VERSION 4.3

User s Guide

Heat Transfer Module

C o n t a c t I n f o r m a t i o n

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Part No. CM020801

H e a t T r a n s f e r M o d u l e U s e r s G u i d e 19982012 COMSOL

Protected by U.S. Patents 7,519,518; 7,596,474; and 7,623,991. Patents pending.

This Documentation and the Programs described herein are furnished under the COMSOL Software License Agreement (www.comsol.com/sla) and may be used or copied only under the terms of the license agree-ment.

COMSOL, COMSOL Desktop, COMSOL Multiphysics, and LiveLink are registered trademarks or trade-marks of COMSOL AB. Other product or brand names are trademarks or registered trademarks of their respective holders.

Version: May 2012 COMSOL 4.3

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C o n t e n t s

C h a p t e r 1 : I n t r o d u c t i o n

About the Heat Transfer Module 12

Why Heat Transfer is Important to Modeling . . . . . . . . . . . . 12

How the Heat Transfer Module Improves Your Modeling. . . . . . . . 13

Heat Transfer Module Physics Interface Guide . . . . . . . . . . . . 13

The Heat Transfer Module Study Capabilities by Interface . . . . . . . 16

Model Builder Options for Physics Feature Node Settings Windows . . . 18

Where Do I Access the Documentation and Model Library? . . . . . . 19

Typographical Conventions . . . . . . . . . . . . . . . . . . . 21

Overview of the Users Guide 26

C h a p t e r 2 : H e a t T r a n s f e r T h e o r y

Theory for the Heat Transfer Interfaces 30

What is Heat Transfer? . . . . . . . . . . . . . . . . . . . . 30

The Heat Equation . . . . . . . . . . . . . . . . . . . . . . 31

A Note on Heat Flux . . . . . . . . . . . . . . . . . . . . . 33

Heat Flux Variables and Heat Sources . . . . . . . . . . . . . . . 35

About the Boundary Conditions for the Heat Transfer Interfaces . . . . 41

Radiative Heat Transfer in Transparent Media . . . . . . . . . . . . 44

Consistent and Inconsistent Stabilization Methods for the Heat Transfer

Interfaces. . . . . . . . . . . . . . . . . . . . . . . . . . 46

References for the Heat Transfer Interfaces . . . . . . . . . . . . . 48

About Infinite Elements 49

Modeling Unbounded Domains . . . . . . . . . . . . . . . . . 49

Known Issues When Modeling Using Infinite Elements. . . . . . . . . 51

About the Heat Transfer Coefficients 53

Heat Transfer Coefficient Theory . . . . . . . . . . . . . . . . 54C O N T E N T S | 3

4 | C O N T E N T SNature of the Flowthe Grashof Number . . . . . . . . . . . . . 55

Available Heat Transfer Coefficients. . . . . . . . . . . . . . . . 56

References for the Heat Transfer Coefficients . . . . . . . . . . . . 60

About Highly Conductive Layers 61

Theory of Out-of-Plane Heat Transfer 64

Equation Formulation . . . . . . . . . . . . . . . . . . . . . 64

Activating Out-of-Plane Heat Transfer and Thickness . . . . . . . . . 65

Theory for the Bioheat Transfer Interface 66

Reference for the Bioheat Interface . . . . . . . . . . . . . . . . 66

Theory for the Heat Transfer in Porous Media Interface 67

C h a p t e r 3 : H e a t T r a n s f e r B r a n c h

The Heat Transfer Interfaces 70

Accessing the Heat Transfer Interfaces via the Model Wizard . . . . . . 70

The Heat Transfer Interface 73

Heat Transfer in Solids . . . . . . . . . . . . . . . . . . . . . 77

Translational Motion . . . . . . . . . . . . . . . . . . . . . 78

Pressure Work . . . . . . . . . . . . . . . . . . . . . . . 79

Opaque . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Heat Transfer in Fluids . . . . . . . . . . . . . . . . . . . . . 80

Viscous Heating . . . . . . . . . . . . . . . . . . . . . . . 83

Heat Source. . . . . . . . . . . . . . . . . . . . . . . . . 84

Radiation in Participating Media . . . . . . . . . . . . . . . . . 85

Infinite Elements . . . . . . . . . . . . . . . . . . . . . . . 86

Manual Scaling . . . . . . . . . . . . . . . . . . . . . . . . 87

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . . 88

Boundary Conditions for the Heat Transfer Interfaces . . . . . . . . . 88

Temperature . . . . . . . . . . . . . . . . . . . . . . . . 89

Thermal Insulation . . . . . . . . . . . . . . . . . . . . . . 90

Outflow . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 91

Heat Flux. . . . . . . . . . . . . . . . . . . . . . . . . . 91

Inflow Heat Flux . . . . . . . . . . . . . . . . . . . . . . . 92

Open Boundary . . . . . . . . . . . . . . . . . . . . . . . 93

Surface-to-Ambient Radiation . . . . . . . . . . . . . . . . . . 93

Periodic Heat Condition . . . . . . . . . . . . . . . . . . . . 94

Boundary Heat Source. . . . . . . . . . . . . . . . . . . . . 94

Heat Continuity . . . . . . . . . . . . . . . . . . . . . . . 94

Pair Thin Thermally Resistive Layer . . . . . . . . . . . . . . . . 95

Thin Thermally Resistive Layer. . . . . . . . . . . . . . . . . . 96

Opaque Surface . . . . . . . . . . . . . . . . . . . . . . . 98

Incident Intensity . . . . . . . . . . . . . . . . . . . . . . . 99

Continuity on Interior Boundary . . . . . . . . . . . . . . . . 100

Line Heat Source . . . . . . . . . . . . . . . . . . . . . . 100

Point Heat Source . . . . . . . . . . . . . . . . . . . . . 101

Convective Cooling . . . . . . . . . . . . . . . . . . . . . 101

Highly Conductive Layer Features 103

Highly Conductive Layer . . . . . . . . . . . . . . . . . . 103

Layer Heat Source . . . . . . . . . . . . . . . . . . . . . 105

Edge Heat Flux or Point Heat Flux . . . . . . . . . . . . . . . 105

Edge Temperature or Point Temperature . . . . . . . . . . . . 106

Edge Surface-to-Ambient or Point Surface-to-Ambient Radiation . . . 107

Out-of-Plane Heat Transfer Features 109

Out-of-Plane Convective Cooling . . . . . . . . . . . . . . . 109

Out-of-Plane Radiation . . . . . . . . . . . . . . . . . . . 110

Out-of-Plane Heat Flux . . . . . . . . . . . . . . . . . . . 111

Change Thickness . . . . . . . . . . . . . . . . . . . . . 112

The Bioheat Transfer Interface 114

Biological Tissue . . . . . . . . . . . . . . . . . . . . . . 115

Bioheat . . . . . . . . . . . . . . . . . . . . . . . . . 116

Boundary Conditions for the Bioheat Transfer Interface . . . . . . . 116

The Heat Transfer in Porous Media Interface 118

Porous Matrix . . . . . . . . . . . . . . . . . . . . . . . 119

Heat Transfer in Fluids . . . . . . . . . . . . . . . . . . . 120C O N T E N T S | 5

6 | C O N T E N T SThermal Dispersion . . . . . . . . . . . . . . . . . . . . . 121

Heat Source. . . . . . . . . . . . . . . . . . . . . . . . 122

C h a p t e r 4 : H e a t T r a n s f e r i n T h i n S h e l l s

The Heat Transfer in Thin Shells Interface 124

Thin Conductive Layer. . . . . . . . . . . . . . . . . . . . 125

Heat Source. . . . . . . . . . . . . . . . . . . . . . . . 126

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 127

Change Thickness . . . . . . . . . . . . . . . . . . . . . 127

Other Boundary Conditions . . . . . . . . . . . . . . . . . 127

Edge and Point Conditions . . . . . . . . . . . . . . . . . . 128

Insulation/Continuity . . . . . . . . . . . . . . . . . . . . 128

Radiation . . . . . . . . . . . . . . . . . . . . . . . . . 128

Change Effective Thickness . . . . . . . . . . . . . . . . . . 129

Edge Heat Source . . . . . . . . . . . . . . . . . . . . . 129

Point Heat Source . . . . . . . . . . . . . . . . . . . . . 130

Theory for the Heat Transfer in Thin Shells Interface 131

About Thin Conductive Shells . . . . . . . . . . . . . . . . . 131

Heat Transfer Equation in Thin Conductive Shell . . . . . . . . . . 131

Thermal Conductivity Tensor Components . . . . . . . . . . . . 132

C h a p t e r 5 : R a d i a t i o n H e a t T r a n s f e r B r a n c h

The Surface-To-Surface Radiation Interface 136

Surface-to-Surface Radiation (Boundary Condition) . . . . . . . . . 138

Opaque . . . . . . . . . . . . . . . . . . . . . . . . . 140

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 140

Reradiating Surface . . . . . . . . . . . . . . . . . . . . . 140

Prescribed Radiosity . . . . . . . . . . . . . . . . . . . . 141

Radiation Group . . . . . . . . . . . . . . . . . . . . . . 142

External Radiation Source . . . . . . . . . . . . . . . . . . 143

The Radiation in Participating Media Interface 145

Radiation in Participating Media . . . . . . . . . . . . . . . . 147

Opaque Surface . . . . . . . . . . . . . . . . . . . . . . 148

Incident Intensity . . . . . . . . . . . . . . . . . . . . . . 149

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 150

The Heat Transfer with Radiation in Participating Media Interface

151

Domain and Boundary Conditions . . . . . . . . . . . . . . . 153

Edge, Point, and Pair Conditions . . . . . . . . . . . . . . . . 153

Theory for the Radiative Heat Transfer Interfaces 154

The Radiosity Method . . . . . . . . . . . . . . . . . . . . 154

View Factor Evaluation . . . . . . . . . . . . . . . . . . . 156

Radiation and Participating Media Interactions . . . . . . . . . . . 157

Radiative Transfer Equation . . . . . . . . . . . . . . . . . . 158

Boundary Condition for the Transfer Equation. . . . . . . . . . . 159

Heat Transfer Equation in Participating Media . . . . . . . . . . . 160

Discrete Ordinates Method . . . . . . . . . . . . . . . . . 160

Theory for the Surface-to-Surface Radiation Interface 162

About Surface-to-Surface Radiation . . . . . . . . . . . . . . . 162

Solving for the Radiosity . . . . . . . . . . . . . . . . . . . 164

About the Surface-to-Surface Radiation Boundary Conditions . . . . . 164

Guidelines for Solving Surface-to-Surface Radiation Problems . . . . . 165

Radiation Group Boundaries . . . . . . . . . . . . . . . . . 166

References for the Surface-to-Surface Radiation Interface . . . . . . 167

C h a p t e r 6 : S i n g l e - P h a s e F l o w B r a n c h

The Single-Phase Flow, Laminar Flow Interface 170

The Laminar Flow Interface . . . . . . . . . . . . . . . . . . 170

Fluid Properties . . . . . . . . . . . . . . . . . . . . . . 173

Volume Force . . . . . . . . . . . . . . . . . . . . . . . 175

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 176C O N T E N T S | 7

8 | C O N T E N T SThe Single-Phase Flow, Turbulent Flow Interfaces 177

The Turbulent Flow, k- Interface . . . . . . . . . . . . . . . 177The Turbulent Flow, Low Re k- Interface . . . . . . . . . . . . 178

Boundary Conditions for the Single-Phase Flow Interfaces 180

Wall. . . . . . . . . . . . . . . . . . . . . . . . . . . 181

Interior Wall . . . . . . . . . . . . . . . . . . . . . . . 184

Inlet . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

Outlet . . . . . . . . . . . . . . . . . . . . . . . . . . 192

Symmetry . . . . . . . . . . . . . . . . . . . . . . . . 195

Open Boundary . . . . . . . . . . . . . . . . . . . . . . 196

Boundary Stress . . . . . . . . . . . . . . . . . . . . . . 198

Periodic Flow Condition . . . . . . . . . . . . . . . . . . . 200

Flow Continuity . . . . . . . . . . . . . . . . . . . . . . 201

Pressure Point Constraint . . . . . . . . . . . . . . . . . . 201

Fan . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

Theory for the Laminar Flow Interface 204

Theory for the Pressure, No Viscous Stress Condition . . . . . . . 204

Theory for the Laminar Inflow Condition . . . . . . . . . . . . 205

Theory for the Laminar Outflow Condition. . . . . . . . . . . . 205

Theory for the Fan Defined on an Interior Boundary . . . . . . . . 206

Theory for the Fan and Grill Inlet and Outlet Condition . . . . . . . 207

Theory for the No Viscous Stress Condition . . . . . . . . . . . 209

Theory for the Turbulent Flow Interfaces 211

Turbulence Modeling . . . . . . . . . . . . . . . . . . . . 211

The k-Turbulence Model . . . . . . . . . . . . . . . . . . 214The Low Reynolds Number k- Turbulence Model . . . . . . . . . 219Inlet Values for the Turbulence Length Scale and Intensity . . . . . . 222

Pseudo Time Stepping for Turbulent Flow Models . . . . . . . . . 222

References for the Single-Phase Flow, Turbulent Flow Interfaces . . . . 223

C h a p t e r 7 : C o n j u g a t e H e a t T r a n s f e r B r a n c h

The Conjugate Heat Transfer Interfaces 226

The Non-Isothermal Flow and Conjugate Heat Transfer, Laminar

Flow Interfaces 228

The Non-Isothermal Flow, Laminar Flow Interface . . . . . . . . . 228

The Conjugate Heat Transfer, Laminar Flow Interface . . . . . . . . 231

The Non-Isothermal Flow and Conjugate Heat Transfer, Turbulent

Flow Interfaces 233

The Turbulent Flow, k- and Turbulent Flow Low Re k- Interfaces. . . 233

Shared Interface Features 236

Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . 236

Wall. . . . . . . . . . . . . . . . . . . . . . . . . . . 240

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 242

Pressure Work . . . . . . . . . . . . . . . . . . . . . . 242

Viscous Heating . . . . . . . . . . . . . . . . . . . . . . 243

Theory for the Non-Isothermal Flow and Conjugate Heat Transfer

Interfaces 244

Turbulent Non-Isothermal Flow Theory . . . . . . . . . . . . . 246

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

250

C h a p t e r 8 : M a t e r i a l s

Material Library and Databases 252

About the Material Databases . . . . . . . . . . . . . . . . . 252

About Using Materials in COMSOL . . . . . . . . . . . . . . . 254

Opening the Material Browser . . . . . . . . . . . . . . . . 257

Using Material Properties . . . . . . . . . . . . . . . . . . 258C O N T E N T S | 9

10 | C O N T E N T SLiquids and Gases Material Database 259

Liquids and Gases Materials . . . . . . . . . . . . . . . . . . 259

References for the Liquids and Gases Material Database . . . . . . . 261

C h a p t e r 9 : G l o s s a r y

Glossary of Terms 264

1I n t r o d u c t i o nThis 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 overview with links to each chapter in this guide.

About the Heat Transfer Module

Overview of the Users Guide 11

12 | C H A P T E RAbou t t h e Hea t T r a n s f e r Modu l e

In this section:

Why Heat Transfer is Important to Modeling

How the Heat Transfer Module Improves Your Modeling

Heat Transfer Module Physics Guide

The Heat Transfer Module Study Capabilities

Show More Physics Options

Where Do I Access the Documentation and Model Library?

Typographical Conventions

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 importance, 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.

Overview of the Physics Interfaces and Building a COMSOL Model in the COMSOL Multiphysics Users Guide

See Also 1 : I N T R O D U C T I O N

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 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 physicsa versatile combination increasing the accuracy and predicting power of your models.

This Users Guide 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 flowconjugate 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 where heat transfer plays an important roleelectronics and power systems, processing and manufacturing, and medical technologyand 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?.

Heat Transfer Module Physics Guide

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

14 | C H A P T E RIf 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 only difference is that Fluid is selected as the Default model in the former case. If Heat transfer in solids is selected as the default model, the interface changes to a Conjugate Heat Transfer interface.

Study Types in the COMSOL Multiphysics Reference Guide

Available Study Types in the COMSOL Multiphysics Users Guide

PHYSICS ICON TAG SPACE DIMENSION

PRESET STUDIES

Fluid Flow

Single-Phase Flow

Single-Phase Flow, Laminar Flow*

spf 3D, 2D, 2D axisymmetric

stationary; time dependent

Turbulent Flow, k- spf 3D, 2D, 2D axisymmetric

stationary; time dependent

Turbulent Flow, Low Re k- spf 3D, 2D, 2D axisymmetric

stationary with initialization; transient with initialization

Non-Isothermal Flow

Laminar Flow nitf 3D, 2D, 2D axisymmetric

stationary; time dependent

Turbulent Flow, k- nitf 3D, 2D, 2D axisymmetric

stationary; time dependent

Note

See Also 1 : I N T R O D U C T I O N

Turbulent Flow, Low Re k- nitf 3D, 2D, 2D axisymmetric

stationary with initialization; transient with initialization

Heat Transfer

Heat Transfer in Solids* ht all dimensions stationary; time dependent

Heat Transfer in Fluids* ht all dimensions stationary; time dependent

Heat Transfer in Porous Media ht all dimensions stationary; time dependent

Bioheat Transfer ht all dimensions stationary; time dependent

Heat Transfer in Thin Shells (also called Thin Conductive Shell)

htsh 3D stationary; time dependent

Conjugate Heat Transfer

Laminar Flow nitf 3D, 2D, 2D axisymmetric

stationary; time dependent

Turbulent Flow, k- nitf 3D, 2D, 2D axisymmetric

stationary; time dependent

Turbulent Flow, Low Re k- nitf 3D, 2D, 2D axisymmetric

stationary with initialization; transient with initialization

Radiation

Heat Transfer with Surface-to-Surface Radiation

ht all dimensions stationary; time dependent

Heat Transfer with Radiation in Participating Media

ht 3D, 2D stationary; time dependent

Surface-to-Surface Radiation rad all dimensions stationary; time dependent

Radiation in Participating Media

rpm 3D, 2D stationary; time dependent

PHYSICS ICON TAG SPACE DIMENSION

PRESET STUDIESA B O U T T H E H E A T TR A N S F E R M O D U L E | 15

16 | C H A P T E RThe Heat Transfer Module Study Capabilities

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

Electromagnetic Heating

Joule Heating* jh all dimensions stationary; time dependent

* This is an enhanced interface, which is included with the base COMSOL package but has added functionality for this module.

PHYSICS ICON TAG SPACE DIMENSION

PRESET STUDIES

Study Types in the COMSOL Multiphysics Reference Guide

Available Study Types in the COMSOL Multiphysics Users GuideSee Also

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

PHYSICS TAG DEPENDENT VARIABLES PRESET STUDIES*

ST

AT

ION

AR

Y

TIM

E D

EP

EN

DE

NT

ST

AT

ION

AR

Y W

ITH

IN

ITIA

LIZ

AT

ION

TR

AN

SIE

NT

WIT

H I

NIT

IAL

IZA

TIO

N

FLUID FLOW>SINGLE-PHASE FLOW

Laminar Flow spf u, p

Turbulent Flow, k- spf u, p, k, ep

Turbulent Flow, Low Re k- 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 1 : I N T R O D U C T I O N

Turbulent Flow, Low Re k- nitf u, p, k, ep, G, T HEAT TRANSFER

Heat Transfer in Solids** ht T

Heat Transfer in Fluids** ht T

Heat Transfer in Porous Media**

ht T

Bioheat Transfer** ht T

Heat Transfer in Thin Shells htsh T 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 Surface-to-Surface Radiation**

ht T, J

Heat Transfer with Radiation in Participating Media**

ht T, I (radiative intensity)

Surface-to-Surface Radiation rad J

Radiation in Participating Media

rpm I (radiative intensity)

HEAT TRANSFER>ELECTROMAGNETIC HEATING

Joule Heating** jh T, V

* 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.

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

PHYSICS TAG DEPENDENT VARIABLES PRESET STUDIES*

ST

AT

ION

AR

Y

TIM

E D

EP

EN

DE

NT

ST

AT

ION

AR

Y W

ITH

IN

ITIA

LIZ

AT

ION

TR

AN

SIE

NT

WIT

H I

NIT

IAL

IZA

TIO

N

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 | 17

18 | C H A P T E RShow More Physics Options

There are several features available on many physics interfaces or individual nodes. This section is a short overview of the options and includes links to the COMSOL Multiphysics Users Guide or COMSOL Multiphysics Reference Guide where additional information is available.

To display additional features for the physics interfaces and feature nodes, click the Show button ( ) on the Model Builder and then select the applicable option.

After clicking the Show button ( ), some sections display on the settings window when a node is clicked and other features 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 physics 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 physics nodes in the Model Builder.

Availability of each feature, and whether it is described for a particular physics node, is based on the individual physics selected. For example, the Discretization, Advanced

The links to the features described in the COMSOL Multiphysics Users Guide and COMSOL Multiphysics Reference 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, select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree.

Important

Tip 1 : I N T R O D U C T I O N

Settings, Consistent Stabilization, and Inconsistent Stabilization sections are often described individually throughout the documentation as there are unique settings.

Where Do I Access the Documentation and Model Library?

A number of Internet resources provide more information about COMSOL Multiphysics, including licensing and technical information. The electronic

SECTION CROSS REFERENCE LOCATION IN COMSOL MULTIPHYSICS USER GUIDE OR REFERENCE GUIDE

Show More Options and Expand Sections

Showing and Expanding Advanced Physics Sections

The Model Builder Window

Users Guide

Discretization Show Discretization

Element Types and Discretization

Users Guide

Finite Elements

Discretization of the Equations

Reference Guide

Discretization - Splitting of complex variables

Compile Equations Reference Guide

Pair Selection Identity and Contact Pairs

Specifying Boundary Conditions for Identity Pairs

Users Guide

Consistent and Inconsistent Stabilization

Show Stabilization Users Guide

Stabilization Techniques

Numerical Stabilization

Reference Guide

Geometry Working with Geometry Users Guide

Constraint Settings Using Weak Constraints Users GuideA B O U T T H E H E A T TR A N S F E R M O D U L E | 19

20 | C H A P T E Rdocumentation, Dynamic Help, and the Model Library are all accessed through the COMSOL Desktop.

T H E D O C U M E N T A T I O N

The COMSOL Multiphysics Users Guide and COMSOL Multiphysics Reference Guide describe all interfaces and functionality included with the basic COMSOL Multiphysics license. These guides also have instructions about how to use COMSOL Multiphysics and how to access the documentation electronically through the COMSOL Multiphysics help desk.

To locate and search all the documentation, in COMSOL Multiphysics:

Press F1 for Dynamic Help,

Click the buttons on the toolbar, or

Select Help>Documentation ( ) or Help>Dynamic Help ( ) from the main menu

and then either enter a search term or look under a specific module in the documentation tree.

T H E M O D E L L I B R A R Y

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.

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 Dynamic

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 users guide. However, if you are using the online help in COMSOL Multiphysics, these links work to other modules, model examples, and documentation sets.

Important 1 : I N T R O D U C T I O N

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 library (including those developed by you), feel free to contact us at info@comsol.com.

C O N T A C T I N G C O M S O L B Y E M A I L

For general product information, contact COMSOL at info@comsol.com.

To receive technical support from COMSOL for the COMSOL products, please contact your local COMSOL representative or send your questions to support@comsol.com. An automatic notification and case number is sent to you by email.

C O M S O L WE B S I T E S

Typographical Conventions

All COMSOL user guides use a set of consistent typographical conventions that make it easier to follow the discussion, understand what you can expect to see on the Graphical User Interface (GUI), and know which data must be entered into various data-entry fields.

In particular, these conventions are used throughout the documentation:

Click text highlighted in blue to go to other information in the PDF. When you are using the online help desk in COMSOL Multiphysics, these links also work to other modules, model examples, and documentation sets.

Main Corporate web site www.comsol.com

Worldwide contact information www.comsol.com/contact

Technical Support main page www.comsol.com/support

Support Knowledge Base www.comsol.com/support/knowledgebase

Product updates www.comsol.com/support/updates

COMSOL User Community 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 | 21

http://www.comsol.com/http://www.comsol.com/contact/http://www.comsol.com/support/http://www.comsol.com/support/knowledgebase/http://www.comsol.com/support/updates/http://www.comsol.com/community/

22 | C H A P T E R A boldface font indicates that the given word(s) appear exactly that way on the COMSOL Desktop (or, for toolbar buttons, in the corresponding tooltip). For example, the Model Builder window ( ) is often referred to and this is the window that contains the model tree. As another example, the instructions might say to click the Zoom Extents button ( ), and this means that when you hover over the button with your mouse, the same label displays on the COMSOL Desktop.

The names of other items on the COMSOL Desktop that do not have direct labels contain a leading uppercase letter. For instance, the Main toolbar is often referred to the horizontal bar containing several icons that are displayed on top of the user interface. However, nowhere on the COMSOL Desktop, nor the toolbar itself, includes the word main.

The forward arrow symbol > is instructing you to select a series of menu items in a specific order. For example, Options>Preferences is equivalent to: From the Options menu, choose Preferences.

A Code (monospace) font indicates you are to make a keyboard entry in the user interface. You might see an instruction such as Enter (or type) 1.25 in the Current density field. The monospace font also is an indication of programming code. or a variable name. An italic Code (monospace) font indicates user inputs and parts of names that can vary or be defined by the user.

An italic font indicates the introduction of important terminology. Expect to find an explanation in the same paragraph or in the Glossary. The names of other user guides in the COMSOL documentation set also have an italic font.

T H E D I F F E R E N C E B E T W E E N N O D E S , B U T T O N S , A N D I C O N S

Node: A node is located in the Model Builder and has an icon image to the left of it. Right-click a node to open a context menu and to perform actions.

Button: Click a button to perform an action. Usually located on a toolbar (the main toolbar or the Graphics toolbar, for example), or in the upper-right corner of a settings window.

Icon: An icon is an image that displays on a window (for example, the Model Wizard or Model Library) or displays in a context menu when a node is right-clicked. Sometimes selecting an item with an icon from a nodes context menu adds a node with the same image and name, sometimes it simply performs the action indicated (for example, Delete, Enable, or Disable). 1 : I N T R O D U C T I O N

K E Y T O T H E G R A P H I C S

Throughout the documentation, additional icons are used to help navigate the information. These categories are used to draw your attention to the information based on the level of importance, although it is always recommended that you read these text boxes.

CautionA Caution icon is used to indicate that the user should proceed carefully and consider the next steps. It might mean that an action is required, or if the instructions are not followed, that there will be problems with the model solution, for example:

ImportantAn Important icon is used to indicate that the information provided is key to the model building, design, or solution. The information is of higher importance than a note or tip, and the user should endeavor to follow the instructions, for example:

NoteA Note icon is used to indicate that the information may be of use to the user. It is recommended that the user read the text, for example:

This may limit the type of boundary conditions that you can set on the eliminated species. The species selection must be carefully done.

Caution

Do not select any domains that do not conduct current, for example, the gas channels in a fuel cell.

Important

Undo is not possible for nodes that are built directly, such as geometry objects, solutions, meshes, and plots.

NoteA B O U T T H E H E A T TR A N S F E R M O D U L E | 23

24 | C H A P T E RTipA Tip icon is used to provide information, reminders, short cuts, suggestions of how to improve model design, and other information that may or may not be useful to the user, for example:

See AlsoThe See Also icon indicates that other useful information is located in the named section. If you are working on line, click the hyperlink to go to the information directly. When the link is outside of the current document, the text indicates this, for example:

ModelThe Model icon is used in the documentation as well as in COMSOL Multiphysics from the View>Model Library menu. If you are working online, click the link to go to the PDF version of the step-by-step instructions. In some cases, a model is only available if you have a license for a specific module. These examples occur in the COMSOL Multiphysics Users Guide. The Model Library path describes how to find the actual model in COMSOL Multiphysics.

Space Dimension IconsAnother set of icons are also used in the Model Builderthe model space dimension is indicated by 0D , 1D , 1D axial symmetry , 2D , 2D axial symmetry

, and 3D icons. These icons are also used in the documentation to clearly list

It can be more accurate and efficient to use several simple models instead of a single, complex one.

Tip

Theory for the Single-Phase Flow Interfaces

The Laminar Flow Interface in the COMSOL Multiphysics Users Guide See Also

Acoustics of a Muffler: Model Library path COMSOL_Multiphysics/Acoustics/automotive_muffler

If you have the RF Module, see Radar Cross Section: Model Library path RF_Module/Tutorial_Models/radar_cross_section

Model 1 : I N T R O D U C T I O N

the differences to an interface, feature node, or theory section, which are based on space dimension.

The following tables are examples of these space dimension icons.

3D models often require more computer power, memory, and time to solve. The extra time spent on simplifying a model is time well spent when solving it.

Remember that modeling in 2D usually represents some 3D geometry under the assumption that nothing changes in the third dimension.

3D

2DA B O U T T H E H E A T TR A N S F E R M O D U L E | 25

26 | C H A P T E ROve r v i ew o f t h e U s e r s Gu i d e

The Heat Transfer Module Users 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 Users Guide.

TA B L E O F C O N T E N T S , G L O S S A R Y , A N D I N D E X

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

H E A T TR A N S F E R T H E O R Y

The Heat Transfer Theory chapter starts with the general theory underlying the heat transfer interfaces used in this module. It then discusses theory about infinite elements, 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.

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

The module includes interfaces for the simulation of heat transfer. As with all other physical descriptions simulated by COMSOL Multiphysics, any description of heat 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 TransferThe 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 transferconduction, convection, and radiationfor 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 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.

As detailed in the section Where Do I Access the Documentation and Model Library? this information is also searchable from the COMSOL Multiphysics software Help menu. Tip 1 : I N T R O D U C T I O N

Bioheat TransferThe Bioheat Transfer Interface section discusses modeling heat transfer within biological tissue using the Bioheat Transfer interface.

Heat Transfer in Thin ShellsHeat Transfer in Thin Shells chapter describes the Thin Conductive Shell interface, which opens after selecting Heat Transfer in Thin Shells in the Model Wizard. It is suitable for solving thermal-conduction problems in thin structures.

Radiative Heat TransferThe The Radiation Heat Transfer Branch chapter describes the Surface-to-Surface Radiation, the Heat Transfer with Surface-to-Surface Radiation, and the Radiation in Participating Media interfaces.

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

The The Conjugate Heat Transfer Branch chapter describes the Non-Isothermal Flow Laminar Flow (nitf) and Turbulent Flow (nitf) interfaces found under 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.

T H E F L U I D F L O W B R A N C H I N T E R F A C E S

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.

M A T E R I A L S

The Materials chapter has details about the Liquids and Gases material database included with this module. O V E R V I E W O F T H E U S E R S G U I D E | 27

28 | C H A P T E R 1 : I N T R O D U C T I O N

2H e a t T r a n s f e r T h e o r y 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 Interfaces

About Infinite Elements

About the Heat Transfer Coefficients

About Highly Conductive Layers

Theory of Out-of-Plane Heat Transfer

Theory for the Bioheat Transfer Interface

Theory for the Heat Transfer in Porous Media Interface 29

30 | C H A P T E RTh eo r y f o r t h e Hea t T r a n s f e r I n t e r f a c e s

This section reviews the theory about the heat transfer equations. For more detailed discussions of the fundamentals of heat transfer, see Ref. 1 and Ref. 3.

The Heat Transfer Interface theory is described in this section:

What is Heat Transfer?

The Heat Equation

A Note on Heat Flux

Heat Flux Variables and Heat Sources

About the Boundary Conditions for the Heat Transfer Interfaces

Radiative Heat Transfer in Transparent Media

Consistent and Inconsistent Stabilization Methods for the Heat Transfer Interfaces

References for the Heat Transfer 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:

ConductionHeat 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.

ConvectionHeat convection (sometimes called heat advection) takes place through the net displacement of a fluid, which transports the heat content in a fluid through the fluids own velocity. The term convection (especially convective cooling 2 : H E A T TR A N S F E R T H E O R Y

and convective heating) also refers to the heat dissipation from a solid surface to a fluid, typically described by a heat transfer coefficient.

RadiationHeat 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:

(2-1)

where

is the density (SI unit: kg/m3)

Cp is the specific heat capacity at constant pressure (SI unit: J/(kgK))

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/m2)

p is pressure (SI unit: Pa)

is the viscous stress tensor (SI unit: Pa)

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

Q contains heat sources other than viscous heating (SI unit: W/m3)

CpTt------- u T+ q :S T

---- T-------

p

pt------ u p+ Q+ +=

S 12--- u u T+ =T H E O R Y F O R T H E H E A T TR A N S F E R I N T E R F A C E S | 31

32 | C H A P T E RFor a detailed discussion of the fundamentals of heat transfer, see Ref. 1.

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:

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

(2-2)

where k is the thermal conductivity (SI unit: W/(mK)). In a solid, the thermal conductivity can be anisotropic (that is, it has different values in different directions). Then k becomes a tensor

and the conductive heat flux is given by

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:

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.Note

t v + 0=

qi kTxi--------=

k

kxx kxy kxzkyx kyy kyzkzx kzy kzz

=

qi kijTxj--------

j=

a:b anmbnmm

n= 2 : H E A T TR A N S F E R T H E O R Y

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:

The Heat Transfer interface with the Heat Transfer in Fluids feature solves this equation for the temperature, T. If the velocity is set to zero, the equation governing pure conductive heat transfer is obtained:

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 conserved property. The conserved property is instead the total energy. There is hence heat flux and energy flux which are similar, but not identical.

This section briefly describes the theory for the variables for Total heat flux and Total energy flux. The approximations made do not affect the computational results, only variables available for results analysis and visualization.

TO T A L E N E R G Y F L U X

The total energy flux for a fluid is equal to (Ref. 4, chapter 3.5)

(2-3)

Above, H0 is the total enthalpy

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

CpTt------- Cpu T+ kT Q+=

CpTt------- k T + Q=

u H0 + k T u qr++

H0 H12--- u u +=T H E O R Y F O R T H E H E A T TR A N S F E R I N T E R F A C E S | 33

34 | C H A P T E R (2-4)

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

(2-5)

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

Solid materials and ideal gases:

Gasliquid:

where 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 in COMSOL 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.

H E A T F L U X

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

(2-6)

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

(2-7)

u H 12--- u u +

k T u qr++

H Href Cp Td

Tref

T

1--- 1T----

T-------

p

+

pd

pref

p

+ +=

Href Cp ref Tref=

Href Cp ref ref Tref pref ref+=

For the evaluation of H to work, it is important that the dependence of Cp, and on the temperature are prescribed either via model input or as a function of the temperature variable. If Cp, or depend on the pressure, the dependency must be prescribed either via model input or by using the variable pA which is the variable for the absolute pressure.

Note

uU k T qr+

H U p---+= 2 : H E A T TR A N S F E R T H E O R Y

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 will be a pressure drop along the channel that drives the flow. Since there is no viscous heating and the walls are isolated, Equation 2-5 will give that HinHout. 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 UinUout, 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 HinHout (first law of thermodynamics) and UinUout (since work has been converted to heat).

Heat Flux Variables and Heat Sources

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 domains, boundaries

dflux Conductive heat flux domains, boundaries

turbflux Turbulent heat flux domains, boundaries

aflux Convective heat flux domain, boundaries

trlflux Translation heat flux domains, boundaries

teflux Total energy flux domains, boundaries

ccflux_u

ccflux_d

ccflux_z

Convective out-of-plane heat flux out-of-plane domains (1D and 2D)

rflux_u

rflux_d

rflux_z

Radiative out-of-plane heat flux out-of-plane domains (1D and 2D), boundaries

q0_u

q0_d

q0_z

Out-of-plane inward heat flux out-of-plane domains (1D and 2D)T H E O R Y F O R T H E H E A T TR A N S F E R I N T E R F A C E S | 35

36 | C H A P T E RD O M A I N H E A T F L U X E S

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

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

For solid domains, for example heat transfer in solids and biological tissue domains, the total heat flux is defined by:

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

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

ntflux Total normal heat flux boundaries

ndflux Normal conductive heat flux boundaries

naflux Normal convective heat flux boundaries

ntrlflux Normal translational heat flux boundaries

nteflux Normal total energy flux boundaries

ccflux Convective heat flux boundaries

Qtot Domain heat source domains

Qbtot Boundary heat source boundaries

Ql Line heat source edges

Qp Point heat source points

TABLE 2-1: HEAT FLUX VARIABLES

VARIABLE NAME GEOMETRIC ENTITY LEVEL

See Radiative Heat Flux to evaluate the radiative heat flux.Tip

tflux trlflux dflux+=

tflux aflux dflux+=

dflux keff T= 2 : H E A T TR A N S F E R T H E O R Y

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

in 2D (dz is the domain thickness)

in 1D (Ac is the cross-section area)

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.

For heat transfer in porous media, keff = keq where keq is the equivalent conductivity defined in the Porous Matrix feature.

Turbulent Heat FluxThe turbulent heat flux variable turbflux enables to access the part of the conductive heat flux that is due to the turbulence.

Convective Heat FluxThe conductive heat flux variable aflux is defined using the internal energy:

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

in 2D (dz is the domain thickness)

in 1D (Ac is the cross-section area)

E is the internal energy defined by:

ECpT for solid domains,

ECpT for ideal gas fluid domains,

dflux dzkeff T=

dflux Ackeff T=

turbflux kT T=

aflux uE=

aflux dzuE=

aflux AcuE=T H E O R Y F O R T H E H E A T TR A N S F E R I N T E R F A C E S | 37

38 | C H A P T E R EHp for other fluid domains.

H is the enthalpy defined by:

HCpT for solid domains,

HCpTp for ideal gas fluid domains,

HCpTp for other fluid domains.

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

Total Energy FluxThe total energy flux, teflux, is defined when viscous heating is enabled:

where the total enthalpy, H0, is defined as:

Radiative Heat FluxIn 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.

O U T - O F - P L A N E D O M A I N F L U X E S

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

Convective Out-of-Plane Heat FluxThe convective out-of-plane heat flux, ceflux, is generated by the Out-of-Plane Convective Cooling feature.

In 2D:

upside:

downside:

teflux uH0 dflux u+ +=

H0 Hu u

2------------+=

Qr qr=

ccflux_u hu Text u T =

ccflux_d hd Text d T = 2 : H E A T TR A N S F E R T H E O R Y

In 1D:

Radiative Out-of-Plane Heat FluxThe radiative out-of-plane heat flux, rflux, is generated by the Out-of-Plane Radiation feature.

In 2D:

upside:

downside:

In 1D:

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

In 2D:

upside:

downside:

In 1D:

B O U N D A R Y H E A T F L U X E S

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.

Total Normal Heat FluxThe variable ntflux is defined by:

ccflux_z hz Text z T =

rflux_u u Tamb u4 T4 =

rflux_d d Tamb d4 T4 =

rflux_z z Tamb z4 T4 =

q0_u hu Text u T =

q0_d hd Text d T =

q0_z hz Text z T =

ntflux mean tflux n=T H E O R Y F O R T H E H E A T TR A N S F E R I N T E R F A C E S | 39

40 | C H A P T E RNormal Conductive Heat FluxThe variable ndflux is defined by:

Normal Convective Heat FluxThe variable naflux is defined by:

Normal Translational Heat FluxThe variable ntrlflux is defined by:

Normal Total Energy FluxThe variable nteflux is defined by:

Radiative Heat FluxOn boundaries the radiative heat flux, rflux, is a scalar quantity defined as:

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

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

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

in 2D (dz is the domain thickness):

in 1D (Ac is the cross section area):

ndflux mean dflux n=

naflux mean aflux n=

ntrlflux mean trlflux n=

nteflux mean teflux n=

rflux Tamb4 T4 G T4 qw+ +=

ccflux h Text T =

ccflux dzh Text T =

ccflux Ach Text T = 2 : H E A T TR A N S F E R T H E O R Y

D O M A I N H E A T S O U R C E S

The sum of the domain heat sources added by different features are available in one variable, Qtot (SI unit: W/m

3). This variable Qtot is the sum of:

Q which is the heat source added by Heat Source and Electromagnetic Heat Source features.

Qmet which is the heat source added by the Bioheat feature.

B O U N D A R Y H E A T S O U R C E S

The sum of the boundary heat sources added by different boundary conditions is available in one variable, Qb,tot (SI unit: W/m

2). This variable Qbtot is the sum of:

Qb which is the boundary heat source added by Boundary heat Source, Electrochemical reaction heat flux and Reaction heat flux boundary conditions.

Qsh which is the boundary heat source added by Boundary Electromagnetic Heat Source boundary condition.

Qs: which is the boundary heat source added by Layer heat source subfeature of Highly conductive layer.

L I N E A N D P O I N T H E A T S O U R C E S

The sum of the line heat sources is available in a variable called Ql (SI unit: W/m).

The sum of the point heat sources is available in a variable called Qp (SI unit: W).

The out-of-plane contributions (convective cooling, heat flux, and radiation), and the blood contribution in Bioheat are considered flux so that they are not added to Qtot.Note

In 2D axisymmetric models, the SI unit for the variable Qp is W/m.

2D AxiT H E O R Y F O R T H E H E A T TR A N S F E R I N T E R F A C E S | 41

42 | C H A P T E RAbout the Boundary Conditions for the Heat Transfer Interfaces

TE M P E R A T U R E A N D H E A T F L U X B O U N D A R Y C O N D I T I O N S

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

while the latter specifies the inward heat flux

where

q is the conductive heat flux vector (SI unit: W/m2) where q = kT.

n is the normal vector of the boundary.

q0 is inward heat flux (SI unit: W/m2), 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 q0hTinfT, where Tinf 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.

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

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).

T T0= on

n q q0= on

The Heat Transfer Module contains a set of correlations for convective cooling and heating. See About the Heat Transfer Coefficients.

See Also 2 : H E A T TR A N S F E R T H E O R Y

Several categories of boundary condition exist in heat transfer. Table 2-2 gives the overriding rules for these groups.

Temperature, Convective Outflow, Open Boundary, Inflow Heat Flux

Thermal Insulation, Symmetry, Periodic Heat Condition

Highly Conductive Layer

Heat Flux, Convective Cooling

Boundary Heat Source, Electrochemical Reaction Heat Flux, Reaction Heat Flux, 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

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

Locate the line that corresponds to 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.

TABLE 2-2: OVERRIDING RULES FOR HEAT TRANSFER BOUNDARY CONDITIONS

A\B 1 2 3 4 5 6 7 8

1-Temperature X X X X

2-Thermal Insulation X X X

3-Highly Conductive Layer

X X

4-Heat Flux X X

5-Boundary heat source

6-Surface-to-surface radiation

X X

7-Opaque Surface X

8-Thin Thermally Resistive Layer

X XT H E O R Y F O R T H E H E A T TR A N S F E R I N T E R F A C E S | 43

44 | C H A P T E R If the corresponding cell is empty A and B contribute. If it contains an X, B overrides A.

Example 1Consider 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 2Consider a boundary where Convective Cooling is applied. Then a Symmetry boundary condition is applied on the same boundary afterward.

Convective Cooling 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 Cooling is overridden by Symmetry.

Radiative Heat Transfer in Transparent Media

This discussion so far has considered heat transfer by means of conduction 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.

Group 4 and group 5 boundary conditions are always contributing. That means that they never override any other boundary condition. But they might be overridden.Important

In Example 2 above, if Symmetry followed by Convective Cooling is added, the boundary conditions contribute.

Note 2 : H E A T TR A N S F E R T H E O R Y

D E R I V I N G T H E R A D I A T I V E H E A T F L U X

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

Consider Figure 2-1. A point 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 is named the irradiation, G. The total outgoing radiative flux is named the radiosity, J. The radiosity is the sum of the reflected radiation and the emitted radiation:

(2-8)

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

(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.

(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 relation:

(2-11)

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

(2-12)

G

,T ,T

J =G + T4

x x

x

xx

J G T4+=

q G J=

q 1 G T4=

1 = =

q G T4 =T H E O R Y F O R T H E H E A T TR A N S F E R I N T E R F A C E S | 45

46 | C H A P T E RThis is the equation used as a radiation boundary condition.

R A D I A T I O N TY P E S

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.

S U R F A C E - T O - A M B I E N T R A D I A T I O N

Surface-to-ambient radiation assumes the following:

The ambient surroundings in view of the surface have a constant temperature, Tamb.

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

(2-13)

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

(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.

Consistent and Inconsistent Stabilization Methods for the Heat Transfer Interfaces

Several of the Heat Transfer interfaces have this advanced option to set the stabilization method parameters. Below is some information pertaining to these options.

G Tamb4

=

q Tamb4 T4 =

Theory for the Surface-to-Surface Radiation Interface

Theory for the Radiative Heat Transfer Interfaces

Radiation and Participating Media Interactions See Also 2 : H E A T TR A N S F E R T H E O R Y

To display this section, click the Show button ( ) and select Stabilization.

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

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 corresponding check box is selected.

Streamline DiffusionStreamline 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 The crosswind diffusion 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.

When Crosswind diffusion is selected, enter a Lower gradient limit glim (SI unit: K/m). The default is 0.01[K]/jh.helem. The variable glim is needed because both Equation 2-15 and Equation 2-16 contain terms of the form 1T, which become singular if T0. Hence, all occurrences of 1T are replaced by 1maxTglim where glim is a measure of a small gradient.

The method in the Heat Transfer interfaces adds the following contribution to the weak formulation (see Codina in Ref. 2):

Show Stabilization in the COMSOL Multiphysics Users Guide

Stabilization Techniques and Numerical Stabilization in the COMSOL Multiphysics Reference GuideSee Also

Continuous Casting: Model Library path Heat_Transfer_Module/Process_and_Manufacturing/continuous_casting

ModelT H E O R Y F O R T H E H E A T TR A N S F E R I N T E R F A C E S | 47

48 | C H A P T E R (2-15)

where R is the PDE residual, is the test function for T, h is the element size, and is defined as

(2-16)

I N C O N S I S T E N T 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 diffusion, select the Isotropic diffusion check box. The field for the tuning parameter id then becomes available. The default value is 0.25; increase or decrease the value of id to increase or decrease the amount of isotropic stabilization.

References for the Heat Transfer Interfaces

1. F.P. Incropera and D.P. DeWitt, Fundamentals of Heat and Mass Transfer, 4th ed., John Wiley & Sons, 1996.

2. R. Codina, Comparison of Some Finite Element Methods for Solving the Diffusion-Convection-Reaction Equation, Comp. Meth.Appl. Mech. Engrg, vol. 156, pp. 185210, 1998.

3. A. Bejan, Heat Transfer, Wiley, 1993.

4. G.K. Batchelor, An Introduction 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.

12---max 0 Ce 2k

h ----------

h RT

----------- T

I u uu 2

--------------- T d

e

e 1=

Nel

T

Cp u T

T 2--------------------------------- T if T 0

0 if T 0=

=

2 : H E A T TR A N S F E R T H E O R Y

7. T. Poinsot and D. Veynante, Theoretical and Numerical Combustion, Second Edition, Edwards, 2005.T H E O R Y F O R T H E H E A T TR A N S F E R I N T E R F A C E S | 49

50 | C H A P T E RAbou t I n f i n i t e E l emen t s

In this section:

Modeling Unbounded Domains

Known Issues When Modeling Using Infinite Elements

Modeling Unbounded Domains

Many environments modeled with finite elements are unbounded or open, meaning that the fields extend toward infinity. The easiest approach to modeling an unbounded domain is to extend the simulation domain far enough that the influence of the terminating boundary conditions at the far end becomes negligible. This approach can create unnecessary mesh elements and make the geometry difficult to mesh due to large differences between the largest and smallest object.

Another approach is to use infinite elements. There are many implementations of infinite elements available, and the elements used in this module are often referred to as mapped infinite elements (see Ref. 1). This implementation maps the model coordinates from the local, finite-sized domain to a stretched domain. The inner boundary of this stretched domain coincides with the local domain, but at the exterior boundary the coordinates are scaled toward infinity.

The principle can be explained in a one-coordinate system, where this coordinate represents Cartesian, cylindrical, or spherical coordinates. Mapping multiple

For more information about this feature, see About Infinite Element Domains and Perfectly Matched Layers in the COMSOL Multiphysics Users Guide.Note 2 : H E A T TR A N S F E R T H E O R Y

coordinate directions (for Cartesian and cylindrical systems only) is just the sum of the individual coordinate mappings.

Figure 2-2: The coordinate transform used for the mapped infinite element technique. The meaning of the different variables are explained in the text.

Figure 2-2 shows a simple view of an arbitrary coordinate system. The coordinate r is the unscaled coordinate that COMSOL Multiphysics draw the geometry in (reference system). The position r0 is the new origin from where the coordinates are scaled, tp is the coordinate from this new origin to the beginning of the scaled region also called the pole distance, and w is the unscaled length of the scaled region. The scaled coordinate, t, approaches infinity when t approaches tpw. To avoid solver issues with near infinite values, it is possible to change the infinite physical width of the scaled region to a finite large value, pw. The true coordinate that the PDEs are formulated in is given by

where t comes from the formula

The following figures show typical examples of infinite element regions that work nicely for each of the infinite element types. These types are:

Stretching in Cartesian coordinate directions, labeled Cartesian

Stretching in cylindrical directions, labeled Cylindrical

Stretching in spherical direction, labeled Spherical

User-defined coordinate transform for general infinite elements, labeled General

r0

t

tp

t

unscaled regionunscaled region scaled region

w

r' r0 t+=

t' tpw

p t tp ---------------------------------=

1tp

pw tp---------------------=A B O U T I N F I N I T E E L E M E N T S | 51

52 | C H A P T E RFigure 2-3: A cube surrounded by typical infinite-element regions of Cartesian type.

Figure 2-4: A cylinder surrounded by typical cylindrical infinite-element regions.

Figure 2-5: A sphere surrounded by a typical spherical infinite-element region.

If other shapes are used for the infinite element regions not similar to the shapes shown in the previous figures, it might be necessary to define the infinite element parameters manually. 2 : H E A T TR A N S F E R T H E O R Y

The poor element quality causes poor or slow convergence for iterative solvers and make the problem ill-conditioned in general. For this reason it is strongly recommended to use swept meshing in the infinite element domains. The sweep direction should be selected the same as the direction of scaling. For Cartesian infinite elements in regions with more than one direction of scaling it is recommended to first sweep the mesh in the domains with only one direction of scaling, then sweep the domains with scaling in two directions, and finish by sweeping the mesh in the domains with infinite element scaling in all three direction.

G E N E R A L S T R E T C H I N G

With manual control of the stretching, the geometrical parameters that defines the stretching are added as Manual Scaling subnodes. These subnodes have no effect unless the type of the Infinite Elements node is set to General. Each Manual Scaling subnode has three parameters:

Scaling direction, which sets the direction from the interface to the outer boundary.

Geometric width, which sets the width of the region.

Coordinate at interface, which sets an arbitrary coordinate at the interface.

When going from any of the other types to the General type, subnodes that represent stretching of the previous type are added automatically.

Known Issues When Modeling Using Infinite Elements

Be aware of the following when modeling with infinite elements:

Use of One Single Infinite Elements NodeUse a separate Infinite Elements node for each isolated infinite element domain. That is, to use one and the same Infinite Elements node, all infinite element domains must be in contact with each other. Otherwise the infinite elements do not work properly.

Element QualityThe coordinate scaling resulting from infinite elements also yields an equivalent stretching or scaling of the mesh that effectively results in a poor element quality. (The element quality displayed by the mesh statistics feature does not account for this effect.)

The poor element quality causes poor or slow convergence for iterative solvers and make the problem ill-conditioned in general. For this reason, it is strongly recommended to use swept meshing in the infinite element domains. The sweep direction should be selected the same as the direction of scaling. For Cartesian infinite A B O U T I N F I N I T E E L E M E N T S | 53

54 | C H A P T E Relements in regions with more than one direction of scaling it is recommended to first sweep the mesh in the domains with only one direction of scaling, then sweep the domains with scaling in two directions, and finish by sweeping the mesh in the domains with infinite element scaling in all three direction.

Complicated ExpressionsThe expressions resulting from the stretching get quite complicated for spherical infinite elements in 3D. This increases the time for the assembly stage in the solution process. After the assembly, the computation time and memory consumption is comparable to a problem without infinite elements. The number of iterations for iterative solvers might increase if the infinite element regions have a coarse mesh.

Erroneous ResultsInfinite element regions deviating significantly from the typical configurations shown in the beginning of this section can cause the automatic calculation of the infinite element parameter to give erroneous result. Enter the parameter values manually if this is the case. See General Stretching.

Use the Same Material Parameters or Boundary ConditionsThe infinite element region is designed to model uniform regions extended toward infinity. Avoid using objects with different material parameters or boundary conditions that influence the solution inside an infinite element region.

R E F E R E N C E F O R I N F I N I T E E L E M E N T S

1. O.C. Zienkiewicz, C. Emson, and P. Bettess, A Novel Boundary Infinite Element, International Journal for Numerical Methods in Engineering, vol. 19, no. 3, pp. 393404, 1983. 2 : H E A T TR A N S F E R T H E O R Y

Abou t t h e Hea t T r a n s f e r C o e f f i c i e n t s

One of the most common boundary conditions when modeling heat transfer is convective cooling or heating whereby a fluid cools 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 convection-cooled surfaces

Extend the model to describe the flow and heat transfer in the cooling 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 interf