interfaces in detail. Each section describes the applicable interfaces in detail and concludes with the underlying theory for the interfaces.

OVERVIEW OF THE USER’S GUIDE

| 15

16 |


2

Heat Transfer Theory This chapter discusses some fundamental heat transfer theory. Theory related to individual interfaces is discussed in those chapters. In this chapter: • • • • • • •

Theory for the Heat Transfer User Interfaces About the Heat Transfer Coefficients About Highly Conductive Layers Theory of Out-of-Plane Heat Transfer Theory for the Bioheat Transfer User Interface Theory for the Heat Transfer in Porous Media User Interface About Handling Frames in Heat Transfer

17

Theory for the Heat Transfer User Interfaces The Heat Transfer Interfacetheory is described in this section. This section reviews the theory about the heat transfer equations in COMSOL Multiphysics ® and heat transfer in general. For more detailed discussions of the fundamentals of heat transfer, see Ref. 1 and Ref. 3. In this section: • • • • • • • • • • •

What is Heat Transfer? The Heat Equation A Note on Heat Flux Heat Flux and Heat Source Variables About the Boundary Conditions for the Heat Transfer User Interfaces Radiative Heat Transfer in Transparent Media Consistent and Inconsistent Stabilization Methods for the Heat Transfer User Interfaces Moist Air Theory About Heat Transfer with Phase Change Theory for the Thermal Contact Feature References for the Heat Transfer User Interfaces

What is Heat Transfer?

Heat transfer is defined as the movement of energy due to a difference in temperature. It is characterized by the following mechanisms: • Conduction —Heat conduction takes place through different mechanisms in

different media. Theoretically it takes place in a gas through collisions of the molecules; in a fluid through oscillations of each molecule in a “cage” formed by its nearest neighbors; in metals mainly by electrons carrying heat and in other solids by molecular motion which in crystals take the form of lattice vibrations known as phonons. Typical for heat conduction is that the heat flux is proportional to the temperature gradient.

18 |

CHAPTER 2: HEAT TRANSFER THEORY

• Convection —Heat convection (sometimes called heat advection) takes place

•

through the net displacement of a fluid, which transports the heat content in a fluid through the fluid’s own velocity. The term convection (especially convective cooling and convective heating) also refers to the heat dissipation from a solid surface to a fluid, typically described by a heat transfer coefficient. Radiation —Heat transfer by radiation takes place through the transport of photons. Participating (or semitransparent) media absorb, emit and scatter photons. Opaque surfaces absorb or reflect them.

The Heat Equation

The fundamental law governing all heat transfer is the first law of thermodynamics, commonly referred to as the principle of conservation of energy. However, internal energy, U , is a rather inconvenient quantity to measure and use in simulations. Therefore, the basic law is usually rewritten in terms of temperature, T . For a fluid, the resulting heat equation is:  --- u   C p  ----T +     T  t

= –

T   p u     q  + : S – ---- -------  -----+     p    T p  t

+

Q

(2-1)

where •

 is the density (SI unit: kg/m 3)

• Cp is the specific heat capacity at constant pressure (SI unit: J/(kg·K)) • T is absolute temperature (SI unit: K) • u is the velocity vector (SI unit: m/s) • q is the heat flux by conduction (SI unit: W/m 2) • p is pressure (SI unit: Pa) •

 is the viscous stress tensor (SI unit: Pa)

• S is the strain-rate tensor (SI unit: 1/s):

S

=

1 ---   u +   u  T  2

• Q contains heat sources other than viscous heating (SI unit: W/m 3)

THEORY FOR THE HEAT TRANSFER USER INTERFACES

| 19

For a detailed discussion of the fundamentals of heat transfer, see Ref. 1. Specific heat capacity at constant pressure is the amount of energy required to raise one unit of mass of a substance by one degree while maintained at constant pressure. This quantity is also commonly referred to as specific heat or specific heat capacity . In deriving Equation 2-1, a number of thermodynamic relations have been used. The equation also assumes that mass is always conserved, which means that density and velocity must be related through:  +   v t

=

0

The heat transfer interfaces use Fourier’s law of heat conduction, which states that the conductive heat flux, q, is proportional to the temperature gradient: qi

 T  x i

= – k --------

(2-2)

where k is the thermal conductivity (SI unit: W/(m·K)). In a solid, the thermal conductivity can be anisotropic (that is, it has different values in dif ferent directions). Then k becomes a tensor k xx k xy k xz k

=

k yx k yy k yz k zx k zy k zz

and the conductive heat flux is given by qi

= –

k j

 T --------

ij  x j

Fourier’s law expect that the thermal conductivity tensor is symmetric. Non symmetric tensor leads to unphysical results.

20 |


The second term on the right of Equation 2-1 represents viscous heating of a fluid. An analogous term arises from the internal viscous damping of a solid. The operation “:” is a contraction and can in this case be written on the following form: a:b

=

a

nm b nm

n m

The third term represents pressure work and is responsible for the heating of a fluid under adiabatic compression and for some thermoacoustic effects. It is generally small for low Mach number flows. A similar term can be included to account for thermoelastic effects in solids. Inserting Equation 2-2 into Equation 2-1, reordering the terms and ignoring viscous heating and pressure work puts the heat equation into a more familiar form: --- +  C p u   T =    k  T  + Q  C p ----T t

The Heat Transfer in Fluids physics solves this equation for the temperature, T . If the velocity is set to zero, the equation governing pure conductive heat transfer is obtained: --- +    – k  T   C p ----T t

=

Q

A Note on Heat Flux

The concept of heat flux is not as simple as it might first appear. The reason is that heat is not a conser ved property. The conserved property is instead the total energy. There is hence heat flux and energy flux that are similar but not identical. This section briefly describes the theory for the variables for Total Energy Flux and Total Heat Flux. The approximations made do not affect the computational results, only variables available for results analysis and visualization. TOTAL ENERGY FLUX

The total energy flux for a fluid is equal to (Ref. 4, chapter 3.5)  u  H 0 +   – k T +   u + q r

(2-3)

Above, H 0 is the total enthalpy


| 21

H 0

=

1 H + ---  u  u  2

where in turn H is the enthalpy. In Equation 2-3  is the viscous stress tensor and qr is the radiative heat flux.  in Equation 2-3 is the force potential. It can be formulated in some special cases, for example, for gravitational effects (Chapter 1.4 in Ref. 4), but it is in general rather difficult to derive. Potential energy is therefore often excluded and the total energy flux is approximated by  u  H + 1---  u  u  – k T +   u + qr 2

(2-4)

For a simple compressible fluid, the enthalpy, H , has the form (Ref. 5) T

H

=

H ref +



p

C p dT +

T ref



1 T   ---  1 + ----  -------  d p    T  

p ref

(2-5)

p

where p is the absolute pressure. The reference enthalpy, H ref , is the enthalpy at reference temperature, T ref , and reference pressure, pref . T ref is 298.15 K and pref is one atmosphere. In theory, any value can be assigned to H ref (Ref. 7), but for practical reasons, it is given a positive value according to the following approximations • •

Solid materials and ideal gases: H ref  C p,ref T ref Gasliquid: H ref  C p,ref ref T ref  pref ref

where the subscript “ref ” indicates that the property is evaluated at the reference state. The two integrals in Equation 2-5 are sometimes referred to as the sensible enthalpy (Ref. 7). These are evaluated by numerical integration. The second integral is only included for gas/liquid since it is commonly much smaller than the first integral for solids and it is identically zero for ideal gases. For the evaluation of H to work, it is important that the dependence of C p, , and  on the temperature are prescribed either via model input or as a function of the temperature variable. If C p, , or  depends on the pressure, that dependency must be prescribed either via model input or by using the variable pA, which is the variable for the absolute pressure. TOTAL HEAT FLUX

The total heat flux vector is defined as (Ref. 6):

22 |


 u U – k T + q r

(2-6)

where U is the internal energy. It is related to the enthalpy via H

=

p U + ---

(2-7)



What is the difference between Equation 2-4 and Equation 2-7? As an example, consider a channel with fully developed incompressible flow with all properties of the fluid independent of pressure and temperature. The walls are assumed to be insulated. Assume that the viscous heating is neglected. This is a common approximation for low-speed flows. There is a pressure drop along the channel that drives the flow. Since there is no viscous heating and the walls are isolated, Equation 2-5 gives that H in  H out. Since everything else is constant, Equation 2-4 shows that the energy flux into the channel is higher than the energy flux out of the channel. On the other hand U inU out, so the heat flux into the channel is equal to the heat flux going out of the channel. If the viscous heating on the other hand is included, then H in  H out (first law of thermodynamics) and U in  U out (since work has been converted to heat). Heat Flux and Heat Source Variables

This section lists some predefined variables that are available to compute heat fluxes and sources. All the variable names start with the physics interface prefix. By default the Heat Transfer interface prefix is ht. As an example, the variable named tflux can be analyzed using ht.tflux (as long as the physics interface prefix is ht). TABLE 2-1: HEAT FLUX VARIABLES VARIABLE

NAME

GEOMETRIC ENTITY LEVEL

tflux

Total Heat Flux Conductive Heat Flux Turbulent Heat Flux Convective Heat Flux Translational Heat Flux Total Energy Flux Radiative Heat Flux

Domains, boundaries

dflux turbflux aflux trlflux teflux

not applicable

Domains, boundaries Domains, boundaries Domain, boundaries Domains, boundaries Domains, boundaries Domains


| 23

TABLE 2-1: HEAT FLUX VARIABLES VARIABLE

NAME


ccflux_u

Convective Out-of-Plane Heat Flux

Out-of-plane domains (1D and 2D)

Radiative Out-of-Plane Heat Flux

Out-of-plane domains (1D and 2D), boundaries

Out-of-Plane Heat Flux

Out-of-plane domains (1D and 2D)

Normal Total Heat Flux, Extrapolated Normal Conductive Heat Flux, Extrapolated Normal Convective Heat Flux Normal Translational Heat Flux Normal Total Energy Flux, Extrapolated Internal Normal Conductive Heat Flux, Extrapolated, Upside Internal Normal Conductive Heat Flux, Extrapolated, Downside Internal Normal Convective Heat Flux, Extrapolated, Upside Internal Normal Convective Heat Flux, Downside Internal Normal Translational Heat Flux, Upside Internal Normal Translational Heat Flux, Downside Internal Total Normal Heat Flux, Upside Internal Total Normal Heat Flux, Downside Internal Normal Total Energy Flux, Extrapolated, Upside

Boundaries

ccflux_d ccflux_z rflux_u rflux_d rflux_z q0_u q0_d q0_z ntflux ndflux naflux ntrlflux nteflux ndflux_u ndflux_d naflux_u naflux_d ntrlflux_u ntrlflux_d ntflux_u ntflux_d nteflux_u

24 |


Boundaries Boundaries Boundaries Boundaries Interior boundaries Interior boundaries Interior boundaries Interior boundaries Interior boundaries Interior boundaries Interior boundaries Interior boundaries Interior boundaries

TABLE 2-1: HEAT FLUX VARIABLES VARIABLE

nteflux_d ndflux_acc ntflux_acc nteflux_acc ndflux_acc_u ndflux_acc_d ntflux_acc_u ntflux_acc_d nteflux_acc_u nteflux_acc_d rflux ccflux Qtot Qbtot Ql Qp

NAME


Internal Normal Total Energy Flux, Extrapolated, Downside Normal Conductive Flux, Accurate Normal Total Heat Flux, Accurate Normal Total Energy Flux, Accurate Internal Normal Conductive Flux, Accurate, Upside Internal Normal Conductive Flux, Accurate, Downside Internal Normal Total Heat Flux, Accurate, Upside Internal Normal Total Heat Flux, Accurate, Downside Internal Normal Total Energy Flux, Accurate, Upside Internal Normal Total Energy Flux, Accurate, Downside Radiative Heat Flux Convective Heat Flux Domain Heat Sources Boundary Heat Sources Line heat source (Line and Point Heat Sources) Point heat source (Line and Point Heat Sources)

Interior boundaries Exterior boundaries Exterior boundaries Exterior boundaries Interior boundaries Interior boundaries Interior boundaries Interior boundaries Interior boundaries Interior boundaries Boundaries Boundaries Domains Boundaries Edges Points

DOMAIN HEAT FLUXES

On domains the heat fluxes are vector quantities. Their definition can vary depending on the active physics nodes and selected properties.


| 25

Total Heat Flux

On domains the total heat flux, tflux, corresponds to the conductive and convective heat flux. For accuracy reasons the radiative heat flux is not included.

See Radiative Heat Flux to evaluate the radiative heat flux. For solid domains, for example heat transfer in solids and biological tissue domains, the total heat flux is defined by: tflux

trlflux + dflux

=

For fluid domains (for example, heat transfer in fluids), the total heat flux is defined by: tflux

aflux + dflux

=

Conductive Heat Flux

The conductive heat flux variable, dflux, is evaluated using the temperature gradient and the effective thermal conductivity: dflux

T

= – k eff

When the out-of-plane property is activated (1D and 2D only) the conductive heat flux is defined as follows: •

In 2D (d z is the domain thickness): dflux

•

T

= – d z k eff

In 1D ( Ac is the cross-section area): dflux

= – Ac k

T

eff

In the general case keff is the thermal conductivity, k. For heat transfer in fluids with turbulent flow, keff = k + kT , where kT is the turbulent thermal conductivity.

26 |


For heat transfer in porous media, keff = keq, where keq is the equivalent conductivity defined in the Heat Transfer in Porous Media feature. The Heat Transfer in Porous Media feature requires one of the following products: Batteries & Fuel Cells Module, CFD Module, Chemical Reaction Engineering Module, Corrosion Module, Electrochemistry Module, Electrodeposition Module, Heat Transfer Module, or Subsurface Flow Module. Turbulent Heat Flux

The turbulent heat flux variable, turbflux, enables access to the part of the conductive heat flux that is due to the turbulence. T

= – k T

turbflux

Convective Heat Flux

The convective heat flux variable,

aflux, is

defined using the internal energy, E:

aflux

=

 u E

When the out-of-plane property is activated (1D and 2D only) the convective heat flux is defined as follows: •

In 2D (d z is the domain thickness): aflux

•

=

d z  u E

=

A c  u E

In1D ( Ac is the domain thickness): aflux

The internal energy, E, is defined by: • E  C pT for solid domains • E  C pT  for ideal gas fluid domains • E  H  p for other fluid domains

where H is the enthalpy defined by Equation 2-5. Translational Heat Flux

Similar to convective heat flux but defined for solid domains with translation. The variable name is trlflux.


| 27

Total Energy Flux

The total energy flux, teflux, is defined when viscous heating is enabled: teflux

 u H 0 + dflux +   u

=

where the total enthalpy, H 0, is defined as H 0

=

uu H + -----------2

Radiative Heat Flux

In participating media, the radiative heat flux, qr, is not available for analysis on domains because it is much more accurate to evaluate the radiative heat source: Qr

=

  qr

OUT-OF-PLANE DOMAIN FLUXES

When the out-of-plane property is activated (1D and 2D only), out-of-plane domain fluxes are defined. If there are no out-of-plane physics features, they are evaluated to zero. Convective Out-of-Plane Heat Flux

The convective out-of-plane heat flux, ceflux, is generated by the Out-of-Plane Convective Heat Flux feature. •

In 2D: upside: ccflux _u

=

downside: ccflux _d •

h u  T

ext u – T 

=

hd  T ext d – T 

In 1D: ccflux _z

=

h z  T ext z – T 

Radiative Out-of-Plane Heat Flux

The radiative out-of-plane heat flux, rflux, is generated by the Out-of-Plane Radiationfeature. •

In 2D: upside: rflux _u

28 |


=

4

4

u   T amb u – T 

downside: rflux _d •

 d   T 4amb d – T 4 

=

In 1D: rflux _z

=

 z   T 4amb z – T 4 


The convective out-of-plane heat flux, q0, is generated by the Out-of-Plane Heat Flux feature. •

In 2D: upside: q0 _u

downside: q0 _d •

hu  T ext u – T 

=

=

h d  T ext d – T 

In 1D: q0 _z

=

h z  T ext z – T 

BOUNDARY HEAT FLUXES

All the domain heat fluxes (vector quantity) are also available as boundary heat fluxes. The boundary heat fluxes are then equal to the mean value of the adjacent domains. In addition normal boundary heat fluxes (scalar quantity) are available on boundaries. Normal Total Heat Flux, Extrapolated

The variable ntflux is defined by: ntflux

=

mean  tflux   n

Normal Conductive Heat Flux, Extrapolated

The variable ndflux is defined by: ndflux

=

mean  dflux   n

=

mean  aflux   n

Normal Convective Heat Flux

The variable naflux is defined by: naflux

Normal Translational Heat Flux

The variable ntrlflux is defined by:


| 29

ntrlflux

mean  trlflux   n

=

Normal Total Energy Flux, Extrapolated

The variable nteflux is defined by: nteflux

mean  teflux   n

=

Radiative Heat Flux

On boundaries the radiative heat flux,

rflux,

4

4

rflux

=

is a scalar quantity defined as: 4

  T amb – T  +   G – T  + q w

where the terms respectively account for surface-to-ambient radiative flux, surface-to-surface radiative flux, and radiation in participating net flux. Convective Heat Flux

Convective heat flux, ccflux, is defined as the contribution from the Convective Heat Flux boundary condition: ccflux

=

h  T ext – T 

When the out-of-plane property is activated (1D and 2D only) the convective heat flux is defined as follows: •

In 2D (d z is the domain thickness): ccflux

•

=

d z h  T ext – T 

=

A c h  T ext – T 

In 1D ( Ac is the cross section area): ccflux

INTERNAL BOUNDARY HEAT FLUXES

The internal normal boundary heat fluxes (scalar quantity) are available on interior boundaries. They are calculated using the upside and the downside value of heat fluxes from the adjacent domains. Internal Normal Conductive Heat Flux, Extrapolated, Upside

The variable ndflux_u is defined by: ndflux_u

=

up  dflux   n

Internal Normal Conductive Heat Flux, Extrapolated, Downside

The variable ndflux_d is defined by:

30 |


ndflux_d

=

down  dflux   n

Internal Normal Convective Heat Flux, Extrapolated, Upside

The variable naflux_u is defined by: naflux_u

=

up  aflux   n

Internal Normal Convective Heat Flux, D ownside

The variable naflux_d is defined by: naflux_d

=

down  aflux   n

Internal Normal Translational Heat Flux, Upside

The variable ntrlflux_u is defined by: ntrlflux_u

=

up  trlflux   n

Internal Normal Translational Heat Flux, Downside

The variable ntrlflux_d is defined by: ntrlflux_d

=

down  trlflux   n

Internal Normal Total Energy Flux, Extrapolated, Upside

The variable nteflux_u is defined by: nteflux_u

=

up  teflux   n

Internal Normal Total Energy Flux, Extrapolated, Downside

The variable nteflux_d is defined by: nteflux_d

=

down  teflux   n

Internal Total Normal Heat Flux, Upside

The variable ntflux_u is defined by: ntflux_u

=

up  tflux   n

Internal Total Normal Heat Flux, Downside

The variable ntlux_d is defined by: ntflux_d

=

down  tflux   n


| 31

ACCURATE FLUXES

Normal Conductive Flux, Accurate

The variable ndflux_acc is defined by: ndflux_acc

= –dep.dflux.T

if the adjacent domain is on the downside

ndflux_acc

= – dep.uflux.T

if the adjacent domain is on the upside

Internal Normal Conductive Flux, Accurate, Downside

The variable ndflux_acc_d is defined by: ndflux_acc_d

=

dep.dflux.T

Internal Normal Conductive Flux, Accurate, Upside

The variable ndflux_acc_u is defined by: ndflux_acc_u

=

dep.uflux.T

Normal Total Heat Flux, Accurate

The variable ntflux_acc is defined by: ntflux_acc

=

ndflux_acc + naflux + ntrlflux

Internal Normal Total Heat Flux, Accurate, Downside

The variable ntflux_acc_d is defined by: ntflux_acc_d

=

ndflux_acc_d + naflux_d

+ ntrlflux_d

Internal Normal Total Heat Flux, Accurate, Upside

The variable ntflux_acc_u is defined by: ntflux_acc_u

=

ndflux_acc_u + naflux_u

+

ntrlflux_u

Normal Total Energy Flux, Accurate

The variable nteflux_acc is defined by: ntflux_acc

=

nteflux – ndflux + ndflux_acc

Internal Normal Total Energy Flux, Accurate, Downside

The variable nteflux_acc_d is defined by: nteflux_acc_d

=

nteflux_d – ndflux_d + ndflux_acc_d

Internal Normal Total Energy Flux, Accurate, Upside

The variable nteflux_acc_u is defined by:

32 |


nteflux_acc_u

=

nteflux_u – ndflux_u + ndflux_acc_u

DOMAIN HEAT SOURCES

The sum of the domain heat sources added by different physics features are available in one variable, Qtot (SI unit: W/m3). This variable Qtot is the sum of: • Q which is the heat source added by

•

Heat Source(described for the Heat Transfer interface and Electromagnetic Heat Source (described for the Joule Heating interface in the COMSOL Multiphysics Reference Manual ) feature. Qmet which is the heat source added by the Bioheat feature. The out-of-plane contributions (convective heat flux, heat flux, and radiation), and the blood contribution in Bioheat are considered flux so that they are not added to Qtot.

BOUNDARY HEAT SOURCES

The sum of the boundary heat sources added by different boundary conditions is available in one variable, Qb,tot (SI unit: W/m2). This variable Qbtot is the sum of: • Qb which is the boundary heat •

•

source added by the Boundary Heat Source

boundary condition. Qsh which is the boundary heat source added by the Boundary Electromagnetic Heat Source boundary condition (described for the Joule Heating interface in the COMSOL Multiphysics Reference Manual ). Qs: which is the boundary heat source added by a Layer Heat Source subfeature of a highly conductive layer.

LINE AND POINT HEAT SOURCES

The sum of the line heat sources is available in a variable called

Ql (SI

The sum of the point heat sources is available in a variable called

unit: W/m).

Qp (SI

In 2D axisymmetric models, the SI unit for the variable

Qp is

unit: W).

W/m.


| 33

About the Boundar y Conditions for the Heat Transfer User Interfaces TEMPERATURE AND HEAT FLUX BOUNDARY CONDITI ONS

The heat equation accepts two basic types of boundary conditions: specified temperature and specified heat flux. The specified temperature is of a constraint type and prescribes the temperature at a boundary: T

=

T 0

on 

while the latter specifies the inward heat flux –n

q

=

q0

on 

where • q is the conductive heat flux vector (SI unit: W/m 2) where q = kT . • n is the normal vector of the boundary. • q0 is inward heat flux (SI unit: W/m 2), normal to the boundary.

The inward heat flux, q0, is often a sum of contributions from different heat transfer processes (for example, radiation and convection). The special case q0  0 is called thermal insulation . A common type of heat flux boundary conditions are those where q0  h·T inf  T , where T inf is the temperature far away from the modeled domain and the heat transfer coefficient, h, represents all the physics occurring between the boundary and “far away.” It can include almost anything, but the most common situation is that h represents the effect of an exterior fluid cooling or heating the surface of solid, a phenomenon often referred to as convective cooling or heating. The Heat Transfer Module contains a set of correlations for convective heat flux and heating. See About the Heat Transfer Coefficients. O V E R R I D I N G M E C H A N I S M F O R H E A T TR A N S F E R B O U N D A R Y C O N D I T I O N S

This section includes information for features that may require additional modules.

34 |


Many boundary conditions are available in heat transfer. Some of them can be associated (for example, Heat Flux and Highly Conductive Layer). Others cannot be associated (for example, Heat Flux and Thermal Insulation). Several categories of boundary condition exist in heat transfer. Table 2-2 gives the overriding rules for these groups. • • • • • • • •

Temperature, Convective Outflow, Open Boundar y, Inflow Heat Flux Thermal Insulation, Symmetry, Periodic Heat Condition Highly Conductive Layer Heat Flux, Convective Heat Flux Boundary Heat Source, Radiation Group Surface-to-Surface Radiation, Re-radiating Surface, Prescribed Radiosity, Surface-to-Ambient Radiation Opaque Surface, Incident Intensity, Continuity on Interior Boundaries Thin Thermally Resistive Layers, Thermal Contact

TABLE 2-2: OVERRIDING RULES FOR HEAT TRANSFER BOUNDARY CONDITIONS A\B

1

2

3

1-Temperature

X

X

X

2-Thermal Insulation

X

X

3-Highly Conductive Layer

X

4-Heat Flux

X

4

5

6

7

8

X X

X X

5-Boundary heat source 6-Surface-to-surface radiation

X

7-Opaque Surface 8-Thin Thermally Resistive Layer

X X

X

X

When there is a boundary condition A above a boundary condition B in the model tree and both conditions apply to the same boundar y, use Table 2-2 to determine if A is overridden by B or not: • •

Locate the line that corresponds to the A group (see above the definition of the groups). In the table above only the first member of the group is displayed. Locate the column that corresponds to the group of B .


| 35

•

If the corresponding cell is empty A and B contribute. If it contains an X, B overrides A. Group 4 and group 5 boundar y conditions are always contributing. That means that they never override any other boundary condition. But they might be overridden.

Example 1

Consider a boundary where Temperature is applied. Then a Surface-to-Surface Radiation boundary condition is applied on the same boundary afterward. • Temperature belongs to group 1. • Surface-to-surface radiation belongs to group 6. •

The cell on the line of group 1 and the column of group 6 is empty so Temperature and Surface-to-Surface radiation contribute.

Example 2

Consider a boundary where Convective Heat Flux is applied. Then a Symmetry boundary condition is applied on the same boundary afterward. • Convective Heat Flux belongs to group 4. • Symmetry belongs to group 2. •

The cell on the line of group 4 and the column of group 2 contains an X so Convective Heat Flux is overridden by Symmetry. In Example 2 above, if Symmetry followed by Convective Heat Flux is added, the boundary conditions contribute.

Radiative Heat Transfer in Transparent Media

This discussion so far has considered heat transfer by means of co nduction and convection. The third mechanism for heat transfer is radiation. Consider an environment with fully transparent or fully opaque objects. Thermal radiation denotes the stream of electromagnetic waves emitted from a body at a certain temperature.

36 |


DERIVING THE RADIATIVE HEAT FLUX

J = G + T 4 G

x

x

   ,T

   ,T

Figure 2-1: Arriving irradiation (left), leaving radiosity (right).

Consider Figure 2-1. A point x is located on a surface that has an emissivity , reflectivity , absorptivity , and temperature T . Assume the body is opaque, which means that no radiation is transmitted through the body. This is true for most solid bodies. The total arriving radiative flux at x is named the irradiation, G. The total outgoing radiative flux x is named the radiosity, J . The radiosity is the sum of the reflected radiation and the emitted radiation: 4

J =  G +   T

(2-8)

The net inward radiative heat flux, q, is then given the difference between the irradiation and the radiosity: q

G – J

=

(2-9)

Using Equation 2-8 and Equation 2-9 J can be eliminated and a general expression is obtained for the net inward heat flux into the opaque body based on G and T . q

=

4

 1 –   G –  T

(2-10)

Most opaque bodies also behave as ideal gray bodies, meaning that the absorptivity and emissivity are equal, and the reflectivity is therefore given from the following r elation: 

=



=

1–

(2-11)

Thus, for ideal gray bodies, q is given by: q

=

4

  G –  T 

(2-12)


| 37

This is the equation used as a radiation boundar y condition. RADIATION TYPES

It is common to differentiate between two types of radiative heat transfer: surface-to-ambient radiation and surface-to-surface radiation . Equation 2-12 holds for both radiation types, but the irradiation term, G, is different for each of them. The Heat Transfer interface supports both types of radiation. SURFACE-TO-AMBIEN T RADIATION

Surface-to-ambient radiation assumes the following: • •

The ambient surroundings in view of the surface have a constant temperature, T amb. The ambient surroundings behave as a blackbody. This means that the emissivity and absorptivity are equal to 1, and zero reflectivity.

These assumptions allows the irradiation to be explicitly expressed as G

=

4

 T amb

(2-13)

Inserting Equation 2-13 into Equation 2-12 results in the net inward heat flux for surface-to-ambient radiation q

=

4

4

  T amb – T 

(2-14)

For boundaries where a surface-to-ambient radiation is specified, COMSOL Multiphysics adds this term to the right-hand side of Equation 2-14.

• •

Theory for the Radiation in Participating Media User Interface Radiation and Participating Media Interactions

Consistent and Inconsistent Stabilization Methods for the Heat Transfer User Interfaces

The different versions of the Heat Transfer interface have this advanced option to set the stabilization method parameters. This section provides information pertaining to these options. To display this section, click the

38 |


Show button

(

) and select Stabilization.

CONSISTENT STABILIZATION

This section contains two consistent stabilization methods: streamline diffusion and crosswind diffusion. These are consistent stabilization methods, which means that they do not perturb the original transport equation. The consistent stabilization methods take effect for fluids and for solids with Translational Motion. A stabilization method is active when the cor responding check box is selected. Continuous Casting: Model Library path Heat_Transfer_Module/ Thermal_Processing/continuous_casting

Streamline Diffusion

Streamline diffusion is active by default and should remain active for optimal performance for heat transfer in fluids or other applications that include a convective or translational term. Crosswind Diffusion

Streamline diffusion introduces artificial diffusion in the streamline direction. This is often enough to obtain a smooth numerical solution provided that the exact solution of the heat equation does not contain any discontinuities. At sharp gradients, however, undershoots and overshoots can occur in the numerical solution. Crosswind diffusion addresses these spurious oscillations by adding diffusion orthogonal to the streamline direction—that is, in the crosswind direction. I N C O N S I S T E NT S T A B I L I Z A T I O N

This section contains one inconsistent stabilization method: isotropic diffusion. Adding isotropic diffusion is equivalent to adding a term to the physical diffusion coefficient. This means that the original problem is not solved, which is why isotropic diffusion is an inconsistent stabilization method. Still, the added diffusion definitely dampens the effects of oscillations, but try to minimize the use of isotropic diffusion. By default there is no isotropic diffusion. To add isotropic dif fusion, select the Isotropic diffusion check box. The field for the tuning parameter id then becomes available. The


| 39

default value is 0.25; increase or decrease the value of id to increase or decrease the amount of isotropic stabilization. See Show Stabilization and Stabilization Techniques in the COMSOL Multiphysics Reference Manual .

Moist Air Theory

For the Heat Transfer in Fluids physics, the moist air functionality is provided to calculate the relative humidity and to deduce if there is condensation. The following theory assumes that the moist air is an ideal gas.

Heat Transfer in Fluids

HUMIDITY

Moi sture Conte nt

The moisture content (also called mixing ratio or humidity ratio) is defined as the ratio of water vapor mass m v to dry air mass ma: x vap

=

m v ------ma

=

p v M v ------------- p a M a

(2-15)

where p v is the water vapor partial pressure, pa is the dry air partial pressure, and M a and M v are the molar mass of dry air and water vapor, respectively. Without condensation, the moisture content is not affected by temperature and pressure. The Moisture content represents a ratio of mass, and it is thus a dimensionless number (SI unit: 1). Relative Humidity

The relative humidity of an air mixture is expressed as follows: 

40 |


=

p v ------- p sat

(2-16)

where p v is the water vapor partial pressure and psat is the saturation pressure of water vapor. According to Dalton’s law, the total pressure of a mixture of gases is the sum of all the partial pressures of each individual gas; that is, p = p v + pa where pa is the dry air partial pressure. The relative humidity formulation is often used to quantify humidity. However, for a same quantity of moisture content, the relative humidity changes with temperature and pressure, so in order to compare different values of , it has to be at the same temperature and pressure conditions. This quantity is very useful to study the condensation as it defines the boundar y between the liquid phase and the vapor phase. In fact, when the relative humidity  reaches unity, it means that the vapor is saturated and that water vapor will condense. The Reference relative humidity (SI unit: 1) is a quantity defined between 0 and 1, where 0 corresponds to dry air and 1 to a water vapor-saturated air. This Reference relative humidity associated to the Reference temperature and the Reference pressure are used to calculate the moisture content. Then the thermodynamical properties of moist air can be deduced through the mixture formula described below. The Reference relative humidity cannot be greater than one, above which value the water vapor is condensing. If the value is greater than one, the Reference relative humidity value is forced to be one. The condensation area cannot be simulated. Specific Humidity

The specific humidity is defined as the ratio of water vapor m v to the total mass mtot = m v + ma:


| 41



=

m v ---------- m tot

(2-17)

As the water vapor only accounts for a few percent in the total mass, the moisture content and the specific humidity are very close: x vap   (only for low values). For bigger values of , the two quantities are more precisely related by: x vap

=



------------1–

Concentration

The concentration is defined by: n v c v = -----V

(2-18)

where n v is the amount of water vapor in mol and V is the total volume. The water vapor concentration is defined in this SI unit: mol/m3. According to the ideal gas hypothesis, the saturation concentration is defined as follows: c sat

=

psat  T  ----------------- RT

SATURATION STATE

The saturation state is reached when the relative humidity reaches one. It means that the partial pressure of the water vapor is equal to the saturation pressure (which depends on the temperature too). The saturation pressure can be defined using the Clausius-Clapeyron formulation of the vaporization-condensation equilibrium. Under ideal gas hypothesis and considering only the gas volume: hfg M v p d p ------- = ------------------2 dT RT

(2-19)

where p is the pressure, T is the temperature, hfg is the latent heat of vaporization, M v is the molar mass of water vapor, and R is the universal gas constant.

42 |


By integrating, you can obtain the saturation pressure equation: sat

=

 h fg M v 1 1  p ref exp  ---------------- --------- – ---------   R  T ref T sat 

(2-20)

where the reference values are: pref  101325 Pa (1 atm), T ref  373.15 K (100 °C), and hfg = 2.26106 J/kg. The temperature and saturation pressure can easily be deduced from this formulation. MOIST AIR PROPERTIES

The thermodynamical properties of moist air can be found with some mixture laws. Preliminary Definitions

The molar fraction of dry air X a and the molar fraction of water vapor X v are defined such as: Molar Fraction

X a

=

na --------n tot

=

n v X v = --------ntot

p a ---- p

=

=

p v ---- p

p –  psat -------------------- p

=

 psat

----------- p

(2-21)

(2-22)

where na and n v are respectively the amount of dry air and water vapor, ntot is the total amount of moist air in mol, where pa and p v are the partial pressure of dry air and water vapor, p is the pressure,  is the relative humidity, and psat is the saturation pressure.

X a  X v  1

Moisture content and relative humidity can be linked with the following expression: Relation Between Relative Humidity And Moisture Content



=

x vap p -------------------------------------- M v psat  -------- + x vap  M a 

(2-23)


| 43

Mixture Proper ties

The thermodynamical properties are built through a mixture formula. The expressions depend on dry air properties and pure steam properties and are balanced by the mass fraction. According to the ideal gas law, the density mixture m expression is defined as follows: Density:

m

=

p --------  M a X a + M v X v  RT

(2-24)

where M a and M v are respectively the molar mass of dry air and water vapor, respectively, and X a and X v are the molar fraction of dry air and water vapor, respectively. The ideal gas assumption sets the compressibility factor and the enhancement factor to the unity. In fact, the accuracy lost by this assumption is small as the pure steam represents a small fraction. Specific heat capacity at constant pressure:

According to Ref. 10, the heat capacity at

constant pressure of a mixture is: c p,m

=

M a M v --------- X a c p,a + -------- X c M m M m v p,v

(2-25)

where M m represents the mixture molar fraction and is defined by M m = X a M a + X v M v and where cp,a and cp,v are the heat capacity at constant pressure of dry air and steam, respectively. Dynamic viscosity:

According to Ref. 9 and Ref. 10, the dynamic viscosity is defined

as follows: m



=

i

=

X i  i --------------------------

a,v j

where ij is given by

44 |


 X 

j ij

=

a,v

(2-26)

1 --2

 ij

=

1 2 --4

 i M j 1 +  -----  -------   j  M i ----------------------------------------------- M i 8  1 + -------  M j

1 --2

Here, a and  v are the dynamic viscosity of dry air and steam, respectively. According to Ref. 9 and Ref. 10, the thermal conductivity of the mixture is defined similarly: Thermal conductivity:

km



=

i

=

X i k i --------------------------

(2-27)

 X 

a,v

j ij

j

=

a,v

where ka and k v are the thermal conductivity of dry air and steam, respectively. Pure Component Properties

The dry air and steam properties used to define the mixture properties are temperature-dependent high-order polynomials. The polynomials have been computed according to Ref. 1 for dry air properties and Ref. 8 for pure steam properties. The steam properties are based on the Industrial Formulation IAPWS-IF97. The valid temperature range is 200 K  T  1200 K for dry air properties and 273.15 K  T  873.15 K for steam properties. Results and Analysis Variables

The following variables are provided to display the related quantities: • • • • •

Moisture content xvap. Vapor mass fraction omega_moist. Concentration of water vapor c. Relative humidity phi. This variable corresponds to the calculated  with the system temperature and pressure). Condensation indicator condInd; this indicator is set to 1 if condensation has been detected (   and 0 if not.


| 45

Functions

Three functions are defined and can be used as feature parameters as well as in post processing. • ht.fluid1.fc(RH,T, pA), where RH is

the relative humidity 0    1 , T is the temperature (SI unit: K) and pA is the pressure (SI unit: Pa). It returns the corresponding water vapor concentration (SI unit: mol/m^3). The concentration computation assumes that the ideal gas assumption is valid

• ht.fluid1.fxvap(RH, T, pA), where RH is the relative humidity 0    1 , T is

the temperature (SI units: K) and pA is the pressure (SI units Pa). It returns the moisture content (SI unit: 1). • ht.fluid1.fpsat(T), where T is the temperature (SI unit: K). It returns the saturation pressure (SI unit: Pa). About Heat Transfer with Phase Change

The Heat Transfer with Phase Change node is used to solve the heat equation after specifying the properties of a phase change material according to the apparent heat capacity formulation. Instead of adding a latent heat L in the energy balance equation when the material reaches its phase change temperature T pc, it is assumed that the transformation occurs in a temperature interval between T pc  T 2 and T pc  T 2. In this interval, the material phase is modeled by a smoothed function, , representing the fraction of phase after transition, which is equal to 0 before T pc  T 2 and to 1 after T pc  T  2. The enthalpy H is expressed by: H =  1 –   H phase1 +  H phase 2

where H phase1 and H phase2 are the enthalpies when the material is in phase 1 or in phase 2, respectively. Differentiating with respect to temperature, this equality provides the following formula for the specific heat capacity: Cp

46 |

=

 1 – 

d

 H p phase1 + 

d T


d

  H p phase2 +  H phase 2 – H phase1  d

d T

d T

which can be rewritten as: Cp

=

 1 Cp phase1 + 2 Cp phase2 +  H phase 2 – H phase1  d 

d T

Here, 1 and 2 are equal to 1   and , respectively. The specific heat capacity is the sum of an equivalent heat capacity Ceq: Ceq

=

 1 Cp phase1 +  2 Cp phase2

and the distribution of latent heat C L: C L  T 

=

 H phase 2 – H phase1  d 

d T

In the ideal case, when  is the Heaviside function (equal to 0 before T pc and to 1 after T pc), d  is the Dirac pulse. d T

Therefore, C L is the enthalpy jump L at temperature T pc that is added when you have a pure substance. The latent heat distribution C L is approximated by C L  T 

=

L



d

d T

so that the total heat per unit mass released during the phase transformation coincides with the latent heat:  T T pc + -------2 C  T  dT = T L  T pc – -------2



L

 T T pc + -------2 d  dT = T  T pc – -------- d T 2



L

The latent heat L may depend on the absolute pressure. It should not depend on the temperature. Finally, the apparent heat capacity Cp, used in the heat equation, is given by: Cp

=

 1 Cp phase1 +  2 Cp phase2 + C L

The equivalent heat conductivity and volumetric heat capacity reduce to:


| 47

k

=

 1 kphase1 +  2 k phase2

and  Cp

=

 1  phase1 Cp phase1 +  2  phase1 Cp phase2

The density is thus given by: 

=

 1  phase1 Cp phase1 + 2  phase1 Cp phase2 ---------------------------------------------------------------------------------------------------1 Cp phase1 + 2 Cp phase2

To satisfy energy and mass conservation in phase change models, particular attention should be paid to the density in time simulations. When the material density is not constant over time, for example, dependent on the temperature, volume change is expected. Moving Mesh User Interface (described in the COMSOL Multiphysics Reference Manual ) has the tools to deform the geometry accordingly. However, if Moving Mesh is not used, it is recommended to set material density to a constant value. Theory for the Thermal Contact Feature

The Thermal Contact feature has correlations to evaluate the joint conductance at two contacting surfaces. The heat fluxes at the upside and downside boundaries depend on the temperature difference according to the relations: – nd –nu

48 |


  – kd  T d 

  –k u  T u 

= –h

= –h

 T u – T d  + rQ fric

 T d – T u  +  1 – r  Qfric

At a microscopic level, contact is made at a finite number of spots as in Figure 2-2.

masp,u

Y

asp,u

Figure 2-2: Contacting surfaces at the microscopic level.

The joint conductance h has three contributions: the constriction conductance, hc, from the contact spots, the gap conductance, hg , due to the fluid at the interstitial space, and the radiative conductance, hr: h

=

hc + hg + hr

SURFACE ASPERITIES

The microscopic surface asperities are characterized by the average height uasp and dasp and the average slope muasp and mdasp. The RMS values asp and masp are (4.16 in Ref. 11):  asp

=

 u2 asp +  d2 asp

m asp

=

mu2 asp

+

md2 asp

CONSTRICTION CONDUCTANCE

Cooper-Mikic-Yovanovich (CMY) Correlation

The Cooper-Mikic Yovanovich (CMY) correlation is valid for isotropic rough surfaces and assumes plastic deformation of the sur face asperities. It relates hc to the asperities and pressure load at the contact interface: hc

=

m asp p 0 . 95 1 . 25kcontact -----------  -------  asp  H c

Here, H c is the microhardness of the softer material, p is the contact pressure and kcontact is the harmonic mean of the contacting surface conductivities: kcontact

=

2k u kd -----------------ku + kd


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When ku (resp. kd) is not isotropic, it is replaced by its normal conductivity nTkun (resp. nTkdn). The relative pressure p H c can be evaluated by specifying H c directly or using the following relation (4.16.1 in Ref. 11) for the relative pressure using c1 and c2, the Vickers correlation coefficient and size index: p ------ H c

1   ---------------------------------- 1 0 071c  . +   p -------------------------------------------------  c c  .   asp   1  1 62 ---------- masp  2

=

2

0

Here 0 is equal to 1 µm. For materials with Brinell hardness between 1.30 and 7.60 GPa, c1 and c2 are given by the correlations below (4.16.1 in Ref. 11): c1 ------ H 0

=

H B H B 2 H B 3 4 . 0 – 5 . 77 -------- + 4 . 0  -------- – 0 . 61  --------  H 0  H 0 H 0 c2

= –

H B 0 . 37 + 0 . 442 -------c1

The Brinell hardness is denoted by H B and H 0 is equal to 3.178 GPa. Mik ic Ela stic Cor rel ation

The Mikic correlation is valid for isotropic rough surfaces and assumes elastic deformations of surface asperities. It gives hc by the following relation: hc

=

0 . 94 masp 2 p 1 . 54k contact -----------  ------------------------   asp  mE contact

Here, Econtact is an effective Young’s modulus for the contact interface, satisfying (4.16.3 in Ref. 11): 1 ------------------ Econtact

=

1 –  u2 1 –  d2 --------------- + -------------- E u E d

where Eu and Ed are the Young’s moduli of the two contacting surfaces and u and d are the Poisson’s ratios.

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GAP CONDUCTANCE

The gap conductance due to interstitial fluid cannot be neglected for high fluid thermal conductivity or high contact pressure. The parallel-plate gap gas correlation assumes that the interstitial fluid is a gas and defines hg by: hg

=

kg -----------------Y + M g

Here kg is the gas conductivity, Y denotes the mean separation thickness (see Figure 2-2), and M g is the gas parameter equal to: M g

=

 



=

kB T g ------------------------2 2  D p g

In these relations,  is the contact thermal accommodation parameter,  is a gas property parameter (equal to 1.7 for air),  is the gas mean free path, kB is the Boltzmann constant, D is the average gas particle diameter, pg is the gas pressure (often the atmospheric pressure), and T g is the gap temperature equal to: T g

=

T u + T d -------------------2

RADIATIVE CONDUCTANCE

At high temperatures, above 600 °C, radiative conductance needs to be considered. The gray-diffuse parallel plate model provides the following formula for hr: hr

=

u  d  --------------------------------- T 3 + T u2 T d + T u T d2 + T d3   u +  d –  u d u

which implies that: hr  T u – T d 

=

u d  --------------------------------- T 4 – T d4  u + d –  u  d u

hr  T d – T u 

=

u d  --------------------------------- T 4 – T u4  u + d –  u  d d

THERMAL FRICTION

The friction heat, Qfric, is partitioned into rQfric and 1  rQfric at the contact interface. If the two bodies are identical, r and 1  r would be 0.5 so that half of the friction heat goes to each surface. However, in the general case where the two bodies


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are made of different materials, the partition rate might not be 0 .5. The Charron’s relation (Ref. 12) defines r as: r

=

1 --------------1 + d

d

 u C p u k u ----------------------- d C p d k d

=

and symmetrically, 1  r is: 1 – r

=

1 --------------1 + u

u

=

 d C p d k d ------------------------ u C p u k u

For anisotropic conductivities, nTkdn (resp. nTkun) replaces kd (resp. ku).

Thermal Contact

Contact Switch: Model Library path Heat_Transfer_Module/ Thermal_Contact_and_Friction/contact_switch

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About the Heat Transfer Coefficients One of the most common boundary conditions when modeling heat transfer is convective cooling or heating whereby a fluid cools or heats a surface by natural or forced convection. In principle, it is possible to model this process in two ways: • •

Use a heat transfer coefficient on the surfaces Extend the model to describe the flow and heat transfer in the surrounding fluid

The second approach is the correct approach if the geometry or the external flow is complicated. The Heat Transfer Module includes the Conjugate Heat Transfer interface for this purpose. However, such a simulations can become costly, both in terms of computational time and memory requirement. The first method is simple, yet powerful and efficient. Convective heat flux is then modeled by specifying the heat flux on the boundaries that interface with the fluid as being proportional to the temperature difference across a fictitious thermal boundary layer. Mathematically, the heat flux is described by the equation –n

  – k  T 

=

h  T inf – T 

where h is a heat transfer coefficient and T inf the temperature of the external fluid far from the boundary. The main difficulty in using heat transfer coefficients is in calculating or specifying the appropriate value of the h coefficient. That coefficient depends on the fluid’s material properties, and the surface temperature—and, for forced-convection, also on the fluid’s flow rate. In addition, the geometrical configuration af fects the coefficient. The Heat Transfer interface provides built-in functions for heat transfer coefficients. For most engineering purposes, the use of these coefficients is an accurate and numerically efficient modeling approach. In this section: • • • •

Heat Transfer Coefficient Theory Nature of the Flow—the Grashof Number Heat Transfer Coefficients — External Natural Convection Heat Transfer Coefficients — Internal Natural Convection

A B O U T T H E H E A T TR A N S F E R C O E F F I C I E N T S

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• •

Heat Transfer Coefficients — External Forced Convection Heat Transfer Coefficients — Internal Forced Convection

• •

The Heat Transfer Interface Theory for the Heat Transfer User Interfaces

Heat Transfer Coefficient Theory

It is possible to divide convection heat flux into four main categories depending on the type of convection conditions (natural or forced) and on the type of geometry (internal or external convection flow). In addition, these four cases can all experience either laminar or turbulent flow conditions, resulting in a total of eight types of convection, as in Figure 2-3. Natural

Forced

External

Internal

Laminar Flow

Turbulent Flow

Figure 2-3: The eight possible categories of convective heat flux.

The difference between natural and forced convection is that in the latter case an external force such as a fan creates the flow. In natural convection, buoyancy forces induced by temperature differences and the thermal expansion of the fluid drive the flow.

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Heat transfer handbooks generally contain a large set of empirical and theoretical correlations for h coefficients. This module includes a subset of them. The expressions are based on the following set of dimensionless numbers: • • • •

The Nusselt number, Nu L(Re, Pr, Ra)  hL/k The Reynolds number, Re L  U L/ The Prandtl number, Pr  C p/k The Rayleigh number, Ra  GrPr  2 gC p T L3/(k)

where • h is the heat transfer coefficient (SI unit: W/(m 2·K)). • L is the characteristic length (SI unit: m). •

T is the temperature difference between surface and external fluid bulk (SI unit: K).

• g is the acceleration of gravity (SI unit: m/s 2). • k is the thermal conductivity of the fluid (SI unit: W/(m·K)). •

 is the fluid density (SI unit: kg/m 3).

• U is the bulk velocity (SI unit: m/s). •

 is the dynamic viscosity (SI unit: Pa·s).

• C p equals the heat •

capacity of the fluid (SI unit: J/(kg·K)).  is the thermal expansivity (SI unit: 1/K)

Further, Gr refers to the Grashof number, which is defined as the ratio between the buoyancy force and the viscous force. Nature of the Flow—the Grashof Number

In cases of externally driven flow, such as forced convection, the flow’s nature is characterized by the Reynolds number, Re, which describes the ratio of the inertial to viscous forces. However, the velocity is largely unknown for internally driven flows such as natural convection. In such cases the Grashof number, Gr, characterizes the flow. It describes the ratio of the internal driving force (buoyancy force) to a viscous force acting on the fluid. Similar to the Reynolds number it r equires the definition of a length scale, the fluid’s physical properties, and the temperature scale (temperature difference). The Grashof number is defined as:


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3

Gr L

g   T s – T 0  L = --------------------------------------

     2

where g is the acceleration of gravity,  is the fluid’s coefficient of volumetric thermal expansion, T s denotes the temperature of the hot surface, T 0 equals the temperature of the surrounding air, L is the length scale,  represents the fluid’s dynamic viscosity, and  is the density. In general, the coefficient of volumetric thermal expansion  is given by 

1    ------  T 

= – ---

p

which for an ideal gas reduces to 

=

1  T

The transition from laminar to turbulent flow occurs at a Gr value of 109; the flow is turbulent for larger values. Heat Transfer Coefficients — External Natural Convection VERTICAL WALL

The correlations are equations 9.26 and 9.27 in Ref. 1:

h

=

    1/4 0 . 67 Ra L  k  -----------------------------------------------------------. + Ra L  10 9 0 68    L 9 / 16 4 / 9 . 0 492k  1 +  -------------------       Cp         2 1/6  k 0 . 387 Ra L   ----  0 .825 + ------------------------------------------------------------ Ra L  10 9 9 16 8 27 / / L . 0 492k    1 +  -------------------        Cp    

(2-28)

where L, the height of the wall, is a correlation input and g    T  p  C p T – T ext L 3 RaL = --------------------------------------------------------------------------k

56 |


(2-29)

where in turn g is the acceleration of gravity equal to 9.81 m/s 2. All material properties are evaluated at T  T ext2. INCLINED WALL

The correlations are equations 9.26 and 9.27 in Ref. 1 (same as for vertical wall):

h

=

    1/4       . Ra 0 67 cos  k  ---- 0 . 68 + ---------------------------------------------------------L  RaL  10 9  L  9 / 16 4 / 9 . 0 492k  1 +  -------------------            C  p    2 1 6 /  k 0 . 387 Ra L   ----  0 .825 + ------------------------------------------------------------ RaL  10 9 . 492k 9 / 16 8 / 27  L   1 +  0------------------    Cp    

(2-30)

where L, the height of the wall, is a correlation input and  is the tilt angle (angle between the wall and the vertical direction,  = 0 for vertical walls). These correlations are valid for 60°    60°. The definition of Raleigh number, Ra L, is analog to these for vertical walls and is given by the following: Ra L

g    T  p  C p T – T ext L 3 = --------------------------------------------------------------------------k

(2-31)

where in turn g denotes the gravitational acceleration, equal to 9.81 m/s 2. For turbulent flow, 1 is used instead of cos  in the expression for h, because this gives better accuracy (see Ref. 3). According to Ref. 1., correlations for inclined walls are only satisfactory for the top side of a cold plate or the down face of a hot plate. Hence, these correlations are not recommended for the bottom side of a cold face and for the top side of a hot plate. The laminar-turbulent transition depends on  (see Ref. 3). Unfortunately, few data is available about this transition. There is some data available in Ref. 3 but this data gives only approximations of this transition, according to the authors. In ad dition, data is only provided for water (Pr around 6). For this reasons, we define a flow as turbulent, independently of  value, when


| 57

g    T  p  C p T – T ext L 3 ---------------------------------------------------------------------------  10 9 k

All material properties are evaluated at T  T ext2. HORIZONTAL PLATE, UPSIDE

The correlations are equations 9.30–9.32 in Ref. 1 but can also be found as equations 7.77 and 7.78 in Ref. 3. If T  T ext, then

h

=

 k 1 / 4 Ra  10 7  ---- 0 . 54 Ra L L L   k 1 / 3 Ra  10 7  ---- 0 . 15 Ra L L  L

(2-32)

k 1/4 ---- 0 . 27 Ra L L

(2-33)

while if T  T ext, then h

=

RaL is given by

Equation 2-29, and L, the plate diameter (defined as area/perimeter, see Ref. 3) is a correlation input. The material data are evaluated at T  T ext2. HORIZONTAL PLATE, DOWNSIDE

Equation 2-32 is used when T  T ext and Equation 2-33 is used when T  T ext. Otherwise it is the same implementation as for Horizontal plate, upside. Heat Transfer Coefficients — Internal Natural Convection NARROW CHIMNEY, PARALLEL PLATES

If RaL  H  L and T  T ext, then h

=

k 1 ----- ------ Ra L H 24

(2-34)

where L, the plate distance, and H , the chimney height, are correlation inputs (equation 7.96 in Ref. 3). Ra L is given by Equation 2-29. The material data are evaluated at T  T ext2.

58 |


Narrow Chimney, Circular Tube If Ra D  H  D, then h

=

k 1 ----- ----------Ra D H 128

where D, the tube diameter, and H , the chimney height, are correlation inputs (table 7.2 in Ref. 3 with Dh  D). Ra D is given by Equation 2-29 with L replaced by D. The material data are evaluated at T  T ext2. Heat Transfer Coefficients — External Forced Convection PLATE, AVERAGED TRANSFER COEFFICIENT

This correlation is an assembly of equations 7.34 and 7.41 in Ref. 1:

h

=

 k 0 . 3387 Pr 1 / 3 Re 1 / 2  2 ---- ------------------------------------------------L --------------- Re  5  10 5  L  1 +  0 . 0468  Pr  2 / 3  1 / 4 L  k 1/3  2 --4 /5 5  L- Pr  0.037ReL – 871  Re L  5  10

(2-35)

where Pr  cpk and Re L  U ext L. L, the plate length and U ext, the exterior velocity are correlation inputs. The material data are evaluated at T  T ext2. PLATE, LOCAL TRANSFER COEFFICIENT

This correlation corresponds to equations 5.79b and 5.131 in Ref. 3:

h

=

 k  ---------------------------------- 0 . 332 Pr 1 / 3 Re x1 / 2 Re x  5  105  max  x eps   k  ---------------------------------0.0296Pr1 / 3 Rex4 / 5 Re x  5  10 5  max  x eps 

(2-36)

where Pr  cpk and Re x  U ext x. x, the position along the plate, and U ext, the exterior velocity are correlation inputs. The material data are evaluated at T  T ext2. Heat Transfer Coefficients — Internal Forced Convection ISOTHERMAL TUBE

This correlation corresponds to equations 8.55 and 8.61 in Ref. 1:


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h

=

 k ---- 3 . 66 Re D  2500  D   k   0 . 14  ---- 0 . 027Re D4 / 5 Pr n  -----------Re D  2500    T   D

(2-37)

where Pr  cpk, Re D  U ext D and n  0.3 if T  T ext and n0.4 if T  T ext. D, the tube diameter and U ext, the exterior velocity, are correlation inputs. All material data are evaluated at T ext except T  which is  evaluated at the wall temperature, T .

60 |


About Highly Conductive Layers This module supports heat transfer in highly conductive layers in 3D, 2D, and 2D axisymmetry. Also see Highly Conductive Layer Nodes. The highly conductive layer feature is efficient for modeling heat transfer in thin layers without the need to create a fine mesh for them. The material in the thin layer must be a good thermal conductor. A good example is a copper trace on a printed circuit board, where the traces are good thermal conductors compared to the board’s substrate material. More generally, the highly conductive layer feature can be applied in a part of a geometry with the following properties: • •

The part is a thin layer compared to the thickness of the adjacent geometry The part is a good thermal conductor compared to the adjacent geometry

Because the layer is very thin and has a high thermal conductivity, you can assume that no variations in temperature and in-plane heat flux exist along the layer’s thickness. Furthermore, think of the difference in heat flux in the layer’s normal direction between its upper and lower face as a heat source or sink that is smeared out along the layer thickness. A significant benefit is that a layer can be represented as a boundary instead of a domain, which simplifies the geometry and reduces the required number of mesh elements. Figure 2-4 shows an example where a highly conductive layer reduces the mesh density significantly.

ABOUT HIGHLY CONDUCTIVE LAYERS

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Copper wire modeled with a mesh

Copper wire represented as a highly conductive layer

Figure 2-4: Modeling a copper wire as a domain (top) requires a denser mesh compared to modeling it as a boundary with a highly conductive layer (bottom).

To describe heat transfer in highly conductive layers, the Highly Conductive Layer feature uses a variant of the heat equation that describes the in-plane heat flux in the layer:  T d s  s C s ------- +  t   – ds ks tT  t

=

q 

–

q + ds QS

= – qs

(2-38)

Here the operator t denotes the del or nabla operator projected onto the plane of the highly conductive layer. The properties in the equation are: •

s is the layer density (kg m3)

• Cs is the layer heat capacity (J (kg·K)) • ks is the layer thermal conductivity at constant pressure (W (m·K)) • ds is the layer thickness (m) • q is the heat flux from the surroundings into the layer (W m2) • q is the heat flux from the layer into the domain (W m2) • Qs represents internal heat sources within the conductive layer (W m3) • qs is the net outflux of heat through the top and bottom faces of the layer (W m2)

With the above boundary equation inserted, the general heat flux boundary condition becomes –n

62 |

q




 T    – d s k tT  on  t s 

= –ds C ------- – s p s t

Theory of Out-of-Plane Heat Transfer Out-of-Plane Heat Transfer Nodes When the object to model in COMSOL Multiphysics® is thin or slender enough along one of its geometry dimensions, there is usually only a small variation in temperature along the object’s thickness or cross section. For such objects, it is efficient to reduce the model geometry to 2D or even 1D and use the out-of-plane heat transfer mechanism. Figure 2-5 shows examples of likely situations where this type of geometry reduction can be applied. q

qup

qdown

Figure 2-5: Geometry reduction from 3D to 1D (top) and from 3D to 2D (bottom).

The reduced geometry does not include all the boundaries of the original 3D geometry. For example, the reduced geometry does not represent the upside and downside surfaces of the plate in Figure 2-5 as boundaries. Instead, heat transfer through these boundaries appears as sources or sinks in the thickness-integrated version of the heat equation used when out-of-plane heat transfer is active.

Out-of-Plane Heat Transfer Nodes

THEORY OF OUT- OF-PLANE HEAT TRANSFER

| 63

Equation Formulation

When out-of-plane heat transfer is enabled, the equation for heat transfer in solids, Equation 3-1 is replaced by  T d z  C p ------- –    d z k  T  t

=

d z Q

(2-39)

where d z is the thickness of the domain in the out-of-plane direction. The equation for heat transfer in fluids, Equation 3-2, is replaced by  --- u  C p d z  ----T   T  + t

=

   d z k  T  + d z Q

(2-40)

The Pressure Work attribute on Solids and Fluids and the Viscous Heating attribute on Fluids are not available when out-of-plane heat transfer is activated. Heat Source nodes that are added to a model with out-of-plane heat transfer enabled are multiplied by the thickness, d z. Boundary conditions are also adjusted. Activating Out-of-Plane Heat Transfer and Thickness

Using a 1D or 2D model, activate the physics features for out-of-plane heat transfer and the thickness property by clicking the main Heat Transfer node and selecting the Out-of-plane heat transfer check box under Physical Model.

64 |


Theory for the Bioheat Transfer User Interface The Bioheat Transfer Interface uses the bioheat equation and the corresponding physics nodes in the Heat Transfer interface. This is used to model heat transfer within biological tissue. This feature uses Pennes’ approximation to represent heat sources from metabolism and blood perfusion. The equation for conductive heat transfer using this approximation:  C p T +    – k T  t

=

 b Cb  b  T b – T  + Q me t

(2-41)

The density , heat capacity Cp, and thermal conductivity k are the thermal properties of the tissue. For a steady-state problem the temperature does not change with time and the first term disappears. To model Equation 2-41 add the Biological Tissue model equation, with a Bioheat feature. The Biological Tissue model provides the left-hand side of Equation 2-41 while the Bioheat node provides the two source terms on the right-hand side of Equation 2-41.

T H E O R Y F O R T H E B I O H E A T T R A N S FE R U S E R I N T E R F A C E

| 65

Theory for the Heat Transfer in Porous Media User Interface The Heat Transfer in Porous Media Interface uses the following version of the heat equation as the mathematical model for heat transfer in porous media (Ref. 14):  --  C p  eq ----T +  C p u   T =    k eq  T  + Q t

(2-42)

with the following material properties: •

 is the fluid density.

• C p is the fluid heat capacity at constant pressure. • • •

•

(C p)eq is the equivalent volumetric heat capacity at constant pressure. keq is the equivalent thermal conductivity (a scalar or a tensor if the thermal conductivities are anisotropic). u is the fluid velocity field, either an analytic expression or a velocity field from a fluid-flow interface. u should be interpreted as the Darcy velocity, that is, the volume flow rate per unit cross-sectional area. The average linear velocity (the velocity within the pores) can be calculated as uL  uL, where L is the fluid’s volume fraction, or equivalently the porosity. Q is the heat source (or sink). Add one or several heat sources as separate physics features.

The equivalent thermal conductivity of the solid-fluid system, keq, is related to the conductivity of the solid kp and to the conductive of the fluid, k by keq

=

 p k p +  L k

The equivalent volumetric heat capacity of the solid-fluid system is calculated by   Cp  eq

=

 p  p Cp p +  L  Cp

Here p denotes the solid material’s volume fraction, which is related to the volume fraction of the liquid L (or porosity) by  L +  p

66 |


=

1

For a steady-state problem the temperature does not change with time, and the first term in the left-hand side of Equation 2-42 disappears.

T H E O R Y F O R T H E H E A T T R A N S F ER I N P O R O U S M E D I A U S E R I N T E R F A C E

| 67

About Handling Frames in Heat Transfer This section discusses heat transfer analysis with moving frames, when spatial and material frames do not coincide. When the Enable conversions between material and spatial frame check box is selected, all heat transfer physics account for deformation effects on heat transfer properties. The entire physics (equations and variables) are defined on the spatial frame. When a moving mesh is detected, the user inputs for certain features are defined on the material and are converted so that all the corresponding variables contain the value on the spatial frame. • • •

The Heat Transfer Interface Domain, Boundary, Edge, Point, and Pair Nodes for the Heat Transfer User Interfaces Theory for the Heat Transfer User Interfaces

Frame Physics Feature Nodes and Definitions

This subsection contains the list of all heat transfer nodes and the corresponding definition frame. Some of the physics require additional licenses, for example, a Heat Transfer Module or a CFD Module. The following explains the different values listed in the definition frame column in Table 2-3, Table 2-4, and Table 2-5: Material: •The

inputs are entered by the user and defined on the material frame. Because the heat transfer variables and equations are defined on the spatial frame, the inputs are internally converted to the spatial frame.

68 |


Spatial: •The

inputs are entered by the user are defined on the spatial frame. No conversion is done.

Material/(Spatial): •For these physics nodes, select from a

menu to decide if the inputs are defined on the material or spatial frame. The default definition frame is the material frame.

(Material)/Spatial: •For these physics nodes, select from a menu to decide if the inputs

are defined on the material or spatial frame. The default definition frame is the spatial frame. N/A:

There is no definition frame for this physics node.

Domain Nodes

TABLE 2-3: DOMAIN PHYSICS NODES FOR FRAMES NODE NAME

DEFINITION FRAME

Heat Transfer in Solids

Material

Translational Motion

Material


Spatial

Biological Tissue

Material

Heat Transfer with Phase Change

Spatial


Material (Solid part) Spatial (Fluid part)

Thermal Dispersion

Spatial

Immobile Fluids

Spatial

Geothermal Heating

Material

Infinite Elements

Spatial

Pressure Work

Spatial

Viscous Heating

Spatial

Heat Source

Material/(Spatial)

Bioheat

Material

Opaque

N/A

Out-of-Plane Convective Heat Flux

Spatial

Out-of-Plane Radiation

Spatial


Spatial

A B O U T H A N D L I N G F R A M E S I N H E A T T R A N S F ER

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TABLE 2-3: DOMAIN PHYSICS NODES FOR FRAMES

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NODE NAME

DEFINITION FRAME


Spatial

Change Thickness

Spatial

Initial Values

Spatial


Boundary Nodes TABLE 2-4: BOUNDARY PHYSICS NODES FOR FRAMES NODE NAME

DEFINITION FRAME

Temperature

Spatial

Thermal Insulation

N/A

Outflow

N/A

Symmetry

N/A

Heat Flux

(Material)/Spatial

Inflow Heat Flux

Spatial

Open Boundary

Spatial

Thin Thermally Resistive Layer

Material

Thermal Contact

Material

Surface-to-Ambient Radiation

Spatial


Spatial

Prescribed Radiosity

Spatial

Reradiating Surface

Spatial

Radiation Group

N/A

Periodic Heat Condition

Spatial

Boundary Heat Source

Material/(Spatial)

Heat Continuity

Spatial

Pair Thin Thermally Resistive Layer

Material

Pair Thermal Contact

Material

Pair Boundar y Heat Source

Material/(Spatial)


Spatial

Highly Conductive Layer

Material

Layer Heat Source

Material

Opaque Surface

Spatial

Continuity on Interior Boundary

Spatial

Edge and Point Nodes TABLE 2-5: EDGE AND POINT NODES FOR FRAMES NODE NAME

DEFINITION FRAME

Line Heat Source

Material/(Spatial)

Point Heat Source

Material


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TABLE 2-5: EDGE AND POINT NODES FOR FRAMES NODE NAME

DEFINITION FRAME

Edge Heat Flux

(Material)/Spatial

Point Heat Flux

Spatial

Temperature

Spatial

Point Temperature

Spatial

Edge Surface-to-Ambient

Spatial

Point Surface-to-Ambient

Spatial

TABLE 2-6: HEAT TRANSFER IN THIN SHELLS NODES

NODE NAME

DEFINITION FRAME

Thin Conductive Layer

Material

Heat Source

Material/(Spatial)

Change Thickness

Spatial

Initial Values

Spatial


Spatial

Out-of-Plane Radiation

Spatial


Spatial

Heat Flux

Spatial/(Material)


Spatial

Temperature

Spatial

Change Effective Thickness

Spatial

Edge Heat Source

Material/(Spatial)

Conversion Between Material and Spatial Frames

This subsection explains how the user inputs are converted. The conversion depends on the dimension of the variables (scalars, vectors, or tensors) and on their density order. DENSITY, HEAT SOURCE, HEAT FLUX

Scalar density variables do not have the same value in the material and in the spatial frame. In heat transfer physics, the following variables are relative scalars of weight one (also called scalar densities): the mass density , the heat source Q, the heat flux q0, the heat transfer coefficient h, and the production/absorption coefficient qs.

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When a feature has its definition frame on the spatial frame, no transformation is done because the user input is defined on the spatial frame. For example, if =500[kg/m^3] is defined in the Heat Transfer in Fluids (definition frame = spatial frame) the variable ht.rho is equal to 500[kg/m^3] (on the spatial frame). When a feature has its definition frame on the material frame, the user input is defined on the material frame so it has to be multiplied by spatial.detInvF to get the corresponding value on the spatial frame. For example, if =500[kg/m^3] is defined in the Heat Transfer in Solids (definition frame = material frame) the variable ht.rho is equal to spatial.detInvF*500[kg/m^3] (on the spatial frame). As a consequence, to evaluate or integrate the mass density on the material frame, the value of ht.rho has to be multiplied by spatial.detF. spatial.detF has

different definitions based on the dimension of the geometric entity where it is evaluated. On domains it corresponds to the local volume change from the material to the spatial frame while it corresponds to local surface or length change on boundaries and edges. spatial.detInvF is the inverse of spatial.detF. VELOCITY VECTOR

The relationship between

u  x y z

and

u  X Y Z

u  x y z

=

is

T

F u  X Y Z

where F is the coordinate transform matrix from the material to the spatial frame: x X y X z X F = x Y y Y z Y x Z y Z z Z

with x X corresponding to the derivative of x with respect to X . THERMAL CONDUCTIVITY

Thermal conductivity is a tensor density. The relationship between the value on the spatial frame and the material frame is k x y z

=

T 1 ----------------- F k X Y Z  F det  F 

where k x y z  is the thermal conductivity tensor in the spatial frame and k X Y Z is the thermal conductivity tensor in the material frame. F is the coordinate transform matrix from the material frame to the spatial frame defined in the paragraph above.


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THERMAL CONDUCTIVIT Y OF HIGHLY CONDUCTIVE LAYER

The same transformations are applied to thermal conductivity but with different transformation matrices. The transformation matrix uses tangential derivatives and is defined as xT X y T X z T X F tang

=

xT Y y T Y z T Y xT Z yT Z z T Z

where xT X corresponds to the tangential derivative x with respect to X , and so on. AXISYMMETRIC GEOMETRIES

In 1D axisymmetric and 2D axisymmetric models an additional conversion is done between the material frame and the spatial frame. The density variables (density, heat source, heat flux, and so forth) are multiplied by R ---r

which corresponds to the ratio of the material first cylindrical coordinate over the spatial one. For example, if you enter a heat source Q =500[W/m^3]in the material frame in axisymmetric cases, the conversion leads to: R Q =500[W/m^3]* ---- spatial.detInvF r

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References for the Heat Transfer User Interfaces 1. F.P. Incropera, D.P. DeWitt, T.L. Bergman and A.S. Lavine, Fundamentals of Heat and Mass Transfer , John Wiley & Sons, Sixth edition, 2006. 2. R. Codina, “Comparison of Some Finite Element Methods for Solving the Diffusion-Convection-Reaction Equation,” Comp. Meth.Appl. Mech. Engrg , vol. 156, pp. 185–210, 1998. 3. A. Bejan, Heat Transfer , John Wiley & Sons, 1993. 4. G.K. Batchelor, An Introducti on to Fluid Dynamics , Cambridge University Press, 2000. 5. R.L. Panton, Incompressible Flow , 2nd ed., John Wiley & Sons, 1996. 6. M. Kaviany, Principles of Convective Heat Transfer , 2nd ed., Springer, 2001. 7. T. Poinsot and D. Veynante, Theoretical and Numerical Combustion , Second Edition, Edwards, 2005. 8. W. Wagner, and H-J Kretzschmar, International Steam Tables , 2nd ed., Springer, 2008. 9. J. Zhang, A. Gupta, and J. Bakera, “Effect of Relative Humidity on the Prediction of Natural Convection Heat Transfer Coefficients,” Heat Transfer Engineering , vol. 28, no. 4, pp. 335–342, 2007. 10. P.T. Tsilingiris, “Thermophysical and Transport Properties of Humid Air at Temperature Range Between 0 and 100 °C,” Energy Conversion and Management , vol. 29, no. 2008, pp. 1098–1110, 2007. 11. A. Bejan et al., Heat Transfer Handbook , Wiley, 2003. 12. F. Charron, Partage de la chaleur entre deux corps frottants [Heat Partition Between Two Rubbing Bodies], Publication Scientifique et Technique du Ministère de l'Air, no. 182, 1943. 13. F.P. Incropera and D.P. DeWitt, Fundamentals of Heat and Mass Transfer , fifth ed. John Wiley & Sons, 2002.

R E F E R E N C E S F O R T H E H E A T TR A N S F E R U S E R I N T E R F A C E S

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14. J. Bear and Y. Bachmat, Introduction to Modeling of Transport Phenomena in Porous Media , Kluwer Academic Publisher, 1990.

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3

The Heat Transfer Branch This chapter details the variety of interfaces found under the Heat Transfer branch ( ) in the Model Wizard and these form the fundamental interfaces in the Heat Transfer Module. It covers all the types of heat transfer—conduction, convection, and radiation—for heat transfer in solids and fluids. For information about surface-to-surface radiation see Radiation Heat Transfer. Transfer. In this chapter: • • • • • •

About the Heat Transfer Interfaces Interfaces The Heat Transfer Interface Highly Conductive Layer Nodes Out-of-Plane Heat Transfer Nodes The Bioheat Transfer Interface The Heat Transfer in Porous Media Interface

77

About the Heat Transfer Interfaces The Heat Transfer interfaces model heat transfer by conduction and convection. Surface-to-ambient radiation effects around edges and boundaries can also be included. The interfaces are available in 1D, 2D, and 3D and for axisymmetric models with cylindrical coordinates in 1D and 2D. The default default dependent variable isis the temperature, T . After selecting a version of the Heat Transfer interface interface in the Model Wizard, default nodes are added under the main node. For example: • •

If Heat Transfer in Solids ( ) is selected, a Heat Transfer (ht) node is added with a default Heat Transfer in Solids model. If Heat Transfer in Fluids ( ) is se select lected ed,, a defa defaul ultt Heat Transfer in Fluids model is added.

The benefit of the different versions of the standard Heat Transfer interface, interface, all with ht as the identifier (see Table 3-1), 3-1), is that it is easy to add the default settings when selecting the interface from the Model Wizard. At any time, right-click the parent node to add a Heat Transfer in Fluids or Heat Transfer in Solids node—the functionality is always available. The interface options are also available from the Heat Transfer interface by selecting a specific check box under the Physical Model section (for surface-to-surface radiation, biological tissue, radiation in participating media, or porous media). See Table 3-1 and 3-1 and Table 3-2. 3-2. TABLE 3-1: THE HEAT TRANSFER TRANSFER (HT) PHYSICS INTERFACE OPTIONS ICON

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NAME

DEFAULT PHYSICAL MODEL

Heat Heat Trans ransffer in Solid olidss

not not appl pplicab icablle

Heat Tr Transfer in in Fl Fluids

not ap applicable


The Heat transfer in porous media check box is selected.

Heat Transfer with Surface-to-Surface Radiation (under the Radiation branch)

The Surface-to-surface radiation check box is selected (which enables the Radiation Settings section).

C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H

TABLE 3-1: THE HEAT TRANSFER TRANSFER (HT) PHYSICS INTERFACE OPTIONS ICON

NAME


Heat Transfer with Radiation in Participating Media (under the Radiation branch)

The Radiation in participating media check box is selected (which enables the Participating Media Settings section).

Bioheat Transfer

The Heat transfer in biological tis tissue check box is selected.

TABLE 3-2: ADDITIONAL HEAT TRANSFER TRANSFER PHYSICS OPTIONS ICON

NAME

ID


Laminar Flow (under the

nitf

See Table 7-1 for details.

Turbulent Flow k- and Turbulent Flow, Low Re k- (under the Conjugate Heat Transfer branch)

nitf

See Table 7-1 for details.


htsh htsh

No Phys Physic ical al Mod Model el sect sectio ion, n, bu butt the the Surface-to-Surface Radiation check box is available to activate the Radiation Settings section.

Surface-to-Surface Radiation (under the Radiation branch)

rad rad

No Phys Physic ical al Mod Model sect sectio ion, n, but th thee Radiation Settings section is automatically available by default.

Radiation in Participating Media (under the Radiation branch)

rpm

not ap applicable

Conjugate Heat Transfer

branch)

Joule Heating (under (under the jh Electromagnetic Heating

No check boxes boxes are selected under under Physical Model.

branch)

A B O U T T H E H E A T T R A N S F ER I N T E R F A C E S

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• • • • • • •

The Heat Transfer Interface The Bioheat Transfer Interface The Heat Transfer in Porous Media Interface The Heat Transfer in Thin Shells User Interface The Conjugate Heat Transfer Branch Radiation Heat Transfer The Joule Heating User Interface in Interface in the COMSOL Multiphysics Reference Manual

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The Heat Transfer Interface The Heat Transfer user interfaces model heat transfer by conduction and convection. Surface-to-ambient radiation effects around edges and boundaries can also be included. The interfaces are available in 1D, 2D, and 3D and for axisymmetric models with cylindrical coordinates in 1D and 2D. The default default dependent variable isis the temperature, T . The Heat Transfer user interfaces include the equations, boundary conditions, and sources for modeling conductive and convective heat transfer and solving for the temperature. After selecting selecting a version version of the physics user interface in the Model Wizard (as described in About in About the Heat Transfer Transfer Interfaces), Interfaces), default nodes are added under the main node. For example: • •

If Heat Transfer in Solids ( ) is selected, a Heat Transfer in Solids (ht) node is added with a default Heat Transfer in Solids model as a subnode. If Heat Transfer in Fluids ( ) is selected, a Heat Transfer in Fluids (ht) node is added with a default Heat Transfer in Fluids model as a subnode.

The benefit of the different versions of the Heat Transfer user interfaces, with ht as the common default identifier (see Table 3-1), 3-1), is that it is easy to add the default settings when selecting the interface from the Model Wizard. At any time, time, right-click the parent node to add a Heat Transfer in Fluids or Heat Transfer in Solids node—the functionality is always available. When this interface is added, default nodes are added to to the Model Builder based based on the selection made in the Model Wizard—Heat Transfer in Solids or Heat Transfer in Fluids, Thermal Insulation (the default boundary condition), and Initial Values. Right-click the node to add other features that implement, for example, boundary Heat Transfer node conditions and sources.

Depending on the version of the interface selected, the default nodes vary var y.

THE HEAT TRANSFER INTERFACE

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INTERFACE IDENTIFIER

The interface identifier is used primarily as a scope prefix for variables defined by the physics user interface. Refer to such interface variables in expressions using the pattern .. In order to distinguish between variables belonging to different physics user interfaces, the identifier string must be unique. Only letters, numbers and underscores (_) are permitted in the Identifier field. The first character must be a letter. The default identifier (for the first interface in the model) is

ht.

DOMAIN SELECTION

The default setting is to include All domains in the model to define heat transfer and a temperature field. To choose specific domains, select Manual from the Selection list. PHYSICAL MODEL

By default, no check boxes are selected for the standard version of the Heat Transfer interface. Click to select any of the available check boxes to activate the other versions of the ht interface as detailed in Table 3-1 and Table 3-2. If required for 2D or 1D models, select the Out-of-plane heat transfer check box and then enter the Thickness of the plane ( d z). The default is 1 m and applies to the entire geometry. If another thickness is specified for some of the domains, use the Change Thickness node.

About the Heat Transfer Interfaces

CONSISTENT STABILIZATION

To display this section, click the Show button ( ) and select Stabilization. The Streamline diffusion check box is selected by default and should remain selected for optimal performance for heat transfer in fluids or other applications that include a convective or translational term. The Crosswind diffusion check box is also selected by default. It provides extra diffusion in the region of sharp gradients. The added diffusion is orthogonal to the streamline diffusion, so streamline diffusion and crosswind diffusion can be used simultaneously.

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I N C O N S I S T E NT S T A B I L I Z A T I O N

To display this section, click the Show button ( diffusion check box is not selected by default.

) and select Stabilization. The Isotopic

DISCRETIZATION

To display this section, click the • •

• •

Show button

(

) and select Discretization.

Select an element order (shape function order) for the Temperature—Quadratic (the default), Linear , Cubic, Quartic, or Quintic. The Compute boundary fluxes check box is selected by default so that COMSOL computes predefined accurate boundary flux variables (with the suffix _acc such as ht.ndflux_acc for the accurate—as opposed to the standard extrapolated— normal conductive heat flux). The Apply smoothing to boundary fluxes check box is selected by default. The smoothing can provide a more well-behaved flux value close to singularities. In the table, specify the Value type when using splitting of complex variables —Real (the default) or Complex.

DEPENDENT VARIABLES

The Heat Transfer user interfaces have a dependent variable for the Temperature T . The dependent variable names can be changed. Editing the name of a scalar dependent variable changes both its field name and the dependent variable name. If a new field name coincides with the name of another field of the same type, the fields will share degrees of freedom and dependent variable names. A new field name must not coincide with the name of a field of another type, or with a component name belonging to some other field. ADVANCED SETTINGS

Add both a Heat Transfer (ht) and Moving Mesh (ale) interface (found under the Mathematics>Deformed Mesh branch in the Model Wizard) then click the Show button ( ) and select Advanced Physics Options to display this section. When the model contains moving mesh, the Enable conversions between material and spatial frame check box is selected by default. This option has no effect when the model does not contain a moving frame since the material and spatial frames are identical in this case. With moving mesh, and when this option is active, the heat transfer features automatically account for deformation effects on heat transfer properties. In particular the ef fects for volume changes on the density are considered. Rotation effects on thermal conductivity of an anisotropic material


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and, more generally, deformation effects on arbitrary thermal conductivity, are also covered. When the Enable conversions between material and spatial frame check box is not selected, the feature inputs (for example, Heat Source, Heat Flux, Boundary Heat Source, and Line Heat Source) are not converted and all are defined on the Spatial frame. • • • • • •

About Handling Frames in Heat Transfer Show More Physics Options Domain, Boundary, Edge, Point, and Pair Nodes for the Heat Transfer User Interfaces Consistent and Inconsistent Stabilization Methods for the Heat Transfer User Interfaces Theory for the Heat Transfer User Interfaces Show Stabilization in the COMSOL Multiphysics Reference Manual

Domain, Boundary, Edge, Point, and Pair Nodes for the Heat Transfer User Interfaces

The Heat Transfer Interface has these domain, boundary, edge, point, and pair nodes and subnodes available (listed in alphabetical order including out-of-plane and highly conductive layer features). To locate and search all the documentation, in COMSOL Multiphysics, select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree.

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• • • • • • • • • • • • • • • • • •

Boundary Heat Source Change Thickness Convective Heat Flux Continuity Edge Heat Flux Edge Surface-to-Ambient Radiation Heat Flux Heat Source Heat Transfer in Fluids Heat Transfer in Solids Heat Transfer with Phase Change Highly Conductive Layer Initial Values Inflow Heat Flux Layer Heat Source Line Heat Source Open Boundary Outflow

• • • • • • • • • • • • • • • • •

Out-of-Plane Convective Heat Flux Out-of-Plane Heat Flux Out-of-Plane Radiation Periodic Heat Condition Point Heat Flux Point Heat Source Point Temperature Point Surface-to-Ambient Radiation Pressure Work Surface-to-Ambient Radiation Symmetry Temperature Thermal Contact Thermal Insulation (the default boundary condition) Thin Thermally Resistive Layer Translational Motion Viscous Heating

For axisymmetric models, COMSOL Multiphysics takes the axial symmetry boundaries into account and automatically adds an Axial Symmetry node to the model that is valid on the axial symmetry boundaries only. The following nodes are also available for some versions of the Heat Transfer interface and described in Radiation Heat Transfer chapter. • •

Incident Intensity Opaque


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• •

Opaque Surface Radiation in Participating Media

Heat Transfer in Solids

The Heat Transfer in Solids node uses the heat equation version in Equation 3-1 as the mathematical model for heat transfer in solids: --- –    k  T   C p ----T t

=

Q

(3-1)

For a steady-state problem the temperature does not change with time and the first term disappears. The equation includes the following material properties: the density  , the heat capacity C p, and the thermal conductivity k (a scalar or a tensor if the thermal conductivity is anisotropic), and a heat source (or sink) Q—one or more heat sources can be added separately. When parts of the model are moving in the material frame, right-click the Heat Transfer in Solids node to add a Translational Motion node to take this into account. If you have the CFD Module, also right-click to add a Pressure Work node. If you have the Heat Transfer Module, also right-click to add Pressure Work or Opaque nodes. The Opaque subnode is automatically added to the entire selection when Surface-to-surface radiation is activated. The selection can be edited. DOMAIN SELECTION

For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically selected and is the same as for the interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific domains or select All domains as required. MODEL INPUTS

This section contains fields and values that are inputs to expressions that define material properties. If such user-defined materials are added, the model inputs appear here. Initially, this section is empty.

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COORDINATE SYSTEM SELECTION

The Global coordinate system is selected by default. The Coordinate system list contains any additional coordinate systems that the model includes (except boundary coordinate systems). The coordinate system is used for interpreting directions of orthotropic and anisotropic thermal conductivity. HEAT CONDUCTION, SOLID

The default setting is to use the Thermal conductivity k (SI unit: W/(m·K)) From material. If User defined is selected, choose Isotropic, Diagonal, Symmetric, or Anisotropic based on the characteristics of the thermal conductivity, and enter another value or expression. The thermal conductivity describes the relationship between the heat flux vector q and the temperature gradient T as in q = kT, which is Fourier’s law of heat conduction. Enter this quantity as power per length and temperature. The components of a thermal conductivity k in the case that it is a tensor (k xx, k yy, and so on, representing an anisotropic thermal conductivity) are available as ht.kxx, ht.kyy, and so on (using the default interface identifier ht). The single scalar mean effective thermal conductivity ht.kmean is the mean value of the diagonal elements k xx, k yy, and k zz.

Fourier’s law expect that the thermal conductivity tensor is symmetric. A non symmetric tensor can lead to unphysical results. THERMODYNAMI CS; SOLID

The default Density  (SI unit: kg/m3) and Heat capacity at constant pressure C p (SI unit: J/(kg·K)) use values From material. Select User defined to enter other values or expressions. The heat capacity at constant pressure describes the amount of heat energy required to produce a unit temperature change in a unit mass. Thermal Diffusivity

In addition, the thermal diffusivity  is defined as k/( C p) (SI unit: m 2/s) is also a predefined quantity. The thermal diffusivity can be interpreted as a measure of thermal inertia (heat propagates slowly where the thermal diffusivity is low, for example). The components of a thermal diffusivity  in the case that it is a tensor (  xx,  yy, and so on,


| 87

representing an anisotropic thermal diffusivity) are available as ht.alphaTdxx, ht.alphaTdyy, and so on (using the default interface identifier ht). The single scalar mean thermal diffusivity ht.alphaTdMean is the mean value of the diagonal elements  xx,  yy, and  zz. The denominator  C p is the effective volumetric heat capacity, and is also available as a predefined quantity, ht.C_eff. •

Axisymmetric Transient Heat Transfer: Model Library path COMSOL_Multiphysics/Heat_Transfer/heat_transient_axi

•

2D Heat Transfer Benchmark with Convective Cooling: Model Library path COMSOL_Multiphysics/Heat_Transfer/heat_convection_2d

Translational Motion

Right-click the Heat Transfer in Solids node to add the Translational Motion node, which provides movement by translation to model heat transfer in solids. It adds the following contribution to the right-hand side of Equation 3-1, defined in the parent node: –

 Cp u  T

The contribution describes the effect of a moving coordinate system that is required to model, for example, a moving heat source. Special care must be taken at boundaries where n·u  0. The Heat Flux boundary condition does not, for example, work at boundaries where n·u  0. DOMAIN SELECTION

From the Selection list, choose the domains to define. By default, the selection is the same as for the Heat Transfer in Solids node that it is attached to, but it is possible to use more than one Heat Translation subnode, each covering a subset of the Heat Transfer in Solids node’s selection.

88 |


TRANSLATIONAL MOTION

Enter component values for x, y, and z (in 3D) for the Velocity field utrans (SI unit: m/ s). Heat Generation in a Disc Brake: Model Library path Heat_Transfer_Module/Thermal_Contact_and_Friction/brake_disc


The Heat Transfer in Fluids model uses the following version of the heat equation as the mathematical model for heat transfer in fluids: --- +  C p u   T =    k  T  + Q  C p ----T t

(3-2)

For a steady-state problem the temperature does not change with time and the first term disappears. This equation includes the following material proper ties, fields, and sources: • Density  (SI unit: kg/m 3) • Heat capacity at constant pressure C p (SI unit: J/(kg·K))—describes the amount of • • • •

heat energy required to produce a unit temperature change in a unit mass. Thermal conductivity k (SI unit: W/(m·K)) —a scalar or a tensor if the thermal conductivity is anisotropic. Velocity field u (SI unit: m/s)—either an analytic expression or a velocity field from a fluid-flow interface. The heat source (or sink) Q—one or more heat sources can be added separately. The Ratio of specific heats  (dimensionless)— the ratio of heat capacity at constant pressure, C p, to heat capacity at constant volume, Cv. When using the ideal gas law to describe a fluid, specifying  is enough to evaluate C p. For common diatomic gases such as air,   1.4 is the standard value. Most liquids have   1.1 while water has   1.0.  is used in the streamline stabilization and in the variables for heat fluxes and total energy fluxes. It is also used if the ideal gas law is applied. See Thermodynamics; Solid.


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Right-click to add Viscous Heating (for heat generated by viscous friction), Opaque, or Pressure Work nodes to the Heat Transfer in Fluids feature.

Heat Transfer by Free Convection: Model Library path COMSOL_Multiphysics/Multiphysics/free_convection

DOMAIN SELECTION


This section has fields and values that are inputs to expressions that define material properties. If such user-defined property groups are added, the model inputs appear here. There are also two standard model inputs— Absolute pressure and Velocity field. The absolute pressure is used in some predefined quantities that include the enthalpy (the energy flux, for example). Absolute pressure is also used if the ideal gas law is applied. See Thermodynamics; Solid. Abs olute Pre ssure

This section controls both the variable as well as any property value (reference pressures) used when solving for pressure. There are usually two ways to calculate the pressure when describing fluid flow and mass and heat transfer. Solve for the absolute

90 |


pressure or a pressure (often denoted gauge pressure) that relates back to the absolute pressure through a reference pressure. Using one or the other option usually depends on the system and the equations being solved for. For example, in a straight incompressible flow problem, the pressure drop over the modeled domain is probably many orders of magnitude less than atmospheric pressure, which, if included, reduces the chances for stability and convergence during the solving process for this variable. In other cases, the absolute pressure may be required to be solved for, such as where pressure is a par t of an expression for gas volume or diffusion coefficients. The default Absolute pressure p A (SI unit: Pa) is User defined and is 1 atm (101,325 Pa). When additional physics interfaces are added to the model, the pressure variables solved can also be selected from the list. For example, if a fluid-flow interface is added you can select Pressure (spf/fp) from the list. When a Pressure variable is selected, the Reference pressure check box is selected by default and the default value of pref is 1[atm] (1 atmosphere). This makes it possible to use a system-based (gauge) pressure as the pressure variable while automatically including the reference pr essure in places where it is required, such as for gas flow governed by the gas law. While this check box maintains control over the pressure variable and instances where absolute pressure is required within this r espective physics interface, it may not with physics interfaces that it is being coupled to. In such models, check the coupling between any interfaces using the same variable. Velocity Field

The default Velocity field u (SI unit: m/s) is User defined. When User defined is selected, enter values or expressions for the components based on space dimension. The defaults are 0 m/s. Or select an existing velocity field in the model (for example, Velocity field (spf/fp1) from a Laminar Flow interface). COORDINATE SYSTEM SELECTION

The Global coordinate system is selected by default. The Coordinate system list contains any additional coordinate systems that the model includes (except boundary


| 91

coordinate systems). The coordinate system is used for interpreting directions of orthotropic and anisotropic thermal conductivity. HEAT CONDUCTION, FLUID

The default Thermal conductivity k (SI unit: W/(m·K)) is taken From material. If User defined is selected, choose Isotropic, Diagonal, Symmetric, or Anisotropic based on the characteristics of the thermal conductivity, and enter another value or expression. The thermal conductivity describes the relationship between the heat flux vector q and the temperature gradient T as in q = kT which is Fourier’s law of heat conduction. Enter this quantity as power per length and temperature. THERMODYNAMIC S, FLUID

The default Density  (SI unit: kg/m3), Heat capacity at constant pressure C p (SI unit: J/(kg·K)), and Ratio of specific heats  (dimensionless) for a general gas or liquid use values From material. Select User defined to enter other values or expressions. Select a Fluid type—Gas/Liquid, Moist air , or Ideal gas. Gas/Liquid

Select Gas/Liquid to specify the Density, the Heat capacity at constant pressure, and the Ratio of specific heats for a general gas or liquid. The default settings are to use data From material. Select User defined to enter another value for the density, heat capacity, or ratio of specific heats. Ideal Gas

Select Ideal gas to use the ideal gas law to describe the fluid. Then: •

Select a Gas constant type—Specific gas constant Rs (SI unit: J/(kg·K)) or Mean molar mass M n (SI unit: kg/mol). For both properties, the default setting is to use the property value from the material. Select User defined to enter another value for either of these material properties. If Mean molar mass is selected, the software uses the universal gas constant R  8.314 J/(mol·K), which is a built-in physical constant.

92 |


•

From the list under Specify Cp or , select Heat capacity at constant pressure C p (SI unit: J/(kg·K)) or Ratio of specific heats  (dimensionless). For both properties, the default setting is to use the property value From material. Select User defined to define another value for either of these material properties. For an ideal gas, specify either C p or the ratio of specific heats,  , but not both since these, in that case, are dependent.

Moi st Air

If Moist air is selected, the thermodynamics properties are defined as a function of the quantity of vapor in the moist air. Four different options are available from the Input quantity list to define the amount of vapor in the moist air: • •

• •

Select Vapor mass fraction (the default) to define the vapor mass fraction (SI unit: kg/kg). Select Concentration to define the concentration of vapor (SI unit: mol/m3). Once this option is selected a Concentration model input is automatically added in the Models Inputs section. Select Moisture content to define the moisture content of the moist air (SI unit: kg/ kg). Select Relative humidity to define the quantity of vapor from a Reference relative humidity (SI unit: 1), a Reference temperature (SI unit: K), and a Reference pressure (SI unit: Pa). These three reference values are used to estimate the mass fraction of vapor, which is used to define the thermodynamic properties of the moist air.

Moist Air Theory

Initial Values

The Initial Values node adds an initial value for the temperature that can serve as an initial condition for a transient simulation or as an initial guess for a nonlinear solver. Right-click to add additional Initial Values nodes.


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DOMAIN SELECTION

For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically selected and is the same as for the interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific domains or select All domains as required. INITIAL VALUES

Enter a value or expression for the initial value of the Temperature T (SI unit: K). The default value is approximately room temperature, 293.15 K (20 ºC). Heat Source

The Heat Source describes heat generation within the domain. You express heating and cooling with positive and negative values, respectively. Add one or more nodes as required—all heat sources within a domain contribute to the total heat source. Specify the heat source as the heat per volume in the domain, as a linear heat source, or as a total heat source (power). DOMAIN SELECTION

From the Selection list, choose the domains to add the heat source to. HEAT SOURCE

Click the General source (the default), Linear source, or Total power button. •

• •

If General source is selected, enter a value for the distributed heat source Q (SI unit: W/m3) when the default option, User defined, is selected. The default is 0 W/m3(that is, no heat source). See also Additional General Source Options. If Linear source (Q  qs·T ) is selected, enter the Production/absorption coefficient qs (SI unit: W/(m3·K)). The default is 0 W/(m 3·K). If Total power is selected, enter the total heat source, Ptot, (SI unit: W). The default is 0 W. In this case Q = Ptot/V, where V is the total volume of the selected domains.

In 3D and 2D axial symmetry, V =

94 |


1 .

In 2D and 1D axial symmetry: V

=



dz 1

where dz is the out-of-plane thickness. If the out-of-plane property is not active, a text field is available to define dz. In 1D: V

=



Ac 1

where Ac is the cross-sectional area. If the out-of-plane property is not active, a text field is available to define Ac. The advantage of writing the source in this second form is that it can be stabilized by the streamline diffusion. The theory covers qs that is independent of the temperature, but some stability can be gained as long as qs is only weakly dependent on the temperature. Additional General Source Options For the general heat source Q, there are predefined heat

sources available (in addition to a User defined heat source) when simulating heat transfer together with electrical or electromagnetic physics user interfaces. Such sources represent, for example, ohmic heating and induction heating. The following options are also available from the General source list above but require additional interfaces and/or licenses as indicated. • •

With the addition of an Electric Currents physics interface, the Total power dissipation density (ec/cucn1) heat source is available from the General source list. With the addition of any version of the Electromagnetic Waves user interface (which requires the RF Module), the Total power dissipation density (emw/wee1) and Electromagnetic power loss density (emw/wee1) heat sources are available from the General source list.


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•

•

•

With the addition of a Magnetic Fields user interface (a 3D model requires the AC/ DC Module), the Electromagnetic heating (mf/al1) heat source is available from the General source list. With the addition of a Magnetic and Electric Fields user interface (which requires the AC/DC Module), the Electromagnetic heating (mef/alc1) heat source is available from the General source list. For the Heat Transfer in Porous Media user interface, with the addition of physics interfaces from the Batteries & Fuel Cells Module, Corrosion Module, or Electrodeposition Module, heat sources from the electrochemical current distribution interfaces are available.

FRAME SELECTION

To display this section, add both a Heat Transfer (ht) and a Moving Mesh (ale) user interface (found under the Mathematics>Deformed Mesh branch in the Model Wizard). Then click the Show button ( ) and select Advanced Physics Options. When the model contains a moving mesh, the Enable conversions between material and spatial frame check box is selected by default on the Heat Transfer interface, which in turn enables this section. Use Frame Selection to select the frame where the input variables are defined. If Spatial is selected, the variables take their values from the edit fields. If Material (the default) is selected, a conversion from the material to the spatial frame is applied to the edit field values . • • •

About Handling Frames in Heat Transfer The Heat Transfer Interface Stabilization Techniques in the COMSOL Multiphysics Reference Manual

Heat Transfer with Phase Change

The Heat Transfer with Phase Change node is used to solve the heat equation after specifying the properties of a phase change material according to the apparent heat capacity formulation. Right-click to add Viscous Heating (for heat generated by viscous friction), Opaque, or Pressure Work nodes to the Heat Transfer with Phase Change node.

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DOMAIN SELECTION

From the Selection list, choose the domains to define where heat transfer with phase change occurs. MODEL INPUTS

This section is the same as for Heat Transfer in Fluids. NUMBER OF TRANSITIONS

To display this section, click the Show button ( ) and select Advanced Physics Options. Choose the Number of phase transitions to model. The default value is 1 and the maximum value is 5. In most cases, only one phase transition is needed to simulate solidification, melting, or evaporation. If you want to model successive melting and evaporation, or any couple of successive phase transformations, choose 2 in the Number of phase transitions list. It is useful to choose 3 or more transitions to handle extra changes of material properties such as in mixtures of compounds, metal alloys, composite materials, or allotropic varieties of a substance. For example, , , and -iron are allotropes of solid iron that can be considered as phases with distinct phase change temperatures. PHASE CHANGE

Enter a Phase change temperature between phase 1 and phase 2 T pc,12 (SI unit: K). The default is 273.15 K. Enter any additional phase change temperatures as per the Number of phase transitions. Enter a Transition interval between phase 1 and phase 2 T 12 (SI unit: K). The default is 10 K. Enter any additional transition intervals as per the Number of phase transitions. The value of T 1  2 must be strictly positive. A value near 0 K corresponds to a behavior close to a pure substance. Enter a Latent heat from phase 1 and phase 2 L12 (SI unit: J/kg). The default is 333 kJ/kg. Enter any additional latent heat values as per the Number of phase transitions. About the Phases

The different phases are ordered according to the temperatures of fusion. Hence, the material properties of phase 1 are valid when T  T pc,1  2 while the material properties of phase 2 hold for T  T pc,1  2.


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When more than one transition is modeled, the number of phases exceeds 2 and new variables are created (for example, T pc, 2  3, T 2  3 or L2  3). The phase change temperatures T pc, j  j  1 are increasing and satisfy T pc 1  2  T pc 2  3  

This defines distinct domains of temperature bounded by T pc, j  1  j and T pc, j  j  1 where the material properties of phase j only apply. In addition, the values of T j  j  1 are chosen so that the ranges between T pc, j  j  1  T j  j  12 and T pc, j  j  1  T j  j  12 do not overlap. If this condition is not satisfied, unexpected behaviors may occur because some phases would never form completely. The values of T j  j  1 must all be strictly positive. PHASE

For each Phase section (based on the Number of phase transitions) select or enter the following: Select a Material, phase [1,2...], which can point to any material in the model. The default uses the Domain material. The defaults for the following use values From material. Or select User defined to enter a different value or expression for each: •

• • •

The default Thermal conductivity kphase[1,2,...] (SI unit: W/(m·K)) uses the material values for phase i. If User defined is selected, choose Isotropic, Diagonal, Symmetric, or Anisotropic based on the characteristics of the thermal conductivity, and enter another value or expression. The default is 1 W/(m·K). 3 3 Density  phase[1,2,...] (SI unit: kg/m ). The default is 1000 kg/m . Heat capacity at constant pressure Cp,phase[1,2,...] (SI unit: J/(kg·K)). The default is 4200 J/(kg·K). Ratio of specific heats  phase[1,2,...] (dimensionless). The default is 1.1

About Heat Transfer with Phase Change

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Thermal Insulation

The Thermal Insulation node is the default boundary condition for all Heat Transfer interfaces. This boundary condition means that there is no heat flux across the boundary: n   k  T 

=

0

This condition specifies where the domain is well insulated. Intuitively this equation says that the temperature gradient across the boundary must be zero. For this to be true, the temperature on one side of the boundary must equal the temperature on the other side. Because there is no temperature difference across the boundary, heat cannot transfer across it. An interesting numerical check for convergence is the numerical evaluation of the thermal insulation condition along the boundary. Another check is to plot the temperature field as a contour plot. Ideally the contour lines are perpendicular to any insulated boundar y. BOUNDARY SELECTION

For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically selected and is the same as for the interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific boundaries or select All boundaries as required. Temperature

Use the Temperature node to specify the temperature somewhere in the geometry, for example, on boundaries. BOUNDARY SELECTION

From the Selection list, choose the boundaries to define. For The Heat Transfer in Thin Shells User Interface, this condition can also be applied to edges and pairs. TEMPERATURE

The equation for this condition is T = T 0 where T 0 is the prescribed temperature on the boundary. Enter the value or expression for the Temperature T 0 (SI unit: K). The default is 293.15 K.


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CONSTRAINT SETTINGS

To display this section, click the Show button ( •

• •

) and select Advanced Physics Options .

By default Classic constraints is selected. To Apply reaction terms on all dependent variables, select All physics (symmetric). Otherwise, select Current physics (internally symmetric) or Individual dependent variables to restrict the reaction terms as required. Select the Use weak constraints check box to replace the standard constraints with a weak implementation. Select the Discontinuous Galerkin constraints button when Classic constraints do not work satisfactorily. The Discontinuous Galerkin constraints option is especially useful to prevent oscillations on inlet boundaries where convection dominates. Unlike the Classic constraints, these constraints do not enforce the temperature on the boundary extremities. This is relevant on fluid inlets where the temperature may not be enforced on the walls at the inlet extremities.

Show More Physics Options

Outflow

The Outflow node provides a suitable boundary condition for convection-dominated heat transfer at outlet boundaries. In a model with convective heat transfer, this condition states that the only heat transfer over a boundary is by convection. The temperature gradient in the normal direction is zero, and there is no radiation. This is usually a good approximation of the conditions at an outlet boundary in a heat transfer model with fluid flow. BOUNDARY SELECTION

In most cases, the Outflow node does not require any user input. If required, select the boundaries that are convection-dominated outlet boundaries.

100 |


Symmetry

The Symmetry node provides a boundary condition for symmetry boundaries. This boundary condition is similar to a Thermal Insulation condition, and it means that there is no heat flux across the boundary. The symmetry condition only applies to the temperature field. It has no effect on the radiosity (surface-to-surface radiation) and on the radiative intensity (radiation in participating media). BOUNDARY SELECTION

In most cases, the node does not require any user input. If required, define the symmetry boundaries.

Heat Flux

Use the Heat Flux node to add heat flux across boundaries and edges. A positive heat flux adds heat to the domain. This feature is not applicable to inlet boundaries.

For inlet boundaries, use the Inflow Heat Flux condition instead.

BOUNDARY OR EDGE SELECTION

From the Selection list, choose the boundaries or edges to define. PAIR SELECTION

If this node is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect. HEAT FLUX

Click to select the General inward heat flux (the default), Inward heat flux, or Total heat flux button.


| 101

General Inward Heat Flux If General inward heat flux q0 (SI unit: W/m2) is selected, it adds to the total flux across the selected boundaries. Enter a value for q0 to represent a heat flux that enters the

domain. For example, any electric heater is well represented by this condition, and its geometry can be omitted. The default is 0 W/m 2. Inward Heat Flux

If Inward heat flux is selected, enter the Heat transfer coefficient h (SI unit: W/(m2·K)). The default is 0 W/(m 2·K). Also enter an External temperature T ext (SI unit: K). The default is 293.15 K. The value depends on the geometry and the ambient flow conditions. Inward heat flux is defined by q0  hT ext  T . For a thorough introduction about how to calculate heat transfer coefficients, see Incropera and DeWitt in Ref. 1. Total Heat Flux

If Total heat flux is selected, enter the total heat flux qtot (SI unit: W) for the total heat flux across the boundaries where the Heat Flux node is active. The default is 0 W. In this case q0 = qtot/A, where A is the total area of the selected boundaries.

In 3D and 2D axial symmetry, A

=

1 .

In 2D and 1D axial symmetry: A

=



dz 1

where dz is the out-of-plane thickness. If the out-of-plane property is not active, a text field is available to define dz.

102 |


In 1D: A

=



Ac 1

where Ac is the cross-sectional area. If the out-of-plane property is not active, a text field is available to define Ac. FRAME SELECTION

The settings are the same for the Heat Source node and described under the Frame Selection section. • •

About Handling Frames in Heat Transfer The Heat Transfer Interface


Use the Surface-to-Ambient Radiation condition to add surface-to-ambient radiation to boundaries. The net inward heat flux from surface-to-ambient radiation is q

=

4

4

  T amb – T 

where  is the surface emissivity,  is the Stefan-Boltzmann constant (a predefined physical constant), and T amb is the ambient temperature. BOUNDARY SELECTION

From the Selection list, choose the boundaries to define. MODEL INPUTS

This section contains fields and values that are inputs to expressions that define material properties. If such user-defined materials are added, the model inputs appear here. Initially, this section is empty. SURFACE-TO-AMBI ENT RADIATION

The default Surface emissivity  (a dimensionless number between 0 and 1) is taken From material. An emissivity of 0 means that the surface emits no radiation at all and an emissivity of 1 means that it is a perfect blackbody.


| 103

Enter an Ambient temperature T amb (SI unit: K). The default is 293.15 K. Continuous Casting: Model Library path Heat_Transfer_Module/ Thermal_Processing/continuous_casting

Periodic Heat Condition

Use the Periodic Heat Condition to add a periodic heat condition to boundaries. Right-click to add a Destination Selection node. BOUNDARY SELECTION

From the Selection list, choose the boundaries to define. In the COMSOL Multiphysics Reference Manual : • •

Periodic Condition and Destination Selection Periodic Boundary Conditions

Boundary Heat Source

The Boundary Heat Source models a heat source (or heat sink) that is embedded in the boundary. When selected as a Pair Boundary Heat Source, it also prescribes that the temperature field is continuous across the pair. BOUNDARY SELECTION

From the Selection list, choose the boundaries to define. PAIR SELECTION

When Pair Boundary Heat Source is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect. BOUNDARY HEAT SOURCE

Click the General source (the default) or Total boundary power button. •

104 |

If General source is selected, enter a value for the boundary heat source Qb (SI unit: W/m2) when the default option, User defined, is selected. A positive Qb is heating and a negative Qb is cooling. The default is 0 W/m 2. For the general boundary heat


•

source Qb, there are predefined heat sources available when simulating heat transfer together with electrical or electromagnetic physics user interfaces. Such sources represent, for example, ohmic heating and induction heating. If Total boundary power is selected, enter the total power (total heat source) Pb, tot (SI unit: W). The default is 0 W. In this case Qb = Pb, tot/A, where A is the total area of the selected boundaries.


=

1 .

In 2D and 1D axial symmetry: A

=



dz 1

where dz is the out-of-plane thickness. If the out-of-plane property is not active, a text field is available to define dz. In 1D: A

=



Ac 1

where Ac is the cross-sectional area. If the out-of-plane property is not active, a text field is available to define Ac. FRAME SELECTION

The settings are the same for the Heat Source node and described under the Frame Selection section.

• •



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Continuity

The Continuity node can be added to pairs. It prescribes that the temperature field is continuous across the pair. Continuity is only suitable for pairs where the boundaries match. BOUNDARY SELECTION

The selection list in this section shows the boundaries for the selected pairs. PAIR SELECTION

When this node is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect. In the COMSOL Multiphysics Reference Manual : • •

Continuity on Interior Boundaries Identity and Contact Pairs

Thin Thermally Resistive Layer

Use the Thin Thermally Resistive Layer node to define the thickness and thermal conductivity of a resistive material located on boundaries. This material can be formed of one or more layers. It can be added to pairs by selecting Pair Thin Thermally Resistive Layer from the Pairs menu. The resistive material can also be defined through the Thermal Resistance: ds R s = ----ks

The heat flux across the Thin Thermally Resistive Layer is defined by

106 |

–nd

  –kd  T d 

T u – T d = – k s -------------------d

–nu

  –ku  T u 

T d – T u = – k s -------------------d


s

s

where the u and d subscripts refer to the upside and the downside of the slit, respectively. When using the Pair Thin Thermally Resistive Layer node, then the u and d subscripts refer to the upside and the downside of the pair, respectively, instead of the slit. Like any pair feature, the Pair Thin Thermally Resistive Layer condition contributes with any other pair feature. However, do not use two conditions on the same pair. In order to model a thin resistive layer made of several materials, use the Multiple layers option, which is available with the Heat Transfer Module. When the material has a multilayer structure ks and ds in the expressions above are replaced by dtot and ktot, which are defined according to Equation 3-3 and Equation 3-4: nl

d tot

d

=

sj

(3-3)

d tot ------------------

(3-4)

j

k tot

=

=

1

nl

dsj

 ------k j

=

1

sj

where nl is the number of layers. BOUNDARY SELECTION


This section contains fields and values that are inputs to expressions that define material properties. If such user-defined materials are added, the model inputs appear here. Initially, this section is empty. PAIR SELECTION

If this node is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect.


| 107

THIN THERMALLY RESISTIVE LAYER

By default the Multiple layers check box is not selected. To define multiple sandwiched thin layers with different thermal conductivities, click to select the check box. Then select the Number of layers to define (1 to 5) and set the properties for each layer selected. •

• •

Select an option from the Solid material (Solid material 1, Solid material 2, and so on) list to assign a material to each layer. The default setting, Boundary material, takes the material from the boundary. For each layer, enter the Layer thickness ds (SI unit: m). The default is 0.0050 m. The default Thermal conductivity ks (SI unit: W/(m·K)) is taken From material, which is then taken from the material selected in Solid material (1, 2, ...). Select User defined to enter another value or expression. The default is 0.01 W/(m·K).

Thermal Contact

The Thermal Contact node defines correlations for the conductance h at the interface of two bodies in contact. It can be added to pairs by selecting Pair Thermal Contact from the Pairs menu. The conductance h is involved in the heat flux across the surfaces in contact according to: – nd –nu

  – kd  T d 

  –k u  T u 

= –h

= –h

 T u – T d  + rQ fric

 T d – T u  +  1 – r  Qfric

where u and d subscripts refer respectively to the upside and downside of the slit. BOUNDARY SELECTION

From the Selection list, choose the boundaries to add a thermal contact condition. MODEL INPUTS


When Pair Thermal Contact is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect.

108 |


CONTACT

Constriction Conductance Correlation

Choose the Constriction conductance correlation—Cooper-Mikic-Yovanovich correlation (the default), Mikic elastic correlation, or User defined. If User defined is selected, enter a value or expression for the Constriction conductance hc (SI unit: W/(m2·K)). The default is 0 W/(m2·K). Gap Conductance Correlation

Choose the Gap conductance correlation—User defined or Parallel-plate gap gas conductance (the second option is only available if Cooper-Mikic-Yovanovich correlation or Mikic elastic correlation is chosen as the Constriction conductance correlation). •

If User defined is selected, enter a value for the Gap conductance hg (SI unit: W/ (m2·K)). The default is 0 W/(m 2·K).

Radiative Conductance Correlation

When the Surface-to-surface radiation check box is selected under the Physical Model section on a physics interface, choose the Radiative conductance correlation—User defined or Gray-diffuse parallel surfaces (the default). •

If User defined is selected, enter a value for the Radiative conductance hT (SI unit: W/ (m2·K)). The default is 0 W/(m 2·K).

CONTACT SURFACE PROPERTIES

This section displays if Cooper-Mikic-Yovanovich correlation or Mikic elastic correlation are chosen under Contact. Enter values for the: • Asperities average height  asp (SI unit: m). The default is 1 µm. • Asperities average slope masp (dimensionless). The default is 0.4. • Contact pressure p (SI unit: Pa). The default is 0 Pa.

When Cooper-Mikic-Yovanovich correlation is selected, choose a Hardness definition— Microhardness (the default), Vickers hardness, or Brinell hardness. •

If Microhardness is selected, enter a value for H c (SI unit: Pa). The default is 3 GPa.


| 109

•

•

If Vickers hardness is selected, enter a value for the Vickers correlation coefficient c1 (SI unit: Pa) and Vickers size index c2 (dimensionless). The defaults are 5 GPa and 0.1, respectively. If Brinell hardness is selected, enter a value for H B (SI unit: Pa). The default is 3 GPa. H B should be between 1.30 and 7.60 GPa.

When Mikic elastic correlation is selected, choose the Contact interface Young’s modulus definition—Weighted harmonic mean (the default) or User defined. • •

If Weighted harmonic mean is selected, enter values or expressions for the Young’s modulus E (SI unit Pa) and Poisson’s ratio  (dimensionless). The defaults are 0. If User defined enter another value or expression for the Contact interface Young’s modulus Econtact (SI unit: Pa). The default is 1 GPa.

GAP PROPERTIES

This section is available when Parallel-plate gap gas conductance is selected as the Gap conductance correlation under Contact. The default Gas thermal conductivity kgap (SI unit: W/(m·K)) is taken From material. If User defined is selected, also choose Isotropic, Diagonal, Symmetric, or Anisotropic based on the characteristics of the gas thermal conductivity, and enter another value or expression. The default is 0.025 W/(m·K). Also enter the following: • Gas pressure pgap (SI unit: Pa). The default value is 1 atm. • Gas thermal accommodation parameter  (dimensionless). The default is 1.7. • Gas fluid parameter and  (dimensionless). The default is 1.7. • Gas particles diameter D (SI unit: m). The default value is 0.37 nm. RADIATIVE CONDUCTANCE

This section is available when Gray-diffuse parallel surfaces is selected as the Radiative conductance correlation under Contact. By default the Surface emissivity  (dimensionless) is taken From material. Select User defined to enter another value or expression. The default is 1. THERMAL FRICTION

Select a Heat partition coefficient definition—Charron’s relation (the default) or User defined. If User defined is selected, enter a value for the Heat partition coefficient r (dimensionless). The default is 0.5.

110 |


Enter a Frictional heat source Qfric (SI unit: W/m2). The default is 0 W/m 2.

Theory for the Thermal Contact Feature

Line Heat Source

The Line Heat Source node models a heat source (or sink) that is so thin that it has no thickness in the model geometry. Select this node from the Edges submenu. The Line Heat Source node is only available in 3D. This is because in 2D it is a boundary and in 1D it is a domain. In theory, the temperature in a line source in 3D is plus or minus infinity (to compensate for the fact that the heat source does not have any volume). The finite element discretization used in COMSOL Multiphysics returns a finite temperature distribution along the line, but that distribution must be interpreted in a weak sense. EDGE SELECTION

From the Selection list, choose the edges to define. LINE HEAT SOURCE

Click the General source (the default) or Total line power button. •

•

When General source is selected, enter a value for the distributed heat source, Ql (SI unit: W/m) in unit power per unit length. Positive Ql is heating while a negative Ql is cooling. The default is 0 W/m. If Total line power is selected, enter the total power (total heat source) Pl,tot (SI unit: W). The default is 0 W.


| 111

FRAME SELECTION


• •


Point Heat Source

The Point Heat Source node models a heat source (or sink) that is so small that it can be considered to have no spatial extension. Select this node from the Points menu.

The Point Heat Source is available in 3D and 2D. In 1D it is not available because points are boundaries there (possibly interior boundaries).

In theory, the temperature in a point source in 2D or 3D is plus or minus infinity (to compensate for the fact that the heat source does not have a spatial extension). The finite element discretization used in COMSOL Multiphysics returns a finite value, but that value must be interpreted in a weak sense. POINT SELECTION

From the Selection list, choose the points to define. POINT HEAT SOURCE

Enter the Point heat source Q p (SI unit: W) in unit power. Positive Q p is heating while a negative Q p is cooling. The default is 0 W. Pressure Work

Right-click the Heat Transfer in Solidsor Heat Transfer in Fluids node to add the Pressure Work subnode. When added under Heat Transfer in Solidsnode, the Pressure Work node adds the following term to the right-hand side of the Heat Transfer in Solids equation:

112 |


 t

(3-5)

– T ----- S el

where Sel is the elastic contribution to entropy . When added under Heat Transfer in Fluids node, the Pressure Work feature adds the following contribution to the right-hand side of the Heat Transfer in Fluids equation: T    p ------ + u   p    T p   t

– ---- -------

(3-6)

The software computes the pressure work using the absolute pressure. DOMAIN SELECTION

From the Selection list, choose the domains to define. By default, the selection is the same as for the parent node (Heat Transfer in Solidsor Heat Transfer in Fluids) it is attached to. PRESSURE WORK

For the Heat Transfer in Solids model, enter a value or expression for the Elastic 3 3 contribution to entropy Ent (SI unit: Jm ·K)). The default is 0 J m ·K). For the Heat Transfer in Fluids model, select a Pressure work formulation—Full formulation (the default) or Low Mach number formulation. The low Mach number formulation excludes the term u ·  p from Equation 3-6, which is small for most flows with a low Mach number. Viscous Heating

Right-click the Heat Transfer in Fluids to add the Viscous Heating node, which adds the following term to the right-hand side of the heat transfer in fluids equation: :S

(3-7)

where  is the viscous stress tensor and S is the strain-rate tensor. Equation 3-7 represents the heating caused by viscous friction within the fluid. DOMAIN SELECTION

From the Selection list, choose the domains to define. By default, the selection is the same as for the Heat Transfer in Fluids feature that it is attached to.


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DYNAMIC VISCOSITY

The Dynamic viscosity  (SI unit: Pa·s) uses the value of the viscosity From material. Select User defined to enter another value or expression. The default is 0 Pa·s. COMSOL uses the dynamic viscosity together with the velocity expressions to compute the viscous stress tensor, . Inflow Heat Flux

Use the Inflow Heat Flux node to model inflow of heat through a virtual domain with a heat source. The temperature at the outer boundary of the virtual domain is known. This boundary condition estimates the heat flux through the system boundar y

–n

  –k  T 



1 q 0  – u  n  ------------------- +    hin –  h ext  u  n un

=

(3-8)



where h in –  h ext

T in

=



T ext

1 T   ---  1 + ----  -------  d p    T   p ext   p A

C p dT +



(3-9)

p

A positive heat flux adds heat to the domain. This feature is applicable to inlet boundaries. The second integral in Equation 3-5 is neglected if the feature is applied to the boundary of a solid domain. BOUNDARY SELECTION

From the Selection list, choose the boundaries to define. INFLOW HEAT FLUX

Select the Inward heat flux (the default) or Total heat flux buttons. • •

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When Inward heat flux is selected, define q0 (SI unit: W/m2) to add to the total flux across the selected boundaries. The default value is 0 W/m 2. When Total heat flux is selected, define qtot. In this case q0  qtot/ A, where A is the total area of the selected boundaries. The default is 0 W.


For either selection, enter a value or expression for the External temperature T ext (SI unit: K) (the default is 273.15 K) and the External absolute pressure pext (SI unit: Pa) (the default is 1 atm).


=

1 .

In 2D and 1D axial symmetry A

=



dz 1

where dz is the out-of-plane thickness. If the out-of-plane property is not active, a text field is available to define dz or Ac. In 1D A

=



Ac 1

where Ac is the cross-sectional area. If the out-of-plane property is not active, a text field is available to define dz or Ac. Open Boundary

The Open Boundary node adds a boundary condition for modeling heat flux across an open boundary; the heat can flow out of the domain or into the domain with a specified exterior temperature. Use this node to limit a modeling domain that extends in an open fashion. BOUNDARY SELECTION

From the Selection list, choose the boundaries to define. OPEN BOUNDARY

Enter the exterior Temperature T 0 (SI unit: K) outside of the open boundary.


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This feature was previously called Convective Cooling. The Convective Heat Flux node adds the following heat flux contribution to its boundaries: h  T ext – T 

where the heat transfer coefficient, h, can be user defined or by using a library of predefined coefficients described in About the Heat Transfer Coefficients. BOUNDARY SELECTION

From the Selection list, choose the boundaries to define. HEAT FLUX

Select a Heat transfer coefficient h (SI unit: W/(m 2·K)) to control the type of convective heat flux to model— User defined (the default), External natural convection, Internal natural convection, External forced convection, or Internal forced convection. • •

For all options, enter an External temperature, T ext (SI unit: K). The default is 293.15 K. For all options (except User defined), follow the individual instructions below and select an External fluid—Air (the default), Transformer oil, or water . If Air is selected, also enter an Absolute pressure, p A (SI unit: Pa). The default is 1 atm.

External Natural Convection

If External natural convection is selected, choose Vertical wall, Inclined wall, Horizontal plate, upside, or Horizontal plate, downside from the list under Heat transfer coefficient. • •

•

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If Vertical wall is selected, enter a Wall height L (SI unit: m). The default is 0 m. If Inclined wall is selected, enter a Wall height L (SI unit: m) and the Tilt angle  (SI unit: rad). The tilt angle is the angle between the wall and the vertical direction,    for vertical walls. The default is 0 rad. If Horizontal plate, upside or Horizontal plate, downside is selected, define the Plate diameter (area/perimeter) L (SI unit: m). L is approximated by the ratio between the surface area and its perimeter. The default is 0 m.


Internal Natural Convection

If Internal natural convection is selected, choose Narrow chimney, parallel plates or Narrow chimney, circular tube from the list under Heat transfer coefficient. • •

If Narrow chimney, parallel plates is selected, enter a Plate distance L (SI unit: m) and a Chimney height H (SI unit: m). The defaults are each 0 m. If Narrow chimney, circular tube is selected, enter a Tube diameter D (SI unit: m) and a Chimney height H (SI unit: m). The defaults are each 0 m.

External Forced Convection

If External forced convection is selected, choose Plate, averaged transfer coefficient or Plate, local transfer coefficient from the list under Heat transfer coefficient. • •

If Plate, averaged transfer coefficient is selected, enter a Plate length L (SI unit: m) and a Velocity, external fluid U ext (SI unit: m/s). The defaults are each 0 m. If Plate, local transfer coefficient is selected, enter a Position along the plate xpl (SI unit: m) and a Velocity, external fluid U ext (SI unit: m/s). The defaults are each 0 m.

Internal Forced Convection

If Internal forced convection is selected, the only option is Isothermal tube. Enter a Tube diameter D (SI unit: m); the default is 0 m, and a Velocity, external fluid U ext (SI unit: m/s); the default is 0 m/s. •

Power Transistor: Model Library path Heat_Transfer_Module/ Power_Electronics_and_Electronic_Cooling/ power_transistor

•

Free Convection in a Water Glass: Model Library path Heat_Transfer_Module/Tutorial_Models_Forced_and_Natural_Convection/ cold_water_glass


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Highly Conductive Layer Nodes In this section: • • • • • • • •

Highly Conductive Layer Layer Heat Source Edge Heat Flux Point Heat Flux Temperature Point Temperature Edge Surface-to-Ambient Radiation Point Surface-to-Ambient Radiation

About Highly Conductive Layers

•

Heat Transfer in a Surface-Mount Package for a Silicon Chip: Model Library path Heat_Transfer_Module/Electronics_and_Power_Systems/ surface_mount_package

•

Copper Layer on Silica Glass: Model Library path Heat_Transfer_Module/Tutorial_Models/copper_layer

Highly Conductive Layer

Use the Highly Conductive Layer node to model heat transfer in thin highly conductive layers on boundaries in 2D and 3D. This feature can also be added to 2D axisymmetric models.

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About Highly Conductive Layers Right-click the Highly Conductive Layer node to add these additional features: • •

Layer Heat Source—to add a layer internal heat source, Qs, within the highly conductive layer. Edge (3D) or Point (2D and 2D axisymmetric) Heat Flux —adds a heat flux through a specified set of boundaries of a highly conductive layer. See Edge Heat Flux and Point Heat Flux. If you also have the Microfluidics Module, the Edge Heat Flux and Point Heat Flux nodes are not available with the Slip Flow interface.

• Edge (3D) or Point (2D and 2D axisymmetric) Temperature —sets a prescribed

•

temperature condition on a specified set of boundaries of a highly conductive layer. See Temperature and Point Temperature. Edge (3D) or Point (2D and 2D axisymmetric) Surface-to-Ambient Radiation —adds a surface-to-ambient radiation for the highly conductive layer. See Edge Surface-to-Ambient Radiation and Point Surface-to-Ambient Radiation.

BOUNDARY SELECTION


This section contains fields and values that are inputs to expressions that define material properties. If such user-defined property groups are added, the model inputs appear here. The Layer thickness ds, displays in this section. The default value is 0.01 m. COORDINATE SYSTEM SELECTION

The Global coordinate system is selected by default. The Coordinate system list contains any additional coordinate systems that the model includes (except boundary coordinate systems). The coordinate system is used for interpreting directions of orthotropic and anisotropic thermal conductivity.

HIGHLY CONDUCTIVE LAYER NODES

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HEAT CONDUCTION

The default Layer thermal conductivity ks (SI unit: W/(m·K)) is taken From material and describes the layer’s ability to conduct heat. If User defined is selected, choose Isotropic, Diagonal, Symmetric, or Anisotropic based on the characteristics of the thermal conductivity, and enter another value or expression. THERMODYNAMICS

The default Layer density  s (SI unit: kg/m3) and Layer heat capacity Cs (SI unit: J/ (kg·K)) are taken From material. Select User defined to enter other values or expressions. The defaults are 0 kg/m 3 and 0 J/ (kg·K), respectively. Enter a value or expression for the Layer thickness ds (SI unit: m). The default is 0.01 m.

If the thickness is zero, the highly conductive layer does not take effect.

About Highly Conductive Layers

Layer Heat Source

Use a Layer Heat Source node to add an internal heat source, Qs, within the highly conductive layer. Add one or more heat sources. Right-click the Highly Conductive Layer node to add this feature. BOUNDARY SELECTION

From the Selection list, choose the boundaries to define. By default, the selection is the same as for the Highly Conductive Layer node.

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LAYER HEAT SOURCE

Select the General source (the default) or Total power button to define Qs. • •

When the General source button is selected, enter a value or expression for Qs (SI unit: W/m3) as a heat source per volume. The default is 0 W/m3. If the Total power button is selected, define the total power Ps,tot (SI unit: W). In this case Qs = Ps,tot/V where V=A·ds with A equal to the area of the boundary selection. The default is 0 W.


=

1 .

In 2D, A

=



dz 1

where dz is the out-of-plane thickness. If the out-of-plane property is not active, a text field is available to define dz. Edge Heat Flux

Use the Edge Heat Flux node for 3D models to add heat flux across boundaries of a highly conductive layer. A positive heat flux adds heat to the layer. Right-click the Highly Conductive Layer node to add this feature.

If you also have the Microfluidics Module, the Edge Heat Flux node is not available with the Slip Flow interface. EDGE SELECTION

From the Selection list, choose the edges to define.


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EDGE HEAT FLUX

Select either the General inward heat flux (the default), Inward heat flux, or Total heat flux buttons. •

•

•

When General inward heat flux q0 (SI unit: W/m2) is selected, it adds to the total flux across the selected edges. Enter a value for q0 to represent a heat flux that enters the layer. For example, any electric heater is well represented by this condition, an d its geometry can be omitted. The default is 0 W/m 2. If Inward heat flux is selected (in the form q0  h·T ext  T , enter the Heat transfer 2 2 coefficient h (SI unit: W/(m ·K)). The default value is 0 W/(m ·K). Enter an External temperature T ext (SI unit: K). The default value is 293.15 K. The value depends on the geometry and the ambient flow conditions. If Total heat flux is selected, enter the total heat flux qtot (SI unit: W). In this case q0 = qtot/ A where A=L·ds with L equal to the length of the edge selection. The default is 0 W.

FRAME SELECTION



Point Heat Flux

Use the Point Heat Flux node for 2D and 2D axisymmetric models to add heat flux across boundaries of a highly conductive layer. A positive heat flux adds heat to the layer. Right-click the Highly Conductive Layer node to add this feature.

If you also have the Microfluidics Module, the Point Heat Flux node is not available with the Slip Flow interface.

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POINT SELECTION

From the Selection list, choose the points to define. HEAT FLUX

Select either the General inward heat flux (the default) or Inward heat flux buttons. •

•

When General inward heat flux q0 (SI unit: W/m2) is selected, it adds to the total flux across the selected points. Enter a value for q0 to represent a heat flux that enters the layer. For example, any electric heater is well represented by this condition, and its geometry can be omitted. The default is 0 W/m2. If Inward heat flux is selected (in the form q0  h·T ext  T , enter the Heat transfer 2 2 coefficient h (SI unit: W/(m ·K)). The default value is 0 W/(m ·K). Enter an External temperature T ext (SI unit: K). The default value is 293.15 K. The value depends on the geometry and the ambient flow conditions.

Temperature

Use the Temperature node to specify the temperature on a set of edges that represent thin boundary surfaces of the highly conductive layer. Right-click the Highly Conductive Layer node to add this feature.

Only edges adjacent to the boundaries can be selected in the parent node.

EDGE SELECTION

From the Selection list, choose the edges to define. PAIR SELECTION

If this node is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect. TEMPERATURE

Enter the value or expression for the Temperature T 0 (SI unit: K). The equation for this condition is T = T 0 where T 0 is the prescribed temperature on the edges. The default is 293.15 K.


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CONSTRAINT SETTINGS

To display this section, click the Show button ( ) and select Advanced Physics Options . To Apply reaction terms on all dependent variables, select All physics (symmetric). Otherwise, select Current physics (internally symmetric) or Individual dependent variables to restrict the reaction terms as required. Select the Use weak constraints check box to replace the standard constraints with a weak implementation. Point Temperature

Use the Point Temperature node to specify the temperature on a set of points that represent thin boundary surfaces of the highly conductive layer. Right-click the Highly Conductive Layer node to add this feature.

Only points adjacent to the boundaries can be selected in the parent node.

POINT SELECTION

From the Selection list, choose the points to define. POINT TEMPERATURE

Enter the value or expression for the Temperature T 0 (SI unit: K). The equation for this condition is T = T 0 where T 0 is the prescribed temperature on the points. The default is 293.15 K. CONSTRAINT SETTINGS

To display this section, click the Show button ( ) and select Advanced Physics Options . To Apply reaction terms on all dependent variables, select All physics (symmetric). Otherwise, select Current physics (internally symmetric) or Individual dependent variables to restrict the reaction terms as required. Select the Use weak constraints check box to replace the standard constraints with a weak implementation.

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Edge Surface-to-Ambient Radiation

Use the Edge Surface-to-Ambient Radiation node to add surface-to-ambient radiation to edges representing boundaries of a highly conductive layer. Right-click the Highly Conductive Layernode to add this feature. The net inward heat flux from surface-to-ambient radiation is q

=

  T 4amb – T 4 

where  is the surface emissivity,  is the Stefan-Boltzmann constant (a predefined physical constant), and T amb is the ambient temperature. EDGE SELECTION

From the Selection list, choose the edges to define. SURFACE-TO-AMBI ENT RADIATION

Enter an Ambient temperature T amb (SI unit: K). The default is 293.15 K. The default Surface emissivity  (a dimensionless number between 0 and 1) is taken From material. An emissivity of 0 means that the surface emits no radiation at all and an emissivity of 1 means that it is a perfect blackbody. The default is 0. Point Surface-to-Ambient Radiation

Use the Point Surface-to-Ambient Radiation node to add surface-to-ambient radiation to points representing boundaries of a highly conductive layer. Right-click the Highly Conductive Layernode to add this feature. The net inward heat flux from surface-to-ambient radiation is q =   T 4amb – T 4  where  is the surface emissivity,  is the Stefan-Boltzmann constant (a predefined physical constant), and T amb is the ambient temperature. POINT SELECTION

From the Selection list, choose the points to define.


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SURFACE-TO-AMBIEN T RADIATION

Enter an Ambient temperature T amb (SI unit: K). The default is 293.15 K. The default Surface emissivity  (a dimensionless number between 0 and 1) is taken From material. An emissivity of 0 means that the surface emits no radiation at all and an emissivity of 1 means that it is a perfect blackbody. The default is 0.

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Out-of-Plane Heat Transfer Nodes The following nodes are available for 1D and 2D Heat Transfer models and in 3D for the Heat Transfer in Thin Shells interface. In this section: • • • •

Out-of-Plane Convective Heat Flux Out-of-Plane Radiation Out-of-Plane Heat Flux Change Thickness

Theory of Out-of-Plane Heat Transfer

Select the Out-of-plane heat transfer check box on a Heat Transfer interface to add these nodes to 2D and 1D models.

For the Heat Transfer in Thin Shells interface, these features are available for 3D models.

Surface Resistor: Model Library path Heat_Transfer_Module/ Thermal_Stress/surface_resistor


Use the Out-of-Plane Convective Heat Flux node to model upside and downside cooling or heating caused by the presence of an ambient fluid.

O U T - O F - P L A N E H E A T TR A N S F E R N O D E S

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This feature adds the following contribution h u  T ext,u – T  + h d  T ext,d – T 

to the right-hand side of Equation 3-10 or Equation 3-11  T d z  C p ------- –    d z k  T  t --- + u   T   C p d z  ----T  t

=

=

d z Q

   d z k  T  + d z Q

(3-10)

(3-11)

Select the Out-of-plane heat transfer check box on a Heat Transfer interface to add these nodes to 2D and 1D models. For the Heat Transfer in Thin Shells interface these features are available in 3D. DOMAIN SELECTION

Select the domains where you want to add an out-of-plane convective heart flux contribution. UPSIDE HEAT FLUX

About the Heat Transfer Coefficients Select a Heat transfer coefficient hu (SI unit: W/(m2·K)) to control the type of convective heat flux to model— User defined (the default), External natural convection, Internal natural convection, External forced convection, or Internal forced convection. If only convective flux is required on the downside, use the default, which sets hu  0. • •

128 |

For all of the options, enter an External temperature, T ext, u (SI unit: K). For all of the options (except User defined), follow the individual instructions in the Heat Flux section described for the Convective Heat Flux feature, then select an External fluid—Air , Transformer oil, or water . If Air is selected, also enter an Absolute pressure, p A (SI unit: Pa). The default is 1 atm.


DOWNSIDE HEAT FLUX

The controls in the Downside Heat Flux section are the same as those in the Upside Heat Flux section except that it is applied to the downside instead of the upside, for example, the Heat transfer coefficient is hd. Out-of-Plane Radiation

The Out-of-Plane Radiation node models surface-to-ambient radiation on the upside and downside. The feature adds the following contribution to the right-hand side of Equation 3-10 or Equation 3-11: 4 4 4 4  u   T amb  u – T  +  d   T amb d – T 

Compare to the equation in the section Surface-to-Ambient Radiation.


Select the domains where you want to add an out-of-plane surface-to-ambient heat transfer contribution. UPSIDE PARAMETERS

The default Surface emissivity  u (a dimensionless number between 0 and 1) is taken From material. Select User defined to enter another value. An emissivity of 0 means that the surface emits no radiation at all while a emissivity of 1 means that it is a perfect blackbody. The default is 0. Enter an Ambient temperature T amb,u (SI unit: K). The default is 293.15 K. DOWNSIDE PARAMETERS

Follow the instructions for the Upside Parameters section to define the downside parameters  d and T amb,d.


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The Out-of-Plane Heat Flux node adds a heat flux q0,u as an upside heat flux and a heat flux q0,d as a downward heat flux to the right-hand side of Equation 3-10 or Equation 3-11: d s q 0 u + d s q 0 d


Select the domains where you want to add an out-of-plane heat flux. UPSIDE INWARD HEAT FLUX

Select between specifying the upside inward heat flux directly or as a convective term using a heat transfer coefficient. The General inward heat flux button is selected by default. Enter a value or expression for the inward (or outward, if the quantity is negative) heat flux through the upside (SI unit: W/m 2) in the q0,u field. The default is 0 W/m2. Click the Inward heat flux button to specify an inward (or outward, if the quantity is negative) heat flux through the upside (SI unit: W/m 2) as hu·(T ext,uT ). Enter a value or expression for the heat transfer coefficient in the hu field (SI unit: W/(m 2·K) and a value or expression for the external temperature in the T ext,u field (SI unit: K). The default value for the external temperature is 293.15 K. DOWNSIDE INWARD HEAT FLUX

The controls in the Downside Inward Heat Flux section are identical to those in the Upside Inward Heat Flux section except that they apply to the downside instead of the upside. Change Thickness

The Change Thickness node makes it possible model domains with another thickness than the overall thickness that is specified in the Heat Transfer interface Physical Model section.

130 |


DOMAIN SELECTION

Select the domains where you want to use a different thickness. CHANGE THICKNESS

Specify a value for the Thickness d z (SI unit: m). The default value is 1 m. This value replaces the overall thickness in the domains that are selected in the Domain Selection section.


| 131

The Bioheat Transfer Interface When Bioheat Transfer is selected under the Heat Transfer branch ( ) in the Model Wizard, Biological tissue is automatically selected as the default physical model and a ) interface is added to the Model Builder . Heat Transfer (ht) ( When this version of the interface is added, these default nodes are added to the Model Builder —Biological Tissue (with a default Bioheat node), Thermal Insulation (the default boundary condition), and Initial Values. All functionality to include both solid and fluid domains is also available. Right-click the Heat Transfer node to add other features that implement boundary conditions and sources. PHYSICAL MODEL

The Heat transfer in biological tissue check box is automatically selected. The rest of the settings as well as the interior and exterior boundary conditions are the same as for The Heat Transfer Interface.

• • • • •

The Heat Transfer Interface Theory for the Bioheat Transfer User Interface Biological Tissue Bioheat Domain, Boundary, Edge, Point, and Pair Nodes for the Heat Transfer User Interfaces

Hepatic Tumor Ablation: Model Library path Heat_Transfer_Module/ Medical_Technology/tumor_ablation

132 |


Biological Tissue

The Biological Tissue node adds the bioheat equation as the mathematical model for heat transfer in biological tissue. See Equation 2-41. Right-click the node to add a Bioheat node. When parts of the model (for example, a heat source) are moving, also right-click to add a Translational Motion node, which includes the effect of the movement by translation that requires a moving coordinate system. The Opaque subnode is automatically added to the entire selection when the Surface-to-surface radiation check box is selected on the Bioheat Transfer interface settings window. The selection can be edited. DOMAIN SELECTION


This section has fields and values that are inputs to expressions that define material properties. If such user-defined property groups are added, the model inputs appear here. Initially, this section is empty. COORDINATE SYSTEM SELECTION

The Global coordinate system is selected by default. The Coordinate system list contains any additional coordinate systems that the model includes (except boundary coordinate systems). The coordinate system is used for interpreting directions of orthotropic and anisotropic thermal conductivity. HEAT CONDUCTION

The default Thermal conductivity k (SI unit: W/(m·K)) uses values From material. If User defined is selected, choose Isotropic, Diagonal, Symmetric, or Anisotropic based on the characteristics of the thermal conductivity and enter another value or expression in the field or matrix. The defaults are 0 W/(m·K).

T H E B I O H E A T TR A N S F E R I N T E R F A C E

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THERMODYNAMICS

The default Density  (SI unit: kg/m3) and Heat capacity at constant pressure C p (SI unit: J/(kg·K)) are taken From material. The heat capacity describes the amount of heat energy required to produce a unit temperature change in a unit mass. If User 3 defined is selected, enter other values or expressions. The defaults are 0 kg/m and 0 J/(kg·K), respectively. Bioheat

A default Bioheat node is added to the Biological Tissue node. This feature provides the source terms that represent blood perfusion and metabolism to model heat transfer in biological tissue using the bioheat equation: b Cb b (T b  T )

Right-click the Biological Tissue node to add more Bioheat subnodes. DOMAIN SELECTION

For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically selected and is the same as for the interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific domains or select All domains as required. BIOHEAT

Enter values or expressions for these properties and source terms: • Arterial blood temperature T b (SI unit: K), which is the temperature at which blood

•

•

134 |

leaves the arterial blood veins and enters the capillaries. T is the temperature in the tissue, which is the dependent variable that is solved for and not a material property. The default tis 310.15 K. Specific heat, blood Cb (SI unit: J/(kg·k)), which describes the amount of heat energy required to produce a unit temperature change in a unit mass of blood. The default is 0 J/(kg·k). 3 3 Blood perfusion rate  b (SI unit: 1/s, which in this case means (m /s)/m ), describes the volume of blood per second that flows through a unit volume of tissue. The default is 0 1/s.


• Density, blood  b (SI unit: kg/m3), which is the mass per volume of blood. The

default is 0 kg/m3.

• Metabolic heat source Qmet (SI unit: W/m3), which describes heat generation from

metabolism. Enter this quantity as the unit power per unit volume. The default is 0 W/m3.

T H E B I O H E A T TR A N S F E R I N T E R F A C E

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The Heat Transfer in Porous Media Interface The Heat Transfer in Porous Media user interface ( ), found under the Heat Transfer branch ( ) in the Model Wizard, is an extension of the generic Heat Transfer interface that includes modeling heat transfer through convection, conduction and radiation, conjugate heat transfer, and non-isothermal flow. When this interface is added, default nodes are added to the Model Builder —Heat Transfer in Porous Media , Thermal Insulation (the default boundary condition), and Initial Values. Right-click the main node to open a context menu and add as many physics features as required to define the equations, properties and boundar y conditions. The ability to define material properties, boundary conditions, and mor e for porous media heat transfer is activated by selecting the Heat transfer in porous media check box. (Figure 3-1).

Figure 3-1: The ability to model porous media heat transfer is activated by selecting the Heat transfer in porous media check box in any Heat Transfer (ht) settings window under Physical Model.

The rest of the settings for this interface are the same as for the Heat Transfer interface.

Phase Change: Model Library path Heat_Transfer_Module/Phase_Change/ phase_change

136 |


• • • •

Show More Physics Options Domain, Boundary, Edge, Point, and Pair Nodes for the Heat Transfer in Porous Media User Interface Theory for the Heat Transfer in Porous Media User Interface Theory for the Heat Transfer User Interfaces

Domain, Boundary, Edge, Point, and Pair Nodes for the Heat Transfer in Porous Media User Interface

The Heat Transfer in Porous Media Interface has these nodes described in this section: Heat Transfer in Porous Media • Thermal Dispersion These domain, boundary, edge, point, and pair nodes are described forThe Heat Transfer Interface (listed in alphabetical order): •

To locate and search all the documentation, in COMSOL Multiphysics, select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree. • • • • • • • •

Boundary Heat Source Continuity Heat Flux Heat Source Heat Transfer with Phase Change Heat Transfer in Solids Line Heat Source Outflow

• • • • • • • •

Periodic Heat Condition Point Heat Source Surface-to-Ambient Radiation Symmetry Temperature Thermal Contact Thermal Insulation Thin Thermally Resistive Layer


The Heat Transfer in Porous Media node is used to specify the thermal properties of a porous matrix. Right-click to add a Thermal Dispersion subnode. The Heat Transfer

THE HEAT TRANSFER IN POROUS MEDIA INTERFACE

| 137

in Porous Media model uses the following version of the heat equation as the mathematical model for heat transfer in fluids:  -- C p ----T +  C p u   T =    k  T  + Q t

(3-12)

For a steady-state problem the temperature does not change with time and the first term disappears. It has these material properties: • Density  (SI unit: kg/m3) • Heat capacity at constant pressure C p (SI unit: J/(kg·K)): This describes the amount • • • •

of heat energy required to produce a unit temperature change in a unit mass. Thermal conductivity k (SI unit: W/(m·K)): A scalar or a tensor if the thermal conductivity is anisotropic. Velocity field u (SI unit: m/s): Either an analytic expression or a velocity field from a fluid-flow interface. The heat source (or sink) Q: One or more heat sources can be added separately. The Ratio of specific heats  (dimensionless): The ratio of heat capacity at constant pressure, C p, to heat capacity at constant volume, Cv. When using the ideal gas law to describe a fluid, specifying  is enough to evaluate C p. For common diatomic gases such as air,   1.4 is the standard value. Most liquids have   1.1 while water has   1.0.  is used in the streamline stabilization and in the variables for heat fluxes and total energy fluxes. It is also used if the ideal gas law is applied. See Thermodynamics, Porous Matrix. If you have the Heat Transfer Module, the Opaque subnode is automatically added to the entire selection when Surface-to-surface radiation is activated. The selection can be edited.

DOMAIN SELECTION

For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically selected and is the same as for the interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific domains or select All domains as required.

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MODEL INPUTS

This section has fields and values that are inputs to expressions that define material properties. If such user-defined property groups are added, the model inputs appear here. There are also two standard model inputs— Absolute pressure and Velocity field. The absolute pressure is used in some predefined quantities that include the enthalpy (the energy flux, for example). Absolute pressure is also used if the ideal gas law is applied. See Thermodynamics, Porous Matrix. Abs olute Pressure

This section controls both the variable as well as any property value (reference pressures) used when solving for pressure. There are usually two ways to calculate the pressure when describing fluid flow and mass and heat transfer. Solve for the absolute pressure or a pressure (often denoted gauge pressure) that relates back to the absolute pressure through a reference pressure. Using one or the other option usually depends on the system and the equations being solved for. For example, in a straight incompressible flow problem, the pressure drop over the modeled domain is probably many orders of magnitude less than atmospheric pressure, which, if included, reduces the chances for stability and convergence during the solving process for this variable. In other cases, the absolute pressure may be required to be solved for, such as where pressure is a par t of an expression for gas volume or diffusion coefficients. The default Absolute pressure p A (SI unit: Pa) is User defined and is 1 atm (101,325 Pa). When additional physics interfaces are added to the model, the pressure variables solved can also be selected from the list. For example, if a fluid-flow interface is added you can select Pressure (spf/fp) from the list.


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When a Pressure variable is selected, the Reference pressure check box is selected by default and the default value of pref is 1[atm] (1 atmosphere). This makes it possible to use a system-based (gauge) pressure as the pressure variable while automatically including the reference pressure in places where it is required, such as for gas flow governed by the gas law. While this check box maintains control over the pressure variable and instances where absolute pressure is required within this r espective physics interface, it may not with physics interfaces that it is being coupled to. In such models, check the coupling between any interfaces using the same variable. Velocity Field

The default Velocity field u (SI unit: m/s) is User defined. When User defined is selected, enter values or expressions for the components based on space dimension. The defaults are 0 m/s. Or select an existing velocity field in the model (for example, Velocity field (spf/fp1) from a Laminar Flow interface). HEAT CONDUCTION, FLUID

The default Thermal conductivity k (SI unit: W/(m·K)) is taken From material. If User defined is selected, choose Isotropic, Diagonal, Symmetric, or Anisotropic based on the characteristics of the thermal conductivity, and enter another value or expression. The thermal conductivity describes the relationship between the heat flux vector q and the temperature gradient T as in q = kT which is Fourier’s law of heat conduction. Enter this quantity as power per length and temperature. THERMODYNAMIC S, FLUID

The default Density  (SI unit: kg/m3), Heat capacity at constant pressure C p (SI unit: J/(kg·K)), and Ratio of specific heats  (dimensionless) for a general gas or liquid use values From material. Select User defined to enter other values or expressions. Select a Fluid type—Gas/Liquid, Moist air , or Ideal gas. Gas/Liquid

Select Gas/Liquid to specify the Density, the Heat capacity at constant pressure, and the Ratio of specific heats for a general gas or liquid. The default settings are to use data

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From material. Select User defined to enter

another value for the density, heat capacity,

or ratio of specific heats. Ideal Gas

Select Ideal gas to use the ideal gas law to describe the fluid. Then: •

Select a Gas constant type—Specific gas constant Rs (SI unit: J/(kg·K)) or Mean molar mass M n (SI unit: kg/mol). For both properties, the default setting is to use the property value from the material. Select User defined to enter another value for either of these material properties. If Mean molar mass is selected, the software uses the universal gas constant R  8.314 J/(mol·K), which is a built-in physical constant.

•

From the list under Specify Cp or , select Heat capacity at constant pressure C p (SI unit: J/(kg·K)) or Ratio of specific heats  (dimensionless). For both properties, the default setting is to use the property value From material. Select User defined to define another value for either of these material properties. For an ideal gas, specify either C p or the ratio of specific heats,  , but not both since these, in that case, are dependent.

Moi st Air

If Moist air is selected, the thermodynamics properties are defined as a function of the quantity of vapor in the moist air. Four different options are available from the Input quantity list to define the amount of vapor in the moist air: • •

• •

Select Vapor mass fraction (the default) to define the vapor mass fraction (SI unit: kg/kg). Select Concentration to define the concentration of vapor (SI unit: mol/m3). Once this option is selected a Concentration model input is automatically added in the Models Inputs section. Select Moisture content to define the moisture content of the moist air (SI unit: kg/ kg). Select Relative humidity to define the quantity of vapor from a Reference relative humidity (SI unit: 1), a Reference temperature (SI unit: K), and a Reference pressure


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(SI unit: Pa). These three reference values are used to estimate the mass fraction of vapor, which is used to define the thermodynamic properties of the moist air.

Moist Air Theory

IMMOBILE SOLIDS

This section contains fields and values that are inputs to expressions that define material properties. The Solid material list can point to any material in the model. Enter a Volume fraction p (dimensionless) for the solid material. HEAT CONDUCTION, POROUS MATRIX

The default Thermal conductivity kp (SI unit: W/(m·K)) uses values From material. If User defined is selected, choose Isotropic, Diagonal, Symmetric, or Anisotropic based on the characteristics of the thermal conductivity and enter another value or expression in the field or matrix. The thermal conductivity of the material describes the relationship between the heat flux vector q and the temperature gradient T as in a solid material and q = kpT , which is Fourier’s law of heat conduction. THERMODYNAMI CS, POROUS MATRIX

The default Density  p (SI unit: kg/m3) uses values From material. If User defined is selected, enter another value or expression. The default Specific heat capacity C p,p (SI unit: J/(kg·K)) uses values From material. If User defined is selected, enter another value or expression. The specific heat capacity describes the amount of heat energy required to produce a unit temperature change in a unit mass of the solid material. The equivalent volumetric heat capacity of the solid-liquid system is calculated from   Cp  eq

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=

 p  p Cp p +  L  Cp

Thermal Dispersion

Right-click the Heat Transfer in Porous Media node to add the Thermal Dispersion node. This adds an extra term ·kdT to the right-hand side of --- +  C p u   T =    k eq  T  + Q   C p  eq ----T t

and specifies the values of the longitudinal and transverse dispersivities. DOMAIN SELECTION

From the Selection list, choose the domains to define. DISPERSIVITIES

Define the Longitudinal dispersivity lo (SI unit: m) and Transverse dispersivity tr (SI unit: m). For the Transverse vertical dispersivity the Thermal Dispersion node defines the tensor of dispersive thermal conductivity kijd

=

 L Cp L D ij

where Dij is the dispersion tensor D ij

=

u u

k l  ijkl ------------

u

and ijkl is the fourth order dispersivity tensor  ijkl

=

lo –  tr  tr  ij  kl + ------------------- ik  jl +  il  jk  2


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4

Heat Transfer in Thin Shells This chapter describes the Heat Transfer in Thin Shells interface found under the Heat Transfer branch

(

) in the Model Wizard.

In this chapter: • •

The Heat Transfer in Thin Shells User Interface Theory for the Heat Transfer in Thin Shells User Interface

145

The Heat Transfer in Thin Shells User Interface The Heat Transfer in Thin Shells (htsh) user interface ( ), found under the Heat ) in the Model Wizard, is suitable for solving thermal-conduction Transfer branch ( problems in thin structures and has the equations, edge and point conditions, and heat sources for modeling heat transfer in thin conductive shell, solving for the temperature. It is available for 3D models The Thin Conductive Layer is the main node. It adds the equation for the temperature and provides a settings window for defining the thermal conductivity, the heat capacity and the density (see Equation 4-1). When this interface is added, these default nodes are also added to the Model Builder — Thin Conductive Layer , Insulation/Continuity (a default boundary condition), and Initial Values. Right-click the Heat Transfer in Thin Shells node to add other features that implement, for example, edge or point conditions, and heat sources. INTERFACE IDENTIFIER

The interface identifier is used primarily as a scope prefix for variables defined by the physics user interface. Refer to such interface variables in expressions using the pattern .. In order to distinguish between variables belonging to different physics user interfaces, the identifier string must be unique. Only letters, numbers and underscores (_) are permitted in the Identifier field. The first character must be a letter. The default identifier (for the first interface in the model) is

htsh.

BOUNDARY SELECTION

The default setting is to include All boundaries in the model to define the dependent variables and the equations. To choose specific boundaries, select Manual from the Selection list. SHELL THICKNESS

Define the Shell thickness ds (SI unit: m) (see Equation 4-1). The default is 0.01 m. SURFACE-TO-SURFACE RADIATION

Select the Surface-to-surface radiation check box to add a Radiation Settings section.

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CHAPTER 4: HEAT TRANSFER IN THIN SHELLS

RADIATION SETTINGS

See The Heat Transfer Interface for details about the Surface-to-surface radiation method and Radiation resolution settings. Modify the Transparent media refractive index it is different from 1 that corresponds to vacuum refractive index and that is a good approximation for air refractive index. Select the Number of wavelength intervals. Default is one which correspond to a diffuse gray radiation model. It is possible to define up to 5 wavelength intervals. When the Number of wavelength intervals is greater than one, a diffuse spectral model is used. It is then possible to define the surface emissivity per spectral band. Select the Surface-to-surface radiation method—Hemicube or Direct area integration. •

•

If Direct area integration is selected, select the Radiation integration order . Sharp angles and small gaps between surfaces may require a higher integration order for accuracy but also more time to evaluate the irradiation. If Hemicube is selected, select the Radiation resolution. The default is 256.

Select the Use radiation groups check box to enable the ability of defining radiation groups. This can speed up the radiation calculations in many cases. Select Linear (the default), Quadratic, Cubic, Quartic or Quintic to define the Discretization level used for the surface radiosity shape function. DEPENDENT VARIABLES

The dependent variable (field variable) is for the Temperature T. The variable name can be changed, but the names of dependent variables must be unique within a model. ADVANCED SETTINGS

Add both a Heat Transfer in Thin Shells (htsh) and Moving Mesh (ale) interface (found under the Mathematics>Deformed Mesh branch in the Model Wizard) then click the ) and select Advanced Physics Options to display this section. Show button ( When the model contains moving mesh, the Enable conversions between material and spatial frame check box is selected by default. This option has no effect when the model does not contain a moving frame since the material and spatial frames are identical in this case. With moving mesh, and when this option is active, the heat transfer physics automatically account for deformation effects

T H E H E A T T R A N S F ER I N T H I N S H E L L S U S E R I N T E R F A C E

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on heat transfer properties. In particular the ef fects for volume changes on the density are considered. Rotation effects on thermal conductivity of an anisotropic material and, more generally, deformation effects on arbitrary thermal conductivity, are also covered. When the Enable conversions between material and spatial frame check box is not selected, the feature inputs (for example, Heat Source, Heat Flux, Boundary Heat Source, and Line Heat Source) are not converted and all are defined on the Spatial frame. DISCRETIZATION

To display this section, click the Show button ( ) on the Model Builder and then select Discretization. Select Quadratic (the default), Linear, Cubic, or Quartic for the Temperature. Specify the Value type when using splitting of complex variable s—Real (the default) or Complex for each of the variables in the table. • • • •

About Handling Frames in Heat Transfer Show More Physics Options Boundary, Edge, Point, and Pair Nodes for the Heat Transfer in Thin Shells User Interface Theory for the Heat Transfer in Thin Shells User Interface

Shell Conduction: Model Library path Heat_Transfer_Module/ Tutorial_Models_Thin_Structure/shell_conduction

Boundary, Edge, Point, and Pair Nodes for the Heat Transfer in Thin Shells User Interface

The Heat Transfer in Thin Shells User Interface has the following boundary, edge, point and pair nodes described (listed in alphabetical order):

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The following are described in this section: • • • • •

Change Effective Thickness Change Thickness Edge Heat Source Initial Values Insulation/Continuity

• • • • •

Heat Flux Heat Source Point Heat Source Surface-to-Ambient Radiation Thin Conductive Layer

These nodes are described for the Heat Transfer interfaces: • • • • •

External Radiation Source Opaque Out-of-Plane Heat Flux Out-of-Plane Convective Heat Flux Out-of-Plane Radiation

• • • •

Prescribed Radiosity Radiation Group Surface-to-Surface Radiation (Boundary Condition) Temperature

When nodes are described for other interfaces, the difference for this interface is that Boundaries are selected instead of Domains.

To locate and search all the documentation, in COMSOL Multiphysics, select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree. Heat Flux

Use the Heat Flux node to add heat flux across boundaries. A positive heat flux adds heat to the domain. The Heat Flux feature adds a heat source (or sink) to edges. It adds a heat flux q  d eq0. BOUNDARY OR EDGE SELECTION

From the Selection list, choose the boundaries or edges to define.


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PAIR SELECTION

If this node is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect. HEAT FLUX

See the Heat Flux node, Heat Flux settings section for The Heat Transfer Interface, which are the same for this interface. FRAME SELECTION

To display this section add both a Heat Transfer in Thin Shells (htsh) and Moving Mesh (ale) interface (found under the Mathematics>Deformed Mesh branch in the Model Wizard). Then click the Show button ( ) and select Advanced Physics Options . The rest of the settings are the same for the Heat Source node as described under the Frame Selection section.


Thin Conductive Layer

The Thin Conductive Layer node adds the heat equation for conductive heat transfer in shells (see Equation 4-1). BOUNDARY SELECTION

For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically selected and is the same as for the interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific boundaries or select All boundaries as required. MODEL INPUTS

This section has fields and values that are inputs to expressions that define material properties. If such user-defined property groups have been added, the model inputs are included here.

150 |


HEAT CONDUCTION

Thermal Conductivity Tensor Components By default, the Thermal conductivity k (SI unit: W/(m·K)) uses values From material. If User defined is selected, choose Isotropic, Diagonal, Symmetric, or Anisotropic based on the characteristics of the thermal conductivity and enter other values or expressions in the field or matrix. THERMODYNAMICS

Specify the Density  (SI unit: kg/m3) and the Heat capacity at constant pressure Cp (SI unit: J/(kg·K)) to describe the amount of heat energy required to produce a unit temperature change in a unit mass. The default settings use values From material for both. If User defined is selected, enter other values or expressions. Heat Source

The Heat Source node adds a thermal source Q. It adds the following contributions to the right-hand side of Equation 4-1: ds Q . The Heat Source describes heat generation within the shell. Express heating and cooling with positive and negative values, respectively. Add one or more nodes as required; all heat sources within a boundary contribute to the total heat source. Specify the heat source as the heat per volume in the domain, as linear heat sourc e, or as a total heat source (power). BOUNDARY SELECTION

From the Selection list, choose the boundaries to define. HEAT SOURCE

See the Heat Source node, Heat Source settings section for The Heat Transfer Interface, which are the same for this interface.


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FRAME SELECTION

The settings are the same for the Heat Flux node Frame Selection section


Initial Values

The Initial Values node adds an initial value for the temperature that can serve as an initial condition for a transient simulation or as an initial guess for a nonlinear solver. If more than one set of initial values is needed, right-click the Initial Values node. BOUNDARY SELECTION

For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically selected and is the same as for the interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific boundaries or select All boundaries as required. INITIAL VALUES

Enter a value or expression for the initial value of the Temperature T . The default is approximately room temperature, 293.15 K (20º C). Also enter a Surface radiosity J (SI unit: W/m2). The default is htsh.Jinit W/m2. This variable is defined under the 4 . Surface-to-Surface Radiation boundary condition as     r T 04  –  1 –  r  T amb Change Thickness

Use the Change Thickness node to give parts of the shell a different thickness than that what is specified on the Heat Transfer in Thin Shells interface Shell Thickness section. BOUNDARY SELECTION

From the Selection list, choose the boundaries to define. CHANGE THICKNESS

Specify a value for the Shell thickness ds (SI unit: m). The default value is 0.01 m. This value replaces the overall thickness for the boundaries that are selected.

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Use the Surface-to-Ambient Radiation condition to add surface-to-ambient radiation to edges. The net inward heat flux from surface-to-ambient radiation is n   – d s k  T T 

=

4

4

  T amb – T 

where  is the surface emissivity,  is the Stefan-Boltzmann constant (a predefined physical constant), and T amb is the ambient temperature. EDGE SELECTION

From the Selection list, choose the edges to define. MODEL INPUTS


If this node is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect. SURFACE-TO-AMBI ENT RADIATION

The default Surface emissivity  (a dimensionless number between 0 and 1) is taken From material. An emissivity of 0 means that the surface emits no radiation at all and an emissivity of 1 means that it is a perfect blackbody. Enter an Ambient temperature T amb (SI unit: K). The default is 293.15 K. Insulation/Continuity

The Insulation/Continuity node is the default edge condition. On external edges, this edge condition means that there is no heat flux across the edge: n   k g  T 

=

0

On internal edges, this edge condition means that the temperature field and its flux is continuous across the edge.


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EDGE SELECTION

For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically selected and is the same as for the interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific edges or select All edges as required. PAIR SELECTION

If this node is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect. Change Effective Thickness

The Change Effective Thickness node models edges with another thickness than the overall thickness that is specified in the Heat Transfer in Thin Shells interface Shell Thickness section. It defines the height of the part of the edge that is exposed to the ambient surroundings. EDGE SELECTION


If this node is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect. CHANGE EFFECTIVE THICKNESS

Enter a value for the Effective thickness de (SI unit: m). The default is 0.01 m. This value replaces the overall thickness in the edges selected in the Edges section. Edge Heat Source

The Edge Heat Source node models a linear heat source (or sink). It adds a heat source q  Ql or q  d eQb. A positive q means heating while a negative q means cooling. EDGE SELECTION



154 |


EDGE HEAT SOURCE

From the Edge heat source type list, select Heat source defined per unit of length (the default) or Heat source defined per unit of area . If Heat source defined per unit of length is selected, click the General source (the default) or Total line power button. •

•

When General source is selected, enter a value for the distributed heat source, Ql (SI unit: W/m) in unit power per unit length. Positive Ql is heating while a negative Ql is cooling. The default is 0 W/m. If Total line power is selected, enter the total power (total heat source) Pl,tot (SI unit: W). The default is 0 W.

If Heat source defined per unit of area is selected, click the General source (the default) or Total boundary power button. • •

If General source is selected, enter the boundary heat source Qb (SI unit: W/m2). A positive Qb is heating and a negative Qb is cooling. The default is 0 W/m 2. If Total boundary power is selected, enter the total power (total heat source) Pb, tot (SI unit: W). The default is 0 W.

FRAME SELECTION

The settings are the same for the Heat Flux node Frame Selection section


Point Heat Source

The Point Heat Source node models a point heat source (or sink). POINT SELECTION

From the Selection list, choose the points to define. POINT HEAT SOURCE

Enter a value or expression for the Point heat source Qp (SI unit: W). A positive Qp means heating while a negative Qp means cooling. The added heat source is equal to Qp. The default is 0 W.


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Theory for the Heat Transfer in Thin Shells User Interface The Heat Transfer in Thin Shells User Interface theory is described in this section: • • •

About Heat Transfer in Thin Shells Heat Transfer Equation in Thin Conductive Shell Thermal Conductivity Tensor Components

About Heat Transfer in Thin Shells

The Heat Transfer in Thin Shells User Interface supports two types of heat transfer: conduction and out-of-plane heat transfer and is suitable for solving thermal-conduction problems in thin structures. Because the thermal conductivity across the shell thickness is very large or the shell is so thin, assume constant temperature through the shell thickness. The Thin Conductive Layer node is the main feature. It adds the equation for the temperature and provides a settings window for defining the thermal conductivity, the heat capacity and the density:  d s C p

T + T   –d s k g TT  t

=

0

(4-1)

Heat Transfer Equation in Thin Conductive Shell

The dependent variable is the temperature T . The interface is defined on 3D faces. The governing equation for heat transfer in thin shells is:  d s C p

T +  T   – d s k g TT  t +

=

d s Q + d s h u  T e x t, u – T  + ds hd  T e x t, d – T  (4-2)

4

Where T is the tangential derivative along the shell and •


• ds is the shell thickness (SI unit: m)

156 |

4

ds u   T 4 amb, u – T  + d s  d   T 4 amb, d – T  + d s q u + d s q d


• Cp is the heat capacity (SI unit: J/(kg·K) • k g is the thermal conductivity (SI unit: W/(m·K) • Q is the heat source (SI unit: W/m 3) • hu and hd are the out-of-plane heat transfer coefficients, upside and downside • • • •

(SI unit: W/(m2·K)) T ext, u and T ext, d are the out-of-plane external temperatures, upside and downside (SI unit: K) u and d are the out-of-plane surface emissivities, upside and downside (SI unit: 1), T amb, u and T amb, d are the out-of-plane ambient temperatures, upside and downside (SI unit: K) qu and qd are the out-of-plane inward heat fluxes, upside and downside (SI unit: W/ m2)

Thermal Conductivity Tensor Components

The thermal conductivity k describes the relationship between the heat flux vector and the temperature gradient TT as in q

= – k g

q

T T

which is Fourier’s law of heat conduction (see also The Heat Equation). The tensor components are specified in the shell local coordinate system, which is defined from the geometric tangent and normal vectors. The local x direction, e xl, is the surface tangent vector t1 and the local z direction, e zl, is the normal vector n. Their cross product defines the third orthogonal direction such that: e x l

=

e y l

=

e z l

=

t1 e x l  e z l

=

n  t1

n

From this, a transformation matrix between the shell’s local coordinate system and the global coordinate system can be constructed in the following way: e x l x e y l x e z l x A

=

e x l y e y l y e z l y e x l z e y l z e z l z

THEORY FOR THE HEAT TRANSFER IN THIN SHELLS USER INTERFACE

| 157

The thermal conductivity tensor, kg, can be expressed as kg

158 |


=

Ak A t

5

R a d i a t i o n H e a t T ra ns fe r This chapter describes the interfaces for modeling radiative heat transfer. The physics interface for modeling radiative heat transfer are available under the Heat ) and described in the About the Heat Transfer Transfer>Radiation branch ( Interfaces section. In this chapter: • • • • • •

The Radiation Branch Versions of the Heat Transfer User Interface The Surface-To-Surface Radiation User Interface Theory for the Surface-to-Surface Radiation User Interface The Radiation in Participating Media User Interface Theory for the Radiation in Par ticipating Media User Interface References for the Radiation User Interfaces

159

The Radiation Branch Versions of the Heat Transfer User Interface • •

The Heat Transfer with Surface-to-Surface Radiation User Interface The Heat Transfer with Radiation in Participating Media User Interface • •

About the Heat Transfer Interfaces The Heat Transfer Interface

The Heat Transfer with Surface-to-Surface Radiation User Interface

Use the Heat Transfer with Surface-to-Surface Radiation (ht) user interface ( ), found under the Radiation branch ( ), to model heat transfer that includes surface-to-surface radiation. It is a Heat Transfer interface with the Surface-to-surface radiation check box selected under Physical Model, which enables the Radiation Settings section. The following default nodes are also added to the Model Builder —Heat Transfer in Solids (with a default Opaque node), Thermal Insulation, and Initial Values. Right-click the node to add other boundary conditions and features. Except for the Radiation Settings section, the rest of the settings are the same as for The Heat Transfer Interface. RADIATION SETTINGS

See the Radiation Settings section for The Surface-To-Surface Radiation User Interface.

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CHAPTER 5: RADIATION HEAT TRANSFER

The Heat Transfer with Radiation in Participating Media User Interface

The Heat Transfer with Radiation in Participating Media (ht) user interface ( ), found under the Heat Transfer>Radiation branch ( ) in the Model Wizard, combines features from the Radiation in Participating Media and Heat Transfer interfaces. This enables the modeling of radiative heat transfer inside a participating medium combined with heat transfer in solids and fluids. This interface solves for radiative intensity and temperature fields. When this interface is added, the Radiation in participating media check box is selected in the Physical Model section of the main Heat Transfer settings window. The following default nodes are also added to the Model Builder —Heat Transfer in Solids, Thermal Insulation, Opaque Surface, Continuity on Interior Boundary, and Initial Values. Right-click the node to add other boundary conditions and features. Except for the Participating Media Settings section (described for the The Radiation in Participating Media User Interface), the rest of the settings are the same as for The Heat Transfer Interface. Domain, Boundary, Edge, Point, and Pair Nodes for the Radiation Branch Versions of the Heat Transfer User Interface

Both The Heat Transfer with Surface-to-Surface Radiation User Interface and The Heat Transfer with Radiation in Participating Media User Interface are versions of the Heat Transfer interface. This means all the nodes are shared with, and d escribed for,

THE RADIATION BRANCH VERSIONS OF THE HEAT TRANSFER USER INTERFACE

| 161

The Heat Transfer Interface, including nodes described for the Highly Conductive Layer Nodes and Out-of-Plane Heat Transfer Nodes. The Heat Transfer with Surface-to-Surface Radiation (ht) interface also has the following nodes available and described for the Surface-to-Surface Radiation interface: • • • • • •

External Radiation Source Opaque Prescribed Radiosity Radiation Group Diffuse Mirror Surface-to-Surface Radiation (Boundary Condition)

The Heat Transfer with Radiation in Participating Media (ht) interface also has the following nodes available: • • • •

Continuity on Interior Boundary Incident Intensity Opaque Surface Radiation in Participating Media

To locate and search all the documentation, in COMSOL Multiphysics, select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree.

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The following are links to the domain, boundary, edge, point, and pair nodes and subnodes (listed in alphabetical order): • • • • • • • • • • • • • • • •

Boundary Heat Source Continuity Convective Heat Flux Edge Heat Flux Edge Surface-to-Ambient Radiation Temperature Heat Flux Heat Source Heat Transfer in Fluids Heat Transfer in Solids Heat Transfer with Phase Change Highly Conductive Layer Inflow Heat Flux Initial Values Line Heat Source Open Boundary

• • • • • • • • • • • • • • •

Outflow Periodic Heat Condition Point Heat Flux Point Heat Source Point Surface-to-Ambient Radiation Point Temperature Pressure Work Surface-to-Ambient Radiation Symmetry Temperature Thermal Contact Thermal Insulation Thin Thermally Resistive Layer Translational Motion Viscous Heating

THE RADIATION BRANCH VERSIONS OF THE HEAT TRANSFER USER INTERFACE

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The Surface-To-Surface Radiation User Interface The Surface-to-Surface Radiation (rad) user interface ( ), found under the Heat ) in the Model Wizard, treats thermal radiation as an Transfer>Radiation branch ( energy transfer between boundaries and external heat sources where the m edium does not participate in the radiation (radiation in transparent media). The process transfers energy directly between boundaries and external radiation sources. The radiation therefore contributes to the boundary conditions rather than to the heat equation itself. Right-click the node to add a Surface-to-Surface Radiation boundary condition or other nodes. For the Surface-to-Surface Radiation user interface, select a Stationary or Time Dependent study as a preset study type. The surface-to-surface radiation is always stationary (that is, the radiation time scale is assumed to be shorter than any other time scale), but the interface is compatible with all standard study types. Absolute (thermodynamical) temperature units must be used. See Specifying Model Equation Settings in the COMSOL Multiphysics Reference Manual . For this user interface, COMSOL Multiphysics works under the assumption that the domain medium does not participate in the radiation process. If the media participate in the radiation, then select The Radiation in Participating Media User Interface. This physics user interface solves only for the surface radiosity. To solve for surface radiosity and temperature, use The Heat Transfer with Surface-to-Surface Radiation User Interface instead. INTERFACE IDENTIFIER

The interface identifier is used primarily as a scope prefix for variables defined by the physics user interface. Refer to such interface variables in expressions using the pattern .. In order to distinguish between variables

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belonging to different physics user interfaces, the identifier string must be unique. Only letters, numbers and underscores (_) are permitted in the Identifier field. The first character must be a letter. The default identifier (for the first user interface in the model) is

rad.

BOUNDARY SELECTION

The default setting is to include All boundaries in the model to define the dependent variables and the equations. To choose specific boundaries, select Manual from the Selection list. RADIATION SETTINGS

This section is alway visible in the Surface-to-Surface Radiation interface. To display this section for any version of the Heat Transfer interface, select the Surface-to-surface radiation check box under Physical Model settings window. Define the Wavelength dependence of emissivity. •

•

•

Keep the default value, Constant, to define a diffuse gray radiation model. In this case, the surface emissivity has the same definition for all wavelength. Still the surface emissivity can depend on other quantities, in par ticular it can be temperature dependent. Select Solar and ambient to define a diffuse spectral radiation model with two spectral bands, one for short wavelengths, [ ,1], (solar radiation) and one for large wavelengths, [1,[, (ambient radiation). It is then possible to define the Intervals endpoint (SI unit m), 1, to adjust the wavelength intervals corresponding to the solar and ambient radiation. The surface properties can then be defined for each spectral band. In particular it is possible to define the solar absorptivity for short wavelengths and the surface emissivity for large wavelengths. Choose Multiple Spectral Bands and set the Number of wavelength intervals value (2 to 5), to define a diffuse spectral radiation model. It is then possible to provide a definition of the surface emissivity for each spectral band. Update Intervals endpoint (SI unit m), 1, 2, ... to define the wavelength intervals [ i-1,i[ for i from 1 to Number of wavelength intervals. Note that the first and the last endpoints, 0 and N (with N equal to the value selected to define of Number of wavelength intervals), are predefined and equal to 0 and  respectively.

Modify the Transparent media refractive index if it is different from 1 that corresponds to vacuum refractive index and that is usually a good approximation for air refractive index.

T H E S U R F A C E - TO - S U R F A C E R A D I A T I O N U S E R I N T E R F A C E

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Also select the Use radiation groups check box to enable the ability to define radiation groups, which can, in many cases, speed up the radiation calculations. Select a Surface-to-surface radiation method—Hemicube (the default) or Direct area integration. See below for descriptions of each method. • •

If Hemicube is selected, select a Radiation resolution—256 is the default. If Direct area integration is selected, select a Radiation integration order —4 is the default.

Select Linear (the default), Quadratic, Cubic, Quartic or Quintic to define the Discretization level used for the surface radiosity shape function. Hemicube

Hemicube is the default method for the heat transfer interfaces. The more sophisticated and general hemicube method uses a z-buffered projection on the sides of a hemicube (with generalizations to 2D and 1D) to account for shadowing effects. Think of it as rendering digital images of the geometry in five different directions (in 3D; in 2D only three directions are needed), and counting the pixels in each mesh element to evaluate its view factor. Its accuracy can be influenced by setting the Radiation resolution of the virtual snapshots. The number of z-buffer pixels on each side of the 3D hemicube equals the specified resolution squared. Thus the time required to evaluate the irradiation increases quadratically with resolution. In 2D, the number of z-buffer pixels is proportional to the resolution property, and thus the time is, as well. For an axisymmetric geometry, Gm and F amb must be evaluated in a corresponding 3D geometry obtained by revolving the 2D boundaries about the axis. COMSOL Multiphysics creates this virtual 3D geometry by revolving the 2D boundary mesh into a 3D mesh. The resolution can be controlled in the azimuthal direction by setting the number of azimuthal sectors, which is the same as the number of elements to a full revolution. Try to balance this number against the mesh resolution in the rz-plane. Direct Area Integration

COMSOL Multiphysics evaluates the mutual irradiation between sur face directly, without considering which face elements are obstructed by others. This means that shadowing effects (that is, surface elements being obstructed in nonconvex cases) are not taken into account. Elements facing away from each other are, however, excluded from the integrals.

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Direct area integration is fast and accurate for simple geometries with no shadowing, or where the shadowing can be handled by manually assigning boundaries to different groups. If shadowing is ignored, global energy is not conserved. Control the accuracy by specifying a Radiation integration order . Sharp angles and small gaps between surfaces may require a higher integration order for accuracy but also more time to evaluate the irradiation. DISCRETIZATION

To display this section, click the

Show button

(


This section is empty for Surface-to-Surface Radiation (rad) interface which define the shape functions discretization in Radiation Settings.

•

About the Heat Transfer Interfaces The Heat Transfer Interface Domain, Boundary, Edge, Point, and Pair Nodes for the Radiation Branch Versions of the Heat Transfer User Interface The Surface-To-Surface Radiation User Interface Theory for the Heat Transfer User Interfaces

•

Thermo-Photo-Voltaic Cell: Model Library path

• • • •

Heat_Transfer_Module/Thermal_Radiation/tpv_cell

•

Free Convection in a Light Bulb: Model Library path Heat_Transfer_Module/Thermal_Radiation/light_bulb

Domain, Boundary, Edge, Point, and Pair Nodes for the Surface-to-Surface Radiation User Interface

The Surface-To-Surface Radiation User Interface has these domain, boundary, edge, point, and pair nodes available (listed in alphabetical order): • • •

External Radiation Source Diffuse Mirror Opaque


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• • •

Prescribed Radiosity Radiation Group Surface-to-Surface Radiation (Boundary Condition)

Surface-to-Surface Radiation (Boundary Condition)

The Surface-to-Surface Radiation boundary condition feature handles radiation with view factor calculation. The feature adds one radiosity shape function per spectral interval to its selection and uses it as surface radiosity. Surface-to-Surface Radiation boundary

condition adds radiative heat source

contribution q

=

G –  e b  T 

on the side of the boundary where the radiation is defined. Where the radiation is defined on both sides the radiative heat source is defined on both sides too. BOUNDARY SELECTION

From the Selection list, choose the boundaries where the boundary condition is applied. MODEL INPUTS

This section has fields and values that are inputs to expressions that define material properties. If such user-defined property groups have been added, the model inputs are included here. There is one standard model input—the Temperature T (SI unit: K). The default is the temperature variable in the Heat Transfer interface or 293.15 K in the Surface-to-Surface Radiation interface. This model input is used in the expression for the blackbody radiation intensity and, when multiple wavelength intervals are used, for the fractional emissive power. The temperature model input is also used to determine the variable that receives the radiative heat source. When the model input does not contain a dependent variable, the radiative heat source is ignored. RADIATION SETTINGS

When Wavelength dependence of emissivity is set to Constant in the interface settings, select a Radiation direction based on the geometric normal (nx, ny, nz)—Opacity

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controlled (the default), Negative normal direction, Positive normal direction,

or Both

sides.

• Opacity controlled requires that each boundary is adjacent to exactly one opaque • • •

domain. Opacity is controlled by the Opaque boundary condition. Select Negative normal direction to specify that the surface radiates in the negative normal direction. Select Positive normal direction if the surface radiates in the positive normal direction. Select Both sides if the surface radiates on both sides.

When Wavelength dependence of emissivity is set to Solar and ambient or Multiple spectral bands in the interface settings, select a Radiation direction for each spectral band—Opacity controlled (the default), Negative normal direction, Positive normal direction, or Both sides, or None. The Radiation direction defines the radiation direction for each spectral band similarly as when Wavelength dependence of emissivity is Constant. Defining a radiation direction for each spectral band make it possible to build models where the transparency/opacity properties defer between spectral bands. This is useful for example to represent that glass is opaque to radiation outside of the 0.3-2.5 µm wavelength range. The additional choice, None is used when adjacent domains are either both transparent or both opaque for a given spectral band. The Wavelength dependence of emissivity is defined in the interface settings, in the Radiation Settings section. AMBIENT

Select Define ambient temperature on each side when the ambient temperature differs between the sides of a boundary. This is needed to define ambient temperature for a surface that radiates on both side and that is exposed to a hot temperature on one side (for example, fire) and to a cold temperature on the other side (for example, external temperature). By default Define ambient temperature on each side is not selected. Enter an Ambient temperature T amb (SI unit: K). The default is 293.15 K. When Define ambient temperature on each side is selected, define the Ambient temperature T amb, u


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and T amb, d on the up and down side respectively. The geometric normal points from the down side to the up side. Set T amb to the far-away temperature in directions where no other boundaries obstruct the view. Inside a closed cavity, the ambient view factor, F amb, is theoretically zero and the value of T amb therefore should not matter. It is, however, good practice to set T amb to T or to a typical temperature value for the cavity surfaces in such cases because that minimizes errors introduced by the finite resolution of the view factor evaluation. SURFACE FRACTIONAL EMISSIVE POWER

This section is available only when Wavelength dependence of emissivity is Solar and ambient or Multiple spectral bands. When the Fractional emissive power is Blackbody/Graybody, the fractional emissive power is automatically computer for each spectral band as a function of the band endpoints and surface temperature. The Wavelength dependence of emissivity is defined in the interface settings, in the Radiation Settings section. When the Fractional emissive power is User defined, define the Fractional emissive power , FEPBi for each spectral band. All fractional emissive powers are expected to be in [0,1] and their sum is expected to be equal to 1. SURFACE EMISSIVITY

In diffuse gray and diffuse spectral radiation models, the surface emissivity and the absorptivity must be equal. For this reason it is equivalent to define the surface emissivity or the absorptivity. The surface emissivity settings are defined per spectral interval.

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When the Radiation direction is Opacity controlled, Negative normal direction, or Positive normal direction for a spectral band, by default, the Surface emissivity  (dimensionless) uses values From material. This is a property of the material surface that depends both on the material itself and the structure of the surface. Make sure that a material is defined at the boundary level (by default materials are defined at the domain level). When the Radiation direction is set to Both sides for a spectral band, define the Material on upside and Material on downside: •

•

The default for both Material on upside and Material on downside use Boundary material. The list contains other options based on the materials defined in the model. Define the Surface emissivity on the upside and downside, respectively. The geometric normal points from the down side to the up side. Set the sur face emissivity to a number between 0 and 1, where 0 represents diffuse mirror and 1 is appropriate for a perfect blackbody. The proper value for a physical material lies somewhere in-between and can be found from tables or measurements.

When the Radiation direction is set to None for a spectral band, no information is needed for this spectral band in the Surface Emissivity section. The Wavelength dependence of emissivity is defined in the interface settings, in the Radiation Settings section. INITIAL VALUES

The surface radiosity initial values are defined per spectral interval. When the Radiation direction is Opacity controlled, Negative normal direction, or Positive normal direction for a spectral band Bi, the default Surface radiosity, J Bi,init (SI unit: 2 W/m ) is defined as J B i init

=

 B i eb  T init  +  1 – B i  eb  T amb 

When Both sides is selected as the Radiation direction, Enter initial values for the Surface radiosity J Bi, init, u and J Bi, init, d (SI unit: W/m2). The default Surface radiosity is ht.JBiinitU and ht.JBiinitD (SI unit: W/m2). J Bi  in it  u

=

 Bi  u eb  T init  +  1 –  Bi  u  eb  T amb u 


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J Bi  in it  d

=

 Bi  d eb  T init  +  1 –  Bi  d  eb  T amb d 

When None is selected as the Radiation direction, no surface radiosity is defined hence no initial value is needed. • • •

• • •

The Wavelength dependence of emissivity is defined in the interface settings, in the Radiation Settings section. In the notation used here, Bi stands for B1, B2,... up to the maximum number of spectral intervals. When the model contains one spectral interval, J Bi, init , J Bi, init, u and J Bi, init, d are named, respectively, J init , J init, u and J init, d Guidelines for Solving Surface-to-Surface Radiation Problems Radiation Group Boundaries Domain, Boundary, Edge, Point, and Pair Nodes for the Surface-to-Surface Radiation User Interface

Opaque

In the Surface-to-Surface Radiation interface, right-click the Surface-to-Surface Radiation (rad) node to add an Opaque node. The Opaque node enables to define the surface-to-surface radiation direction on boundaries surrounding the domains where the Opaque node is defined. When the Radiation direction is defined by Opacity controlled in surface-to-surface boundary features, surface-to-surface radiation propagates in non-opaque domains. Alternatively the Radiation direction can be defined using the normal orientation or on both sides of boundaries. In this case the Opaque node is ignored. DOMAIN SELECTION

From the Selection list, choose the domains to define as opaque.

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OPAQUE

When Wavelength dependence of emissivity is set to Constant in the user interface settings, this section is empty. The opacity is then defined for all wavelengths. When Wavelength dependence of emissivity is set to Solar and ambient or Multiple spectral bands in the interface settings, select the spectral bands for which the opacity is defined by selecting corresponding Opaque on spectral band i check box. By default the Opaque feature is active for all spectral bands. The Wavelength dependence of emissivity is defined in the interface settings, in the Radiation Settings section.

Diffuse Mirror

The Diffuse Mirror node is a variant of the surface-to-surface radiation node with a surface emissivity equal to zero. Diffuse mirror surfaces are common as approximations of a surface that is well insulated on one side and for which convection effects can be neglected on the opposite (radiating) side. It resembles a mirror that absorbs all irradiation and then radiates it back in all directions. The node adds radiosity shape function for each spectral band to its selection and uses it as surface radiosity. The radiative heat flux on a diffuse mirror boundary is zero. BOUNDARY SELECTION

From the Selection list, choose the boundaries where this boundary condition is applied. MODEL INPUTS

This section has fields and values that are inputs to expressions that define material properties. If such user-defined property groups have been added, the model inputs are included here. There is one standard model input—the Temperature T (SI unit: K). The default is the temperature variable in the heat transfer physic interface or 293.15 K in the


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surface-to-surface physic interface. It is used in the blackbody radiation intensity expression. The Radiation Settings, Ambient, and Initial Values sections are the same as for the Surface-to-Surface Radiation (Boundary Condition).

Prescribed Radiosity

Use the Prescribed Radiosity node to specify radiosity on the boundary for each spectral band. Radiosity can be defined as blackbody or graybody radiation . A user-defined surface radiosity expression can also be defined. BOUNDARY SELECTION

From the Selection list, choose the boundaries where this boundary condition is applied. MODEL INPUTS

This section has fields and values that are inputs to expressions that define material properties. If such user-defined property groups have been added, the model inputs are included here. There is one standard model input—the Temperature T (SI unit: K). The default is the temperature variable in the Heat Transfer interface or 293.15 K in the Surface-to-Surface Radiation interface. It is used in the blackbody radiation intensity expression. RADIATION DIRECTION

When Wavelength dependence of emissivity is set to Constant in the interface settings, select a Radiation direction based on the geometric normal (nx, ny, nz)—Opacity controlled (the default), Negative normal direction, Positive normal direction, or Both sides. • Opacity controlled requires that each boundary is adjacent to exactly one opaque •

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domain. Opacity is controlled by the Opaque boundary condition. Select Negative normal direction to specify that the surface radiates in the negative normal direction.


• •

Select Positive normal direction if the surface radiates in the positive normal direction. Select Both sides if the surface radiates on both sides.

When Wavelength dependence of emissivity is set to Solar and ambient or Multiple spectral bands in the interface settings, select a Radiation direction for each spectral band—Opacity controlled (the default), Negative normal direction, Positive normal direction, or Both sides, or None. The Radiation direction defines the radiation direction for each spectral band similarly as when Wavelength dependence of emissivity is Constant. The Wavelength dependence of emissivity is defined in the interface settings, in the Radiation Settings section. RADIOSITY

Radiosity does not directly affect the boundary condition on the boundar y

where it is specified, but rather how that boundary affects others through radiation. Select a Radiosity expression—Graybody radiation (the default), Blackbody radiation, or User defined. Blackbody Radiation

When Blackbody radiation is selected it sets the surface radiosity expression corresponding to a blackbody. •

•

When Wavelength dependence of emissivity is set to Constant in the interface settings, it defines J = eb(T ) when radiation is defined on one side or Ju = eb(T u) and Jd = eb(T d) when radiation is defined on both sides. When Wavelength dependence of emissivity is set to Solar and ambient or Multiple spectral bands, it defines for each spectral band JBi = FEPBi(T)eb(T ) when radiation is defined on one side or


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JBi,d = FEPBi,d(T)eb(T u) and Jd = FEPBi,d(T)eb(T d) when radiation is defined on both sides. •

•

When the temperature varies across a pair (for example with a Thin Thermally Resistive Layer condition is active on the same boundary), the temperature used to define the radiosity is evaluated on the side were the surface radiation is defined. The blackbody hemispherical total emissive power is defined by ebT n 2T 4

Graybody Radiation

When Graybody radiation is selected it sets the surface radiosity expression corresponding to a graybody. By default, the Surface emissivity  (dimensionless) is defined From material. In this case, make sure that a material is defined at the boundar y level (materials are defined by default at the domain level). If User defined is selected for the Surface emissivity, enter another value for  . If Wavelength dependence of emissivity is set to Constant in the interface settings, when radiation is defined on one side, define the Surface emissivity  to set J = eb(T ), or • when radiation is defined on both sides, define the Material on upside, the Surface emissivity  u, Material on downside and the Surface emissivity  d on the upside and downside, respectively. The surface radiosity on upside and downside is then defined by Ju = ueb(T u) and Jd = deb(T d) respectively. If Wavelength dependence of emissivity is set to Solar and ambient or Multiple spectral bands, for all spectral bands, •

• •

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when radiation is defined on one side for Bi spectral band, define the Surface emissivity  Bi to set JBi = FEPBiBieb(T ), or when radiation is defined on both sides for Bi spectral band, define the Material on upside, the Surface emissivity  Bi,u, Material on downside and the Surface emissivity  Bi, d on the upside and downside, respectively. The surface radiosity on upside and


downside is then defined by Ju = FEPBi(T) Bi,ueb(T u) and Jd = FEPBi(T) Bi,deb(T d) respectively. The surface emissivity to a number between 0 and 1, where 0 represents diffuse mirror and 1 is appropriate for a perfect blackbody. The proper value for a physical material lies somewhere in-between and can be found from tables or measurements. User Defined

If Wavelength dependence of emissivity is set to Constant in the interface settings and Radiosity expression is set to User defined, it sets the surface radiosity expression to J = J0, which specifies how the radiosity of a boundary is evaluated when that boundar y is visible in the calculation of the irradiation onto another boundary in the model. Enter a Surface radiosity expression, J 0 (SI unit: W/m2). The default is 0 W/m 2. When the Radiation direction is set to Both sides (under Radiation Settings) also define the surface Radiosity expression J 0,u and J 0,d on the upside and downside, respectively. The geometric normal points from the downside to the upside. If Wavelength dependence of emissivity is Solar and ambient or Multiple spectral bands, similar settings are available for each spectral band. Radiation Group

Add a Radiation Group to a Surface-to-Surface Radiation (rad) interface or any version of a Heat Transfer (ht) interface where the Surface-to-surface radiation check box is selected. Select the Use radiation groups check box under Radiation Settings. By default the check box is not selected, which means that all radiative boundaries belong to the same radiation group. The Radiation Group node enables you to specify radiation groups to speed up the radiation calculations and gather boundaries in a radiation problem that have a chance to see one another. When the Use radiation groups check box is selected, the node is automatically added to the Model Builder and contains all boundaries selected in the Surface-to-Surface Radiation (Boundary Condition), or a Prescribed Radiosity feature.


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When the Wavelength dependence of emissivity is Solar and ambient or Multiple spectral bands, the radiation groups are defined per spectral band. BOUNDARY SELECTION

From the Selection list, choose the boundaries that belong to the same radiation group. This selection should contain any boundary that is selected in a Surface-to-Surface Radiation, a Diffuse Mirror , or a Prescribed Radiosity node and that has a chance to see one of the boundary that is already selected in the Radiation Group. RADIATION GROUP

When the Wavelength dependence of emissivity is Constant, the radiation group is valid for all wavelengths, this is empty. empty. When the Wavelength dependence of emissivity is set to Solar and ambient or Multiple spectral bands, the radiation group is define for all spectral bands by default. Clear Radiation group defined on spectral band i check boxes to remove Bi spectral band from this radiation group.

Radiation Group Boundaries

External Radiation Source

The External Radiation Source node is selected from the Global submenu and is available for 2D and 3D models in the Surface-to-Surface Radiation (rad) interface or in any version of a Heat Transfer (ht) interface where the Surface-to-surface radiation check box is selected. Add an External Radiation Source to define an external radiation source as a point or directional radiation source with view factor calculation. Each External Radiation Source feature contributes to the incident radiative heat flux on all spectral bands, GBi, on all the boundaries where a Surface-to-surface or Diffuse mirror boundary boundary condition is active. The source contribution, Gext,Bi, is equal to the product of the view factor of

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the source by the source radiosity. radiosity. For radiation sources sour ces located on a point, Gext,Bi=F ext,Bi Ps,Bi. For directional radiative source Gext,Bi = F ext,Bi q0,s. The number of spectral bands is defined in the interface settings, in the Radiation Settings. Settings. When only one spectral band is defined, the Bi subscript in variable names is removed. The external radiation sources are ignored on the boundaries when neither Surface-to-Surface Radiation nor Diffuse Mirror is is active. In particular they are not contributing on boundaries where Surface-to-Surface Ambient is active. Because the Surface-to-surface radiation check box cannot be selected with in 2D, External Radiation Source is not available Out-of-plane heat transfer in in this case. SOURCE

Select a Source position—Point coordinate (the default) or Infinite distance. In 3D, Solar position is also available as an option Point Coordinate

If Point coordinate is selected, define the Source location xs (SI unit: m) and the Source power P Ps (SI unit: W, default is 0 W). The source radiates uniformly in all directions. If Wavelength dependence of emissivity is Solar and ambient or Multiple spectral bands, set the Source power definition to Blackbody or User defined. When Blackbody is selected, enter the Source temperature, T s (SI unit: K, default is 5780K), to define the source power on the spectral band Bi as P as Ps,Bi= FEPBi(T s )Ps where FEPBi(T s ) is the fractional black body emissive power over Bi interval at T s. When User defined is selected, enter an expression to define the source power on each spectral band Bi, P Bi, Ps,Bi (SI unit: W, default is 0 W). xs should not belong to any surface where a Surface-to-surface or Diffuse

boundary Mirror boundary

condition is active.


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The Wavelength dependence of emissivity is defined in the interface settings, in the Radiation Settings section. Settings section. Infinite Distance

If Infinite distance is selected, define the Incident radiation direction is (dimensionless) and the Source heat flux q0,s (SI unit: W/m2). The default is 0 W/m 2. If Wavelength dependence of emissivity is Solar and ambient or Multiple spectral bands, set the Source heat flux definition to Blackbody or User defined. When Blackbody is selected, enter the Source temperature, T s (SI unit: K, default is 5780K), to define the source heat flux on the spectral band Bi as q0,s,Bi= q0,s FEPBi(T s ) where FEPBi(T s ) is the fractional black body emissive power over Bi interval at T s. When User defined is selected, enter an expression to define the source heat flux on each spectral band Bi, q0,s,Bi (SI unit: W, default is 0 W). Solar Position Solar position is available for 3D models. When this option is

selected use it to estimate the external radiative heat source due to the sun. North, West, and the up directions correspond to the x, y, y, and z directions, respectively. In the Location table define the: • Latitude, a decimal value, positive in the northern hemisphere; the default is

•

•

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Greenwich UK latitude, 51.479; as the value is expected to represent degrees and as the model’s unit for angles maybe ma ybe different (SI unit for angle is radian), the value should be enter without any unit to avoid double conversion; Longitude, a decimal value, positive at the East of the Prime Meridian; the default is Greenwich UK longitude, 0.01064; as the value is expected to represent degrees and as the model’s unit for angles maybe different (SI unit for angle is radian), the value should be enter without any unit to to avoid double conversion; and Time zone, the number of hours to add to UTC to get local time; the default is Greenwich UK time zone, 0.


In the Date table, enter the: • Day, the default is 01; as the value is expected to represent days and as the model’s

unit for time maybe different (SI unit for time is second), the value should be enter without any unit to avoid double conversion; conversion; • Month, the default is 6, which is June; as the value is expected to represent months and as the model’s unit for time maybe different (SI unit for time is second), the value should be enter without any unit unit to avoid double conversion; and • Year , the default is 2012. As the value is expected to represent years and as the model’s unit for time maybe different (SI unit for time is second), the value should be enter without any unit to avoid double conversion. The solar position is accurate for a date between 2000 and 2199. In the Local time table, enter the: • Hour , the default is 12; as the value is expected to

•

•

represent hours and as the model’s unit for time maybe different (SI unit for time is second), the value should be enter without any unit to avoid double conversion; conversion; Minute, the default is 0; as the value is expected to represent minutes and as the model’s unit for time maybe different (SI unit for time is second), the value should be enter without any unit to avoid double conversion; and Second, the default is 0. The sun position is updated if the location, date, or local time changes during a simulation. In particular for transient analysis, if the unit system for the time is in seconds (the default), the time change can be taken into account by adding t to the Second field in the Local time table.

In the Solar irradiance field I s (SI unit: W/m2) define the incident radiative intensity coming from the sun. The default is 1000 W/m 2. I s represents the heat flux received from the sun by a surface perpendicular to the sun rays. When surfaces are not perpendicular to the sun rays the heat flux received from the sun depends on the incident angle. If Wavelength dependence of emissivity is Solar and ambient or Multiple spectral bands, the solar irradiance is divided among all spectral bands Bi as q0,s,Bi= q0,s FEPBi(T sun ) where FEPBi(T sun ) is the fractional black body emissive power over Bi interval at T sun=5780K.


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Theory for the Surface-to-Surface Radiation User Interface The Surface-To-Surface Radiation User Interface theory Interface theory is described in this section: • • • • • • • •

Wavelength Dependence of Surface Emissivity and Absorptivity Wavelength Absorptivity The Radiosity Method for Diffuse-Gray Surfaces The Radiosity Method for Diffuse-Spectral Surfaces View Factor Evaluation About Surface-to-Surface Radiation Guidelines for Solving Surface-to-Surface Radiation Problems Radiation Group Boundaries References for the Radiation User Interfaces

Wavelength Dependence of Sur face Emissivity and Absorptivity

The surface properties proper ties for radiation, the emissivity and absorptivity can be dependent on the angle of emission/absorption, surface temperature or radiation wavelength. The surface-to-surface radiation feature in the Heat Transfer module implements the radiosity method that enable arbitrary temperature dependence and assumes that the emissivity and absorptivity is independent of the angle of emission.absorption. It is also possible to account for wavelength dependency of the surface emissivity and absorptivity. PLANK SPECTRAL DISTRIBUTION

The Planck’s distribution of emissive power for a blackbody in vacuum is given as a function of the surface temperature and of wavelength. The blackbody hemispherical emissive power is noted eb, (  ,T) and defined as e b    T 

=

2 n 2C1 -----------------------------C2  ------

5 T   e  – 1  

182 |


(5-1)

were •

 is the wavelength in vacuum 2

• C1

=

hc 0

• C2

=

hc 0  k

• • •

h is the Planck's constant k is the Boltzmann constant c0 is the speed of the light in vacuum

The graphs below show the hemispherical spectral emissive power for a blackbody at 5780K (sun black body temperature) and for a black body at 300K. The dotted vertical lines delimit the visible wavelength (0.4 to 0.7µm).

Figure 5-1: Planck distribution of a blackbody at 57 80K

THEORY FOR THE SURFACE-TO-SURFACE RADIATION USER INTERFACE

| 183

Figure 5-2: Planck distribution of a blackbody at 300K

The integral of eb, (  ,T) over a spectral band represents the power radiated on the spectral band and is defined by



2 e   T  d  1 b 

=





F 1 T   2 T

0

e b    T  d

=

F 1 T   2 T n 2  T 4

where F  T   T is the fractional black body emissive power, 1

2

2

F 1 T   2 T =

e     T  d ----------------------------------------   e     T d 1

0

b

b

One can notice that and that F 1 T   2 T = F 0   2 T – F 0   1 T and F 0  

184 |


=

1

Note also that e b  T 

=





0

e b    T  d

=

n 2  T 4

as defined for a black surface by the Stefan-Boltzmann law. DIFFUSE-GRAY SURFACES

Diffuse-gray surfaces correspond to the hypothesis that surface properties are independent of the radiation wavelength and of the angle between the sur face normal and the radiation direction. The assumption that the surface emissivity is independent of the radiation wavelength is often valid when most of the radiative power is concentrated on a relatively narrow spectral band. This is likely the case when the radiation is emitted by surface at temperatures in limited range.

This setting is rarely applicable if there is a solar radiation.


| 185

SOLAR AND AMBIENT SPECTRAL BANDS

When solar radiation is part of the model, it is possible to enhance diffuse-gray surface model by considering two spectral bands: one for short wavelengths and one for large wavelengths. It is interesting to notice that about 97% of the radiated power • •

from a blackbody at 5800K is at wavelengths of 2.5µm or shorter. from a blackbody at 700K is at wavelengths of 2.5µm or longer.

Figure 5-3: Normalized Planck distribution of blackbodies at 700K and 5800K

Many problems have a solar load, but the peak temperatures are below 700K. In such cases, it is appropriate to use a two-band approach with a solar band (for wavelengths shorter than 2.5µm) and a ambient band (for wavelengths above 2.5µm). For each surface, properties are then described in terms of a solar absorptivity and an emissivity.

186 |


Solar irradiation,  < 2.5 µm

Re-radiation to surroundings,  > 2.5 µm

Figure 5-4: Absorption of solar radiation and emission to surroundings

By splitting the bands at the default of 2.5um, the fraction of absorbed solar radiation on each surface is defined primarily by the solar absorptivity. The re-radiation at longer wavelengths (objects below ~700K) and the re-absorption of this radiation is defined primarily via the emissivity Emissivity

Wavelength

Figure 5-5: Solar and ambient spectral band approximation of the surface emissivity by a constant per band emissivity


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GENERAL DIFFUSE-SPECTRAL SURFACES

Diffuse-spectral surfaces correspond to the hypothesis that surface properties are wavelength dependent but independent of the angle between the surface normal and the radiation direction. The heat transfer module enables to define constant surface properties per spectral bands (with up to 5 spectral bands) and to adjust spectral intervals endpoints. Emissivity

1

3

2

Wavelength

The multiple spectral bands approach is used in cases when the surface emissivity varies significantly over the bands of interest. The Radiosity Method for Diffuse-Gray Surfaces

The radiation interacts with convective and conductive heat transfer through the source term in the Heat Flux and Boundary Heat Source boundary conditions. By definition, this source must be the difference between incident radiation and radiation leaving the surface. According to Equation 2-9 it is given by q

=

G – J

where • G is the incoming radiative heat flux, or irradiation (SI unit: W/m2). •

J is the total outgoing radiative flux, or radiosity (SI unit: W/m2).

The irradiation, G, at a point can in general be written as a sum according to: G

=

G m + G ex t + Gam b

where • Gm is the mutual irradiation, coming from other boundaries in the m odel

(SI unit: W/m2).

188 |


(5-2)

W/m2). Gext is the sum of the products, for each external source, of the external heat sources view factor by the corresponding source radiosity. For radiation sources located on a point, Gext =F ext(xs) Ps . For directional radiative source G ext =F ext(is) q0,s .

• Gext is the irradiation from external radiation sources (SI unit:

G ex t

=

G

ex P t s+

G

ex t q 0 s

4

is the ambient irradiation. F amb is an ambient view factor whose value is equal to the fraction of the field of view that is not covered by other boundaries. Therefore, by definition, 0  F amb  1 must hold at all points. T amb is the assumed far-away temperature in the directions included in F amb. The radiosity, J , is the sum of the reflected irradiation and the emitted irradiation. For diffuse-gray surface, J is defined by: • G am b

=

n 2 F am b  T am b

J =  G +  e b  T 

where •

 is the surface reflectivity which is equal to 1-  for diffuse-gray surfaces.

•

 is the surface emissivity, a dimensionless number in the range 0    1. Diffuse-gray surface hypothesis corresponds to surfaces where  is independent on the radiation wavelength.

• • •

4

is the blackbody hemispherical total emissive power. n is the transparent media refractive index.  is the Stefan-Boltzmann constant (a predefined physical constant equal to 5.670400·108 W/(m2·T4)). T is the surface temperature (SI unit: K)

• e b  T 

=

n 2  T

The Surface-To-Surface Radiation User Interface includes these radiation types: •

Surface-to-Surface Radiation is the default radiation type. It requires accurate evaluation of the mutual irradiation, Gm. The incident radiation at one point on the boundary is a function of the exiting radiation, or radiosity , J (W/m2), at every other point in view. The radiosity, in turn, is a function of Gm, which leads to an implicit radiation balance: J =  1 –   G +  e b  T 

•

=

 1 –    Gm  J  + Gex t + Gamb  +  eb  T 

(5-3)

Diffuse mirror is a variant of the Surface-to-Surface Radiation radiation type with  = 0. Reradiation surfaces are common as an approximation of a surface that is well


| 189

•

insulated on one side and for which convection effects can be neglected on the opposite (radiating) side (see Ref. 4). It resembles a mirror that absorbs all irradiation and then radiates it back in all directions. Prescribed radiosity makes it possible to specify graybody radiation . The radiosity expressions is then eb(T ). A user-defined surface radiosity expression can also be defined.

The Surface-to-Surface Radiation interface treats the radiosity J as a shape function unless J is prescribed. The Radiosity Method for Diffuse-Spectral Surfaces

For a general diffuse spectral surface: J =

    T  eb    T  d =0 



where •

(  ,T) is the hemispherical spectral surface emissivity, a dimensionless number in the range 0    1. Diffuse-spectral surface corresponds to a surface where  is dependent on the radiation wavelength and surface temperature.

T is the surface temperature (SI unit: K) • eb, (  ,T) is the black body hemispherical emissive power defined by Equation 5-1. The Surface-To-Surface Radiation User Interface assumes that the surface emissivity and opacity properties are constant per spectral band. It defines N spectral band (N=2 when solar and ambient radiation model is used), •

B i

=

  i – 1  i  i   1 .. N  with  0

=

0 and  N = 

The surface properties can then be defined per spectral band: • •

Surface emissivity on Bi: i  T  =    T      i – 1  i  Ambient irradiation on Bi, assuming that the ambient fractional emissive power corresponds to these of a blackbody at T amb: G am b i

•

190 |

=

i

 =

i–1

Gam b    d

=

F i

–

1

T   i T F am b e b  T am b 

External radiation sources on Bi with q 0 s i and Ps i the external radiation source heat flux and source power over Bi i   1 .. N  :


G ex t i

=



i G    d  = i 1 ex t

=

–

F ex t i  i s  q 0 s i

or G ex t i

=

i

 =

i–1

Gex t    d

=

F ex t i  i s  P s i

When the external source fractional emissive power corresponds to these of a blackbody at T ext, external radiation sources can be defined on B i from the external radiation source heat flux and source power over all wavelengths, q0 s and Ps : G ex t i

=

–

1

T   i T  i s  q 0 s

F ex t i F  i

–

1

F ex t i F i

or G ex t i

=

T   i T  i s  P s

The Surface-To-Surface Radiation User Interface includes these radiation types: •

Surface-to-Surface Radiation is the default radiation type. The incident radiation over Bi spectral band at one point on the boundary is a function of the existing radiation, or radiosity , J i (W/m2), at every other point in view. The radiosity, in turn, is a function of Gm,i, which leads to an implicit radiation balance: J i

•

•

=

 1 – i  Gi +  i eb  T 

=

 1 –  i   Gm i  J i  + G ex t i + Gamb i  + i eb  T  (5-4)

Diffuse mirror is a variant of the Surface-to-Surface Radiation radiation type with i = 0. Reradiation surfaces are common as an approximation of a surface that is well insulated on one side and for which convection effects can be neglected on the opposite (radiating) side (see Ref. 4). It resembles a mirror that absorbs all irradiation and then radiates it back in all directions. Prescribed radiosity makes it possible to specify the surface radiation for each spectral band. Using graybody radia tion definition, the radiosity is then F  T   T e b  T  . A user-defined surface radiosity expression can also be defined. 1

2

The Surface-to-Surface Radiation interface treats the radiosity J i as a shape function unless J i is prescribed.


| 191

View Factor Evaluation

The strategy for evaluating view factors is central to any radiation simulation. Loosely speaking, a view factor is a measure of how much influence the radiosity at a given part of the boundary has on the irradiation at some other part. The quantities Gm and F amb in Equation 5-3 are not strictly view factors in the traditional sense. F amb is the view factor of the ambient portion of the field of view, which is considered to be a single boundary with constant radiosity J amb

=

e b  T am b 

Gm, on the other hand, is the integral over all visible points of a differential view factor

times the radiosity of the corresponding source point. In the discrete model, think of it as a product of a view factor matrix and a radiosity vector. This is, however, not necessarily the way the calculation is performed. A separate evaluation is performed for each unique point where Gm or F amb is requested, typically for each quadrature point during solution. Differential view factors are normally computed only once, the first time they are needed, and then stored in memory until next time the model definition or the mesh is changed. The Heat Transfer Module supports two surface-to-surface radiation methods, which are selected in the Radiation Settings section from the Heat Transfer interface. View factors are always calculated directly from the mesh, which is a polygonal, flat-faceted representation of the geometry. To improve the accuracy of the radiative heat transfer simulation, the mesh must be refined rather than raising the element order. VIEW FACTOR FOR EXTERNAL RADIATION SOURCES

In 3D, the view factor for a point at finite distance is given by 2

cos    4  r 

where  is the angle between the normal to the irradiated surface and the direction of the source, and r is the distance from the source. For a source at infinity, the view factor is given by cos . In 2D the view factor for a point at finite distance is given by cos    2  r 

192 |


and the view factor for a source at infinity is cos . SOLAR POSITION

The sun is the most common example of an external radiation source. The position of the sun is necessary to determine the direction of the corresponding external radiation source. The direction of sunlight (zenith angle and the solar elevation) is automatically computed from the latitude, longitude, time zone, date, and time using similar method as described in Ref. 6. The estimated solar position is accurate for a date between year 2000 and 2199, due to an approximation used in the Julian Day calculation. The zenith angle ( zen) and the azimuth ( azi) angles of the sun are converted into a direction vector (is x, is y, is z) in Cartesian coordinates assuming that the North, the West, and the up directions correspond to the x, y, and z directions, respectively, in the model. The relation between azi, zen and (is x, is y, is z) is given by:  az i  sin  ze n  is y = sin  az i  sin  ze n  is z = – cos  ze n 

is x

= – cos

R A D I A T I O N I N A X I S Y M M E T R IC G E O M E T R I E S

For an axisymmetric geometry, Gm and F amb must be evaluated in a corresponding 3D geometry obtained by revolving the 2D boundaries about the axis. COMSOL Multiphysics creates this virtual 3D geometry by revolving the 2D boundary mesh into a 3D mesh. The resolution can be controlled in the azimuthal direction by setting the number of azimuthal sectors, which is the same as the number of elements to a full revolution. Try to balance this number against the mesh resolution in the rz-plane. This number, Azimuthal sectors, is accessible from the Radiation Settings section in physics interfaces for heat transfer. Select between the hemicube and the direct area integration methods also in axial symmetry. Their settings work the same way as in 3D. While Gm and F amb are in fact evaluated in a full 3D, the number of points where they are requested is limited to the quadrature points on the boundary of a 2D geometry. The savings compared to a full 3D simulation are therefore substantial despite the full 3D view factor code being used.


| 193

About Surface-to-Sur face Radiation

Surface-to-surface radiation is more complex than those topics discussed in the section Radiative Heat Transfer in Transparent Media. It includes radiation from both the ambient surroundings and from other surfaces. A generalized equation for the irradiative flux is: G

=

G m + G ex t + G am b

(5-5)

where • Gm is the mutual irradiation (SI unit: W/m 2) arriving from other surfaces in the • •

modeled geometry, Gext is irradiation from external sources (SI unit: W/m 2). G am b = F am b e b  T am b  is the ambient irradiation (SI unit: W/m 2), F amb is the ambient view factor and T amb is the assumed far-away temperature in the directions included in F amb. F amb describes the portion of the view from each point that is covered by ambient conditions. n is the transparent media refractive index. Gm on the other hand is determined from the geometr y and the local temperatures of the surrounding boundaries.

The following sections derive the equations for Gm, Gext and Gamb for a general 3D case. Consider a point x on a surface as in Figure 5-6. Point x can see points on other surfaces as well as the ambient surrounding. Assume that the points on the other surfaces have a local radiosity, J', while the ambient surrounding has a constant temperature, T amb.

194 |


Figure 5-6: Example geometry for surface-to-surface radiation.

The mutual irradiation at point x is given by the following surface integral: Gm

=

 –n'  r   n  r 

J ' dS  ----------------------------------- r 4

S'

The heat flux that arrives from x' depends on the local radiosity J ' projected onto x . The projection is computed using the normal vectors n and n' along with the vector r , which points from x to x' . The irradiation from external radiation sources is the sum of the ### The ambient view factor, F amb, is determined from the integral of the surrounding surfaces S', here denoted as F ', determined from the integral below: F amb

=

1 – F '

=

1–

 –n'  r   n  r 

dS  ----------------------------------- r 4

S'

The two last equations plug into Equation 5-5 to yield the final equation for irradiative flux.


| 195

The equations used so far apply to the general 3D case. 2D geometries results in simpler integrals. For 2D the resulting equations for the mutual irradiation and ambient view factor are Gm

=

 –n'  r   n  r 

J ' dS   -----------------------------------------2 r  3

(5-6)

S'

F am b

=

1–

 –n'  r   n  r 

dS   -----------------------------------------2 r  3

S'

where the integral over S' denotes the line integral along the boundaries of the 2D geometry. In axisymmetric geometries, the irradiation and ambient view factor cannot be computed directly from a closed-form expression. Instead, a virtual 3D geometry must be constructed, and the view factors evaluated according to Equation 5-6. Guidelines for Solving Surface-to-Surface Radiation Problems

The following guidelines are helpful when selecting solver settings for models that involve surface-to-surface radiation: •

•

196 |

Surface-to-surface radiation makes the Jacobian matrix of the discrete model partly filled as opposed to the usual sparse matrix. The additional nonzero elements in the matrix appear in the rows and columns corresponding to the radiosity degrees of freedom. It is therefore common practice to keep the element order of the radiosity variable, J , low. By default, linear Lagrange elements are used irrespective of the shape-function order specified for the temperature. When you need to increase the resolution of your temperature field, it might be wor th considering raising the order of the temperature elements instead of refining the mesh. The Assembly block size parameter (found in the Advanced section of the solver feature) can have a major influence on memory usage during the assembly of problems where surface-to-surface radiation is enabled. It may be useful to c onsider a block size as small as 100. Using a smaller block size also leads to more frequent updates of the progress bar.


Radiation Group Boundaries

The Radiation Group node is only available when the Use radiation groups check box is selected under Radiation Settings. By default this check box is not selected, which means that all radiative boundaries belong to the same radiation group. For radiation problems, a boundary grouping can be applied to save computational time. A radiation group can be defined using a Radiation Group node; see Radiation Group for details. A default group contains all boundaries selected in a Surface-to-Surface Radiation, Diffuse Mirror, or Prescribed Radiosity node. When a node is added to another radiation group, it is overridden in the default group. Then this boundary can be added to other radiation groups without being overridden by the manually added radiation groups. Be careful when grouping boundaries in axisymmetric geometries. The grouping cannot be based on which boundaries have a free view toward each other in the 2D geometry. Instead, consider the full 3D geometry, obtained by revolving the model geometry about the z axis, when defining groups. For example, parallel vertical boundaries must typically belong to the same group in 2D axisymmetric models, but to different groups in a planar model using the same 2D geometry. Figure 5-7 shows four examples of possible boundary groupings. On boundaries that have no number, the user has NOT set a node among the Surface-to-Surface Radiation, Diffuse Mirror, and Prescribed Radiosity nodes. These boundaries do not irradiate other boundaries, neither do other boundaries irradiate them. On boundaries that belong to one or more radiation group, the user has set a node among the Surface-to-Surface Radiation, Diffuse Mirror, and Prescribed Radiosity nodes. The numbers on each boundary specify different groups to which the boundar y belongs. To obtain optimal computational performance, it is good practice to specify as many groups as possible as opposed to specifying few but large groups. For example, in


| 197

Figure 5-7, case (b) is more efficient than case (d). A

B 1

1

1

2

1

12

C

2 123

1

D

1

inefficient boundary grouping

1 1

1 2

2

2

2

Figure 5-7: Examples of radiation group boundaries.


1

1

1

198 |

2

3

3

1

1

1

The Radiation in Participating Media User Interface The Radiation in Participating Media (rpm) user interface ( ), found under the Heat ) in the Model Wizard, enables the modeling of radiative Transfer>Radiation branch ( heat transfer inside a participating medium. This interface solves for radiative intensity field. Solves only for radiation variables. In order to solve for radiation and temperature, use a Heat Transfer interface. When the interface is added, these default nodes are also added to the Model Builder — Radiation in Participating Media, Wall, Continuity on Interior Boundary, and Initial Values. Right-click the main node to add boundary conditions or other features. To display this section select the Radiation in participating media check box under Physical Model on any version of the Heat Transfer interface settings window. INTERFACE IDENTIFIER

The interface identifier is used primarily as a scope prefix for variables defined by the physics user interface. Refer to such interface variables in expressions us ing the pattern .. In order to distinguish between variables belonging to different physics user interfaces, the identifier string must be unique. Only letters, numbers and underscores (_) are permitted in the Identifier field. The first character must be a letter. The default identifier (for the first interface in the model) is

rpm.

DOMAIN SELECTION

The default setting is to include All domains in the model to define the dependent variables and the equations. To choose specific domains, select Manual from the Selection list.

THE RADIATION IN PARTICIPATING MEDIA USER INTERFACE

| 199

PARTICIPATING MEDIA SETTINGS

Refractive Index

Define the Refractive index nr (dimensionless) of the participating media.

The same refractive index is used for the whole model.

Performance Index

Select a Performance index Pindex from the list. Select a value between 0 and 1 that modifies the strategy used to define automatic solver settings. The default is 0.4. With small values, a robust setting for the solver is expected. With large values (up to 1), less memory is needed to solve the model. Discrete Ordinates Method

Select the Discrete ordinates method order from the list. This order defines the discretization of the radiative intensity direction. In 3D, S2, S4, S6, and S8 generate 8, 24, 48, and 80 directions, respectively. The default is S4.

In 2D, S2, S4, S6, and S8 generate 4, 12, 24, and 40 directions, respectively. Discretization Level

Select Linear (the default), Quadratic, Cubic, Quartic, or Quintic to define the Discretization level. DISCRETIZATION


200 |


Show button

(


This section is empty for Radiation in Participating Media (rpm) interface which define the shape functions discretization in Discretization Level.. • • • • • • •

• •

About the Heat Transfer Interfaces The Heat Transfer Interface About Handling Frames in Heat Transfer Theory for the Heat Transfer User Interfaces Theory for the Radiation in Participating Media User Interface Show More Physics Options Domain, Boundary, Edge, Point, and Pair Nodes for the Radiation in Participating Media User Interface Radiative Heat Transfer in Finite Cylindrical Media : Model Library path Heat_Transfer_Module/Tutorial_Models/cylinder_participating_media Radiative Heat Transfer in a Utility Boiler: Model Library path Heat_Transfer_Module/Thermal_Radiation/boiler

Domain, Boundary, Edge, Point, and Pair Nodes for the Radiation in Participating Media User Interface

The Radiation in Participating Media User Interface has these domain, boundary, edge, point, and pair nodes available and described here: • • • •

Radiation in Participating Media Continuity on Interior Boundary Incident Intensity Opaque Surface

These nodes are described for other interfaces: •

Opaque To locate and search all the documentation, in COMSOL Multiphysics, select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree.


| 201


The Radiation in Participating Media node uses the radiative transfer equation   I   

=

  I b  T  –  I    + ------s 4



4

I        d

0

where • I ) is the radiative intensity in the

 direction

• T is the temperature •

,     s and s are absorption, extinction, and scattering coefficients

• I b is the blackbody radiative intensity •

 is the scattering phase function

The following equation is the blackbody radiation intensity and n is the refractive index: 2

I b  T 

=

4

n  T -----------------



It also adds the radiative heat source term in the heat transfer equation: Q r

=

  qr

=

  G – 4  T 4 

DOMAIN SELECTION

From the Selection list, choose the domains to define. MODELS INPUTS

This section contains fields and values that are inputs to expressions that define material properties. If such user-defined materials are added, the model inputs appear here. There is one standard model input—the Temperature T (SI unit: K). The default is to use the heat transfer dependent variable. ABSORPTION

The default Absorption coefficient  (SI unit: 1/m) uses the value From material. The absorption coefficient defines the amount of radiation,  I , that is absorbed by the medium. If User defined is selected, enter another value or expression.

202 |


SCATTERING

The default Scattering coefficient s (SI unit: 1/m) uses the value From material. If User defined is selected, enter another value or expression. The default is 0. Select the Scattering type—Isotropic, Linear anisotropic, or Polynomial anisotropic. This provides three options to define the scattering phase function using the cosine of the scattering angle, 0: • Isotropic (the default) corresponds • •

to the scattering phase function   0  = 1 . If Linear anisotropic is selected, it defines the scattering phase function as    0  = 1 + a1  0 . Enter the Legendre coefficient a1. If Polynomial anisotropic is selected, it defines the scattering phase function 12

  0 

=

1+

 a P m

m

=

m 0 

1

Enter the Legendre coefficients a1, …, a12 as required. INITIAL VALUES

For The Heat Transfer with Radiation in Participating Media User Interface, an Initial Values section is added, which has a default Radiative 2 intensity of ht.Ibinit (SI unit: W/m ). Opaque Surface

The Opaque Surface node is available for The Heat Transfer with Radiation in Participating Media User Interface version of the Heat Transfer interface. It is also available for The Radiation in Participating Media User Interface. The Opaque Surface node defines a boundary opaque to radiation. The Opaque Surface node prescribes incident intensities on a boundary and accounts for the net radiative heat flux, qw, that is absorbed by the surface.


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BOUNDARY SELECTION

For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically selected and is the same as for the interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific boundaries or select All boundaries as required. PAIR SELECTION

If this node is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect. MODELS INPUTS

This section contains fields and values that are inputs to expressions that define material properties. If such user-defined materials are added, the model inputs appear here. There is one standard model input—the Temperature T (SI unit: K). The default is 293.15 K and is used in the black-body radiative intensity expression. The boundary temperature definition can differ from the that of the temperature in the adjacent domain. WALL SETTINGS

Select a Wall type to define the behavior of the wall— Gray wall or Black wall. Gray Wall

If Gray wall is selected the default Surface emissivity e value is taken From material (a material defined on the boundaries). Select User defined to enter another value or expression. Enter a Diffusive reflectivity d. Both are dimensionless numbers between 0 and 1 that satisfy the relation d  w  1. By default d  1  w. In this case, an emissivity of 0 means that the surface emits no radiation at all and that all outgoing radiation is diffusely reflected by this boundary. An emissivity of 1 means that the surface is a perfect black body, outgoing radiation is fully absorbed on this boundar y. If d  1  w, it means that the wall is not opaque and that a part of the outgoing radiative intensity goes through the wall without being reflected nor absorbed. Radiative intensity (W/m 2 in SI units) along incoming discrete directions on this boundary is defined by

204 |


I i bnd

=

  w I b  T  + ----d-- q out 

Black Wall

If Black wall is selected, no user input is required and the radiative intensity along the incoming discrete directions on this boundary is defined by I i bnd

=

I b  T 

Values of radiative intensity along outgoing discrete directions are not prescribed. Incident Intensity

The Incident Intensity node is available for The Heat Transfer with Radiation in Participating Media User Interface version of the Heat Transfer interface. It is also available for The Radiation in Participating Media User Interface. Use an Incident Intensity node to specify the radiative intensity along incident directions on a boundar y. BOUNDARY SELECTION

From the Selection list, choose the boundaries to define. PAIR SELECTION



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INCIDENT INTENSITY

Enter a Boundary radiation intensity I wall (SI unit: W/m2). This represents the value of radiative intensity along incoming discrete directions. Values of radiative intensity on outgoing discrete directions are not prescribed. The components of each discrete ordinate vector can be used in this expression. The syntax is interfaceIdentifier.sx, interfaceIdentifier.sy, interfaceIdentifier.sz where interfaceIdentifier is the physics interface identifier. By default, the Heat Transfer interface identifier is ht so ht.sx, ht.sy, and ht.sz correspond to the components of discrete ordinate vectors. Continuity on Interior Boundary

The Continuity on Interior Boundary node enables intensity conservation across internal boundaries. It is the default boundary condition for all internal boundaries. BOUNDARY SELECTION

For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically selected and is the same as for the interface. PAIR SELECTION

When this node is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect.

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Theory for the Radiation in Participating Media User Interface Radiation and Participating Media Interactions

Figure 5-8: Example of interactions between participating media and radiation.

In some applications the medium is not completely transparent and the radiation rays interact with the medium. Let I  denote the radiative intensity traveling in a given direction, . Different kinds of interactions are observed: • • •

Absorption: The medium absorbs a fraction of the incident radiation. The amount of absorbed radiation is  I  where  is the absorption coefficient. Emission: The medium emits radiation in all directions. The amount of emitted radiative intensity is equal to  I b, where I b is the blackbody radiation intensity . Scattering: A part of the radiation coming from a given direction is scattered in other directions. The scattering properties of the medium are described by the scattering phase function i  j, which gives the probability that a ray coming from one direction i is scattered into the direction  j. The phase function (i,  j) satisfies

THEORY FOR THE RADIATION IN PAR TICIPATING MEDIA USER INTERFACE

| 207

1 -----4

      d i

i =

1

4

Radiative intensity in a given direction is attenuated and augmented by scattering: - It is attenuated because a part of incident radiation in this direction is scattered into other directions. The amount of radiation attenuated by scattering is s I . - It is augmented because a part of radiative intensity coming from other directions is scattered in all direction, including the direction we are looking at. The amount of radiation augmented by scattering is obtained by integrating scattering coming form all directions i: 

s -----4

 I        d i

i

i

4

Radiative Transfer Equation

The balance of the radiative intensity including all contributions (propagation, emission, absorption, and scattering) can now be formulated. The general radiative transfer equation can be written as (see Ref. 1)   I   

=

  I b  T  –  I    + ------s 4



4

I        d

0

where • I  is the radiative intensity at a

given position following the  direction

• T is the temperature •

,  s, s are absorption, extinction, and scattering coefficients, respectively 2

I b  T 

=

4

n  T -----------------



(5-7)

Equation 5-7 is the blackbody radiation intensity, and n is the refractive index of the media   '   is the phase function that gives the probability that a ray from the  direction is scattered into the  direction. The phase function’s definition is material dependent and its definition can be complicated. It is common to use approximate •

208 |


scattering phase functions that are defined using the cosine of the scattering angle, 0. The current implementation handles: •

Isotropic phase function:   '  

•

0

=

1 + a1 0

=

1

Linear anisotropic phase function:   0 

•

=

Polynomial anisotropic up to the 12th order: 12

  0 

1+

=

 a P    n

n

n

0

1

=

where Pn are nth-order Legendre polynomials. Legendre polynomials can be defined by the Rodriguez formula: Pk  x 

k

k 2 1 ----------- d   x – 1   k k 2 k! d x

=

A quantity of interest is the incident radiation, denoted G and defined by 4

G

=



I   

0

Boundary Condition for the Transfer Equation

For gray walls, corresponding to opaque surfaces reflecting diffusively and emitting, the radiative intensity I bnd entering participating media along the  direction is I bnd   

=

  w I b  T  + ----d-- q out 

for all  such that n    0

where 2

I b  T  • •

=

4

n  T -----------------



(5-8)

Equation 5-8 is the blackbody radiation intensity and n is the refractive index w is the surface emissivity, which is in the range [ 0, 1]


| 209

•

d  1  w is the diffusive reflectivity

• n is the outward normal vector • qout is the heat flux

striking the wall: qout

=

n

 j

w j I j n   j

0

For black walls w  1 and d  0. Thus I bnd  I bT . Heat Transfer Equation in Participating Media

Heat flux in gray media is defined by q r  n

=



4

I     n 

0

Heat flux divergence can be defined as a function of G and T (see Ref. 1): Q r

=

  q r

=

  G – 4n  T 4 

In order to couple radiation in participating media, radiative heat flux is taken into account in addition to conductive heat flux: q  qc  qr. The heat transfer equation reads --- +  u    T   C p  ----T  t

= –

  p --- -------  ------ +  u    p    qc + q r  + : S – -T    T p   t

+

Q

and is implemented using following form:  --  C p  ----T +  u    T  t

210 |

= –


T   p    q c +   G – 4n  T 4  + : S – ---- -------  -----+  u    p    T p  t

+

Q

Discrete Ordinates Method

The discrete ordinates method is implemented for 3D and 2D geometries.

Radiative intensity is defined for any direction , because the angular space is continuous. In order to treat radiative intensity equation numerically, the angular space is discretized. The SN approximation provides a discretization of angular space into n  N  N  2 in 3D (or n  N  N  22 in 2D) discrete directions. It consists of a set of directions and quadrature weights. Several sets are available in the literature. A set should satisfy first, second, and third moments (see Ref. 1); it is also recommended that the quadrature fulfills the half moment for vectors of Cartesian basis. Since it is not possible to fulfill exactly all these conditions, accuracy should be improved when N increases. Following the conclusion of Ref. 3, the implementation uses LSE symmetric quadrature for S2, S4, S6, and S8. LSE symmetric quadrature fulfills the half, first, second, and third moments. Thanks to angular space discretization, integrals over directions are replaced by numerical quadratures of discrete directions: 4



I    d 

0

n

 w I

j j

j

1

=

Depending on the value of N , a set of n dependent variables has to be defined and solved for I 1, I 2, …, I n. Each dependent variable obeys the equation i  I i

=

  I b  T  –  I i + ------s 4

n

 w I      

j

j j

=

j

i

1

with the boundary condition


| 211

I i bnd

=

  w I b  T  + ----d-- qout 

for all i such that n   i  0

Discrete Ordinates Method Implementation in 2D

For a given index i, let’s define 2 indexes, i+ and i-, so that •

 i+ and i- have the same components in the ( x, y) plane and

•

i+ and i- have opposite component in z direction.

Assuming that a model is invariant in the z direction, we can write the DOM form of the radiative transfer equation in two directions, i+ and i-:  i+  I i+

=

n

  I b  T  –  I i+ + ------s 4

 w I     j j

j

 i-  I i-

=

  I b  T  –  I i- + ------s 4

=

j

i

+



1

n

 w I      

j

j j

=

j

i

-

1

By summing the two above equations and introducing I ˜i which is equal to I i- and I i + (which are equal in 2D) we get: ˜ 2  i  I i

=

n

s ˜ + -----2  I b  T  – 2  I i 4

 w I       j j

j

=

j

i

+

 +    j  i-  

1

which can be rewritten as: i  I ˜i

=

  I b  T  –  I ˜i + ------s 8

n

 w I      

j

j j

=

j

i

+

 +    j i-  

1

In addition if   i  j  can be rewritten as a function of i   j , as it is in COMSOL Multiphysics implementation, then    j+ i+ 

=

   j- i-  and    j-  i+ 

=

   j+  i- 

In addition I j-    j -  i+  + I j+    j+  i- 

212 |


=

˜      2 I + j j i

=

˜     2 I + j j i

so we can simplify above equation: ˜ i   I ˜i

  I b  T  –  I ˜i + ------s 4

=

n

 w I ˜      

j

j i

=

j

(5-9)

i

1

with ˜

i =

1 i  1 0

since the z component of  I ˜i is null in 2D. One can also notice that

 with w˜ i

=

4

I    d  

0

n  2

n

n  2

 w I  w I j j =

j

=

1

-

j j -

j

=

1

+

w j + I + j

=

 w˜ I ˜

i i

j

=

(5-10)

1

2w i .

Using results from Equation 5-9 and Equation 5-10 we can formulate DOM in 2D using only radiative intensities, I ˜i , on half of the 3D DOM directions, ˜ i , except for the scattering term. In other expressions than the scattering term, the z component of the radiative intensities I i and of the discrete directions i can by ignored (or set to zero) and the weight wi, multiplied by 2.


| 213

References for the Radiation User Interfaces 1. M.F. Modest, Radiative Heat Transfer , 2nd ed., Academic Press, San Diego, California, 2003. 2. R. Sieger, J. Howell, Thermal Radiation Heat Transfer , 4th ed., Taylor & Francis, New York, 2002. 3. W.A. Fiveland, “The Selection of Discrete Ordinate Quadrature Sets for Anisotropic Scattering,” Fundamentals of Radiation Transfer , HTD, vol. 160, ASME, 1991. 4. F.P. Incropera and D.P. DeWitt, Fundamentals of Heat and Mass Transfer , 5th ed., John Wiley and Sons, 2002. 5. J.R. Welty, C.E. Wicks, and R.E. Wilson, Fundamentals of Momentum, Heat, and Mass Transfer , 3rd ed., John Wiley and Sons, 1983. 6. http://www.esrl.noaa.gov/gmd/grad/solcalc

214 |


6

The Single-Phase Flow Branch The Heat Transfer Module extends the CFD capability of COMSOL

Multiphysics® by adding turbulence modeling and support for low Mach number compressible flows. This enables modeling of forced or temperature gradient-driven flows in both laminar and turbulent regimes. This chapter describes the fluid flow groups under the Fluid Flow>Single-Phase Flow branch ( ) in the Model Wizard. In this chapter: • • • •

The Laminar Flow and Turbulent Flow User Interfaces Theory for the Laminar Flow User Interface Theory for the Turbulent Flow User Interfaces References for the Single-Phase Flow, User Interfaces

215

The Laminar Flow and Turbulent Flow User Interfaces In this section: • • •

The Laminar Flow User Interface The Turbulent Flow, k- User Interface The Turbulent Flow, Low Re k- User Interface For 2D axisymmetric models, COMSOL Multiphysics takes the axial symmetry boundaries (at r  0) into account and automatically adds an Axial Symmetry node to the model that is valid on the axial symmetr y boundaries only.

• •

Domain, Boundary, Pair, and Point Nodes for Single-Phase Flow Theory for the Laminar Flow User Interface

The Laminar Flow User Interface

The Laminar Flow (spf) user interface ( ), found under the Single-Phase Flow branch ( ) in the Model Wizard, has the equations, boundary conditions, and volume forces for modeling freely moving fluids using the Navier-Stokes equations, solving for the velocity field and the pressure. The main node is Fluid Properties, which adds the Navier-Stokes equations and provides an interface for defining the fluid material and its properties. When this interface is added, these default nodes are also added to the Model Builder — Fluid Properties, Wall (the default boundary condition is No slip), and Initial Values. Right-click the Laminar Flow node to add other nodes that implement, for example, boundary conditions and volume forces. INTERFACE IDENTIFIER

The interface identifier is used primarily as a scope prefix for variables defined by the physics user interface. Refer to such interface variables in expressions using the pattern

216 |

CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

.. In order

to distinguish between variables belonging to different physics user interfaces, the identifier string must be unique. Only letters, numbers and underscores (_) are permitted in the Identifier field. The first character must be a letter. The default identifier (for the first interface in the model) is

spf.

PHYSICAL MODEL

By default the interface uses the Compressible flow (Ma<0.3) formulation of the Navier-Stokes equations. Select Incompressible flow to use the incompressible (constant density) formulation. Turbulence Model Type

By default, None is selected as the Turbulence model type. The flow state in a fluid flow model, however, is not always known beforehand. Selecting an option in this section switches between available Single-Phase Flow (spf) interfaces. For example, this interface changes to The Turbulent Flow, k- User Interface when the Turbulence model type selected is RANS (Reynolds-averaged Navier– Stokes). DEPENDENT VARIABLES

These dependent variables (fields) are defined for this inter face— Velocity field u (SI unit: m/s) and its components, and Pressure p (SI unit: Pa). If required, edit the field, component, and dependent variable names. Editing the name of a scalar dependent variable changes both its field name and the dependent variable name. If a new field name coincides with the name of another field of the same type, the fields will share degrees of freedom and dependent variable names. A new field name must not coincide with the name of a field of another type, or with a component name belonging to some other field. Component names must be unique within a model except when two fields share a common field name. CONSISTENT STABILIZATION


Show button

(

) and select Stabilization.

The consistent stabilization methods are applicable to the Navier-Stokes equations— Streamline diffusion and Crosswind diffusion. These check boxes are selected by default. If required, click to clear one or both of the Streamline diffusion and Crosswind diffusion check boxes. Observe that P1+P1 elements require Streamline diffusion to be active. If

T H E L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R I N T E R F A C E S

| 217

you deactivate Streamline diffusion, make sure that your model uses P2+P1 elements or higher. I N C O N S I S T E NT S T A B I L I Z A T I ON

To display this section, click the Show button ( ) and select Stabilization. By default, the Isotropic diffusion check box is not selected for the Navier-Stokes equations. Click to select as required. ADVANCED SETTINGS

To display this section, click the Show button ( ) and select Advanced Physics Options . Normally these settings do not need to be changed. Select the Use pseudo time stepping for stationary equation form check box to add pseudo time derivatives to the equation when the Stationary equation form is used. When selected, also choose a CFL number expression—Automatic (the default) or Manual. Automatic sets the local CFL number (from the Courant–Friedrichs–Lewy condition) to the built-in variable CFLCMP which in turns trigger a PID regulator for the CFL number. If Manual is selected, enter a Local CFL number CFLloc (dimensionless). Pseudo Time Stepping for Laminar Flow Models and About Pseudo Time Stepping in the COMSOL Multiphysics Reference Manual DISCRETIZATION

To display this section, click the Show button ( ) and select Discretization. It controls the discretization (the element types used in the finite element formulation). From the Discretization of fluids list select the element order for the velocity components and the pressure: P1+P1 (the default), P2+P1, or P3+P2. • P1+P1 (the default) means linear elements for both the velocity components and the

pressure field. This is the default element order for the Laminar Flow and Turbulent Flow interfaces. Linear elements are computationally cheaper than higher-order elements and are also less prone to introducing spurious oscillations, thereby improving the numerical robustness. P1+P1 elements require streamline diffusion to be a numerically valid discretization. Make sure that Streamline Diffusion in the Consistent Stabilization section is selection when using P1+P1 elements.

218 |


• P2+P1 means second-order elements for the velocity components and linear •

elements for the pressure field. P3+P2 means third-order elements for the velocity components and second-order elements for the pressure field. This can add additional accuracy but it also adds additional degrees of freedom compared to P2+P1 elements.

Specify the Value type when using splitting of complex variables —Real (the default) or Complex for each of the variables in the table.

•

Show More Physics Options Domain, Boundary, Pair, and Point Nodes for Single-Phase Flow Theory for the Laminar Flow User Interface

•

Flow Past a Cylinder: Model Library path COMSOL_Multiphysics/

• •

Fluid_Dynamics/cylinder_flow

•

Terminal Falling Velocity of a Sand Grain: Model Library path COMSOL_Multiphysics/Fluid_Dynamics/falling_sand

The Turbulent Flow, k-  User Interface

The Turbulent Flow, k- (spf) user interface ( ), found under the Single-Phase ) in the Model Wizard, has the equations, boundary Flow>Turbulent Flow branch ( conditions, and volume forces for modeling turbulent flow using the Reynolds averaged Navier-Stokes (RANS) equations, solving for the mean velocity field, the pressure, and the standard k- model, solving for the turbulent kinetic energy k and the rate of dissipation of turbulent kinetic energy . The main feature is Fluid Properties, which adds the Navier-Stokes equations and the transport equations for k and , and provides an interface for defining the fluid material and its properties. When this interface is added, these default nodes are also added to the Model Builder —Fluid Properties, Wall (the default boundary condition is Wall functions), and Initial Values. Except where included below, see The Laminar Flow User Interface for all the other settings.


| 219

PHYSICAL MODEL

For this interface, the Turbulence model type defaults to RANS and the Turbulence model defaults to k- . This enables the Turbulence Model Parameters section. TURBULENCE MODEL PARAMETERS

Turbulence model parameters are optimized to fit as many flow types as possible, but for some special cases, better performance can be obtained by tuning the model parameters. For this interface the parameters are C e1, C e2, C,  k,  e,  v, and B. DEPENDENT VARIABLES

These dependent variables (fields) are defined for this inter face : • Velocity field u (SI unit: m/s) and its components • Pressure p (SI unit: Pa) • Turbulent kinetic energy k (SI unit: m 2/s2) • Turbulent dissipation rate ep (SI unit: m 2/s3) ADVANCED SETTINGS

To display this section, click the Show button ( ) and select Advanced Physics Options . The Turbulence variables scale parameters subsection is available when the Turbulence model type is set to RANS. In addition to the settings described for the Laminar Flow inter face, enter a value for U scale (SI unit: m/s) (the default is 1 m/s) and Lfact (dimensionless) (the default is 0.035) under the Turbulence variables scale parameters subsection. The U scale and Lfact parameters are used to calculate absolute tolerances for the turbulence variables. The scaling parameters must only contain numerical values, units or parameters defined under Global Definitions. The scaling parameters cannot contain variables. The parameters are used when a new default solver for a transient study step

220 |


is generated. If you change the parameters, the new values take effect the next time you generate a new default solver • •

The Laminar Flow User Interface About Pseudo Time Stepping in the COMSOL Multiphysics Reference Manual

• •

Theory for the Turbulent Flow User Interfaces Show More Physics Options

Turbulent Flow Over a Backward Facing Step: Model Library path Heat_Transfer_Module/Verification_Models/turbulent_backstep

The Turbulent Flow, Low Re k-  User Interface

The Turbulent Flow, Low Re k - (spf) user interface ( ), found under the Single-Phase Flow>Turbulent Flow branch, has the equations, boundary conditions, and volume forces for modeling turbulent flow using the Reynolds averaged Navier-Stokes (RANS) equations, solving for the mean velocity field, the pressure, and the AKN low-Reynolds number k- model, solving for the turbulent kinetic energy k and the rate of dissipation of turbulent kinetic energy . The interface also includes a wall distance equation that solves for the reciprocal wall distance. The Low Reynolds number k- interface requires a Wall Distance Initialization study step in the study previous to the stationary or time dependent study step. For study information, see Stationary with Initialization, Transient with Initialization, and Wall Distance Initialization in the COMSOL Multiphysics Reference Manual. PHYSICAL MODEL

For this interface, the Turbulence model type defaults to RANS and the Turbulence model defaults to Low Reynolds number k- . This enables the Turbulence Model Parameters section.


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TURBULENCE MODEL PARAMETERS

Turbulence model parameters are optimized to fit as many flow types as possible, but for some special cases, better performance can be obtained by tuning the model parameters. For this interface the parameters are C e1, C e2, C,  k,  e, and  v. DEPENDENT VARIABLES

These dependent variables (fields) are defined for this inter face: • Velocity field u (SI unit: m/s) and its components • Pressure p (SI unit: Pa) • Turbulent kinetic energy k (SI unit: m 2/s2) • Turbulent dissipation rate ep (SI unit: m 2/s3) • Reciprocal wall distance G (SI unit: 1/m)

See The Laminar Flow User Interface and The Turbulent Flow, k-  User Interface for all the other settings.

• •

The Laminar Flow User Interface About Pseudo Time Stepping in the COMSOL Multiphysics Reference Manual

• •

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Theory for the Turbulent Flow User Interfaces Show More Physics Options


Domain, Boundary, Pair, and Point Nodes for Single-Phase Flow

The following nodes are for all interfaces found under the Fluid Flow>Single-Phase Flow branch ( ) in the Model Wizard. Other interfaces also share many of these domain, boundary, pair, and point nodes (listed in alphabetical order): • • • • • • • •

Boundary Stress Fan Flow Continuity Fluid Properties Grille Initial Values Inlet Interior Fan

• • • • • • • •

Interior Wall Open Boundary Outlet Periodic Flow Condition Pressure Point Constraint Symmetry Volume Force Wall

For 2D axisymmetric models, COMSOL Multiphysics takes the axial symmetry boundaries (at r  0) into account and adds an Axial Symmetry node to the model that is valid on the axial symmetry boundaries only. The theory about most boundary conditions is found in P.M. Gresho and R.L. Sani, Incompressible Flow and the Finite Element Method, Volume 2: Isothermal Laminar Flow , John Wiley & Sons, 2000. To locate and search all the documentation, in COMSOL, select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree.


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Fluid Properties

The Fluid Properties node adds the momentum equations solved by the interface, except for volume forces which are added by the Volume Force feature. The node also provides an interface for defining the material properties of the fluid. For the Turbulent Flow interfaces, the Fluid Properties node also adds the equations for the turbulence transport equations. MODEL INPUTS

Edit input variables to the fluid-flow equations if required. For fluid flow, these are typically introduced when a material requiring inputs has been applied. Abs olute Pre ssure

This input appears when a material requires the absolute pressure as model input. The absolute pressure input controls the pressure used to evaluate material properties, but it also relates to the value of the pressure field. There are usually two ways to calculate the pressure when describing fluid flow. Solve for the absolute pressure or a pressure (often denoted gauge pressure) that relates back to the absolute pressure through a reference pressure. Using one or the other option usually depends on the system and the equations being solved for. For example, in a straight incompressible flow problem, the pressure dr op over the modeled domain is probably many orders of magnitude less than atmospheric pressure, which, if included, reduces the chances for stability and convergence during the solving process for this variable. In other cases, the absolute pressure may be required to be solved for, such as where pressure is a part of an expression for gas volume or diffusion coefficients. The default Absolute pressure p A (SI unit: Pa) is p pref where p defaults to the pressure variable from the Navier-Stokes equations and pref to 1[atm] (1 atmosphere  101,325 Pa). The default setting is hence consistent with solving for a gauge pressure. If the pressure field instead is an absolute pressure field, clear the check box.

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Reference pressure

To model an incompressible fluid, set Absolute pressure p A is set to User defined and enter the desired pressure level in the edit field. The default value is 1[atm]. This makes it possible to use a system-based (gauge) pressure as the pressure variable while automatically including the reference pr essure in places where it is required, such as for gas flow governed by the gas law. While this check box maintains control over the pressure variable and instances where absolute pressure is required within this r espective physics interface, it may not with physics interfaces that it is being coupled to. In such models, check the coupling between any interfaces using the same variable.

To define the Absolute Pressure, see the settings for the Heat Transfer in Fluids node. FLUID PROPERTIES

The default Density  (SI unit: kg/m 3) uses the value From material. Select User defined to enter a different value or expression. The default Dynamic viscosity  (SI unit: Pa·s) uses the value From material and describes the relationship between the shear rate and the shear stresses in a fluid. Intuitively, water and air have a low viscosity, and substances often described as thick (such as oil) have a higher viscosity. Select User defined to define a different value or expression. Using a built-in variable for the shear rate magnitude, spf.sr, makes it possible to define arbitrary expressions of the dynamics viscosity as a function of the shear rate. MIXING LENGTH LIMIT

This section is available for the Turbulent Flow, k- interface, because an upper limit on the mixing length is required.


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Select how the Mixing length limit lmix,lim (SI unit: m) is defined—Automatic (default) or Manual: •

•

If Automatic is selected, the mixing length limit is automatically evaluated as the shortest side of the geometry bounding box. If the geometry is, for example, a complicated system of slim entities, this measure can give too big a result. In this case, it is recommended that it is defined manually. Select Manual to define a different value or expression. The default is 1 (that is, one unit length of the model unit system).

DISTANCE EQUATION

This section is available for a Turbulent Flow, Low Reynolds number k-  interface since a Wall Distance interface is included. Select how the Reference length scale lref (SI unit: m) is defined— Automatic (default) or Manual: •

•

If Automatic is selected, the wall distance is automatically evaluated to one tenth of the shortest side of the geometry bounding box. This is usually quite accurate but it can sometimes give too great a value if the geometry consists of several slim entities. In this case, it is recommended that it is defined manually. Select Manual to define a different value or expression for the wall distance. The default is 1 m.

lref controls the result of the distance equation. Objects that are much smaller than lref will effectively be diminished while the distance to objects much larger than lref will be

accurately represented. Volume Force

The Volume Force node specifies the volume force F on the right-hand side of the incompressible flow equation. Use it, for example, to incorporate the ef fects of gravity in a model. 

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u + u  u t


=

   – p I +   u +  u  T   + F

If several volume force nodes are added to the same domain, then the sum of all contributions are added to the momentum equations. VOLUME FORCE

Enter the components of the Volume force F (SI unit: N/m3). The defaults for all components are 0 N/m3. The Boussinesq Approximationin the COMSOL Multiphysics Reference Manual

Initial Values

The Initial Values node adds initial values for the velocity field and the pressure that can serve as an initial condition for a transient simulation or as an initial guess for a nonlinear solver. COORDINATE SYSTEM SELECTION

The Global coordinate system is selected by default. The Coordinate system list contains any additional coordinate systems that the model includes. INITIAL VALUES

Enter values or expressions for the initial value of the Velocity field u (SI unit: m/s) and the Pressure p (SI unit: Pa). The default values are 0 m/s and 0 Pa, respectively. In the Turbulent Flow interfaces, initial values for the turbulence variables are also specified. By default these are specified using the predefined variables defined by the expressions in Initial Values. For the k- and Low Reynolds number k- turbulence models, also define the Turbulent 2 2 2 kinetic energy k (SI unit: m /s ) and the Turbulent dissipation rate ep (SI unit: m / s3). The default values are spf.kinit and spf.epinit. For the Low Reynolds number k- turbulence model, define the Reciprocal wall distance G (SI unit: 1/m). The default is spf.G0.


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Wall

The Wall node includes a set of boundary conditions describing the fluid flow condition at a wall. • • • • •

No Slip (the default for laminar flow, and the Low Reynolds number k- turbulence model) Slip Sliding Wall Moving Wall Leaking Wall

In addition to the Slip condition, the following are also available for a k-  turbulence model: • • •

Wall Functions (the default for turbulent flow with a k-  turbulence model) Sliding Wall (Wall Functions) Moving Wall (Wall Functions) In the COMSOL Multiphysics Reference Manual: • • •

Slip Sliding Wall Moving Mesh User Interface

BOUNDARY SELECTION

For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically selected and is the same as for the interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific boundaries or select All boundaries as required. BOUNDARY CONDITION

Select a Boundary condition for the wall. No Slip

is the default boundary condition for a stationary solid wall (and for the Low Reynolds number k-turbulence model). The condition prescribes u = 0, that is, that the fluid at the wall is not moving. No slip

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Slip

The Slip condition prescribes a no-penetration condition, u·n. It hence implicitly assumes that there are no viscous effects at the slip wall and hence, no boundary layer develops. From a modeling point of view, this may be a reasonable approximation if the important effect of the wall is to prevent fluid from leaving the domain. Sliding Wall

The Sliding wall boundary condition is appropriate if the wall behaves like a conveyor belt; that is, the surface is sliding in its tangential direction. The wall does not have to actually move in the coordinate system. For 3D models, enter the components of the Velocity of the sliding wall uw (SI unit: m/s). If the velocity vector entered is not in the plane of the wall, COMSOL Multiphysics projects it onto the tangential direction. Its magnitude is adjusted to be the same as the magnitude of the vector entered. For 2D models, the tangential direction is unambiguously defined by the direction of the boundar y, but the situation becomes more complicated in 3D. For this reason, this boundary condition has slightly different definitions in the different space dimensions. Enter the components of the Velocity of the tangentially moving wall U w (SI unit: m/s). Moving Wall

If the wall moves, so must the fluid. Hence, this boundar y condition prescribes u = uw. Enter the components of the Velocity of moving wall uw (SI unit: m/s). Specifying this boundary condition does not automatically cause the associated wall to move. An additional Moving Mesh interface needs to be added to physically track the wall movement in the spatial reference frame. Leaking Wall

Use this boundary condition to simulate a wall where fluid is leaking into or leaving through a perforated wall u = ul. Enter the components of the Fluid velocity ul (SI unit: m/s).


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Wall Functions

The Wall functions boundary condition applies wall functions to solid walls in a turbulent flow. Wall functions are used to model the thin region near the wall with high gradients in the flow variables. Sliding Wall (Wall Functions)

The Sliding wall (wall functions) boundary condition applies wall functions to a wall in a turbulent flow where the velocity magnitude in the tangential direction of the wall is prescribed. The tangential direction is determined in the same manner as in the Sliding Wall feature. Enter the component values or expressions for the Velocity of sliding wall uw (SI unit: m/s). The defaults are 0 m/s. Moving Wall (Wall Funct ions)

Specifying this boundary condition does not automatically cause the associated wall to move. The Moving wall (wall functions) boundary condition applies wall functions to a wall in a turbulent flow with prescribed velocity uw. Enter the component values or expressions for the Velocity of moving wall uw (SI unit: m/s). The defaults are 0 m/s. CONSTRAINT SETTINGS

To display this section, click the Show button (

) and select Advanced Physics Options .

For the No Slip, Moving Wall, and Leaking Wall boundary conditions, select an option from the Apply reaction terms on: list—All physics (symmetric) or Individual dependent variables. The other types of wall boundary conditions with constraints use Individual dependent variables constraints only. Select the Use weak constraints check box (not available for the Sliding Wall condition) to use weak constraints and create dependent variables for the corresponding Lagrange multipliers.

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Inlet

The Inlet node includes a set of boundary conditions describing the fluid flow condition at an inlet. The Velocity boundary condition is the default. In most cases the inlet boundary conditions are available, some of them slightly modified, in the Outlet type as well. This means that there is nothing in the mathematical formulations to prevent a fluid from leaving the domain through boundaries where the Inlet type is specified. Theory for the Inlet Boundary ConditionIn the COMSOL Multiphysics Reference Manual: • •

Theory for the Pressure, No Viscous Stress Boundary Condition Theory for the Normal Stress Boundary Condition

BOUNDARY CONDITION

Select a Boundary condition for the inlet— Velocity (the default), Pressure, No Viscous Stress, Laminar Inflow , or Normal Stress. After selecting a Boundary Condition from the list, a section with the same name displays underneath. For example, if Velocity is selected, a Velocity section displays where further settings are defined for the velocity. For the Velocity, Pressure, no viscous stress, and Normal stress sections, also enter the turbulent flow settings as described in More Boundary Condition Settings for the Turbulent Flow User Interfaces.


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VELOCITY

The Velocity boundary condition is available for the Inlet and Outlet boundary nodes. •

•

•

Select Normal inflow velocity (the default) to specify a normal inflow velocity magnitude u = nU0 where n is the boundary normal pointing out of the domain. Enter the velocity magnitude U 0 (SI unit: m/s). The default is 0 m/s. If Velocity field is selected, it sets the velocity equal to a given velocity vector u0 when u = u0. Enter the velocity components u0 (SI unit: m/s) to set the velocity equal to a given velocity vector. Also enter the turbulent flow settings as described in More Boundary Condition Settings for the Turbulent Flow User Interfaces.

PRESSURE, NO VISCOUS STRESS

The Pressure, no viscous stress boundary condition is available for the Inlet and Outlet boundary nodes. It specifies vanishing viscous stress along with a Dirichlet condition on the pressure. Enter the Pressure p0 (SI unit: Pa) at the boundary. The default is 0 Pa. Depending on the pressure field in the rest of the domain, an inlet boundary with this condition can become an outlet boundary. Also enter the turbulent flow settings as described in More Boundary Condition Settings for the Turbulent Flow User Interfaces. NORMAL STRESS

The Normal stress boundary condition is available for the Inlet, Outlet, Open Boundary , and Boundary Stress nodes. Enter the magnitude of Normal stress f 0 (SI unit: N/m2). This implicitly specifies that p  f 0 . The default is 0 N/m 2. Also enter the turbulent flow settings as described in More Boundary Condition Settings for the Turbulent Flow User Interfaces. LAMINAR INFLOW

The Laminar inflow boundary condition is available for the Inlet node. Select a flow quantity for the inlet—Average velocity (the default), Flow rate, or Entrance pressure. •

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When Average velocity is selected, enter an Average velocity U av (SI unit: m/s). The default is 0 m/s.


• •

If Flow rate is selected, enter the Flow rate V 0 (SI unit: m3/s). The default is 0 m3/s. If Entrance pressure is selected, enter the Entrance pressure pentr (SI unit: Pa) at the entrance of the fictitious channel outside of the model. The default is 0 Pa.

Then for any selection, specify the entrance length and constraints: •

Enter the Entrance length Lentr (SI unit: m) to define the length of the inlet channel outside the model domain. The Entrance length value must be large enough so that the flow can reach a laminar profile. For a laminar flow, Lentr should be significantly greater than 0.06Re D, where Re is the Reynolds number and D is the inlet length scale (this formula is exact if D is the diameter of a cylindrical pipe and approximate for other geometries). For turbulent flow the equivalent expression is 4.4 Re1/6 D. The default is 1 m.

•

Select the Constrain outer edges to zero (for 3D models) or Constrain endpoints to zero (for 2D and 2D axisymmetric models) check box to force the laminar profile to go to zero at the bounding points or edges of the inlet channel. Otherwise the velocity is defined by the boundary condition of the adjacent boundary in the model. For example, if one end of a boundar y with a laminar inflow condition connects to a slip boundary condition, then the laminar profile has a maximum at that end.

CONSTRAINT SETTINGS

To display this section, click the Show button ( ) and select Advanced Physics Options. Select the Use weak constraints check box to use weak constraints and create dependent variables for the corresponding Lagrange multipliers. When Velocity or Pressure, No Viscous Stress are selected as the Boundary condition, and to Apply reaction terms on all dependent variables, select All physics (symmetric). Or select Individual dependent variables to restrict the reaction terms as required. Outlet

The Outlet node includes a set of boundar y conditions describing fluid flow conditions at an outlet. The Pressure, no viscous stress boundary condition is the default. Other options are based on individual licenses. Selecting appropriate outlet conditions for the Navier-Stokes equations is not a trivial task. Generally, if there is something interesting


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happening at an outflow boundary, extend the computational domain to include this phenomenon. All of the formulations for the Outlet type are also available, possibly slightly modified, in other boundary types as well. This means that there is nothing in the mathematical formulations to prevent a fluid from entering the domain through boundaries where the Outlet boundary type is specified.

Additional Theory for the Outlet Boundary Condition

BOUNDARY CONDITION

Select a Boundary condition for the outlet—Pressure, No Viscous Stress (the default), Velocity , No Viscous Stress, Pressure, Laminar Outflow , or Normal Stress. The Pressure, no viscous stress, Velocity, and Normal stress boundary conditions are described for the Inlet node. Pressure

The Pressure boundary condition prescribes only a Dirichlet condition for the pressure p = p0. Enter the Pressure p0 (SI unit: Pa) at the boundary. While this boundary condition is flexible and seldom produces artifacts on the boundary (compared to Pressure, no viscous stress), it can be numerically unstable. Theoretically, the stability is guaranteed by using streamline diffusion for a flow with a cell Reynolds number Rec  uh 2 1 (h is the local mesh element size). It does however work well in most other situations as well. No Viscous Stress

The No Viscous Stress condition specifies vanishing viscous stress on the outlet. This condition does not provide sufficient information to fully specify the flow at the outlet and must at least be combined with pressure constraints on adjacent points.

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If No viscous stress is selected, it prescribes vanishing viscous stress:     u +   u  T  – 2---     u  I n   3    u +   u  T  n

=

=

0

0

using the compressible and the incompressible formulation respectively. This condition can be useful in some situations because it does not impose any constraint on the pressure. A typical example is a model with volume forces that give rise to pressure gradients that are hard to prescribe in advance. To make the model numerically stable, combine this boundary condition with a point constraint on the pressure. Laminar Outflow

This section displays when Laminar outflow is selected as the Boundary condition. Select a flow quantity to specify for the inlet: • • •

If Average velocity is selected, enter an Average velocity U av (SI unit: m/s). If Flow rate is selected, enter the Flow rate V 0 (SI unit: m3/s). If Exit pressure is selected, enter the Exit pressure pexit (SI unit: Pa) at the end of the fictitious channel following the outlet.

Then specify the Exit length and Constrain endpoints to zero parameters: Enter the Exit length Lexit (SI unit: m) to define the length of the fictitious channel after the model domain. The Exit length value must be large enough so that the flow can reach a laminar profile. For a laminar flow, Lexit should be significantly greater than 0.06Re D, where Re is the Reynolds number and D is the outlet length scale (this formula is exact if D is the diameter of a cylindrical pipe and approximate for other geometries). For turbulent flow the equivalent expression is 4.4 Re1/6 D. The default is 1 m. Select the Constrain outer edges to zero (3D models) or Constrain endpoints to zero (2D models) check box to force the laminar profile to go to zero at the bounding points or edges of the inlet channel. Otherwise the velocity is defined by the boundar y condition of the adjacent boundary in the model. For example, if one end of a boundary with a


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Laminar inflow condition connects to a Slip boundary condition, then the laminar profile has a maximum at that end. CONSTRAINT SETTINGS

To display this section, click the Show button ( ) and select Advanced Physics Options . Select the Use weak constraints check box to use weak constraints and create dependent variables for the corresponding Lagrange multipliers. When Velocity , Pressure, No Viscous Stress, or Pressure are selected as the Boundary condition, and to Apply reaction terms on all dependent variables, select All physics (symmetric). Or select Individual dependent variables to restrict the reaction terms as required. Symmetry

The Symmetry node adds a boundary condition that describes symmetry boundaries in a fluid flow simulation. The boundary condition for symmetry boundaries prescribes no penetration and vanishing shear stresses. The boundary condition is a combination of a Dirichlet condition and a Neumann condition: un

=

 – p I +     u +   u  T  – 2---     u  I  n    3

0, un

=

 – p I +    u +   u T   n

0,

=

=

0

0

for the compressible and the incompressible formulation respectively. The Dirichlet condition takes precedence over the Neumann condition, and the above equations are equivalent to the following equation for both the compressible and incompressible formulation: un

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=

0,

K

=


K –  K  nn  u +  uT  n

=

0

BOUNDARY SELECTION

From the Selection list, choose the boundaries to define. For 2D axial symmetry, a boundary condition does not need to be defined. For the symmetry axis at r  0, the software automatically provides a condition that prescribes ur  0 and vanishing stresses in the z direction and adds an Axial Symmetry node that implements this condition on the axial symmetr y boundaries only. CONSTRAINT SETTINGS

To display this section, click the Show button ( ) and select Advanced Physics Options. Select the Use weak constraints check box to use weak constraints and create dependent variables for the corresponding Lagrange multipliers. Open Boundary

The Open Boundary node adds boundary conditions that describe boundaries that are open to large volumes of fluid. Fluid can both enter and leave the domain on boundaries with this type of condition. BOUNDARY CONDITIONS

Select a Boundary condition for the open boundaries—Normal Stress (the default) or No Viscous Stress. These options are described for the Inlet and Outlet nodes, respectively Also enter the additional settings described in More Boundary Condition Settings for the Turbulent Flow User Interfaces.

Boundary Stress

The Boundary Stress node adds a boundary condition that represents a very general class of conditions also known as traction boundary conditions.


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BOUNDARY CONDITION

Select a Boundary condition for the boundary stress—General stress (the default), Normal Stress (described for the Inlet node), or Normal stress, normal flow. Also enter the settings described in More Boundary Condition Settings for the Turbulent Flow User Interfaces. General Stress

When General stress is selected, enter the components for the Stress F (SI unit: N/ m2).The total stress on the boundary is set equal to a given stress F:  – p I +     u +   u  T  – 2---     u  I  n    3  – p I +    u +   u  T   n

=

=

F

F

using the compressible and the incompressible formulation respectively. This boundary condition implicitly sets a constraint on the pressure that for 2D flows is p

=

 un 2  ---------- – n  F n

(6-1)

If unn is small, Equation 6-1 states that p  n·F. Also enter the turbulent flow settings as described in More Boundary Condition Settings for the Turbulent Flow User Interfaces. Normal Stress, Normal Flow

If Normal stress, normal flow is selected, enter the magnitude of the Normal stress f 0 (SI unit: N/m2). Also enter the turbulent flow settings as described in More Boundary Condition Settings for the Turbulent Flow User Interfaces . In addition to the stress condition set in the Normal Stress condition, the Normal stress, normal flow condition also prescribes that there must be no tangential velocities on the boundary:  – p I +     u +   u  T  – 2---     u  I  n    3  – p I +    u +   u  T   n

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= – f 0 n

= – f 0 n

,

,

tu

tu

=

0

=

0

using the compressible and the incompressible formulation respectively. This boundary condition also implicitly sets a constraint on the pressure that for 2D flows is p

=

 un 2  ---------- + f 0 n

(6-2)

If unn is small, Equation 6-2 states that p  f 0. CONSTRAINT SETTINGS

To display this section, click the Show button ( ) and select Advanced Physics Options. Select the Use weak constraints check box to use weak constraints and create dependent variables for the corresponding Lagrange multipliers. If Normal Stress, Normal Flow is selected as the Boundary condition, then to Apply reaction terms on all dependent variables, select All physics (symmetric). Or select Individual dependent variables to restrict the reaction terms as required. Periodic Flow Condition

The Periodic Flow Condition splits its selection in two groups: one source group and one destination group. Fluid that leaves the domain through one of the destination boundaries enters the domain over the corresponding source boundary. This corresponds to a situation where the geometry is a periodic part of a larger geometry. If the boundaries are not parallel to each other, the velocity vector is automatically transformed. If the boundaries are curved, it is recommended to only include two boundaries.

No input is required when Compressible flow (Ma<0.3) is selected as the Compressibility under the Physical Model section for the interface. Typically when a periodic boundary condition is used with a compressible flow the pressure is the same at both boundaries and the flow is driven by a volume force.


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PRESSURE DIFFERENCE

When Incompressible flow is selected as the Compressibility under the Physical Model section for the interface, this section is available. Enter a value or expression for the pressure difference, psrc  pdst (SI unit: Pa). This pressure difference can, for example, drive the flow in a fully developed channel flow. The default is 0 Pa. To set up a periodic boundary condition select both boundaries in the Periodic Flow Condition node. COMSOL automatically assigns one boundary as the source and the other as the destination. To manually set the destination selection, add a Destination Selection node to the Periodic Flow Condition node. All destination sides must be connected. CONSTRAINT SETTINGS

To display this section, click the Show button ( ) and select Advanced Physics Options . Select the Use weak constraints check box to use weak constraints and create dependent variables for the corresponding Lagrange multipliers. Periodic Boundary Conditions in the COMSOL Multiphysics Reference Manual

Fan

Use the Fan node to define the flow direction (inlet or outlet) and the fan parameters on exterior boundaries. Use the Interior Fan node for interior boundaries.

This node is not available for the Turbulent Flow interfaces.

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FLOW DIRECTION

Select a Flow direction—Inlet or Outlet. After a boundary is selected, an arrow displays in the Graphics window to indicate the selected flow direction. To update the arrow if the selection changes, click any node in the Model Builder and then click the Fan node again to update the Graphics window. PARAMETERS

When Inlet is selected as the Flow direction, enter the Input pressure pinput (SI unit: Pa) to define the pressure at the fan input. The default is 0 Pa. When Outlet is selected as the Flow direction, enter the Exit pressure pexit (SI unit: Pa) to define the pressure at the fan outlet. The default is 0 Pa. For either flow direction, select a Static pressure curve to specify a fan curve—Linear (the default), Static pressure curve data, or User defined. Linear

For both Inlet and Outlet flow directions, if Linear is selected, enter values or expressions for the Static pressure at no flow pnf (SI unit: Pa) (the default is 100 Pa) and the Free delivery flow rate V 0,fd (SI unit: m3/s) (the default is 0.01 m 3/s). The static pressure curve is equal to the static pressure at no flow rate when V 0  0 and equal to 0 when the flow rate is larger than the free delivery flow rate. User Defined

Select User defined to enter a different value or expression for the Static pressure curve. The flow rate across the selection where this boundar y condition is applied is defined by phys_id .V0 where phys_id is the physics interface identifier (for example, phys_id is spf by default for laminar single-phase flow). In order to avoid unexpected behavior, the function used for the fan curve is the maximum between the user defined function and 0. Static Pressure Curve Data

Select Static pressure curve data to enter or load data under the Static Pressure Curve Data section that displays. The interpolation between points given in the table is


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defined using the Interpolation function type list in the Static Pressure Curve Interpolation section. STATIC PRESSURE CURVE DATA

This section is available when Static pressure curve data is selected as the Static pressure curve. In the table, enter values or expressions the Flow rate and Static pressure curve (or click the Load from File button ( ) under the table to import a text file). STATIC PRESSURE CURVE INTERPOLATION

This section is available when Static pressure curve data is selected as the Static pressure curve. Select the Interpolation function type—Linear (the default), Piecewise cubic, or Cubic spline. The extrapolation method is always a constant value. In order to avoid problems with an undefined function, the function used for the boundary condition is the maximum between the interpolated function and 0. Then specify the Units for the Flow rate and the Static pressure curve. UNITS

This section is available when Static pressure curve data is selected as the Static pressure 3 curve. Select Units for the Flow rate (the default SI unit is m /s) and Static pressure curve (the default SI unit is Pa).

Theory for the Fan and Grille Boundary Conditions

Interior Fan

The Interior Fan node represents interior boundaries where a fan condition is set using the fan pressure curve to avoid an explicit representation of the fan. The Interior Fan defines a boundary condition on the slit. That means that the pressure and the velocity can be discontinuous across this boundary. One side represents a flow inlet; the other side represents the fan outlet. The fan boundary condition ensures that the mass flow rate is conserved between its inlet and outlet:

242 |




inlet

u  n +



u  n

=

0

outlet

This boundary condition acts like a Pressure, No Viscous Stress boundary condition on each side of the fan. The pressure at the fan outlet is fixed so that the mass flow rate is conserved. On the fan inlet the pressure is set to the pressure at the fan outlet minus the pressure drop due to the fan. The pressure drop due to the fan is defined by the static pressure curve, which is usually a function of the flow rate. To define a fan boundary condition on an exterior boundary, use the Fan node instead.

This node is not available for the Turbulent Flow interfaces.

BOUNDARY SELECTION

From the Selection list, choose the boundaries to define. INTERIOR FAN

Define the Flow direction by selecting Along normal vector (the default) or Opposite to normal vector . This defines which side of the boundary is considered the fan’s inlet and outlet. After a boundary is selected, an arrow displays in the Graphics window to indicate the selected flow direction. To update the arrow if the selection changes, click any node in the Model Builder and then click the Interior fan node again to update the Graphics window.

The rest of the settings for this section are the same as for the Fan node. See Linear, Static Pressure Curve Data, and User Defined for details.

Theory for the Fan Defined on an Interior Boundary


| 243

Interior Wall

The Interior Wall boundary condition includes a set of boundar y conditions describing the fluid flow condition at an interior wall. It is similar to the Wall boundary condition available on exterior boundaries except that it applies on both sides of an internal boundar y. It allows discontinuities (velocity, pressure, or turbulence) across the boundary. Use the Interior Wall boundary condition to avoid meshing thin structures by using no-slip conditions on interior curves and surfaces instead. You can also prescribe slip conditions and conditions for a moving wall. The Interior Wall boundary condition is only available for single-phase flow. It is compatible with laminar and turbulent flows. BOUNDARY SELECTION

From the Selection list, choose the boundaries to define. The interior wall condition can only be applied to interior boundaries. BOUNDARY CONDITION

Select a Boundary condition—No slip (the default), Slip, or Moving wall. No Slip

is the default boundary condition for a stationary solid wall. The condition prescribes u = 0 on both sides of the boundary; that is, the fluid at the wall is not moving. No slip

Slip

The Slip condition prescribes a no-penetration condition, u·n0. It hence implicitly assumes that there are no viscous effects at both sides of the slip wall and hence, no boundary layer develops. From a modeling point of view, this can be a reasonable approximation if the important effect of the wall is to prevent fluid from leaving the domain.

244 |


Moving Mov ing Wall

If the wall moves, so must the fluid on both sides of the wall. Hence, this boundary condition prescribes u = uw. Enter the components of the Velocity of moving wall uw (SI unit: m/s). Specifying this boundary condition does not automatically cause the associated wall to move. An additional Moving Mesh interface needs to be added to physically track the wall movement in the spatial reference r eference frame. In the COMSOL Multiphysics Reference Manual: • •

Slip Moving Mesh User Interface

CONSTRAINT SETTINGS

To display this section, click the Show but button ( ) and se select Advanced Physics Options. For the No slip and Moving wall boundary conditions, and to Apply reaction terms on all dependent variables, select All physics (symmetric). Otherwise, select Current physics (internally symmetric) or Individual dependent variables to restrict the reaction terms as required. Grille

The Grille node models the pressure drop caused by having a grille that covers the inlet or outlet. This node is not available for the Turbulent Flow interfaces. See Fan Fan for for all of the settings for the Laminar Flow interface, except for Quadratic loss, which is described here. BOUNDARY SELECTION

From the Selection list, choose the boundaries to define. PARAMETERS

If Quadratic loss is selected as the Static pressure curve, enter the Quadratic loss 7 7 coefficient to define qlc (SI unit: kg/m ). The default value is 0 kg/m . qlc defines the


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static pressure curve that is a piecewise quadratic function equal to 0 when flow rate is < 0, equal to V 02qlc when flow rate is > 0.


Flow Continuity

The Flow Continuity node is suitable for pairs where the boundaries match; it prescribes that the flow field is continuous across the pair. A Wall A Wall subnode subnode is added by default and it applies to the parts of the pair boundaries where a source boundary lacks a corresponding destination boundary and vice versa. The Wall feature can be overridden by any other boundar y condition that applies to exterior boundaries. Right-click the Flow Continuity node to add additional subnodes. In the COMSOL Multiphysics Reference Manual: • •

Continuity on Interior Boundaries Identity and Contact Pairs

Pressuree Point Constraint Pressur

The Pressure Point Constraint node adds a pressure constraint at a point. If it is not possible to specify the pressure level using a boundar y condition, the pressure must be set in some other way, for example, by specifying a fixed pressure pr essure at a point. PRESSURE CONSTRAINT

Enter a point constraint for the Pressure p0 (SI unit: Pa). The default is 0 Pa. CONSTRAINT SETTINGS

To display this section, click the Show but button ( ) and select Advanced Physics Options . To Apply reaction terms on all dependent variables, select All physics (symmetric). Or select Individual dependent variables to restrict the reaction terms as required. r equired. Select the Use weak constraints check box to replace the standard constraints with a weak implementation.

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More Boundary Condition Settings for the Turbulent Flow User Interfaces

For the Inlet, Open Boundary Boundar y, and Boundary Boundar y Stress features, the following settings are also required for the Turbulent Flow interfaces. The first sections (Turbulent ( Turbulent Intensity and Turbulence Length Scale Parameters and Parameters and Boundary Stress Turbulent Boundary Type)) provide further information about the boundary conditions, and the additional Type settings information is described under Boundary Condition. Condition. Turbulent Intensity and Turbulence Length Scale Parameters The Turbulent intensity I T and Turbulence length scale LT values are related to the

turbulence variables via the following equations, Equation 6-3 for 6-3 for the Inlet and Equation 6-4 for 6-4 for the Open Boundary: Inlet k

Open Boundary k

=

=

2 3 --  U I T  , 2

2 3 --  I T U ref  , 2



=

3  4 k 3 / 2

C

---------- L T

3  4



=

(6-3) 3 2 -2-

C   3  I T U ref   ----------- ----------------------------  L T  2 

(6-4)

For the Open Boundary and Boundary Stress options, and with any turbulent flow interface, inlet conditions for the turbulence variables also need to be specified. These conditions are used on the parts of the boundary where u·n  0, that is, where flow enters the computational domain. Boundary Stress Turbulent Boundary Type

For Boundary Stress, first select a Turbulent boundary type to apply to the turbulence variables—Open boundary (the default), Inlet, or Outlet. • •

•

If Open boundary is selected, then expect parts of the boundary to be an outlet and parts of the boundary to be an inlet. Select Inlet when it is expected that the whole boundary boundar y is an inlet. Under Exterior turbulence, the same options to specify turbulence variables are available for the Open boundary option is available. The difference is that, for the Inlet option, is applied to the whole boundar y. Select Outlet when it is expected that the whole boundary is an outflow. Homogeneous Neumann conditions are applied to the turbulence variables (that is, for k and ) k  n

=

0

  n

=

0


| 247

BOUNDARY CONDITION

For Open Boundary and Boundary Stress>Open boundary, the following is under the Exterior turbulence subsection. For Boundary Stress, first select a Turbulent boundary type to apply to the turbulence variables—Open boundary (the default), Inlet, or Outlet. Then for Open boundary and Inlet continue entering the following parameters. Select the Specify turbulence length scale and intensity button (the default) to enter values or expressions for the: • Turbulent intensity I T (dimensionless) • Turbulence length scale LT (SI unit: m) • Reference velocity scale U ref (SI unit: m/s). This is available for most options

(excluding Velocity Velocity for the Inlet node). If the Specify turbulence variables button is selected, enter values or expressions for the: • Turbulent kinetic energy k0 (SI unit: m 2/s2) • Turbulent dissipation rate,  0 (SI unit: m 2/s3)

The default values are different for Inlet, Open Boundar y, and Boundary Boundar y Stress. See Table 6-1. 6-1. Also, for recommendations of physically sound values see Inlet Values for the Turbulence Length Scale and Turbulent Intensity . TABLE 6-1: DEFAULT VALUES FOR THE TURBULEN T INTERFACES NAME AND UNIT

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VARIABLE

INLET

OPEN BOUNDARY

BOUNDARY STRESS

Turbulent intensity (dimensionless)

I T

0.05

0.005

0.01

Turbulence length scale (m)

LT

0.01

0.1

0.1

Reference velocity scale (m/s)

U ref

1

1

1


TABLE 6-1: DEFAULT VALUES FOR THE TURBULENT TURBULENT INTERF ACES NAME AND UNIT

VARIABLE

INLET

OPEN BOUNDARY

BOUNDARY STRESS

Turbulent kinetic energy (m2/s2)

k0

0.005

2.5 x 10-3

1 x 10-2

Turbulent dissipation rate (m2/s3)

 0

0.005

1.1 x 10-4

1 x 10-3


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Theory for the Laminar Flow User Interface For the basic laminar flow theory, see Theory of Laminar Flow in in the COMSOL Multiphysics Reference Manual. This section discusses the theory related to the advanced features available with this module and for laminar flow. Also see Theory for the Turbulent Flow User Interfaces. Interfaces . In this section: • • • • •

Theory for the Inlet Boundary Condition Additional Theory for the Outlet Boundary Condition Additional Theory for the Outlet Boundary Condition Theory for the Fan and Grille Boundary Conditions Non-Newtonian Flow: The Power Law and the Carreau Model

Theory for the Inlet Boundary Condition LAMINAR INFLOW

In order to prescribe an inlet velocity profile, this boundary condition adds a weak form contribution corresponding to one-dimensional Navier-Stokes Navier-Stokes equations projected on the boundary boundar y. The applied condition corresponds to the situation shown in Figure 6-1: 6-1: a fictitious domain of length Lentr is assumed to be attached to the inlet of the computational domain. This boundary condition uses the assumption that flow in this fictitious domain is fully ful ly develop dev eloped ed lamina lam inarr flow f low . The “wall” boundary conditions for the fictitious domain is inherited from the real domain, , unless the

250 |


option to constrains outer edges or endpoints to zero is selected in which case the fictitious “walls” will be no-slip walls. pentr

 Lentr

Figure 6-1: An example of the physical situation simulated when using the Laminar inflow boundary condition.  is the actual computational domain while the dashed domain is a fictitious domain.

If an average inlet velocity or inlet volume flow is specified instead of the pressure, COMSOL Multiphysics adds an ODE that calculates a pressure, pentr, such that the desired inlet velocity or volume flow is obtained. Also see Inlet for the node settings. In the COMSOL Multiphysics Reference Manual : •

• • •

Prescribing Inlet and Outlet Conditions Theory for the Pressure, No Viscous Stress Boundary Condition Theory for the Normal Stress Boundary Condition

Additional Theor y for the Outlet Boundary Condition LAMINAR OUTFLOW

In order to prescribe an outlet velocity profile, this boundary condition adds a weak form contribution corresponding to one-dimensional Navier-Stokes equations projected on the boundary. The applied condition corresponds to the situation shown in Figure 6-2: assume that a fictitious domain of length Lexit is attached to the outlet of the computational domain. This boundary condition uses the assumption that flow in this fictitious domain is fully developed laminar flow . The “wall” boundary conditions for the fictitious domain is inherited from the real domain, , unless the

THEORY FOR THE LAMINAR FLOW USER INTERFACE

| 251

option to constrains outer edges or endpoints to zero is selected in which case the fictitious “walls” will be no-slip walls. 

pexit Lexit

Figure 6-2: An example of the physical situation simulated when using Laminar outflow boundary condition.  is the actual computational domain while the dashed domain is a fictitious domain.

If the average outlet velocity or outlet volume flow is specified instead of the pressure, the software adds an ODE that calculates pexit such that the desired outlet velocity or volume flow is obtained. NO VISCOUS STRESS

For this module, and in addition to the Theory for the Pressure, No Viscous Stress Boundary Condition (described in the COMSOL Multiphysics Reference Manual ), the Viscous stress condition sets the viscous stress to zero:     u +   u  T  – 2---     u  I n   3     u +   u  T   n

=

=

0

0

using the compressible and the incompressible formulation, respectively. The condition is not a sufficient outlet condition since it lacks information about the outlet pressure. It must hence be combined with at pressure point constraints on one or several points or lines surrounding the outlet.

252 |


This boundary condition is numerically the least stable outlet condition, but can still be beneficial if the outlet pressure is nonconstant due to, for example, a nonlinear volume force. Also see Outlet for the node settings. In the COMSOL Multiphysics Reference Manual : •

• • •

Prescribing Inlet and Outlet Conditions Theory for the Pressure, No Viscous Stress Boundary Condition Theory for the Normal Stress Boundary Condition

Theory for the Fan Defined on an Interior Boundary

In this case, the inlet and outlet of the device are both interior boundaries (see Figure 6-3). The boundaries are called dev_in and dev_out. The boundary conditions are described as follows: •

•

The inlet of the device is an outlet boundar y condition for the modeled domain. For this outlet boundary condition, on dev_in, the value of the pressure variable is set to the sum of the mean value of the pressure on dev_out and the pressure drop across the device. The pressure drop is calculated from a lumped curve using the flow rate evaluated on dev_in. For the inlet boundary condition, on dev_out, the pressure value is set so that the flow rate is equal on dev_in and dev_out. An ODE is added to compute the pressure value. In both cases, the boundary condition implementation specifies vanishing viscous stress along with a Dirichlet condition on the pressure.

See Interior Fan for node settings.


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Figure 6-3: A device between two boundaries. The red arrows represent the flow direction, the cylindrical part represents the device (that should be not be part of the model), and the two cubes are the domain that are modeled with a particular inlet boundary condition to account for the device.


Fans, pumps, or grilles (devices) can be represented using lumped curves implemented as boundary conditions. These simplifications also imply some assumptions. In particular, it is assumed that a given boundary can only be either an inlet or an outlet. Such a boundary should not be a mix of inlets/outlets, nor should it change during a simulation. Manufacturers usually provide curves that describe the static pressure as a function of flow rate for a fan.

See Fan and Grille for node settings.

DEFINING A DEVICE AT AN INLET

In this case, the device’s inlet is an external boundary, represented by the external circular boundary of the green domain on Figure 6-4. The device’s outlet is an interior face situated between the green and blue domains in Figure 6-4. The lumped curve gives the flow rate as a function of the pressure difference between the external

254 |


boundary and the interior face. This boundary condition implementation specifies vanishing viscous stress along with a Dirichlet condition on the pressure. The Fan boundary condition sets the following conditions:     u +   u  T  – 2---     u  I n   3 T

u +  u n

=

=

0,

0,

p

p

=

=

pinput + p fan  V 0 

pinput + p fan  V 0 

(6-5)

(6-6)

The Grille boundary condition sets the following conditions:     u +   u  T  – 2---     u  I n   3 T

u +  u n

=

0,

=

0,

p

p

=

=

pinput – p grill  V 0 

pinput – pgrill  V 0 

(6-7)

(6-8)

where V 0 is the flow rate across the boundary, pinput is the pressure at the device’s inlet, and pfanV 0) and pgrille(V 0) are the static pressure functions of flow rate for the fan and the grille. Equation 6-5 and Equation 6-7 correspond to the compressible formulation. Equation 6-6 and Equation 6-8 correspond to the incompressible formulation. In 2D the thickness in the third direction, Dz, is used to define the flow rate. Fans are modeled as rectangles in this case.


| 255

Figure 6-4: A device at the inlet. The arrow represents the flow direction, the green circle represents the device (that should not be part of the model), and the blue cube represents the modeled domain with an inlet boundary condition described by a lumped curve for the attached device. DEFINING A DEVICE AT AN OUTLET

In this case (see Figure 6-5), the fan’s inlet is the interior face situated between the blue (cube) and green (circle) domain while its outlet is an external boundary, here the circular boundary of the green domain. The lumped curve gives the flow rate as a function of the pressure difference between the interior face and the external boundary. This boundary condition implementation specifies vanishing viscous stress along with a Dirichlet condition on the pressure. The Fan boundary condition sets the following conditions:     u +   u  T  – 2---     u  I n   3 T

 u +  u   n

=

0,

0,

=

p

=

p

=

pex t – p fa n  V 0 

pex t – pfa n  V 0 

(6-9)

(6-10)

The Grille boundary condition sets the following conditions:     u +   u  T  – 2---     u  I n   3    u +   u  T  n

256 |

=


0,

=

0,

p

p

=

=

pinput + p grill  V 0 

pinput + p grill  V 0 

(6-11)

(6-12)

where V 0 is the flow rate across the boundary, pext is the pressure at the device outlet, and pfan(V 0), p vacuum pump(V 0), and pgrille(V 0) are the static pressure function of flow rate for the fan, the vacuum pump, and the grille. Equation 6-9, Equation 6-10, and Equation 6-11 correspond to the compressible formulation. Equation 6-10, Equation 6-11, and Equation 6-12 correspond to the incompressible formulation. In 2D the thickness in the third direction, Dz, is used to define the flow rate. Fans are modeled as rectangles in this case.

Figure 6-5: A fan at the outlet. The arrow represents the flow direction, the green circle represents the fan (that should not be part of the model), and the blue cube represents the modeled domain with an outlet boundary condition described by a lumped curve for the attached fan.

Non-Newtonian Flow: The Power Law and the Carreau Model

The viscous stress tensor is directly dependent on the shear rate tensor and can be written as: 

=

·

–



2 ---     u  I 3 =

·


| 257

using the compressible and incompressible formulation, respectively. Here the engineering strain-rate tensor defined by: ·

=

·

denotes

  u +   u  T 

Its magnitude, the shear rate, is: ·

·

=

=

1· · --- : 2

where the contraction operator “:” is defined by a:b

=

a

nm b nm

n m

For a non-Newtonian fluid , the dynamic viscosity is assumed to be a function of the shear rate: 

=

  · 

The Laminar Flow interfaces have the following predefined models to prescribe a non-Newtonian viscosity—the power law and the Carreau model. POWER LAW

The power law is an example of a generalized Newtonian model. It prescribes 

=

· m n – 1

(6-13)

where m and n are scalars that can be set to arbitrary values. For n  1, the power law describes a shear thickening (dilatant) fluid. For n  1, it describes a shear thinning (pseudoplastic) fluid. A value of n equal to one gives the expression for a Newtonian fluid . Equation 6-13 predicts infinite viscosity at zero shear rate. This is however never the case physically. Instead, most fluids have constant viscosity for shear rates smaller than 10-2 s-1 (Ref. 1). Since infinite viscosity also makes models using Equation 6-13 difficult to solve, COMSOL implements the power law as 

=

· · m max   min  n – 1

(6-14)

where · min is a lower limit for the evaluation of the shear rate magnitude. The default value for · min is 10-2 s-1, but it can be changed in the equation view.

258 |


CARREAU MODEL

The Carreau expression gives the viscosity by the following four-parameter equation 

=

n – 1 · 2 ----------------  1 +     2

 +   0 –  inf

(6-15)

where  is a parameter with units of time, 0 is the zero shear rate viscosity, inf is the infinite shear rate viscosity, and n is a dimensionless parameter. This expression is able to describe viscosity for mostly stationary polymer flow.


| 259

Theory for the Turbulent Flow User Interfaces The Single-Phase Flow, Turbulent Flow interfaces theory is described in this section: • • • • • • • •

Turbulence Modeling The k- Turbulence Model The Low Reynolds Number k- Turbulence Model Inlet Values for the Turbulence Length Scale and Turbulent Intensity Theory for the Pressure, No Viscous Stress Boundary Condition Solvers for Turbulent Flow Pseudo Time Stepping for Turbulent Flow Models References for the Single-Phase Flow, User Interfaces

Theory for the Laminar Flow User Interface

Turbulence Modeling

Turbulence is a property of the flow field and it is mainly characterized by a wide range of flow scales: the largest occurring scales, which depend on the geometry, the smallest quickly fluctuating scales, and all the scales in between. The tendency for an isothermal flow to become turbulent is measured by the Reynolds number Re

=

-----------UL 

(6-16)

where  is the dynamic viscosity,  the density, and U and L are velocity and length scales of the flow, respectively. Flows with high Reynolds numbers tend to become turbulent and this is the case for most engineering applications. The Navier-Stokes equations can be used for turbulent flow simulations, although this would require a large number of elements to capture the wide range of scales in the flow. An alternative approach is to divide the flow in large resolved scales and small unresolved scales. The small scales are then modeled using a turbulence model with

260 |


the goal that the model is numerically less expensive than resolving all present scales. Different turbulence models invoke different assumptions on the unresolved scales resulting in different degree of accuracy for different flow cases. This module includes Reynolds-averaged Navier-Stokes (RANS) models which is the model type most commonly used for industrial flow applications. REYNOLDS-AVERAGED NAVIER-STOKES (RANS) EQUATIONS

The information below assumes that the flow fluid is incompressible and Newtonian in which case the Navier-Stokes equations take the form: 

u + u    u t

=

T

  – pI +  u +  u  + F   u

=

(6-17)

0

Once the flow has become turbulent, all quantities fluctuate in time and space. It is seldom worth the extreme computational cost to obtain detailed information about the fluctuations. An averaged representation often provides suf ficient information about the flow. The Reynolds-averaged representation of turbulent flows divides the flow quantities into an averaged value and a fluctuating part, 

=

 + 

where  can represent any scalar quantity of the flow. In general, the mean value can vary in space and time. This is exemplified in Figure 6-6, which shows time averaging of one component of the velocity vector for nonstationary turbulence. The unfiltered flow has a time scale t1. After a time filter with width t2  t1 has been applied, there is a fluctuating part, ui, and an average part, U i. Because the flow field also varies

T H E O R Y F O R T H E TU R B U L E N T F L O W U S E R I N T E R F A C E S

| 261

on a time scale longer than t2, U i is still time dependent but is much smoother than the unfiltered velocity ui.

Figure 6-6: The unfiltered velocity component u i, with a time scale t1, and the averaged velocity component, U i, with time scale t2.

Decomposition of flow fields into an averaged part and a fluctuating part, followed by insertion into the Navier-Stokes equation, then averaging, gives the Reynolds-averaged Navier-Stokes (RANS) equations:  U +  U   U +     u'  u'  = –  P +      U +   U  T  + F t   U = 0

(6-18)

where U is the averaged velocity field and  is the outer vector product. A comparison with Equation 6-17 indicates that the only difference is the appearance of the last term on the left-hand side of Equation 6-18. This term represents interaction between the fluctuating velocities and is called the Reynolds stress tensor. This means that to obtain the mean flow characteristics, information about the small-scale structure of the flow is needed. In this case, that information is the correlation between fluctuations in different directions. EDDY VISCOSITY

The most common way to model turbulence is to assume that the turbulence is of a purely diffusive nature. The deviating part of the Reynolds stress is then expressed by    u'  u'  – --- trace  u'  u'  I 3

262 |


   U +   U  T 

= – T

where T is the eddy viscosity , also known as the turbulent viscosity. The spherical part can be written --3

trace  u'  u'  I

=

2 ---  k 3

where k is the turbulent kinetic energy. In simulations of incompressible flows, this term is included in the pressure, but when the absolute pressure level is of impor tance (in compressible flows, for example) this term must be explicitly included. TURBULENT COMPRESSIBLE FLOW

If the Reynolds average is applied to the compressible form of the Navier-Stokes, terms of the form  u 

appear and need to be modeled. To avoid this, a density-based average, known as the Favre average, is introduced: t + T

u˜i

=

1 1 --- lim --- T  T



 (x,) u i (x,) d

(6-19)

t

It follows from Equation 6-19 that  u˜ i

=

 ui

(6-20)

and a variable, ui, is decomposed in a mass-averaged component, u˜ i , and a fluctuating component, ui, according to ui

=

˜ +u  u i i

(6-21)

Using Equation 6-20 and Equation 6-21 along with some modeling assumption for compressible flows (Ref. 8), Equation 6-19 can be written in the form   ˜  ------ + -------   u i  t  xi  u˜ i u˜ i ˜ ------- +  u j -------t  x j

=

=

0

 u˜ k    ----u-˜---i ----u-˜-- j- 2--- -------- p ------------  ij –  u j  u i  + F i – + + –     x i  x j    x j  xi  3  xk 

(6-22)

The Favre-averaged Reynolds stress tensor is modeled using the same argument as for incompressible flows:


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–

 u j  u i 

=

 u˜ i u˜ j 2  u˜ k   T  -------- + -------- – ---   T --------- +  k  ij   x j  xi 3   xk 

where k is the turbulent kinetic energy. Comparing Equation 6-22 to its incompressible counterpart (Equation 6-18), it can be seen that except for the term –

 2  3  k ij

the compressible and incompressible formulations are exactly the same, except that the free variables are u˜ i instead of U i

=

ui

More information about modeling turbulent compressible flows is in Ref. 2 and Ref. 8. The turbulent transport equations are used in their fully compressible formulations (Ref. 9). The k-  Turbulence Model

The k- model is one of the most used turbulence models for industrial applications. This module includes the standard k- model (Ref. 2). This introduces two additional transport equations and two dependent variables: the turbulent kinetic energy, k, and the dissipation rate of turbulence energy, . Turbulent viscosity is modeled by 2

 T =  C k-----

(6-23)

where C is a model constant. The transport equation for k reads:    ----k-- +  u  k =      + ------T k + Pk –  t k

(6-24)

where the production term is P k

=

 T  u :  u +  u  T  – 2---    u  2 – 2---  k   u

The transport equation for  reads:

264 |


3

3

(6-25)

   ----- +  u   =      + ------T  t 

+



2

C  1 --- P – C  2  ----k k k

(6-26)

The model constants in Equation 6-23, Equation 6-24, and Equation 6-26 are determined from experimental data (Ref. 2) and the values are listed in Table 6-2. TABLE 6-2: MODEL CONSTANTS CONSTANT

VALUE

C C1

0.09

C2

1.92

k 

1.0

1.44

1.3

MIXING LENGTH LIMIT

Equation 6-24 and Equation 6-26 cannot be implemented directly as written. There is, for example, nothing that prevents division by zero. The equations are instead implemented as suggested in Ref. 10. The implementation includes an upper limit on lim : the mixing length, l mix l mix

=

3/2 lim  max  C k-----------  lmix  

(6-27)

lim should not be The mixing length is used to calculated the turbulent viscosity. l mix active in a converged solution but is merely a tool to obtain convergence. REALIZABILITY CONSTRAINTS

The eddy-viscosity model of the Reynolds stress tensor can be written  ui u j

= –

2  T S ij

+

2 ---  k  ij 3

where ij is the Kronecker delta and Sij is the strain-rate tensor. The diagonal elements of the Reynolds stress tensor must be nonnegative, but calculating T from Equation 6-23 does not guarantee this. To assert that  u i ui  0 i

the turbulent viscosity is subjected to a realizability constraint. The constraint for 2D and 2D axisymmetry is:


| 265

k 2  T  -----------------------

(6-28)

3 S ij S ij

and for 3D and 2D axisymmetry with swirl flow it reads: k  T  -------------------------- 6 S ij Sij

(6-29)

Swirl flow is not available with the Heat Transfer Module. Combining equation Equation 6-28 with Equation 6-23 and the definition of the mixing length gives a limit on the mixing length scale: l mix 

k 2 --------------------3 S S ij

(6-30)

ij

Equivalently, combining Equation 6-29 with Equation 6-23 and Equation 6-27 gives: 1 k l mix  ------- ------------------6 Sij S ij

(6-31)

This means there are two limitations on lmix: the realizability constraint and the imposed limit via Equation 6-27. The effect of not applying a realizability constraint is typically excessive turbulence production. The effect is most clearly visible in stagnation points. To avoid such artifacts, the realizability constraint is always applied for the RANS models. More details can be found in Ref. 5, Ref. 6, and Ref. 7. MODEL LIMITATIONS

The k- turbulence model relies on several assumptions, the most important of which is that the Reynolds number is high enough. It is also important that the turbulence is in equilibrium in boundary layers, which means that production equal dissipation. These assumptions limit the accuracy of the model because they are not always true. It does not, for example, respond correctly to flows with adverse pressure gradients that can result in under-predicting the spatial extension of recirculation zones (Ref. 2). Furthermore, in the description of rotating flows, the model often shows poor agreement with experimental data (Ref. 3). In most cases, the limited accuracy is a fair

266 |


trade-off for the amount of computational resources saved compared to more complicated turbulence models. WALL FUNCTIONS

The flow close to a solid wall is for a turbulent flow and is ver y different compared to the free stream. This means that the assumptions used to derive the k- model are not valid close to walls. While it is possible to modify the k- model so that it describes the flow in wall regions (see The Low Reynolds Number k- Turbulence Model), this is not always desirable because of the very high resolution requirements that follow. Instead, analytical expressions are used to describe the flow at the walls. These expressions are known as wall functions. The wall functions in COMSOL Multiphysics are such that the computational domain is assumed to start a distance w from the wall (see Figure 6-7).

Mesh cells

w Solid wall

Figure 6-7: The computational domain starts a distance w from the wall for wall functions.

The distance w is automatically computed so that + =  u     w  w

where uC1/4k is the friction velocity, which becomes 11.06. This corresponds to the distance from the wall where the logarithmic layer meets the viscous sublayer (or to some extent would meet if there was not a buffer layer in between). w is limited from below so that it never becomes smaller than half of the height of the boundar y


| 267

+ can become higher than 11.06 if the mesh is relatively mesh cell. This means that  w coarse.

Always investigate the solution to check that w is small compared to the + is 11.06 on most of the dimension of the geometry. Also check that  w walls. + is much higher over a significant part of the walls, the accuracy If  w might become compromised. Both the wall lift-off, w, and the wall + , are available as results and analysis variables. lift-off in viscous units,  w

The boundary conditions for the velocities are a no-penetration condition u  n = 0 and a shear stress condition n   –  n    nn

= –

-----max  C 1 / 4 k u    u --u

u

where =

  u +  u  T 

is the viscous stress tensor and u

=

u ----------------------------1 ----- ln  w + B  v

where in turn,  v, is the von Kárman constant (default value 0.41) and B is a constant that by default is set to 5.2. The turbulent kinetic energy is subject to a homogeneous Neumann condition n  k = 0 and the boundary condition for  reads: 

=

C 3 / 4 k 3 / 2 ---------------------- v  w

See Ref. 10 and Ref. 11 for further details. INITIAL VALUES

The default initial values for a stationary simulation are ( Ref. 10),

268 |


k

=

u=0 p = 0 10    2  ------------------------------   0 . 1  l lim  mix

=

3/2 C  kinit ----------------------lim 0 . 1  l mix

lim is the mixing length limit. For time dependent simulations, the initial value where l mix for k is instead

k

=

  ------------------------------2    0 . 1  l lim  mix

SCALING FOR TIME-DEPENDENT SIMULATIONS

The k- equations are derived under the assumption that the flow has a high enough Reynolds number. If this assumption is not fulfilled, both k and  will have very small magnitudes and chaotically in the manner that the relative values of k and  can change relatively much just because of small changes in the flow field. A time-dependent simulation of a turbulent flow can include a period when the flow is not fully turbulent. A typical example is the star tup phase when for example an inlet velocity or a pressure difference is gradually increased. To sort out numerical fluctuations in k and  during such periods, the default time-dependent solver for the k- model employs unscaled absolute tolerances for k and . The tolerances are set to k scale

=

 0 . 01U scale  2

3 / 2   L scale = 0 . 09ksclae fact  l bb min 

(6-32)

where U scale and Lfact are input parameters available in the Advanced section of the physics interface node. Their default values are 1 ms and 0.035 respectively. lbb,min is the shortest side of the geometry bounding box. Equation 6-32 is closely related to the expressions for k and  on inlet boundaries (see Equation 6-37). The practical implication of Equation 6-32 is that variations in k and  smaller than kscale and scale respectively, will be regarded will be regarded as numerical noise.


| 269

The Low Reynolds Number k-  Turbulence Model

When the accuracy provided by wall functions in the k- model is not enough, a so called low Reynolds number model can be used. “Low Reynolds number” refers to the region close to the wall where the viscous effects dominate. Most low Reynolds number k- models adapt the turbulence transport equations by introducing damping functions. This module includes the AKN model (after the inventors Abe, Kondoh, and Nagano; Ref. 12). The AKN k- model for compressible flows reads (Ref. 9 and Ref. 12):       + ------T k + Pk –  k 2   ----- +  u   =      + ------T  + C 1 --- P – f  C  2  ----t  k k k   ----k-- +  u  k t

=

(6-33)

where P k

=

 T  u :  u +  u  T  – 2---    u  2 – 2---  k   u 3

3

2

 T =  f  C k----- 5 – Rt  200   * 2 f  =  1 – e –l  14    1 + ----------- R 3 / 4 e 

(6-34)

2

f  l*

=

=

1 – e

t 2  3 . 1  2   1 – 0 . 3 e – Rt  6 . 5  

–l*

  u  l w    Rt =  k 2     u  =      1 / 4

and C1

=

1 .5 C  2

=

1 .9 C 

=

0 . 09  k

=

1 . 4 

=

1.4

(6-35)

Also, lw is the distance to the closest wall. Realizability Constraints are applied to the low Reynolds number k- model. WALL DISTANCE

The wall distance variable, lw, is provided by a mathematical Wall Distance interface that is included when using the low Reynolds number k- model. The solution to the wall distance equation is controlled using the parameter lref . The distance to objects

270 |


larger than lref is represented accurately, while objects smaller than lref will effectively be diminished by appearing to be farther away than they actually are. This is a desirable feature in turbulence modeling since small objects would get too large an impact on the solution if the wall distance were measured exactly. The most convenient way to handle the wall distance variable is to solve for it in a separate study step. A Wall Distance Initialization study type is provided for this purpose and should be added before the actual Stationary or Transient study step. In the COMSOL Multiphysics Reference Manual: • • • •

The Wall Distance User Interface Stationary with Initialization Transient with Initialization Wall Distance Initialization

WALL BOUNDARY CONDITIONS

The damping terms in the equations for k and  allows a no slip condition to be applied to the velocity, that is u0. Since all velocities must disappear on the wall, so must k. Hence, k0 on the wall. The correct wall boundary condition for  is 2        k  n  2

where n is the wall normal direction. That condition is however numerically very unstable. Instead,  is not solved for in the cells adjacent to a solid wall and the analytical relation 

=

 k 2 --- ----2  l w

(6-36)

is prescribed in those cells. Equation 6-36 can be derived as the first term in a series expansion of 2        k  n  2

For the expansion to be a valid, it is required that l c*  0 . 5


| 271

l c* is the distance, measured in viscous units, from the wall to the

center of the wall adjacent cell. The boundary variable Dimensionless distance to cell center is available to ensure that the mesh is fine enough. Observe though that it is unlikely that a solution is obtained at all if l c* » 0 . 5 INLET VALUES FOR THE TURBULENCE LENGTH SCALE AND INTENSITY

The guidelines given in Inlet Values for the Turbulence Length Scale and Turbulent Intensity for selecting turbulence length scale, LT, and the turbulence intensity, IT, apply also to the low-Reynolds number k- model. INITIAL VALUES

The low-Reynolds number k- model has the same default initial guess as the standard lim replaced by l . k- model (see Initial Values) but with l mix ref The default initial value for the wall distance equations (which solves for the reciprocal wall distance) is 2lref . In some cases, specially for stationary solutions, a fast way to convergence is to first solve the model using an ordinary k- model and then use that solution as initial guess for the low-Reynolds number k- model. The procedure is then as follows: Solve the model using the k- model. 2 Switch to the low-Reynolds number k- model. 3 Add a new Stationary with Initialization study. 4 In the Wall Distance Initialization study step, set Values of variables not solved for to Solution from the first study. This is to propagate the solution from the first study down to the second step in the new study. 5 Solve the new study. 1

SCALING FOR TIME-DEPENDENT SIMULATIONS

The low-Reynolds number k- model uses absolute scales of the same type as the k- model (see Scaling for Time-Dependent Simulations). S

272 |

=


2 S ij S ij

 = 2  ij  ij

Inlet Values for the Turbulence Length Scale and Turbulent Intensity

If inlet data for the turbulence variables are not available, crude approximations for k and  can be obtained from the following formulas: k

=

2 3 ---  U I T  2

(6-37)

3  4 k 3 / 2

 = C ---------- L T

where I T is the turbulence intensity and LT is the turbulent length scale. A value of 0.1% is a low turbulence intensity I T. Good wind tunnels can produce values of as low as 0.05%. Fully turbulent flows usually have intensities between five and ten percent. The turbulent length scale LT is a measure of the size of the eddies that are not resolved. For free-stream flows these are typically very small (in the order of centimeters). The length scale cannot be zero, however, because that would imply infinite dissipation. Use Table 6-3 as a guideline when specifying LT (Ref. 4) where lw is the wall distance, and + lw

=

l w  l *

TABLE 6-3: TURBULENT LENGTH SCALES FOR TWO-DIMENSIONAL FLOWS FLOW CASE

LT

L

Mixing layer

0.07 L

Layer width

Plane jet

0.09 L

Jet half width

Wake

0.08 L

Wake width

Axisymmetric jet

0.075 L

Jet half width

Boundary layer ( p   x  0)

 l w  1 – exp  –l w+  26  

– Viscous sublayer and log-layer – Outer layer

0.09 L

Pipes and channels

0.07 L

(fully developed flows)

Boundary layer thickness Pipe radius or channel half width


| 273

Theory for the Pressure, No Viscous Stress Boundary Condition

For this module, and in addition to the Theory for the Pressure, No Viscous Stress Boundary Condition (described in the COMSOL Multiphysics Reference Manual ), the turbulent intensity I T, turbulence length scale LT, and reference velocity scale U ref values are related to the turbulence variables via

k

=

2 3 ---  I T U ref  , 2

3  4



=

3 2 -2

C   3  I T U ref   ----------- ----------------------------  LT  2 

,



=

I T U ref 3 ---------------------------2   0*  1 / 4 L T

For recommendations of physically sound values see Inlet Values for the Turbulence Length Scale and Turbulent Intensity . Also see Inlet and Outlet for the node settings. Solvers for Turbulent Flow

The non-linear system that Navier-Stokes and the turbulence transport equations constitute can become ill-conditioned if solved using a fully coupled solver. Turbulent flows are therefore solved using a segregated approach (Ref. 13): Navier-Stokes in one group and the turbulence transport equations in another. For each iteration in the Navier-Stokes group, two or three iterations are made for the turbulence transport equations. This is necessary to make sure that the very non-linear source terms in the turbulence transport equations are in balance before making another iteration for the Navier-Stokes group. The default stationary solver is solved using pseudo time stepping. This both improves the robustness of the non-linear iterations as well as the condition number for the linear equation system. The latter is specially important for large 3D models where iterative solvers must be applied.

274 |


The default iterative solver for the turbulence transport equations is GMRES accelerated by Geometric Multigrid. The default smoother is SOR Line. Pseudo Time Stepping for Turbulent Flow Models In the COMSOL Multiphysics Reference Manual : • • • •

Multigrid Stationary Solver Iterative SOR Line

Pseudo Time Stepping for Turbulent Flow Models

Pseudo time stepping is by default applied to the turbulence equations for stationary problems, for both 2D and 3D models. The turbulence equations use the same  t˜ as the momentum equations. The default manual expression for CFLloc is, for 2D models: 1 . 3 min niterCMP-1 9  + if niterCMP  25 9  1 . 3 min niterCMP – 25 9  0  +

if niterCMP  50 90  1 . 3 min niterCMP – 50 9  0  and for 3D models: 1 . 3 min niterCMP-1 9  + if niterCMP  30 9  1 . 3 min niterCMP – 30 9  0  +

if niterCMP  60 90  1 . 3 min niterCMP – 60 9  0 


| 275

References for the Single-Phase Flow, User Interfaces 1. R.P. Chhabra and J.F. Richardson, “Non-Newtonian Flow and Applied Rheology”, 2:nd ed, Elsivier, 2008. 2. D.C. Wilcox, Turbulence Modeling for CFD , 2nd ed., DCW Industries, 1998. 3. D.M. Driver and H.L. Seegmiller, “Features of a Reattaching Turbulent Shear Layer in Diverging Channel Flow,” AIAA Journal , vol. 23, pp. 163–171, 1985. 4. H.K. Versteeg and W. Malalasekera, An Introduction to Comp utational Fluid Dynamics , Prentice Hall, 1995. 5. A. Durbin, “On the k- Stagnation Point Anomality,” International Journal of Heat and Fluid Flow , vol. 17, pp. 89–90, 1986. 6. A, Svenningsson, Turbulence Transport Modeling in Gas Turbine Related Applications ,” doctoral dissertation, Department of Applied Mechanics, Chalmers University of Technology, 2006. 7. C. H. Park and S.O. Park, “On the Limiters of Two-equation Turbulence Models,” International Journal of Computational Fluid Dynamics , vol. 19, no. 1, pp. 79– 86, 2005. 8. J. Larsson, Numerical Simulation of Turbulent Flows for Turbine Blade Heat Transfer , doctoral dissertation, Chalmers University of Technology, Sweden, 1998. 9. L. Ignat, D. Pelletier, and F. Ilinca, “A Universal Formulation of Two-equation Models for Adaptive Computation of Turbulent Flows,” Computer Methods in Applied Mechanics and Engineering , vol. 189, pp. 1119–1139, 2000. 10. D. Kuzmin, O. Mierka, and S. Turek, “On the Implementation of the k- Turbulence Model in Incompressible Flow Solvers Based on a Finite Element Discretization,” International Journal of Computing Science and Mathematics , vol. 1, no. 2–4, pp. 193–206, 2007. 11. H. Grotjans and F.R. Menter, “Wall Functions for General Application CFD Codes,” ECCOMAS 98, Proceedings of the Four th European Computational Fluid Dynamics Conference , John Wiley & Sons, pp. 1112–1117, 1998.

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12. K. Abe, T. Kondoh, and Y. Nagano, “A New Turbulence Model for Predicting Fluid Flow and Heat Transfer in Separating and Reattaching Flows—I. Flow Field Calculations,” International Journal of Heat and Mass Transfer , vol. 37, no. 1, pp. 139–151, 1994. 13. M. Vázquez, M. Ravachol, F. Chalot, and M. Mallet, “The Robustness Issue on Multigrid Schemes Applied to the Navier-Stokes Equations for Laminar and Turbulent, Incompressible and Compressible Flows,” International Journal for Numerical Methods in Fluids , vol. 45, pp. 555–579, 2004.

REFERENCES FOR THE SINGLE-PHASE FLOW, USER INTERFACES

| 277

278 |


7

The Conjugate Heat Transfer Branch The Heat Transfer Module has interfaces for conjugate heat transfer, which are also under the Fluid Flow branch as Non-Isothermal Flow inter faces. The Non-Isothermal Flow Laminar Flow (nitf) and Turbulent Flow (nitf) interfaces are identical to the Conjugate Heat Transfer interfaces found under the Heat Transfer branch. This chapter discusses applications involving the Conjugate Heat Transfer branch ( ). In this chapter: • • • •

About the Conjugate Heat Transfer User Interfaces The Non-Isothermal Flow and Conjugate Heat Transfer, Laminar Flow and Turbulent Flow User Interfaces Theory for the Non-Isothermal Flow and Conjugate Heat Transfer User Interfaces References for the Non-Isothermal Flow and Conjugate Heat Transfer User Interfaces

279

About the Conjugate Heat Transfer User Interfaces In this section: • • •

Selecting the Right User Interface The Non-Isothermal Flow Options Conjugate Heat Transfer Options

Selecting the Right User Interface

There are several variations of the same predefined multiphysics interface (all with the interface identifier nitf), that combine the heat equation with either laminar flow or turbulent flow. The advantage of using the multiphysics interfaces—compared to adding the individual interfaces separately—is that predefined couplings are available in both directions. In particular, interfaces use the same definition of the density, which can therefore be a function of both pressure and temperature. Solving this coupled system of equation usually requires numerical stabilization, which the predefined multiphysics interface also sets up. The user interfaces found under the Fluid Flow>Non-Isothermal Flow ( ) and Heat ) branches are multiphysics interfaces and contain Transfer>Conjugate Heat Transfer ( the features for modeling fluid flow, which can be laminar, turbulent, or Stokes flow, in combination with heat transfer. The settings vary only by one or two default settings (see Table 7-1), which are selected during Model Wizard selection, or from a check box or list under the Physical Model section for the interface. Figure 7-1 is an example that compares the two settings windows. TABLE 7-1: THE NON-ISOTHERMAL FLOW AND CONJUGATE HEAT TRANSFER PHYSICAL MODEL DEFAULT SETTINGS*

280 |

USER INTERFACE (NITF)

TURBULENCE MODEL TYPE

TURBULENCE MODEL

HEAT TRANSPORT TURBULENCE MODEL

Non-Isothermal Flow, Laminar Flow

None

N/A

N/A

Non-Isothermal Flow, Turbulent Flow, k-

RANS

k-

Kays-Crawford

C H A P T E R 7 : T H E C O N J U G A T E H E A T T R A N S F ER B R A N C H

TABLE 7-1: THE NON-ISOTHERMAL FLOW AND CONJUGATE HEAT TRANSFER PHYSICAL MODEL DEFAULT SETTINGS* USER INTERFACE (NITF)

TURBULENCE MODEL TYPE

TURBULENCE MODEL

HEAT TRANSPORT TURBULENCE MODEL

Non-Isothermal Flow, Turbulent Flow, Low Re k-

RANS

Low Reynolds number k-

Kays-Crawford

Conjugate Heat Transfer, Laminar Flow

None

N/A

N/A

Conjugate Heat Transfer, Turbulent Flow, k- 

RANS

k-

Kays-Crawford

Conjugate Heat Transfer, Turbulent Flow, Low Re k-

RANS

Low Reynolds number k-

Kays-Crawford

*For all the interfaces, the Neglect initial term (Stokes flow) check box is not selected by default.

Figure 7-1: On the left is the settings window for the Non-Isothermal Flow, Turbulent Flow interface. You can model laminar and turbulent flow, or Stokes flow, in combination with

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heat transfer. On the right is the settings window for the Conjugate Heat Transfer, Turbulent Flow, Low Reynold’s k-  interface. Choose to model laminar and turbulent flow in combination with heat transfer.

The next sections are a brief overview of each of the interfaces. The Non-Isothermal Flow Options

Different types of flow require different equations to describe them. If the type of flow to model is known, then select it directly from the Model Wizard. However, when you are not certain of the flow type, or because it is difficult to reach a solution easily, you can start instead with a simplified model and add complexity as you build the model. Usually you start with the simplest-to-set-up physics interface, which in most cases in non-isothermal flow is the Non-Isothermal Flow, Laminar Flow interface. In other cases, you may know exactly how a fluid behaves and which equations, models, or physics interfaces best describe it, but because the model is so complex it is difficult to reach an immediate solution. Simpler assumptions may need to be made to solve the problem, and other interfaces may be better to fine-tune the solution process for the more complex problem. This can be the case when you know that the flow is essentially turbulent in nature, but you would first solve it for laminar conditions in order to build knowledge of the system and provide a good initial guess for the turbulent flow simulation. The various forms of the Non-Isothermal Flow interfaces are, by default, found under the Fluid Flow branch. N O N - I S O T H E R M A L F L O W, L A M I N A R F L O W

The Non-Isothermal Flow, Laminar Flow User Interface ( ) is used primarily to model slow-moving flow in environments where energy transport is also an important part of the system and application, and must coupled or connected to the fluid flow in some way. Processes where natural convection are an important component are classic areas for such modeling. The interface solves the Navier-Stokes equations together with an energy balance assuming heat flux through convection and conduction. The density term is assumed to be affected by temperature and flow is always assumed to be compressible. Stokes’ law (creeping flow) can be activated from the Non-Isothermal Flow, Laminar Flow interface if wanted.

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N O N - I S O T H E R M A L F L O W , TU R B U L E N T F L O W

The Turbulent Flow, k- and Turbulent Flow Low Re k- User Interfaces( ) model flows that are relatively fast-moving and geometries that change significantly to induce disorder, vortices, and eddies. Once again, the interfaces are also set up assuming that energy transport is an important part of the system and application and must be coupled or connected to the fluid flow in some way. Process or component cooling are classic examples. For this reason, the interface includes added functionality for calculating the added dispersion of heat transfer due to turbulence. This is represented by one of the Kays-Crawford or Extended Kays-Crawford Turbulence heat transport models, or by including your own turbulent Prandtl number. In addition to the properties for the different turbulence models mentioned in Theory for the Non-Isothermal Flow and Conjugate Heat Transfer User Interfaces, an additional important aspect is that the reward in terms of accuracy for using low-Reynolds number models is even higher in non-isothermal flow simulations. The reason is that the local equilibrium assumption on which the wall functions rely is seldom fulfilled when there are temperature gradients present. This is particularly relevant for applications in non-isothermal flow where the heat flux at solid-liquid interfaces is important to the final solution. Conjugate Heat Transfer Options

The various forms of the Conjugate Heat Transfer interfaces are, by default, found under the Heat Transfer branch. These are used to set up and model heat transfer throughout a fluid in collaboration with a solid where heat is transferred by conduction. CONJUGATE HEAT TRANSFER, LAMINAR FLOW

The Conjugate Heat Transfer, Laminar Flow User Interface ( ) is used primarily to model slow-moving flow in environments where temperature and energy transport are also an important part of the system and application, and must coupled or connected to the fluid-flow in some way. Processes where natural convection are an important component are classic areas for such modeling. The interface solves the Navier-Stokes equations together with an energy balance assuming heat flux thr ough convection and conduction. The density term is assumed to be affected by temperature and flow is always assumed to be compressible. Stokes’ law (creeping flow) can be activated from the Conjugate Heat Transfer, Laminar Flow interface if required. See Table 7-1 for details.

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There are different versions of the Conjugate Heat Transfer, Turbulent Flow interfaces, and each use the Reynolds-Averaged Navier-Stokes (RANS) equations, solving for the mean velocity field and pressure, along with the k-  model. The Turbulent Flow, k- and Turbulent Flow Low Re k- User Interfaces ( ) are used to model flows that are relatively fast-moving and/or geometries that change significantly to induce disorder, vortices, and eddies. The interfaces are set up assuming that temperature and energy transport are also an important part of the system and application, and must be coupled or connected to the fluid-flow in some way. Process or component cooling are classic examples. Each interface includes added functionality for calculating the added dispersion of heat transfer due to turbulence. This is represented by one of the Kays-Crawford or Extended Kays-Crawford Turbulence heat transport models, or by including your own turbulent Prandtl number.

The Heat Transfer Branch

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The Non-Isothermal Flow and Conjugate Heat Transfer, Laminar Flow and Turbulent Flow User Interfaces In this section: • • •

The Non-Isothermal Flow, Laminar Flow User Interface The Conjugate Heat Transfer, Laminar Flow User Interface The Turbulent Flow, k-  and Turbulent Flow Low Re k- User Interfaces • • • •

About the Conjugate Heat Transfer User Interfaces Selecting the Right User Interface Domain, Boundary, Edge, Point, and Pair Nodes Settings for the NITF User Interfaces Theory for the Non-Isothermal Flow and Conjugate Heat Transfer User Interfaces

The Non-Isothermal Flow, Laminar Flow User Interface

The Non-Isothermal Flow version of the Laminar Flow (nitf) user interface ( ), found under the Fluid Flow>Non-Isothermal Flow branch ( ) of the Model Wizard, is a predefined multiphysics coupling consisting of a single-phase flow interface, using a compressible formulation, in combination with a Heat Transfer interface. When this interface is added, these default nodes are also added to the Model Builder — Non-Isothermal Flow, Fluid, Wall, Thermal Insulation, and Initial Values. Right-click any node to add other features that implement, for example, boundary conditions, volume forces, or heat sources. INTERFACE IDENTIFIER

The interface identifier is used primarily as a scope prefix for variables defined by the physics user interface. Refer to such interface variables in expressions us ing the pattern

T H E N O N - I S O T H E R M A L F L O W A N D C O N J U G A T E H E A T T R A N S F E R, L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R

..

In order to distinguish between variables belonging to different physics user interfaces, the identifier string must be unique. Only letters, numbers and underscores (_) are permitted in the Identifier field. The first character must be a letter. The default identifier (for the first interface in the model) is

nitf.

DOMAIN SELECTION

The default setting is to include All domains in the model to define the dependent variables and the equations. To choose specific domains, select Manual from the Selection list. PHYSICAL MODEL

Define interface properties to control the overall type of model. Neglect Iner tial Term (Stokes Flow)—All Interfaces

Select the Neglect inertial term (Stokes flow) check box to model flow at very low Reynolds numbers where the inertial term in the Navier-Stokes equations can be neglected. Instead use the linear Stokes equations. This flow type is referred to as creeping flow or Stokes flow and can occur in microfluidics (and MEMS devices), where the flow length scales are very small. Turbulence Model Type

By definition, no turbulence model is needed when studying laminar flows. The default Turbulence model type is None. However, if the default Turbulence model type selected is RANS, the additional turbulence model settings are made available. The node is still called Non-Isothermal Flow or Conjugate Heat Transfer with a number added at the end of the name to indicate the change. To enable surface-to-surface radiation, select the Surface-to-surface radiation check box. This adds a Radiation Settings section. To enable radiation in participating media, select the Radiation in participating media check box. This adds a Participating media Settings section. To enable the Biological Tissue feature, select the Heat Transfer in biological tissue check box.

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RADIATION SETTINGS

This section is available when the Surface-to-surface radiation check box is selected. See Radiation Settings as described for The Heat Transfer Interface. PARTICIPATING MEDIA SETTINGS

This section is available when the Radiation in participating media check box is selected. See Participating Media Settings as described for The Heat Transfer Interface. DEPENDENT VARIABLES

The dependent variables (field variables) are for the Velocity field u (SI unit: m/s), the Pressure p (SI unit: Pa), and the Temperature T (SI unit: K). The default values are 0 m/s for the velocity, 0 Pa for the pressure, and 293.15 K for the temperature. The names can be changed but the names of fields and dependent variables must be unique within a model. For turbulence modeling and heat radiation, there are additional dependent variables for the turbulent dissipation rate, turbulent kinetic energy, reciprocal wall distance, and surface radiosity. DISCRETIZATION


Show button

(


Select a Discretization of fluids—P1+P1 (the default), P2+P1, or P3+P2. The first term describes the element order for the velocity components, and the second term is the order for the pressure. The element order for the temperature is set to follow the velocity order, so the temperature order is 1 for P1+P1, 2 for P2+P1, and 3 for P3+P2. Select a Surface radiosity—Linear (the default), Quadratic, Cubic, Quartic, or Quintic (2D axisymmetric and 2D models only). Specify the Value type when using splitting of complex variables —Real (the default) or Complex.


C O N S I S T E N T A N D I N C O N S I S T E NT S T A B I L I Z A T I O N

To display this section, click the Show button ( unique to this interface are listed below. • •

) and select Stabilization. Any settings

The consistent stabilization methods are applicable to the Heat and flow equations. The Isotropic diffusion inconsistent stabilization method can be activated for both the Heat equation and the Navier-Stokes equations.

ADVANCED SETTINGS

To display this section, click the Show button ( ) and select Advanced Physics Options . Normally these settings do not need to be changed. The Use pseudo time stepping for stationary equation form check box is active per default for the Non-Isothermal Flow interface. It adds pseudo time derivatives to the equation momentum and heat equations when the Stationary equation form is used. When selected, also choose a CFL number expression—Automatic (the default) or Manual. Automatic sets the local CFL number (from the Courant–Friedrichs–Lewy condition) to the built-in variable CFLCMP which in turns trigger a PID regulator for the CFL number. If Manual is selected, enter a Local CFL number CFLloc. By default the Enable conversions between material and spatial frames check box is selected. • • • • •

About Handling Frames in Heat Transfer Show More Physics Options Domain, Boundary, Edge, Point, and Pair Nodes Settings for the NITF User Interfaces For settings window details for the Heat Transfer in Solids feature, see The Heat Transfer Interface About Pseudo Time Stepping in the COMSOL Multiphysics Reference Manual

•

The Conjugate Heat Transfer, Laminar Flow User Interface

Viscous Heating in a Fluid Damper: Model Library path Heat_Transfer_Module/Buildings_and_Constructions/fluid_damper

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The Conjugate Heat Transfer, Laminar Flow User Inter face

The Conjugate Heat Transfer version of the Laminar Flow (nitf) user interface ( ), found under the Heat Transfer>Conjugate Heat Transfer branch ( ), is a predefined multiphysics coupling consisting of a single-phase flow interface, using a compressible formulation, in combination with a Heat Transfer interface. When this interface is added, these default nodes are also added to the Model Builder —Conjugate Heat Transfer , Heat Transfer in Solids (with domain selection set to All domains), Thermal Insulation, Wall, Fluid (with empty initial domain selection), and Initial Values. Right-click any node to add other features that implement, for example, boundary conditions, volume forces, or heat sources. • • • •

Show More Physics Options The Non-Isothermal Flow, Laminar Flow User Interface Domain, Boundary, Edge, Point, and Pair Nodes Settings for the NITF User Interfaces The Heat Transfer Interface for settings window details for the Heat Transfer in Solids feature.

Viscous Heating in a Fluid Damper: Model Library path Heat_Transfer_Module/Buildings_and_Constructions/fluid_damper

The Turbulent Flow, k-  and Turbulent Flow Low Re k-  User Interfaces

These predefined multiphysics couplings consist of a turbulent flow interface, using a compressible formulation, in combination with a Heat Transfer interface. Most of the setting options are the same as for The Non-Isothermal Flow, Laminar Flow User Interface, except where noted below. Right-click any node to add other features that implement, for example, boundary conditions, volume forces, or heat sources. PHYSICAL MODEL

The default Turbulence model for the Turbulent flow, k- interface is k- . For the Turbulent flow, Low Re k- interface it is Low Reynolds number k- .


For all the turbulent interfaces, the default Turbulence model type is RANS and the default Heat transport turbulence model is Kays-Crawford. Other Heat transport turbulence model options are Extended Kays-Crawford or User-defined turbulent Prandtl number . The Extended Kays-Crawford model requires a Reynolds number at infinity. That input is given in the Model Inputs section of the Fluid feature node. It is always possible to specify a user-defined model for the turbulence Prandtl number. Enter the user-defined value or expression for the turbulence Prandtl number in the Model Inputs section of the Fluid feature node. TURBULENCE MODEL PARAMETERS

Turbulence model parameters are optimized to fit as many flow types as possible, but for some special cases, better performance can be obtained by tuning the model parameters. For a description of the turbulence model and the included model parameters see Theory for the Turbulent Flow User Interfaces. DEPENDENT VARIABLES

The dependent variables (field variables) are for the Velocity field u (SI unit: m/s), the Pressure p (SI unit: Pa), and the Temperature T (SI unit: K). The default values are 0 m/s for the velocity, 0 Pa for the pressure, and 293.15 K for the temperature. The names can be changed but the names of fields and dependent variables must be unique within a model. For turbulence modeling and heat radiation, there are additional dependent variables for the transported turbulence properties and also a dependent variable for Reciprocal wall distance if the Low-Reynolds number k- model or Spalart-Allmaras model is employed.

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CONSISTENT AND INCONSISTENT STABILIZATION

To display this section, click the Show button ( unique to this interface are listed below. • •

• •

) and select Stabilization. Any settings

The consistent stabilization methods are applicable to the Heat and flow equations and the Turbulence Equations. When the Crosswind diffusion check box is selected, enter a Tuning parameter Ck for one or both of the Heat and flow equations and Turbulence Equations. The default for the Heat and flow equations is 0.5, and 1 for the Turbulence equations. The Isotropic diffusion inconsistent stabilization method can be activated for the Heat equation, Navier-Stokes equations, and the Turbulence equations. By default there is no isotropic diffusion selected. If required, select the Isotropic diffusion check box and enter a Tuning parameter id for one or all of Heat equation, Navier-Stokes equations, or Turbulence equations. The defaults are 0.25.

ADVANCED SETTINGS

To display this section, click the Show button ( ) and select Advanced Physics Options. Normally these settings do not need to be changed. The Use pseudo time stepping for stationary equation form check box adds pseudo time derivatives not only to the equation momentum and heat equations when the Stationary equation form is used, but to the turbulence equations as well. The Turbulence variables scale parameters subsection contains the parameters U scale and Lfact that are used to calculate absolute tolerances for the turbulence variables. The section is only visible when Turbulence model type in the Physical Model section is set to RANS. The scaling parameters must only contain numerical values, units or parameters defined under Global Definitions. The scaling parameters cannot contain variables. The parameters are used when a new default solver for a transient study step is generated. If you change the parameters, the new values will take effect the next time you generate a new default solver.


By default the Enable conversions between material and spatial frames check box is selected. • • • •

About Handling Frames in Heat Transfer Show More Physics Options Domain, Boundary, Edge, Point, and Pair Nodes Settings for the NITF User Interfaces Turbulent Non-Isothermal Flow Theory

Turbulent Flow Through a Shell-and-Tube Heat Exchanger: Model Library path Heat_Transfer_Module/Heat_Exchangers/ turbulent_heat_exchanger

Domain, Boundary, Edge, Point, and Pair Nodes Settings f or the NITF User Interfaces

All the versions of the Non-Isothermal Flow and Conjugate Heat Transfer interfaces have shared domain, boundary, edge, point, and pair physics features based on the selections made for the model. Also because these are all multiphysics interfaces, almost every physics node is shared with, and described for, other interfaces. Below are links to the domain, boundary, edge, point, and pair physics as indicated. To locate and search all the documentation, in COMSOL Multiphysics, select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree. These features are described in this section: • • • • •

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Fluid Initial Values Interior Wall Open Boundary Pressure Work


• • • •

Symmetry, Flow Symmetry, Heat Viscous Heating Wall

These features are described for the Single Phase Flow inter face (listed in alphabetical order): • • • • •

Boundary Stress Interior Fan Flow Continuity Inlet Interior Wall

• • • •

Outlet Periodic Flow Condition Pressure Point Constraint Volume Force

These physics are described for the Heat Transfer interface (listed in alphabetical order): • • • • • • • • •

Boundary Heat Source Convective Heat Flux Continuity Heat Flux Heat Source Heat Transfer in Solids Highly Conductive Layer Inflow Heat Flux Line Heat Source • • •

• • • • • • • •

Outflow Periodic Heat Condition Point Heat Source Surface-to-Ambient Radiation Temperature Thermal Contact Thermal Insulation Thin Thermally Resistive Layer

The Heat Transfer Interface The Non-Isothermal Flow and Conjugate Heat Transfer, Laminar Flow and Turbulent Flow User Interfaces Theory for the Non-Isothermal Flow and Conjugate Heat Transfer User Interfaces

Fluid

The Fluid node adds both the momentum equations and the temperature equation but without volume forces, heat sources, pressure work, or viscous heating. You can add volume forces and heat sources as separate features, and Viscous Heating and Pressure


Work can be added as subnodes. When the turbulence model type is set to RANS, the Fluid node also adds the equations for k and . DOMAIN SELECTION


This section controls both the variable as well as any property value (reference pressures) used when solving for pressure. There are usually two ways to calculate the pressure when describing fluid flow and mass and heat transfer. Solve for the absolute pressure or a pressure (often denoted gauge pressure) that relates back to the absolute pressure through a reference pressure. Using one or the other option usually depends on the system and the equations being solved for. For example, in a straight incompressible flow problem, the pressure drop over the modeled domain is probably many orders of magnitude less than atmospheric pressure, which, if included, reduces the chances for stability and convergence during the solving process for this variable. In other cases, the absolute pressure may be required to be solved for, such as where pressure is a par t of an expression for gas volume or diffusion coefficients.

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The default Absolute pressure p (SI unit: Pa) is Pressure (nitf/fluid). The Reference pressure check box is selected by default and the default value of pref is 1[atm] ((101,325 Pa). This makes it possible to use a system-based (gauge) pressure as the pressure variable while automatically including the reference pr essure in places where it is required, such as for gas flow governed by the gas law. While this check box maintains control over the pressure variable and instances where absolute pressure is required within this r espective physics interface, it may not with physics interfaces that it is being coupled to. In such models, check the coupling between any interfaces using the same variable. HEAT CONDUCTION

The default Thermal conductivity k (SI unit: W/(m·K)) uses values From material. If User defined is selected, choose Isotropic, Diagonal, Symmetric, or Anisotropic based on the characteristics of the thermal conductivity and enter other values or expressions in the field or matrix. The thermal conductivity describes the relationship between the heat flux vector q and the temperature gradient T as in q = kT which is Fourier’s law of heat conduction. Enter this quantity as power per length and temperature. When the turbulence model type is set to RANS, the conductive heat flux includes the turbulent contribution: q = k+T I )T where k is the thermal conductivity tensor, I the identity matrix and T the thermal turbulent conductivity defined by C p  T T = -------------Pr T

THERMODYNAMICS

Select a Fluid type—Gas/Liquid (the default), Ideal Gas, or Moist Air. The Heat capacity at constant pressure Cp describes the amount of heat energy required to produce a unit temperature change in a unit mass. The Ratio of specific heats is the ratio of heat capacity at constant pressure, Cp, to heat capacity at constant volume, Cv.


Gas/Liquid

If Gas/Liquid is selected as the Fluid type, properties of a non-ideal gas or liquid can be used. By default the Density  (SI unit: kg/m3), Heat capacity at constant pressure C p (SI unit: J/(kg·K)), and Ratio of specific heats  (dimensionless) use values From material. Select User defined to enter other values or expressions. Ideal Gas

If Ideal gas is selected as the Fluid type, the ideal gas law is used to describe the fluid. For an ideal gas the density is defined in the following equation where p A is the absolute pressure, and T the temperature: 

=

M n p A --------------- RT

=

p A ---------- Rs T

Select a Gas constant type—Specific gas constant Rs (SI unit: J/(kg·K)) or Mean molar mass M n (SI unit: kg/mol). In both cases, the default uses data From material. Select User defined to enter other values or expressions. If Mean molar mass is selected, the universal gas constant R  8.314 J/(mol·K), which is a built-in physical constant, is also used. Select an option from the Specify Cp or  list—Heat capacity at constant pressure Cp (SI unit: J/(kg·K)) or Ratio of specific heats  (dimensionless). For both options, the default uses the value From material. Select User defined to enter another value or expression. For an ideal gas, you choose to specify either Cp or , but not both since they are, in that case, dependent. When using the ideal gas law to describe a fluid, specifying  is enough to evaluate Cp. For common diatomic gases such as air,   1.4 is the standard value. Most liquids have   1.1 while water has   1.0.  is used in the streamline stabilization and in the results and analysis variables for heat fluxes and total energy fluxes. It is also used in the ideal gas law.

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Moi st Air

If Moist air is selected as the Fluid type, the thermodynamics properties are defined as a function of the quantity of vapor in the moist air. Select an Input quantity: (the default)  (SI unit: dimensionless, (kg of water vapor) / (total mass in kg = kg of dry air + kg of water vapor)) 3 Concentration (SI unit: mol/m )

• Vapor mass fraction •

When Concentration is selected, a Concentration model input is automatically added in the Models Inputs section. • Moisture content x vap (SI unit: dimensionless, kg of water vapor/ kg of dry air). •

Select Relative humidity to define the quantity of vapor from the following reference values, which are used to estimate the mass fraction of vapor that is used to define the thermodynamic properties of the moist air: - Reference relative humidity  ref (dimensionless). The default is 0. - Reference temperature T ref (SI unit: K). The default is 293.15 K. - Reference pressure pref (SI unit: Pa). The default is 1 atm.

DYNAMIC VISCOSITY

This section is not available if Moist air is selected as the Fluid type. The dynamic viscosity describes the relationship between the shear rate and the shear stresses in a fluid. Intuitively, water and air have a low viscosity, and substances often described as thick, such as oil, have a higher viscosity. Non-Newtonian fluids have a shear-rate dependent viscosity. Examples of non-Newtonian fluids include yoghurt, paper pulp, and polymer suspensions.


The default Dynamic viscosity  (SI unit: Pa·s) uses values From material. Or select User defined to use a built-in variable for the shear rate magnitude, spf.sr, which makes it possible to define arbitrary expressions of the dynamic viscosity. If you also have the following modules, additional options ar e available— Non-Newtonian power law and Non-Newtonian Carreau model. • • •

CFD Module Heat Transfer Module plus the Microfluidics Module, or Microfluidics Module plus either the CFD Module or Heat Transfer Module

Non-Newtonian Flow: The Power Law and the Carreau Model

Non-Newtonian Power Law

This option is available with the CFD Module or the Heat Transfer Module plus the Microfluidics Module. If Non-Newtonian power law is selected, enter the Power law model parameter m and Model parameter n (both dimensionless). This selection uses the power law as the viscosity model for a non-Newtonian fluid where the following equation defines dynamic viscosity: 

=

· · m  max    min  n – 1

where · is the shear rate and · min is a lower limit for the shear rate evaluation. · min is per default 1·10-2 1s. Non-Newtonian Carreau Model

This option is available with the CFD Module or the Heat Transfer Module plus the Microfluidics Module.

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If Non-Newtonian Carreau model is selected, enter these Carreau model parameters: • • •

The Zero shear rate viscosity 0 (SI unit: Pa·s) The Infinite shear rate viscosity inf (SI unit: Pa·s) The Model parameters  (SI unit: s) and n (dimensionless)

This selection uses the Carreau model as the viscosity model for a non-Newtonian fluid where the following equation defines the dynamic viscosity: 

=

n – 1 · 2 ----------------  1 +     2

 +   0 –  inf

MIXING LENGTH LIMIT (TURBULENCE MODELS ONLY)

This section is available for the k- model, which needs an upper limit on the mixing length. Select a Mixing length limit—Automatic (the default) or Manual. •

If Automatic is selected, the mixing length limit is automatically evaluated as: lim l mix

•

=

0 . 5l bb

(7-1)

where lbb is the shortest side of the geometry bounding box. If the geometry is for example a complicated system of very slender entities, Equation 7-1 tends to give a lim manually. result that is too large. Then define l mix lim If Manual is selected, enter a value or expression for the Mixing length limit l mix (SI unit: m).

DISTANCE EQUATION (TURBULENCE MODELS ONLY)

This section is available for the low-Reynolds number k- model, which needs the distance to the closest wall. Select a Reference length scale—Automatic (the default) or Manual. •

If Automatic is selected, the reference length scale is automatically evaluated as:


l ref = 0 . 1l bb

•

(7-2)

where lbb is the shortest side of the geometry bounding box. If the geometry is for example a complicated system of very slender entities, Equation 7-1 tends to give a result that is too large. Then define l ref manually. If Manual is selected, enter a value or expression for the Reference length scale l ref (SI unit: m).

Wall

For laminar flow, the low Reynolds number k- turbulence model, the Wall node is identical to the single-phase flow settings (the Boundary condition defaults to No slip). In these cases, continuity of the temperature is enforced on internal walls separating a fluid and solid domain.

The settings below are for the k- turbulence model.

About the The rmal Wall Fun ction

Whenever wall functions are used, there is a theoretical gap between the solid wall and the computational domain of the fluid. This gap is often ignored in so much that it is ignored when the computational geometry is drawn, but it must nevertheless be considered in the equations for the temperature field. Figure 0-1 shows the difference between internal and external walls. The approach is slightly different depending on what type of wall the condition applies to. Any wall feature that utilizes wall functions automatically detects internal and external walls. On internal walls, there are two temperatures, one for the solid, T s, and one for the fluid, T f . If a temperature is prescribed to an internal wall, the constraint is applied to the temperature for the solid, that is, to T s. On external walls, the temperature T is the temperature of the fluid while the wall temperature is represented by the dependent variable T w. T w is a variable that is solved for and the equation for T w is q wf = q tot

where qtot is the total heat flux prescribed to the boundar y.

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External wall Inflow

Internal wall

Fluid

Outflow

Solid

Figure 7-2: A simple example that includes both an external wall and an internal wall.

If a temperature is prescribed to an external wall, the constraint is applied to the wall temperature T w. Any other heat boundary condition applied to an external wall is wrong in the sense that it acts on the fluid temperature, T s, instead of the wall temperature, T w. BOUNDARY CONDITION

When using the k- turbulence model, the Boundary condition defaults to Wall functions. The other options available are Slip, Sliding wall (wall functions), and Moving wall (wall functions). If any one of these options are selected—Wall functions, Sliding wall (wall functions), or Moving wall (wall functions)—the wall functions for the temperature field is also prescribed, which is called a thermal wall functions. • •

If Sliding wall (wall functions) is selected, enter the coordinates for the Velocity of sliding wall uw (SI unit: m/s). If Moving wall (wall functions) is selected, enter the coordinates for the Velocity of moving wall uw (SI unit: m/s). Turbulence model parameters are optimized to fit as many flow types as possible, but for some special cases, better performance can be obtained by tuning the model parameters. For a description of the turbulence model and the included model parameters see Theory for the Turbulent Flow User Interfaces.


About the Thermal Wall Function

Interior Wall

For laminar flow, the low Reynolds number k- turbulence model, the Interior Wall node is identical to the single-phase flow settings (the Boundary condition defaults to No slip). In these cases, continuity of the temperature is enforced across internal walls separating two fluid domains.

The settings below are for the k- turbulence model.

About the The rmal Wall Fun ction

Whenever wall functions are used, there is a theoretical gap between the internal wall and the computational domain of the fluid. This gap is often ignored in so much that it is ignored when the computational geometry is drawn, but it must nevertheless be considered in the equations for the temperature field. On internal walls, there are three temperatures, one for the wall, T w, and one for the fluid on up and down sides of the wall, T f, u and T f, d. If a temperature is prescribed to an internal wall, the constraint is applied to the temperature for the wall, that is, to T w. BOUNDARY CONDITION

When using the k- turbulence model, the Boundary condition defaults to Wall functions. The other options available are Slip, and Moving wall (wall functions). If any one of these options are selected— Wall functions or Moving wall (wall functions)— the wall functions for the temperature field is also prescribed, wh ich is called a thermal wall functions.

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If Moving wall (wall functions) is selected, enter the coordinates for the Velocity of moving wall uw (SI unit: m/s). Turbulence model parameters are optimized to fit as many flow types as possible, but for some special cases, better performance can be obtained by tuning the model parameters. For a description of the turbulence model and the included model parameters see Theory for the Turbulent Flow User Interfaces.

About the Thermal Wall Function

Initial Values

The Initial Values node adds initial values for the velocity field, the pressure, and temperature that can serve as an initial condition for a transient simulation or as an initial guess for a nonlinear solver. For turbulent flow there are also initial values for the turbulence model variables. DOMAIN SELECTION

For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically selected and is the same as for the interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific domains or select All domains as required. INITIAL VALUES

Enter values or expressions for the initial value of the Velocity field u (SI unit: m/s), the Pressure p (SI unit: Pa), and the Temperature T (SI unit: K). The default values are 0 m/s for the velocity, 0 Pa for the pressure, and 293.15 K for the temperature. For turbulence models, default initial values for the turbulence variables, defined for each turbulence model in Theory for the Turbulent Flow User Interfaces, are applied. Open Boundary

Use the Open Boundary node to set up heat and momentum transport across boundaries where both convective inflow and outflow can occur. The node specifies a


fluid flow condition, together with an exterior temperature to be applied on the par ts of the boundary where fluid flows into the domain. The direction of the flow across the boundar y is typically calculated by a Fluid Flow branch interface and is entered as Model Inputs. BOUNDARY SELECTION

From the Selection list, choose the boundaries to define. BOUNDARY CONDITION

From the Boundary Condition list, choose a fluid flow condition for the open boundary the boundaries to apply the open boundary condition. Select • •

Select Normal stress (the default) and enter the normal stress f 0 (SI unit: N/m2). This implicitly specifies that p  f 0 Select No viscous stress to prescribe a vanishing normal viscous stress on the boundary.

EXTERIOR TEMPERATURE

Enter a value or expression for the external temperature (SI unit: K). Pressure Work

Right-click the Heat Transfer in Solidsor Fluid node to add this subnode. When added under a Heat Transfer in Solids node, the Pressure Work node adds the following term to the right-hand side of the Heat Transfer in Solids equation:  t

– T ----- S el

(7-3)

where Sel is the elastic contribution to entropy . When added under Fluid node, the Pressure Work feature adds the following contribution to the right-hand side of the Heat Transfer in Fluids equation: T    p ------ + u   p    T p   t

– ---- -------

The software computes the pressure work using the absolute pressure.

304 |


(7-4)

DOMAIN SELECTION

From the Selection list, choose the domains to define. By default, the selection is the same as for the parent node to which it is attached (Heat Transfer in Solidsor Fluid node). PRESSURE WORK

For the Heat Transfer in Solids model, enter a value or expression for the Elastic 3 3 contribution to entropy Ent (SI unit: Jm ·K)). The default is 0 J m ·K). For the Fluid model, select a Pressure work formulation—Full formulation (the default), or Low Mach number formulation. The latter excludes the term u ·  p from Equation 7-4, which is small for most flows with a low Mach number. Viscous Heating

The Viscous Heating subnode adds the following term to the right-hand side of the heat transfer in fluids equation: :S

(7-5)

Here  is the viscous stress tensor and S is the strain rate tensor. Equation 7-5 represents the heating caused by viscous friction within the fluid. DOMAIN SELECTION

From the Selection list, choose the domains to define. By default, the selection is the same as for the Fluid node that it is attached to. Symmetry, Heat

The Symmetry, Heat node provides a boundary condition for symmetry boundaries. This boundary condition is similar to a Thermal Insulation condition, and it means that there is no heat flux across the boundary. If you apply a Symmetry, Heat feature to a boundary adjacent to a fluid domain, you should also consider adding a Symmetry, Flow node to that boundary since physically,


you cannot have symmetry in the temperature without having symmetry in the velocity and pressure as well. The symmetry condition only applies to the temperature field. It has no effect on the radiosity (surface-to-surface radiation) and on the radiative intensity (radiation in participating media). BOUNDARY SELECTION

In most cases, the node does not require any user input. If required, define the symmetry boundaries.

Symmetry, Flow

The Symmetry, Flow node adds a boundary condition that describes symmetry boundaries in a fluid flow simulation. The boundary condition for symmetry boundaries prescribes no penetration and vanishing shear stresses. The boundar y condition is a combination of a Dirichlet condition and a Neumann condition: un

=

 – p I +     u +   u  T  – 2---     u  I  n    3

0, un

=

 – p I +    u +   u T   n

0,

=

=

0

0

for the compressible and the incompressible formulation respectively. The Dirichlet condition takes precedence over the Neumann condition, and the above equations are equivalent to the following equation for both the compressible and incompressible formulation: un

=

0,

K

=

K –  K  nn  u +  uT  n

=

0

If you apply a Symmetry, Flow feature to a boundary, the boundary should also be supplemented with a Symmetry, Heat node because it is not possible to have symmetry in the velocity and pressure without having symmetry in the temperature as well.

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BOUNDARY SELECTION

From the Selection list, choose the boundaries to define. For 2D axial symmetry, a boundary condition does not need to be defined. For the symmetry axis at r  0, the software automatically provides a condition that prescribes ur  0 and vanishing stresses in the z direction and adds an Axial Symmetry node that implements this condition on the axial symmetr y boundaries only. CONSTRAINT SETTINGS

To display this section, click the Show button ( ) and select Advanced Physics Options. Select the Use weak constraints check box to use weak constraints and create dependent variables for the corresponding Lagrange multipliers.


Theory for the Non-Isothermal Flow and Conjugate Heat Transfer User Interfaces In industrial applications it is common that the density of a process fluid varies. These variations can have a number of different sources but the most common one is the presence of an inhomogeneous temperature field. This module includes the Non-Isothermal Flow predefined multiphysics coupling to simulate systems where density varies with temperature. Other situations where the density might vary includes chemical reactions, for instance where reactants associate or dissociate. The Non-Isothermal Flow and Conjugate Heat Transfer interfaces contain the fully compressible formulation of the continuity equation and momentum equations:  ------ +     u  t

=

0

u  ------- +  u   u = –  p +       u +   u  T  – 2---     u  I t 3

(7-6) +

F

where •


• u is the velocity vector (SI unit: m/s) • p is pressure (SI unit: Pa) •

 is the dynamic viscosity (SI unit: Pa·s)

• F is the body force vector (SI unit: N/m 3)

It also solves the heat equation, which for a fluid is --- +  u    T   C p  ----T  t

= –

  p --- -------  ------ +  u    p    q  + : S – -T    T p   t

where in addition to the quantities above • Cp is the specific heat capacity at

308 |

constant pressure (SI unit: J/(kgK))


+

Q

• T is absolute temperature (SI unit: K) • q is the heat flux by conduction (SI unit: W/m 2) •

 is the viscous stress tensor (SI unit: Pa)

• S is the strain-rate tensor (SI unit: 1/s)

S

=

1 ---   u +   u  T  2

• Q contains heat sources other than viscous heating (SI unit: W/m 3)

The pressure work term T    p ---- ------- ------ +  u    p    T p   t

and the viscous heating term :S are not included by default because they are commonly negligible. These can, however, be added as subnodes to the Fluid node. For a detailed discussion of the fundamentals of heat transfer in fluids, see Ref. 3. The interface also supports heat transfer in solids: E --- = –    q  – T ------- + Q  C p ----T t t

where E is the elastic contribution to entropy (SI unit: J/(m 3·K)). As in the case of fluids, the pressure work term E T ------t

is not included by default but must be added as a subfeature. • • •

The Heat Equation Turbulent Non-Isothermal Flow Theory References for the Non-Isothermal Flow and Conjugate Heat Transfer User Interfaces

T H E O R Y F O R T H E N O N - I S O T H E R M A L F L O W A N D C O N J U G A T E H E A T TR A N S F E R U S E R I N T E R F A C E S

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Turbulent Non-Isothermal Flow Theory

Turbulent energy transport is conceptually more complicated than energy transpor t in laminar flows since the turbulence is also a form of energy. Equations for compressible turbulence are derived using the Favre average. The Favre ˜ average of a variable T is denoted T and is defined by ˜  T T = -------



where the bar denotes the usual Reynolds average. The full field is then decomposed as T

˜ T + T ''

=

With these notations the equation for total internal energy, e, becomes ˜ u ˜ u -----   ˜ ----------i i  e+ t   2 

+

 u i'' ui'' ------------------2 

+

˜ u˜ i u  i ˜ h ˜ + -----------------   u  x j  j  2 

 u j'' ui'' u i''  -------  – q j –  u j'' h'' +  ij u i'' – --------------------------  x j  2

+

˜

 u i'' ui'' 2 

+ u j -------------------

 ˜  -------  u  x j i ij

–

=

(7-7)

 u i'' u j''  

where h is the enthalpy. The vector q j

= –

T  ------ x j

(7-8)

is the laminar conductive heat flux and ij

=

2  S ij

–

2 u k ---  --------- ij 3  x k

is the laminar, viscous stress tensor. Notice that the thermal conductivity is denoted . The modeling assumptions are in large part analogous to incompressible turbulence modeling. The stress tensor –

 u i'' u'' j

is model with the Boussinesq approximation :

310 |


˜ ˜ 1 u k  2 ---------2  T  S ij –  ij – ---  k  ij  x 3   3 k

–  u i'' u'' j =  T ij =

(7-9)

where k is the turbulent kinetic energy, which in turned is defined by k

=

1 ---  u i'' ui'' 2

(7-10)

The correlation between u j'' and h'' in Equation 7-7 is the turbulent transport of heat. It is model analogous to the laminar conductive heat flux ˜

 u j'' h''

˜

 T C p T T = q T j = –  T ------- = – -------------- ------ x j Pr T  x j

(7-11)

The molecular diffusion, ij u i''

and turbulent transport term,  u j'' u i'' ui''  2

are modeled by a generalization of the molecular diffusion and turbulent transport term found in the incompressible k equation  u j'' u i'' ui''  ij ui'' – --------------------------2

=

T  k  +  -----------  k  x j

(7-12)

Inserting Equation 7-8, Equation 7-9, Equation 7-10, Equation 7-11 and Equation 7-12 into Equation 7-7 gives ˜ u ˜ u -----   ˜ ----------i i  +k  e+  t   2

˜ u ˜ u  i i ˜ ˜    = + -------  u j h + ----------- + k   x j   2  T  k   ˜  -------  – q j – q T j +   + ------------ + -------  u   +  T ij     x j   k  x j  x j i ij

(7-13)

The Favre average can also be applied to the momentum equation, which, using Equation 7-9, can be written ----- ˜    u i  + -------   u˜ j u˜ i  t  x j

= –

p  ------- + -------   ij  x j  x j

+

 T ij 

(7-14)


| 311

Taking the inner product between u˜ i and Equation 7-14 results in an equation for the resolved kinetic energy, which can be subtracted from Equation 7-13 with the following result: -----    ˜ +   + ------   ˜  ˜ +   = – ----u-˜-- j- + e k u e k p t  x j j  x j  T  k   ˜  -------  – q j – q T j +   + ------------ + -------  u   +  T ij     x j   k  x j  x j i ij

(7-15)

where the relation e˜

=

h˜ + p  

has been used. According to Wilcox (Ref. 1), it is usually a good approximation to neglect the contributions of k for flows with Mach numbers up to the supersonic range. This gives the following approximation of Equation 7-15 is u˜ j  -----   ˜  ------  ˜ ˜   ˜    T   + e + u j e = – p -------- + -------  – q j – q T j  + -------  u ij t  x j  x j  x j  x j i ij

(7-16)

Larsson (Ref. 2) suggests to make the split ij

=

˜ ij +  ij ''

Since ˜ ij »  ij ''

for all applications of engineering interest, it will follow that  ij  ˜ ij

and consequently ˜ -----   ˜  + ------   ˜ ˜  = – ----u-˜-- j- + ------    +   ------T  e u e p T  x  t  x j j  x j  x j  j

where

312 |


+

 ˜ ˜ Tot  -------  u  x j i ij

(7-17)

˜ Tot ij

=

˜  ˜ 2 u k  ---------  +  T   2 S ij –   3  x k ij 

Equation 7-17 is completely analogous to the laminar energy equation and can be expanded using the same theory (see for example Ref. 3): ˜ ˜  T  T  ˜  C p  ------- + u j -------  x j  t

=

˜    T  -------    +  T  -------  x j   x j

˜ ˜ +  ij S ij

˜ T    p ˜ p  ------ + u j ------– ---- ------˜  t  x j  T p

which is the temperature equation solved in the turbulent Non-Isothermal Flow and Conjugate Heat Transfer interfaces. TURBULENT CONDUCTIVIT Y

Kays-Crawford

This is a relatively exact model for PrT, still simple. In Ref. 4, it is compared to other models for PrT and found to be good for most kind of turbulent wall bounded flows except for liquid metals. The model is given by PrT

=

C p  T 2  1 0 . 3 C p  T      0 . 3C  ----------------------------------------------+ – 0 . 3 --------------  1 – e –      2P r Pr  T T p

T

 –1  (7-18) 

Pr T   

where the Prandtl number at infinity is PrT  0.85 and  is the conductivity. Extended Kays-Crawford

Weigand and others (Ref. 5) suggested an extension of Equation 7-18 to liquid metals by introducing PrT 

=

100  0 . 85 + -------------------------------0 . 888 C p  Re 

where Re, the Reynolds number at infinity must be provided either as a constant or as a function of the flow field. This is entered in the Model Inputs section of the Fluid feature. TEMPERATURE WALL FUNCTIONS

Analogous to the single-phase flow wall functions (see Wall Functions described for the Wall boundary condition), there is a theoretical gap between the solid wall and the computational domain of the fluid also for the temperature field. This gap is often ignored in so much that it is ignored when the computational geometr y is drawn.


| 313

The heat flux between the fluid with temperature T f and a wall with temperature T w, is:  Cp C 1 / 4 k 1 / 2  T w – T f  q wf = ----------------------------------------------------------T +

where  is the fluid density, Cp is the fluid heat capacity, C is a turbulence modeling constant, and k is the turbulent kinetic energy. T  is the dimensionless temperature and is given by (Ref. 6): + Pr  w

T +

=

+  + for  w w1

 15 Pr 2 / 3 – 500 +  +  + ---------- for  w1 w w2  +  w2

Pr + + ----- ln  w 

+  + for  w2 w

where in turn  w  C1 / 2 k +  w = ----------------------------- +  w2

=



10 10 --------PrT

+  w1

=

10 -------------

Pr1 / 3 Cp  Pr = ---------

Pr  = 15 Pr 2 / 3 – -------T--  1 + ln  1000 ---------  PrT 2 where in turn  is the thermal conductivity, and  is the von Karman constant equal to 0.41. The computational result should be checked so that the distance between the computational fluid domain and the wall, w, is almost everywhere small compared to any geometrical quantity of interest. The distance w is available as a variable on boundaries.

314 |


References for the Non-Isothermal Flow and Conjugate Heat Transfer User Interfaces 1. D.C. Wilcox, Turbulence Modeling for CFD , 2nd ed., DCW Industries, 1998. 2. J. Larsson, Numerical Simulation of Turbulent Flows for Turbine Blade Heat Transfer , Doctoral Thesis for the Degree of Doctor of Philosophy, Chalmers University of Technology, Sweden, 1998. 3. R.L. Panton, Incompressible Flow , 2nd ed., John Wiley & Sons, Inc., 1996. 4. W.M. Kays, “Turbulent Prandtl Number — Where Are We?”, ASME Journal of Heat Transfer , 116, pp. 284–295, 1994. 5. B. Weigand, J.R. Ferguson, and M.E. Crawford, “An Extended Kays and Crawford Turbulent Prandtl Number Model,” International Journal of Heat and Mass Transfer , vol. 40, no. 17, pp. 4191–4196, 1997. 6. D. Lacasse, È. Turgeon, and D. Pelletier, “On the Judicious Use of the k— Model, Wall Functions and Adaptivity,” International Journal of Thermal Sciences, vol. 43, pp. 925–938, 2004.

REFERENCES FOR THE NON-ISOTHERMAL FLOW AND CONJUGATE HEAT TRANSFER USER INTERFACES

| 315

316 |


8

Glossary This Glossary of Terms contains application-specific terms used in the Heat Transfer Module software and documentation. For information about terms relating to finite element modeling, mathematics, geometry, and CAD, see the glossary in the COMSOL Multiphysics Reference Manual . For references to more information about a term, see the Index in this or other manuals.

317

Glossary of Terms The condition of exhibiting properties with different values when measured in different directions. anisotropy

An alternative form of the heat equation that incorporates the effects of blood perfusion, metabolism, and external heating. The equation describes heat transfer in tissue. bioheat equation

A blackbody is a surface that absorbs all incoming radiation; that is, it does not reflect radiation. The blackbody also emits the maximum possible radiation. blackbody

Heat conduction takes place through different mechanisms in different media. Theoretically, conduction takes place through collisions of molecules in a gas, through oscillations of each molecule in a “cage” formed by its nearest neighbors in a fluid, and by the electrons carrying heat in metals or by molecular motion in other solids. Typical for heat conduction is that the heat flux is proportional to the temperature gradient. conduction

Heat advection takes place through the net displacement of a fluid, which translates the heat content in a fluid through the fluid's own velocity. advection

The term convection is used for the heat dissipation from a solid sur face to a fluid, where the heat transfer coefficient and the temperature difference across a fictitious film describes the flux. convection

discrete ordinates method

This order defines the discretization of the radiative

intensity direction. A dimensionless factor between 0 and 1 that specifies the ability of a surface to emit radiative energy. The value 1 corresponds to an ideal surface, which emits the maximum possible radiative energy. emissivity

heat capacity

See specific heat .

A highly conductive layer is a thin layer on a boundary. It has much higher thermal conductivity than the material in the adjacent domain. This allows for the assumption that the temperature is constant across the layer’s thickness. highly conductive layer

318 |

CHAPTER 8: GLOSSARY

The general Heat Transfer physics interface supports heat transfer in highly conductive layers. irradiation

The total radiation that arrives at a surface.

The equations for the momentum balances coupled to the equation of continuity for a Newtonian incompressible fluid are often referred to as the Navier-Stokes equations. The most general versions of Navier-Stokes equations do however describe fully compressible flows. Navier-Stokes equations

An opaque body does not transmit any radiative heat flux, that is, the surface of an opaque body has a transmissivity equal to 0. opaque material

Heat transfer by radiation takes place through the transport of photons, which can be absorbed or reflected on solid surfaces. The Heat Transfer Module includes surface-to-surface radiation, which accounts for effects of shading and reflections between radiating surfaces. It also includes surface-to-ambient radiation where the ambient radiation can be fixed or given by an arbitrary function. radiation

participating media

A media that can absorb, emit, and scatter thermal radiation.

The total radiation that leaves a surface, that is, both the emitted and the reflected radiation. radiosity

Refers to the quantity that represents the amount of heat required to change one unit of mass of a substance by one degree. It has units of energy per mass per degree. This quantity is also called specific heat or specific heat capacity . specific heat

specific heat capacity

See specific heat .

“Thin” means that the shell is thin enough, or has high enough thermal conductivity, to allow for the assumption that the temperature is constant across the shell’s thickness. See also highly conductive layer . thin conductive shell

A transparent body transmits radiative heat flux, that is, the surface of a transparent body has a transmissivity greater than 0. transparent material

The definition of thermal conductivity is given by Fourier’s law, which relates the heat flux to the temperature gradient. In this equation, the thermal conductivity is the proportional constant. thermal conductivity

G L O S S A R Y O F TE R M S

| 319

320 |

CHAPTER 8: GLOSSARY

I n d e x 1D and 2D models

nitf interfaces 292

out-of-plane heat transfer 156

radiation in participating media 201

3D models

spf 223

thin conductive shells 146 A

absolute pressure 91, 139, 224– 225, 295

boundary selection 10

acceleration of gravity 55

boundary stress (node) 237

accurate flux variables 32

Brinell hardness 50

advanced settings 9

bulk velocity 55

AKN model 270

buoyancy force 55

apparent heat capacity 46

C

Carreau model 259, 299

arterial blood temperature 134

cell Reynolds number 234

axisymmetric geometries 166, 193, 197

CFL number

azimuthal sectors 193 B

surface-to-surface radiation 167

bioheat (node) 134 bioheat transfer interface 132 theory 65 biological tissue 133

pseudo time stepping, and 218 CFL number, pseudo time stepping, and 288

change effective thickness (node) 154 change thickness (node)

black walls 205

heat transfer in thin shells 152

blackbody intensity 208– 209


blackbody radiation 175

characteristic length 55

blackbody radiation intensity 209

Charron’s relation 52

blackbody radiation intensity, definition

Clausius-Clapeyron formulation 42

207

blood, bioheat properties 133 boundary conditions heat equation, and 34 heat transfer coefficients, and 53 radiation groups 197

coefficient of volumetric thermal expansion 56 conductive heat flux variable 26 conjugate heat transfer interface theory 308 conjugate heat transfer interfaces 285

boundary heat source (node) 104

conservation of energy 19

boundary heat source variable 33

consistent stabilization settings 9

boundary nodes

constraint settings 9

bioheat interface 132 heat transfer 84 heat transfer in porous media 137 heat transfer in thin shells 148

continuity (node) heat transfer 106 continuity on interior boundary (node) 206

convection, natural and forced 54

INDEX|

i

convective heat flux (node) 116

nitf interfaces 292

convective heat flux variable 27, 30


convective out-of-plane heat flux varia-

spf interfaces 223

ble 28


Cooper-Mikic Yovanovich (CMY) correlation 49

domain selection 10 E

coordinate system selection 10

edge heat flux (node) 121

crosswind diffusion

edge heat source (node) 154

definition 39

edge nodes

crosswind diffusion, consistent stabiliza-

heat transfer 84

tion method 39

heat transfer in porous media 137

curves, fan 254 D


Dalton’s law 41

nitf interfaces 292

del operators 62


density, blood 135


dev_in and dev_out variables 253

edge selection 10

diffuse gray radiation model 165

edge surface-to-ambient radiation

diffuse mirror (node) 173

(node) 125

diffuse spectral radiation model 165

edges

diffuse-gray surface 188– 1189 89

heat flux 121

diffuse-spectral 190

temperature 123

diffuse-spectral surface 190

effective volumetric heat capacity 88

dimensionless distance to cell center var-

elastic contribution to entropy 113, 304

iable 272

elevation 193

Dirac pulse 47

emailing COMSOL 12

direct area integration, axisymmetric ge-

emission, radiation and 207

ometry and 193

equation view 9

direct area integration, radiation settings

equivalent thermal conductivity 66 equivalent volumetric heat capacity 66

165

Dirichlet condition 255

evaluating view factors 192

discrete ordinates method (DOM) 211

exit length 235

discretization 9

expanding sections 9

dispersivities, porous media 143

external radiation source (node) 178

documentation 11 domain heat source variable 33 domain nodes heat transfer 84 heat transfer in porous media 137

ii | I N D E X

eddy viscosity 262

F

fan (node) 240 fan curves inlet boundary condition 241 theory 254 Favre average 263, 310

first law of thermodynamics 19

heat transfer coefficients

flow continuity (node) 246


fluid (node) 293

theory 54

fluid flow

heat transfer in fluids (node) 89

selecting interfaces 280

heat transfer in participating media inter-

turbulent flow theory 260

face 161

fluid properties (node) 224

heat transfer in porous media (node) 137

Fourier’s law 20

heat transfer in porous media interface

frames, moving 68 G

136

theory 66

Galerkin constraints 100

heat transfer in solids (node) 86

gap conductance 51

heat transfer in thin shells interface 146

general stress (boundary stress condi-

theory 156

tion) 238

heat transfer interfaces 78, 81

geometric entity selection 10

selecting 280

Grashof number 55

theory 18

gravity 55

heat transfer with phase change (node)

gray walls 204

96

graybody radiation 175, 190– 1191 91

heat transfer with surface-to-surface ra-

gray-diffuse parallel plate model 51

diation interface 160

grille (node) 245

Heaviside function 47

grouping boundaries 197

hemicubes, axisymmetric geometry and

guidelines, solving surface-to-surface ra-

193

diation problems 196

hemicubes, radiation settings 165 H

heat equation, highly conductive layers

hide button 9

and 62

highly conductive layer (node) 118

heat flux (node) 101

highly conductive layers, defined 61

heat transfer in thin shells 149 heat flux, theory 21 heat source (node) heat transfer 94 heat transfer in thin shells 151 heat sources defining as total power 94, 105, 111, 155

edges, thin shells 154 highly conductive layers 120 line and point 111 point, thin shells 155

I

incident intensity (node) 205 inconsistent stabilization settings 9 inflow heat flux (node) 114 initial values (node) heat transfer 93 heat transfer in thin shells 152 nitf interfaces 303 spf interfaces 227 inlet (boundary stress condition) 247 inlet (node) 231 insulation/continuity insulation/continuity (node) 153

INDEX|

iii

interior fan (node) 242

mechanisms of heat transfer 18

interior wall (node) 302

metabolic heat source 135

spf interfaces 244

Mikic elastic correlation 50

internal boundary heat flux variables 30

Model Library 12

Internet resources 11

Model Library examples

isotropic diffusion, inconsistent stabiliza-

bioheat transfer interface 132

tion methods 40 K

consistent stabilization 39 convective heat flux 117

Karman constant 314

heat transfer in fluids 90

Kays-Crawford models 283, 313


k-epsilon turbulence model 219, 264

heat transfer in solids 88

knowledge base, COMSOL 13

heat transfer in thin shells 148 L

laminar flow (nitf) interface, conjugate

heat transfer with surface-to-surface

heat transfer 289

radiation 167

laminar flow (nitf) interface, non-isother-

highly conductive layers 118

mal flow 285

laminar flow 219

laminar flow interface 216

nitf interfaces 288– 2289 89

laminar inflow (inlet boundary condition)

out-of-plane convective heat flux 127

232


laminar outflow (outlet boundary condi-

surface-to-ambient radiation 104

tion) 235

thermal contact 52

Latitude 180

thermodynamics 88

latitude 193

translational motion 89

layer heat source (node) 120

turbulent flow, k-epsilon (nitf) inter-

leaking wall, wall boundary condition 229

face 292

line heat source (node) 111

turbulent flow, k-epsilon (spf) 221

line heat source variable 33

moist air 93, 141

local CFL number 218, 275, 288

moisture content 93, 141

Longitude 180

moving frames 68

longitude 193

moving mesh, heat transfer and 48

low re k-epsilon turbulence model 221

moving wall (wall functions), boundary

low Reynolds number

condition 230

k-epsilon turbulence theory 270

moving wall, wall boundary condition

neglect inertial term 286 M

229, 245

material frame

MPH-files 12

heat transfer 73 mean effective thermal conductivity 87 mean effective thermal diffusivity 87

iv | I N D E X

mutual irradiation 194 N

nabla operators 62 natural and forced convection 54

Neumann condition 268 Newtonian model 258 no slip, interior wall boundary condition 244

pair boundary heat source (node) 104 pair nodes heat transfer 84 heat transfer in porous media 137

no slip, wall boundary condition 228


no viscous stress (outlet boundary con-

nitf interfaces 292

dition) 234 non-isothermal flow interface theory 308

radiation in participating media 201 spf interfaces 223 surface-to-surface radiation 167

non-isothermal flow interfaces 285

pair selection 10

non-Newtonian power law and Carreau

pair thermal contact (node) 108

model 298 normal conductive heat flux variable 29 normal convective heat flux variable 29 normal stress (boundary condition) 232 normal stress, normal flow (boundary stress condition) 238

O

P

pair thin thermally resistive layer (node) 106

participating media, radiative heat transfer 207 Pennes’ approximation 65 perfusion rate, blood 134

normal total energy flux variable 30

periodic flow condition (node) 239

normal translational heat flux variable 29

periodic heat condition (node) 104

Nusselt number 55

phase transitions 97

open boundary (boundary stress condition) 247 open boundary (node) heat transfer 115 single-phase flow 237 outflow (node) 100 outlet (boundary stress condition) 247 outlet (node) 233 out-of-plane convective heat flux (node) 127

out-of-plane heat flux (node) 130 out-of-plane heat transfer change thickness 130 general theory 63 thin shells theory 156 out-of-plane inward heat flux variable 29 out-of-plane radiation (node) 129 override and contribution 9

point heat flux (node) 122 point heat source (node) 155 heat transfer 112 point heat source variable 33 point nodes heat transfer 84 heat transfer in porous media 137 heat transfer in thin shells 148 nitf interfaces 292 radiation in participating media 201 spf interfaces 223 surface-to-surface radiation 167 point selection 10 point surface-to-ambient radiation (node) 125 point temperature (node) 124 points heat flux 122

INDEX|

v

temperature 124

radiosity method 188

power law, non-Newtonian 298

RANS

power law, single-phase flow theory 258

theory, single-phase flow 261

Prandtl number 55, 283, 313

ratio of specific heats 89, 138

prescribed radiosity (node) 174

Rayleigh number 55

pressure (outlet boundary condition)

Refractive index 200 refractive index 189, 208– 2209 09

234

pressure point constraint (node) 246

relative humidity 93, 141

pressure work (node)

Reynolds number 55

heat transfer 112

extended Kays-Crawford 313

nitf interfaces 304

low, turbulence theory 270

pressure, no viscous stress (inlet and

turbulent flow theory 260

outlet boundary conditions) 232

Reynolds stress tensor 262, 265

pseudo time stepping

Reynolds-averaged Navier-Stokes. See

advanced settings 218, 288, 291

RANS.

turbulent flow theory 275 pseudoplastic fluids 258 pumps, lumped curves and 254 R

radiation axisymmetric geometries, and 166, 193, 197

participating media 207 radiation group (node) 177

S

scalar density variables, frames and 72 scattering, radiation and 207 sectors, azimuthal 193 selecting conjugate heat transfer interfaces 280 heat transfer interfaces 78, 81 non-isothermal flow interfaces 280

radiation groups 197

settings windows 9

radiation in participating media (node)

shear rate magnitude variable 225

heat transfer 202 radiation in participating media interface 199

shear thickening fluids 258 shell thickness 146 shells, conductive 156

radiation intensity, for blackbody 207

show button 9

radiation, out-of-plane 129

single-phase flow

radiative conductance 51 radiative heat flux variable 30 radiative heat, theory 36 radiative out-of-plane heat flux variable 28

vi | I N D E X

Rodriguez formula 209

turbulent flow theory 260 single-phase flow interface laminar flow 216 sliding wall (wall functions), boundary condition 230

radiative transfer equation 208

sliding wall, wall boundary condition 229

radiosity 189

slip, wall boundary condition 229, 244

radiosity expressions 175

Solar position 193

solving surface-to-surface radiation problems 196

heat transfer 18

source terms, bioheat 134

heat transfer coefficients 54

spatial frame


heat transfer and 73


specific heat capacity, definition 20

non-isothermal flow 308

specific heat, blood 134


spf.sr variable 225


stabilization settings 9

turbulent flow k-epsilon model 260

stabilization techniques

turbulent flow low re k-epsilon model

crosswind diffusion 39 static pressure curves 241 strain-rate tensors 113 streamline diffusion, consistent stabilization methods 39 sun position 181 surface emissivity 189 surface-to-ambient radiation (node) 103, 153

edges and points 125

260

thermal conductivity components, thin shells 157 thermal conductivity, frames and 73 thermal conductivity, mean effective 87 thermal contact (node) 108 theory 48 thermal diffusivity 87 thermal dispersion (node) 143 thermal expansivity 55

surface-to-surface radiation (node) 168

thermal friction 51

surface-to-surface radiation interface

thermal insulation (node) 99

164

theory 194

T

heat equation definition 19

thin conductive layer (node) 150 thin conductive layers, definition 61

swirl flow 266

thin thermally resistive layer (node) 106

symmetry (node) 236

Time zone 180

heat transfer 101

time zone 193

symmetry, flow (node) 306

total energy flux variable 28

symmetry, heat (node) 305

total heat flux 102

technical support, COMSOL 12 temperature (node) 99, 123 tensors Reynolds stress 265 strain-rate 113 viscous stress theory 257 theory bioheat transfer 65 conjugate heat transfer interface 308

total heat flux variable 26 total normal heat flux variable 29 total power 94, 105, 111, 155 traction boundary conditions 237 translational heat flux variable 27 translational motion (node) 88 turbulence models k-epsilon 219, 264 low re k-epsilon 221

INDEX|

vii

Heat Transfer Module Users Guide with comsol.

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