Heat Transfer Module - Montana Tech High Performance Cluster

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 1998–2012 COMSOL

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

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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 User’s 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 . . . . . . . . . . . . . . . . 54

C O N T E N T S | 3

4 | C O N T E N T S

Nature of the Flow—the 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 . . . . . . . . . . . . . . . . . . . 120

C O N T E N T S | 5

6 | C O N T E N T S

Thermal 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. . . . . . . . . . . . . . . . . . . . . . . . 176

C O N T E N T S | 7

8 | C O N T E N T S

The Single-Phase Flow, Turbulent Flow Interfaces 177

The Turbulent Flow, k- Interface . . . . . . . . . . . . . . . 177

The 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 . . . . . . . . . . . . . . . . . . 214

The Low Reynolds Number k- Turbulence Model . . . . . . . . . 219

Inlet 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 . . . . . . . . . . . . . . . . . . 258

C O N T E N T S | 9

10 | C O N T E N T S

Liquids 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

1

I n t r o d u c t i o n

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 overview with links to each chapter in this guide.

• About the Heat Transfer Module

• Overview of the User’s Guide

11

12 | C H A P T E R

Abou 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 User’s 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 physics—a versatile combination increasing the accuracy and predicting power of your models.

This User’s 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 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 where 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?.

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 R

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 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 User’s 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


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


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


Note

See Also


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


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


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


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


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


PRESET STUDIES


16 | C H A P T E R

The 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.


PRESET STUDIES

• Study Types in the COMSOL Multiphysics Reference Guide

• Available Study Types in the COMSOL Multiphysics User’s 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


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


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


18 | C H A P T E R

Show 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 User’s 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 User’s 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


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

User’s Guide

Discretization • Show Discretization

• Element Types and Discretization

User’s 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

User’s Guide

Consistent and Inconsistent Stabilization

Show Stabilization User’s Guide

• Stabilization Techniques

• Numerical Stabilization

Reference Guide

Geometry Working with Geometry User’s Guide

Constraint Settings Using Weak Constraints User’s Guide


20 | C H A P T E R

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

Important


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 [email protected].

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 [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.

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


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 node’s context menu adds a node with the same image and name, sometimes it simply performs the action indicated (for example, Delete, Enable, or Disable).


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.

Note


24 | C H A P T E R

TipA 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 User’s 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 Builder—the 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 User’s 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


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

2D


26 | C H A P T E R

Ove r v i ew o f t h e U s e r ’ s Gu i d e

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 User’s 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 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 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


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

2

H 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 R

Th 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:

• 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.

• 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

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.

• 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:

(2-1)

where

• is the density (SI unit: kg/m3)

• 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/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 R

For 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 Fourier’s 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/(m·K)). 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 kxz

kyx kyy kyz

kzx kzy kzz

=

qi kijTxj--------

j–=

a:b anmbnm

m

n=


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 H 12--- u u +=


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--- 1 T

----

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---+=


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)


36 | C H A P T E R

D 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–=


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=


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 H u u2

------------+=

Qr qr=

ccflux_u hu Text uT– =

ccflux_d hd Text d T– =


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=


40 | C H A P T E R

Normal 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– =


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/m3). 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/m2). 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 Axi


42 | C H A P T E R

About 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 q0h·TinfT, 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


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 X


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


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– =


46 | C H A P T E R

This 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


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 User’s 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_castingModel


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. 185–210, 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

----------- Tˆ

I u uu 2

---------------– T d

e

e 1=

Nel

–

Tˆ

Cp u T

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

0 if T 0=

=


7. T. Poinsot and D. Veynante, Theoretical and Numerical Combustion, Second Edition, Edwards, 2005.


50 | C H A P T E R

Abou 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 User’s Guide.Note


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 R

Figure 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.


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 R

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.

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. 393–404, 1983.


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 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. Convection cooling is then modeled by specifying the heat flux on the boundaries that interface with the cooling fluid as being proportional to the temperature difference across a fictitious thermal boundary layer. Mathematically, the heat flux is described by the equation

where h is a heat transfer coefficient and Tinf 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 cooling fluid, the fluid’s material properties, and the surface temperature—and, for forced-convection cooling, also on the fluid’s flow rate. In addition, the geometrical configuration affects 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

• Available Heat Transfer Coefficients

• References for the Heat Transfer Coefficients

n– k– T h Tinf T– =

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 | 55

56 | C H A P T E R

Heat Transfer Coefficient Theory

It is possible to divide convection cooling 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 2-6.

Figure 2-6: The eight possible categories of convective cooling.

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.

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

• The Nusselt number, NuL(Re, Pr, Ra)hL/k

• The Reynolds number, ReLU L/

• The Prandtl number, PrCp/k

• The Rayleigh number, RaGr Pr 2 gCp T L3/(k)

Natural Forced

External

Internal

Laminar Flow

Turbulent Flow


where

• h is the heat transfer coefficient (SI unit: W/(m2·K)).

• L is the characteristic length (SI unit: m).

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

• g is the acceleration of gravity (SI unit: m/s2).

• k is the thermal conductivity of the fluid (SI unit: W/(m·K)).

• is the fluid density (SI unit: kg/m3).

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

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

• Cp 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 requires the definition of a length scale, the fluid’s physical properties, and the temperature scale (temperature difference). The Grashof number is defined as:

where g is the acceleration of gravity, is the fluid’s coefficient of volumetric thermal expansion, Ts denotes the temperature of the hot surface, T0 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

GrLg Ts T0– L3

2--------------------------------------=


58 | C H A P T E R

which for an ideal gas reduces to

The transition from laminar to turbulent flow occurs at a Gr value of 109; the flow is turbulent for larger values.

Available Heat Transfer Coefficients

E X T E R N A L N A T U R A L C O N V E C T I O N

Vertical WallThe correlations are equations 9.26 and 9.27 in Ref. 1:

(2-17)

where L, the height of the wall, is a correlation input and

(2-18)

where in turn g is the acceleration of gravity equal to 9.81 m/s2. All material properties are evaluated at TText2.

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

1---

T-------

p

–=

1 T=

h

kL---- 0.68

0.67RaL1 4/

1 0.492kCp

------------------- 9 16/

+ 4 9/----------------------------------------------------------+

RaL 109

kL---- 0.825

0.387RaL1 6/

1 0.492kCp

------------------- 9 16/

+ 8 27/-------------------------------------------------------------+

2

RaL 109

=

RaL

g T p Cp T Text– L3

k---------------------------------------------------------------------------=


(2-19)

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, RaL, is analog to these for vertical walls and is given by the following:

(2-20)

where in turn g denotes the gravitational acceleration, equal to 9.81 m/s2.

For turbulent flow, 1 is used instead of cos( ) in the expression for h, since this gives better accuracy (see Ref. 2).

The laminar-turbulent transition depends on (see Ref. 2). Unfortunately, few data is available about this transition. There is some data available in Ref. 2 but this data gives only approximations of this transition, according to the authors. In addition, data is only provided for water (Pr around 6). For this reasons, we define a flow as turbulent, independently of value, when

All material properties are evaluated at TText2.

h

kL---- 0.68

0.67 Racos L 1 4/

1 0.492kCp

------------------- 9 16/

+ 4 9/----------------------------------------------------------+

RaL 109

kL---- 0.825

0.387RaL1 6/

1 0.492kCp

------------------- 9 16/

+ 8 27/-------------------------------------------------------------+

2

RaL 109

=

RaL


k---------------------------------------------------------------------------=

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.Note


k--------------------------------------------------------------------------- 109


60 | C H A P T E R

Horizontal Plate, UpsideThe correlations are equations 9.30–9.32 in Ref. 1 but can also be found as equations 7.77 and 7.78 in Ref. 2.

If TText, then

(2-21)

while if T Text, then

(2-22)

RaL is given by Equation 2-18, and L, the plate diameter (defined as area/perimeter, see Ref. 2) is a correlation input. The material data are evaluated at TText2.

Horizontal Plate, DownsideEquation 2-21 is used when T Text and Equation 2-22 is used when T Text. Otherwise it is the same implementation as for Horizontal plate, upside.

I N T E R N A L N A T U R A L C O N V E C T I O N

Narrow Chimney, Parallel PlatesIf RaL HL and T Text, then

(2-23)

where L, the plate distance, and H, the chimney height, are correlation inputs (equation 7.96 in Ref. 2). RaL is given by Equation 2-18. The material data are evaluated at TText2.

Narrow Chimney, Circular TubeIf RaDHD, then

where D, the tube diameter, and H, the chimney height, are correlation inputs (table 7.2 in Ref. 2 with DhD). RaD is given by Equation 2-18 with L replaced by D. The material data are evaluated at TText2.

h

kL----0.54RaL

1 4/ RaL 107

kL----0.15RaL

1 3/ RaL 107

=

h kL----0.27RaL

1 4/=

h kH----- 1

24------RaL=

h kH----- 1

128----------RaD=


E X T E R N A L F O R C E D C O N V E C T I O N

Plate, Averaged Transfer CoefficientThis correlation is an assembly of equations 7.34 and 7.41 in Ref. 1:

(2-24)

where Prcpk and ReLUextL. L, the plate length and Uext, the exterior velocity are correlation inputs. The material data are evaluated at TText2.

Plate, Local Transfer CoefficientThis correlation corresponds to equations 5.79b and 5.131 in Ref. 2:

(2-25)

where Prcpk and RexUextx. x, the position along the plate, and Uext, the exterior velocity are correlation inputs. The material data are evaluated at TText2.

I N T E R N A L F O R C E D C O N V E C T I O N

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

(2-26)

where Prcpk, ReDUextD and n0.3 if TText and n0.4 if T Text. D, the tube diameter and Uext, the exterior velocity, are correlation inputs. All material data are evaluated at Text except T which is evaluated at the wall temperature, T.

h2k

L----

0.3387Pr1 3/ ReL1 2/

1 0.0468 Pr 2 3/+ 1 4/--------------------------------------------------------------- ReL 5 10 5

2kL----Pr1 3/ 0.037ReL

4 5/ 871– ReL 5 10 5

=

h

kmax x eps ----------------------------------0.332Pr1 3/ Rex

1 2/ Rex 5 10 5

kmax x eps ----------------------------------0.0296Pr1 3/ Rex

4 5/ Rex 5 10 5

=

h

kD----3.66 ReD 2500

kD----0.027ReD

4 5/ Prn T ------------ 0.14

ReD 2500

=


62 | C H A P T E R

References for the Heat Transfer Coefficients

1. F.P. Incropera and D.P. DeWitt, Fundamentals of Heat and Mass Transfer, Fifth ed. John Wiley & Sons, 2002.

2. A. Bejan, Heat Transfer, John Wiley & Sons, 1993.


Abou t H i g h l y C ondu c t i v e L a y e r s

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

This module supports heat transfer in highly conductive layers in 2D, 2D axisymmetry, and 3D.

2D

2D Axi

3D

A B O U T H I G H L Y C O N D U C T I V E L A Y E R S | 63

64 | C H A P T E R

elements. Figure 2-7 shows an example where a highly conductive layer reduces the mesh density significantly.

Figure 2-7: 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:

(2-27)

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 (kgm3)

• 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 (Wm2)

• q is the heat flux from the layer into the domain (Wm2)

• Qs represents internal heat sources within the conductive layer (Wm3)

• qs is the net outflux of heat through the top and bottom faces of the layer (Wm2)

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

Copper wire modeled

Copper wire represented as ahighly conductive layer

with a mesh

dssCsTt------- t dsks Tt– + q q– dsQS+ q– s= =


n– q dssCp sTt-------– t– dsks Tt– on =

Highly Conductive Layer FeaturesSee Also

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Th eo r y o f Ou t - o f - P l a n e Hea t T r a n s f e r

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 2-8 shows examples of likely situations where this type of geometry reduction can be applied.

Figure 2-8: 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 2-8 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.

Equation Formulation

When out-of-plane heat transfer is enabled, the equation for heat transfer in solids, Equation 3-1 is replaced by

(2-28)

Out-of-Plane Heat Transfer FeaturesSee Also

qdown

qup

q

dzCpTt------- – dzkT dzQ=


where dz 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

(2-29)

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.

Activating Out-of-Plane Heat Transfer and Thickness

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

CpdzTt------- u T+ dzkT dzQ+=

Heat Source features that are added to a model with out-of-plane heat transfer enabled are multiplied by the thickness, dz. Boundary conditions are also adjusted.Note

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Th eo r y f o r t h e B i o h e a t T r a n s f e r I n t e r f a c e

The Bioheat Transfer Interface uses the bioheat equation and the corresponding features 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:

(2-30)

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 2-30 add the Biological Tissue model equation, with a Bioheat feature. The Biological Tissue model provides the left-hand side of Equation 2-30 while the Bioheat feature provides the two source terms on the right-hand side of Equation 2-30.

Reference for the Bioheat Interface

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

C p tT k T– + b Cbb Tb T– Qmet+=


Th eo r y f o r t h e Hea t T r a n s f e r i n Po r ou s Med i a I n t e r f a c e

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:

(2-31)

with the following material properties:

• is the fluid density.

• Cp is the fluid heat capacity at constant pressure.

• (Cp)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 uLuL, 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 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

The equivalent volumetric heat capacity of the solid-fluid system is calculated by

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

For a steady-state problem the temperature does not change with time, and the first term in the left-hand side of Equation 2-31 disappears.

Cp eqTt------- Cpu T+ keqT Q+=

keq pkp Lk+=

Cp eq ppCp p LCp+=

L p+ 1=

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3

T h e H e a t T r a n s f e r B r a n c h

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 the The Radiation Heat Transfer Branch.

In this chapter:

• About the Heat Transfer Interfaces

• The Heat Transfer Interface

• Heat Transfer Interface Advanced Features

• Highly Conductive Layer Features

• Out-of-Plane Heat Transfer Features

• The Bioheat Transfer Interface

• The Heat Transfer in Porous Media Interface

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Abou t t h e Hea t T r a n s f e r I n t e r f a c e s

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 suitable for modeling heat transfer in solids and fluids, porous media, and biological tissue. The interfaces are available in 1D, 2D, and 3D and for axisymmetric models with cylindrical coordinates in 1D and 2D. The default dependent variable is the temperature, T.

There are Heat Transfer interfaces displayed in the Model Builder with the same name but with different icons and default models. After selecting a Heat Transfer interface in the Model Wizard, default settings are added under the main node. For example, if Heat Transfer in Solids ( ) is selected, a Heat Transfer node is added with a default Heat Transfer in Solids model. If Heat Transfer in Fluids ( ) is selected, a Heat Transfer

in Fluids model is added instead, but the parent nodes are both called Heat Transfer. Any interface based on the main Heat Transfer feature has the suffix ht.

Heat Transfer in Solids and Heat Transfer in FluidsUse the Heat Transfer in Solids (ht) ( ) to model mainly heat transfer in solid materials. A default Heat Transfer in Solids model is added, but all functionality for including fluid domains is also available.

This interface has Surface-to-surface radiation and radiation in participating media check boxes to model radiation.

Use the Heat Transfer in Fluids (ht) ( ) to model mainly heat transfer in fluid materials. A default Heat Transfer in Fluids model is added, but all functionality for including solid domains is also available.

This interface has Surface-to-surface radiation and radiation in participating media check boxes to model radiation.

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.Tip

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

Heat Transfer in Porous MediaUse the Heat Transfer in Porous Media (ht) ( ) to model mainly heat transfer in porous materials. The Porous Matrix and Heat Transfer in Fluids models are added, but all functionality for including solid domains is also available.

Heat Transfer with Surface-to-Surface RadiationUse the Heat Transfer with Surface-to-Surface Radiation (ht) ( ), found under the Radiation branch ( ), to model heat transfer that includes surface-to-surface radiation. It is a Heat Transfer interface with the Physical Model>Surface-to-surface

radiation check box selected, which enables the Radiation Settings section. A Heat

Transfer in Solids default model with surface-to-surface radiation is added and available as a boundary condition, but all functionality to include both solid and fluid domains is also available. All available features are as described in Domain, Boundary, Edge, Point, and Pair Features for the Heat Transfer Interfaces.

Bioheat TransferUse the Bioheat Transfer (ht) ( ) to model heat transfer in biological tissue. A Biological Tissue default model is added, but all functionality to include both solid and fluid domains is also available. See The Bioheat Transfer Interface.

Joule HeatingSelect Joule Heating (jh) ( ), found under the Electromagnetic Heating subbranch ( ), to combine all features from the Electric Currents interface with the Heat Transfer interface for modeling Joule heating (also called resistive heating or ohmic heating). See The Joule Heating Interface in the COMSOL Multiphysics User’s Guide.

Conjugate Heat TransferSelect the Conjugate Heat Transfer (nitf) ( ), Laminar Flow ( ) or Conjugate Heat

Transfer, Turbulent Flow ( ) interfaces to use a predefined multiphysics coupling

Radiative Heat Transfer in a Utility Boiler: Model Library path Heat_Transfer_Module/Process_and_Manufacturing/boiler

Model

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consisting of a Single-Phase Flow interface, using a compressible formulation, in combination with a Heat Transfer interface. See The Conjugate Heat Transfer Branch.

Surface-to-Surface RadiationSelect Surface-to-Surface Radiation (rad) ( ), found under the Radiation branch ( ), to use a model that treats heat transfer by surface-to-surface radiation as a process that transfers energy directly between boundaries. The radiation therefore contributes to the boundary conditions rather than to the heat equation itself. See The Surface-To-Surface Radiation Interface. This physics interface solves only for the radiation variable. To solve for radiation and temperature, use a Heat Transfer interface instead.

Radiation in Participating MediaSelect Radiation in Participating Media (rpm) ( ), found under the Radiation branch ( ), to model radiative heat transfer inside a participating medium. This physics interface solves for radiative intensity field. See The Radiation in Participating Media Interface. This physics interface solves only for radiation variables. In order to solve for radiation and temperature, use a Heat Transfer interface.

Heat Transfer in Thin ShellsSelect Heat Transfer in Thin Shells (htsh) ( ) to model conductive heat transfer in thin thermally conducting shells. The Thin Conductive Shell interface is added when this is selected. A Thin Conductive Layer default model, and all functionality for modeling heat conduction and out-of-plane radiation and convective cooling is added. See The Heat Transfer in Thin Shells Interface.

The Non-Isothermal Flow (nitf) ( ), Laminar Flow ( ) and Non-Isothermal Flow, Turbulent Flow ( ) interfaces, found under the Fluid Flow branch, are identical to the Conjugate Heat Transfer interfaces. The only difference is that Fluid is selected as the default model. If Heat transfer in solids is selected as the default model, the interface changes to a Conjugate Heat Transfer interface. To change the default model, select the Heat Transfer interface node and locate the Physical Model section in the settings window.

Note


Th e Hea t T r a n s f e r I n t e r f a c e

The Heat Transfer (ht) interface is available in many forms and each one has the equations, boundary conditions, and sources for modeling conductive and convective heat transfer, and solving for the temperature.

When this interface is added, default nodes are added to the Model Builder 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 Heat Transfer node to add other features that implement, for example, boundary conditions and sources.

I N T E R F A C E I D E N T I F I E R

The interface identifier is a text string that can be used to reference the respective physics interface if appropriate. Such situations could occur when coupling this interface to another physics interface, or when trying to identify and use variables defined by this physics interface, which is used to reach the fields and variables in expressions, for example. It can be changed to any unique string in the Identifier field.

The default identifier (for the first interface in the model) is ht.

D O M A I N S E L E C T I O N

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.

Depending on the version of the Heat Transfer interface selected, these default nodes may be different.

Note

About the Heat Transfer InterfacesSee Also

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P H Y S I C A L M O D E L

Select a Default model— Heat transfer in solids or Heat transfer in fluids.

• Select the Surface-to-surface radiation check box to add a Radiation Settings section on the Heat Transfer interfaces.

• Select the Radiation in participating media check box to add a Participating Media

Settings section.

• Select the Heat transfer in porous media check box to add Porous Matrix nodes for modeling the thermal properties of the immobile solids in the porous matrix.

• Select the Heat Transfer in biological tissue check box to make Biological Tissue the default model. See the Physical Model section in The Bioheat Transfer Interface for details.

R A D I A T I O N S E T T I N G S

To display this section select the Surface-to-surface radiation check box under Physical

Model on any version of the Heat Transfer interface settings window.

If required for 1D or 2D models, select the Out-of-plane heat transfer check box and then enter the Thickness of the plane (dz). 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 feature.

If Heat Transfer with Surface-to-Surface Radiation (ht) is selected from the Model Wizard, the Surface-to-surface radiation check box selected by default. This enables the Radiation Settings section as described. All other available features are the same as described for the Heat Transfer interface.

• Thermo-Photo-Voltaic Cell: Model Library path Heat_Transfer_Module/Electronics_and_Power_Systems/tpv_cell

• Cavity Radiation: Model Library path Heat_Transfer_Module/

Verification_Models/cavity_radiation

1D

2D

Note

Model


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.

For either method, 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.

HemicubeHemicube 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 Famb 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 IntegrationCOMSOL Multiphysics evaluates the integrals in Equation 2-10 and Equation 2-11 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.

P A R T I C I P A T I N G M E D I A S E T T I N G S

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.

Define the Refractive index of the participating media, n.

Select the Discrete ordinates method order from the list. This order defines the discretization of the radiative intensity direction.

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.Note

The same refractive index is used for the whole model.

Note

In 3D, S2, S4, S6, and S8 generate 8, 24, 48, and 80 directions, respectively.


3D

2D


Select Linear (the default), Quadratic, Cubic, Quartic, or Quintic to define the discretization level of the Radiative intensity fields.

D E P E N D E N T V A R I A B L E S

The Heat Transfer interface has a dependent variable for the Temperature T. For surface-to-surface radiation, there is a dependent variable for the Surface radiosity J. The dependent variable names can be changed but the names of fields and dependent variables must be unique within a model.

A D V A N C E D S E T T I N G S

To display this section, click the Show button ( ) and select Advanced Physics Options. Normally these settings do not need to be changed.

Performance IndexSelect 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 0.5. 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.

D I S C R E T I Z A T I O N

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

Select a Temperature—Quadratic (the default), Linear, Cubic, or Quartic. Select an element order for the Surface Radiosity—Linear (the default), Quadratic, Cubic, or Quartic. 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 N T S T A B I L I Z A T I O N

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. 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.


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If Crosswind diffusion is selected, enter a Lower gradient limit glim (SI unit: K/m). The default is 0.01[K]/ht.helem.

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

The Heat Transfer Interface has these domain, boundary, edge, point, and pair conditions available (listed in alphabetical order):

• Boundary Heat Source

• Continuity on Interior Boundary

• Convective Cooling

• Continuity

• Heat Flux

• Heat Source

• Heat Transfer in Fluids

• Heat Transfer in Solids


• Initial Values

• Inflow Heat Flux

• Line Heat Source

• Open Boundary

• Outflow

• Pair Boundary Heat Source

• Pair Thin Thermally Resistive Layer

• Periodic Heat Condition


• Domain, Boundary, Edge, Point, and Pair Features for the Heat Transfer Interfaces

• Consistent and Inconsistent Stabilization Methods for the Heat Transfer Interfaces


• Show Stabilization in the COMSOL Multiphysics User’s Guide

See Also


• Point Heat Source

• Pressure Work

• Reaction Heat Flux

• Surface-to-Ambient Radiation

• Symmetry

• Temperature

• Thermal Insulation (the default boundary condition)

• Thin Thermally Resistive Layer

• Translational Motion

• Viscous Heating

Heat Transfer in Solids

The Heat Transfer in Solids model uses the heat equation version in Equation 3-1 as the mathematical model for heat transfer in solids:

(3-1)

For a steady-state problem the temperature does not change with time and the first term disappears. It has these material properties: density , heat capacity Cp, thermal conductivity k (a scalar or a tensor if the thermal conductivity is anisotropic), and Q, which is the heat source (or sink)—one or more heat sources can be added separately.

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.

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.

1D Axi

2D Axi

Tip

CpTt------- – kT Q=


82 | C H A P T E R

When parts of the model (for example, a heat source) are moving, right-click the Heat

Transfer in Solids node to add a Translational Motion feature to take this into account.

Add Pressure Work or Opaque to the Heat Transfer in Solids node as required.


From the Selection list, choose the domains to define the heat transfer. The default is to use all domains.

M O D E L I N P U T S

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.

C O O R D I N A T E S Y S T E M S E L E C T I O N

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.

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

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 = kT, 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 (kxx, kyy, and so on) are available as ht.kxx, ht.kyy, and so on (using the default Heat Transfer interface identifier ht). The single scalar mean effective thermal conductivity ht.kmean is the mean value of the diagonal elements kxx, kyy, and kzz.

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

Model


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

The default Density (SI unit: kg/m3) and Heat capacity at constant pressure Cp (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.

Translational Motion

The Translational Motion node provides movement by translation to model heat transfer in solids. It adds the following contribution to the right-hand side of Equation 3-1:

The contribution describes the effect of a moving coordinate system that is required to model, for example, a moving heat source.

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

Model

– Cpu T

Special care must be taken at boundaries where n·u0. The Heat Flux boundary condition does not, for example, work at boundaries where n·u0.

Heat Generation in a Disc Brake: Model Library path Heat_Transfer_Module/Tutorial_Models/brake_disc

Caution

Model


84 | C H A P T E R


From the Selection list, choose the domains to prescribe a translational motion.

TR A N S L A T I O N A L M O T I O N

Enter component values for x, y, and z (in 3D) for the Velocity field utrans (SI unit: m/s).

Heat Transfer in Fluids

The Heat Transfer in Fluids model uses the following version of the heat equation as the mathematical model for heat transfer in fluids:

(3-2)

For a steady-state problem the temperature does not change with time and the first term disappears. It has these material properties:

• The density ()

• The fluid heat capacity at constant pressure (Cp)—describes the amount of heat energy required to produce a unit temperature change in a unit mass

• The fluid thermal conductivity (k)—a scalar or a tensor if the thermal conductivity is anisotropic

• The fluid velocity field (u)—can be 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

Also, the ratio of specific heats is defined. It is the ratio of heat capacity at constant pressure, Cp, to heat capacity at constant volume, Cv. When using the ideal gas law to

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.Note

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

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

Note


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 variables for heat fluxes and total energy fluxes. It is also used if the ideal gas law is applied. See Thermodynamics.


From the Selection list, choose the domains to define the heat transfer.


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 PressureEnter the Absolute pressure pA (SI unit: Pa). The default is atmosphere pressure, 1 atm (101,325 Pa).

This section controls both the variable as well as any property value (reference pressures) used when solving for pressure. There are usually two ways of calculating 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 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,

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

Model

Absolute pressure is also used if the ideal gas law is applied. See Thermodynamics.

Note


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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 absolute pressure model input is controlled by both a drop-down list and a check box within this section. Use the User defined option to manually define the absolute pressure in a system. This is the default setting.

The pressure variables solved for by a fluid-flow interface can also be used, which is selected from the list as, for example, Pressure spf/fp. Selecting a pressure variable also activates a check box for defining the reference pressure, where 1[atm] (1 atmosphere) is the default value. 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 respective physics interface, it may not with physics interfaces that being coupled to. In such models, check the coupling between any interfaces using the same variable.

Velocity FieldFrom the Velocity field list, select an existing velocity field in the model (for example, Velocity field (spf/fp1) from a Laminar Flow interface) or select User defined to enter values or expressions for the components of the Velocity field (SI unit: m/s).




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 = kT which is Fourier’s law of heat conduction. Enter this quantity as power per length and temperature.



The default Density (SI unit: kg/m3), Heat capacity at constant pressure CP (SI unit: J/(kg·K)), and Ratio of specific heats (unitless) for a general gas or liquid use values From material. Select User defined to enter other values or expressions.

Select a Fluid type:

• 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 the material. Select User defined to enter another value for the density, heat capacity, or ratio of specific heats.

• Select Ideal gas to use the ideal gas law to describe the fluid. In this case, specify the thermodynamics properties by selecting a gas constant type and selecting between entering the heat capacity at constant pressure or the ration of specific heats:

From the list under Gas constant type, select Specific gas constant to specify the specific gas constant Rs, or select Mean molar mass to specify the mean molar mass Mn. 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. For both properties, the default setting is to use the property value from the material. Select User defined to define another value for either of these material properties.

From the list under Specify Cp or , select Heat capacity at constant pressure to specify the heat capacity at constant pressure Cp, or select Ratio of specific heats to specify the ratio of specific heats . For both properties, the default setting is to use the property value from the material. Select User defined to define another value for either of these material properties.

Heat Source

The Heat Source describes heat generation within the domain. 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 linear heat source, or as a total heat source (power).

For an ideal gas, specify either Cp or the ratio of specific heats, , but not both since these, in that case, are dependent.

Tip


88 | C H A P T E R


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

H E A T S O U R C E

Click the General source, Linear source, or Total power button.

• If General source is selected, enter a value for the distributed heat source Q (SI unit: W/m3).

• If Total power is selected, enter the total power (total heat source) Ptot (SI unit: W).

• If Linear source (Qqs·T) is selected, enter the Production/absorption coefficient qs (SI unit: W/(m3·K)).

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.


From the Selection list, choose the domains to define an initial value.

For the Heat Transfer in Porous Media interface, and for the Batteries & Fuel Cells Module, Corrosion Module, or Electrodeposition Module, when General Source is selected, heat sources from the electrochemical current distribution interfaces are also made available in the list under Heat Source. Choose the appropriate one or User defined to defined your own.

Note

The advantage of writing the source in this 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.

Stabilization Techniques in the COMSOL Multiphysics Reference Guide

Tip

See Also


I N I T I A L V A L U E S

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).

Temperature

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

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

From the Selection list, choose the boundaries to apply a temperature.

TE M P E R A T U R E

The equation for this condition is T = T0 where T0 is the prescribed temperature on the boundary. Enter the value or expression for the Temperature T0 (SI unit: K). The default is 293.15 K.

C O N S T R A I N T S E T T I N G S

To display this section, click the Show button ( ) and select Advanced Physics Options. By default Classic constraints is selected. If required, select the Use weak constraints check box.

Use Discontinuous Galerkin constraints as an alternative to the Classic constraints when it does not work satisfactorily. This 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.

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:

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

n kT 0=


90 | C H A P T E R

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 boundary.


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.


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

Symmetry

The Symmetry node provides a boundary condition for symmetry boundaries. This boundary condition is similar to an insulation condition, and it means that there is no heat flux across the boundary.

The default Thermal Insulation feature does not require any user input. If required, add more features by right-clicking the Heat Transfer node and selecting the boundaries to apply the thermal insulation. Note

The symmetry condition only applies for the temperature field. It has no effect on the radiosity (surface-to-surface radiation) and on the radiative intensity (radiation in participating media).Note



Heat Flux

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

B O U N D A R Y O R E D G E S E L E C T I O N

From the Selection list, choose the boundaries or edges to add the heat flux contribution.

P A I R S E L E C T I O N

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

H E A T F L U X

Click one of the General inward heat flux, Inward heat flux, or Total heat flux buttons.

• 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.

• If Inward heat flux is selected, enter the Heat transfer coefficient h (SI unit: W/(m2·K)). The default is 0. Also enter an External temperature Text (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 this equation:

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

Note

For inlet boundaries, use the Inflow Heat Flux condition instead. For the Thin Conductive Shell interface, the Heat Flux feature adds a heat source (or sink) to edges. It adds a heat flux qdeq0.Tip


92 | C H A P T E R

For a thorough introduction about how to calculate heat transfer coefficients, see Incropera and DeWitt in Ref. 1.

• 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.

Surface-to-Ambient Radiation

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

where is the surface emissivity, is the Stefan-Boltzmann constant (a predefined physical constant), and Tamb is the ambient temperature.


From the Selection list, choose the boundaries to add surface-to-ambient radiation contribution.


The default Surface emissivity e (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 Tamb (SI unit: K). The default is 293.15 K.

Periodic Heat Condition

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

q0 h Text T– =

q Tamb4 T4

– =

Continuous Casting: Model Library path Heat_Transfer_Module/

Process_and_Manufacturing/continuous_castingModel



From the Selection list, choose the boundaries to add a periodic heat condition.

Boundary Heat Source

The Boundary Heat Source node models a heat source (or heat sink) that is embedded in the boundary.


From the Selection list, choose the boundaries to apply the heat source.

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

Click the General source 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.

• If Total boundary power is selected, enter the total power (total heat source) Pb, tot (SI unit: W).

Pair Boundary Heat Source

The Pair Boundary Heat Source node models a heat source (or heat sink) that is embedded in the boundary. It also prescribes that the temperature field is continuous across the pair.


From the Selection list, choose the boundaries to apply the heat source.

Periodic Condition is also described in the COMSOL Multiphysics User’s Guide:

• Periodic Condition

• Destination Selection

• Using Periodic Boundary Conditions

• Periodic Boundary Condition Example

See Also


94 | C H A P T E R


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.

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

Click the General source 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.

• If Total boundary power is selected, enter the total power (total heat source) Pb, tot (SI unit: W).

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.


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


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

Pair Thin Thermally Resistive Layer

Use the Pair 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.

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

In the COMSOL Multiphysics User’s Guide:

• Identity and Contact Pairs

• Specifying Boundary Conditions for Identity PairsSee Also


where u and d subscript refer respectively to the upside and the downside of the pair.

When the material has a multi-layer structure ks and ds in the expressions above are replaced by dtot and ktot which are defined according to following expressions:

(3-3)

(3-4)

where nl is the number of layers.


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


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

P A I R T H I N T H E R M A L L Y R E S I S T I V E L A Y E R

Enter a value or expression for the Layer thickness ds (SI unit: m).

The default is to use the Thermal conductivity ks (SI unit: W/(m·K)) From material. Select User defined to enter another value or expression.

nd kdTd– – ksTu Td–

ds--------------------–=

nu kuTu– – ksTd Tu–

ds--------------------–=

dtot dsj

j 1=

nl

=

ktotdtot

dsjksj-------

j 1=

nl

------------------=

The Multiple layers check box enables the definition of multiple sandwiched thin layers with different thermal conductivities.

By default Multiple layers is not selected. Define the properties of a single layer.

Note


96 | C H A P T E R

When Multiple layers is selected, define the properties of a multiple layers structure.

Select the number of layer to define (1 to 5) and set following properties for each:

• Assign a material to each layer by selecting the appropriate material in the Solid

material i list.

• Enter a value or expression for the Layer thickness dsi (SI unit: m).

• The default is to use the Thermal conductivity ksi (SI unit: W/(m·K)) From material. This gets the thermal conductivity from the material selected in Solid material i. Select User defined to enter another value or expression.

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.

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

where u and d subscript refer respectively to the upside and the downside of the slit.

When the material has a multi-layer structure ks and ds in the expressions above are replaced by dtot and ktot which are defined according to following expressions:

(3-5)

(3-6)

where nl is the number of layers.

nd kdTd– – ksTu Td–

ds--------------------–=

nu kuTu– – ksTd Tu–

ds--------------------–=

dtot dsj

j 1=

nl

=

ktotdtot

dsj

ksj-------

j 1=

nl

------------------=



From the Selection list, choose the boundaries to add a thermally resistive layer.


The temperature model input used to evaluate the material properties is equal to the mean temperature on interior boundaries.

T H I N T H E R M A L L Y R E S I S T I V E L A Y E R

Enter a value or expression for the Layer thickness ds (SI unit: m).

The default is to use the Thermal conductivity ks (SI unit: W/(m·K)) From material. Select User defined to enter another value or expression.

When Multiple layers is selected, define the properties of a multiple layers structure.

Select the number of layer to define (1 to 5) and set following properties for each:

• Assign a material to each layer by selecting the appropriate material in the Solid

material i list.

• Enter a value or expression for the Layer thickness dsi (SI unit: m).

• The default is to use the Thermal conductivity ksi (SI unit: W/(m·K)) From material. This gets the thermal conductivity from the material selected in Solid material i. Select User defined to enter another value or expression.

The Multiple layers check box enables the definition of multiple sandwiched thin layers with different thermal conductivities.

By default Multiple layers is not selected. Then define the properties of a single layer.

Note


98 | C H A P T E R

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 feature from the Edges menu.

E D G E S E L E C T I O N

From the Selection list, choose the edges to apply the heat source.

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

Click the General source or Total power button.

• If General source is selected, enter a value for the distributed heat source, Ql, in unit power per unit length (SI unit: W/m). Positive Ql is heating while a negative Ql is cooling.

• If Total power is selected, enter the total power (total heat source) Pl,tot (SI unit: W)

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 feature from the Points menu.

The Line Heat Source node is available in 3D only because a line in 2D is a boundary and a domain in 1D.

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 returns a finite temperature distribution along the line, but that distribution must be interpreted in a weak sense.

3D

The Point Heat Source is available in 2D and 3D. It is not available in 1D since points are boundaries (possibly internal boundaries) there.

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 returns a finite value, but that value must be interpreted in a weak sense.

2D

3D


PO I N T S E L E C T I O N

From the Selection list, choose the points to apply the heat source.

PO I N T H E A T S O U R C E

Enter the quantity Qp in unit power (SI unit: W). Positive Qp is heating while a negative Qp is cooling.

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.


From the Selection list, choose the boundaries to activate the continuity on interior boundaries.


If Continuity on Interior Boundary is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect.

Reaction Heat Flux

The Reaction Heat Flux feature is available only with the Plasma Module and is documented in the Plasma Module User’s Guide.

Note


100 | C H A P T E

Hea t T r a n s f e r I n t e r f a c e Ad v an c ed F e a t u r e s

For the Heat Transfer Module, several advanced features are available with this interface. In addition to the nodes described in The Heat Transfer Interface, this section details these nodes and subnodes (listed in alphabetical order):

• Incident Intensity

• Infinite Elements


• Opaque

• Opaque Surface

• Open Boundary

• Pressure Work

• Radiation in Participating Media

• Viscous Heating


The Radiation in Participating Media node uses the radiative transfer equation

• About the Heat Transfer Interfaces

• Domain, Boundary, Edge, Point, and Pair Features for the Heat Transfer Interfaces



See Also

Tip

I s Ib T I s –s4------ I s

0

4

+=

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

where

• Is) is the radiative intensity at the position s position in the direction

• T is the temperature

• , , s are absorption, extinction, and scattering coefficients

• Ib is the blackbody radiative intensity

It also adds the radiative heat source term in the heat transfer equation:


From the Selection list, choose the domains to define radiation in participating media.

M O D E L S I N P U T S

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 dependant variable.

A B S O R P T I O N

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.

S C A T T E R I N G

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 Nonlinear anisotropic.

• Isotropic (the default) and corresponds to the scattering phase function .

• If Linear anisotropic is selected, it defines the scattering phase function as . Enter the Legendre coefficient a1.

• If Nonlinear anisotropic is selected, it defines the scattering phase function

Qr q r G 4T4– = =

0 1=

0 1 a10+=

H E A T TR A N S F E R I N T E R F A C E A D V A N C E D F E A T U R E S | 101

102 | C H A P T E

Enter the Legendre coefficients a1, …, a12 as required.

Opaque

Right-click the Heat Transfer in Solids or Heat Transfer in Fluids node to add the Opaque feature. By default all the domains are transparent. Surface to surface boundary conditions can only be set at the interface between transparent and opaque domains.


From the Selection list, choose the domains to add this feature. By default, the selection is the same as for the Heat Transfer in Solids or Heat Transfer in Fluids node it is attached to.

Infinite Elements

0 1 amPm 0

m 1=

12

+=

The Opaque node is available for the Heat Transfer with Surface-to-Surface

Radiation version of the Heat Transfer interface. It is also available for The Surface-To-Surface Radiation Interface.Note

Cavity Radiation: Model Library path Heat_Transfer_Module/

Verification_Models/cavity_radiationModel

For the Infinite Elements node, see About Infinite Element Domains and Perfectly Matched Layers in the COMSOL Multiphysics User’s Guide.

Note


Opaque Surface

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.


From the Selection list, choose the boundaries to define the wall condition.


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


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.

WA L L S E T T I N G S

Select a Wall type to define the behavior of the wall—Gray wall or Black wall.

Gray WallIf 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.

The Opaque Surface node is available for The Heat Transfer with Radiation in Participating Media Interface version of the Heat Transfer interface. It is also available for The Radiation in Participating Media Interface.Note

The boundary temperature definition can differ from the that of the temperature in the adjacent domain.

Tip


104 | C H A P T E

Both are dimensionless numbers between 0 and 1 that satisfy the relation dw1. By default d1w. 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 boundary. If d1w, 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/m2 in SI units) along incoming discrete directions on this boundary is defined by

Black WallIf Black wall is selected, no user input is required and the radiative intensity along the incoming discrete directions on this boundary is defined by

Values of radiative intensity along outgoing discrete directions are not prescribed.

Incident Intensity

Use an Incident Intensity node to specify the radiative intensity along incident directions on a boundary.


From the Selection list, choose the boundaries to define the radiative intensity along incident directions.


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

Ii bnd wIb T d

------qout+=

Ii bnd Ib T =

The Incident Intensity node is available for The Heat Transfer with Radiation in Participating Media Interface version of the Heat Transfer interface. It is also available for The Radiation in Participating Media Interface.Note


I N C I D E N T I N T E N S I T Y

Enter a Boundary radiation intensity Iwall (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.

Pressure Work

Right-click the Heat Transfer in Solids or Heat Transfer in Fluids node to add this subnode. The Pressure Work node adds the following term to the right-hand side of Equation 3-7:

(3-7)

where Sel is the Elastic Contribution to Entropy. Compared to

(this is also described in the theory for Laminar Flow), the part that corresponds to sound wave propagation is neglected. The reason is the energy in the sound waves are almost always negligible compared to the contribution from Equation 3-7. The software computes the pressure work using the absolute pressure.


From the Selection list, choose the domains to add pressure work. By default, the selection is the same as for the Heat Transfer in Solids or Heat Transfer in Fluids node it is attached to.

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.

Tip

T t-----Sel–

ut------- u u+ pI– + F+=

Theory for the Laminar Flow Interface in the COMSOL Multiphysics User’s Guide.

See Also


106 | C H A P T E


Enter a value or expression for the Elastic contribution to entropy Ent (SI unit: Jm3·K)). The default is 0.

Viscous Heating

The Viscous Heating node adds the following term to the right-hand side of the Heat

Transfer in Fluids equation:

(3-8)

where is the viscous stress tensor and S is the strain-rate tensor. Equation 3-8 represents the heating caused by viscous friction within the fluid.


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

D Y N A M I C V I S C O S I T Y

Specify the Dynamic viscosity (SI unit: Pa·s). The default setting is to use the value of the viscosity from the material. Select User defined to define another value for the viscosity. 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 boundary

(3-9)

where

(3-10)

:S

n kT– – q0 u n– 1

u n------------------- hin h– u n+=

hin h– Cp TdT

Tin

1--- 1 T

----

T-------

p

+

pdp

pA

+=


A positive heat flux adds heat to the domain. The Inflow Heat Flux feature is applicable to inlet boundaries.

The second integral in Equation 3-7 is neglected if the Inflow Heat Flux feature is applied to the boundary of a solid domain.


From the Selection list, choose the boundaries to add the heat flux contribution.

I N F L O W H E A T F L U X

Enter a value or expression for each of the following properties.

• Select the Inward heat flux or Total heat flux button.

- 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.

- When Total heat flux is selected, define qtot. In this case q0qtot/A, where A is the total area of the selected boundaries.

In 3D and 2D axial symmetry, .

In 2D and 1D axial symmetry, , 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, , 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.

2D Axi

3D

A 1=

2D

1D Axi

A dz 1=

1D

A Ac 1=


108 | C H A P T E

• External temperature Text (SI unit: K). The default value is 273.15 K.

• External absolute pressure pext (SI unit: Pa). The default value is 1 atm.

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.


From the Selection list, choose the boundaries to model as open boundaries.

E X T E R I O R T E M P E R A T U R E

Enter the exterior Temperature T0 (SI unit: K) outside of the open boundary.

Convective Cooling

The Convective Cooling node adds the following heat flux contribution to its boundaries:

where h can be defined by the user or by using a library of predefined coefficients described in the section About the Heat Transfer Coefficients.


From the Selection list, choose the boundaries to add a convective cooling contribution. For information about selecting boundaries.

h Text T–

• Power Transistor: Model Library path Heat_Transfer_Module/

Electronics_and_Power_Systems/power_transistor

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

Model


H E A T F L U X

Select a Heat transfer coefficient h (SI unit: W/(m2·K)) to control the type of convective cooling to model—User defined (the default), External natural convection, Internal natural convection, External forced convection, or Internal forced convection.

• For all of the options, enter an External temperature, Text (SI unit: K).

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

External Natural ConvectionSelect Vertical wall, Inclined wall, Horizontal plate, upside, or Horizontal plate, downside

from the list.

• If Vertical wall is selected, enter a Wall height L (SI unit: m).

• If Inclined wall is selected, enter a Wall height L (SI unit: m) and the Tilt angle (the angle between the wall and the vertical direction, for vertical walls).

• 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 defaults are 0.

Internal Natural ConvectionSelect Narrow chimney, parallel plates or Narrow chimney, circular tubes from the list.

• 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 0.

• If Narrow chimney, circular tubes is selected, enter a Tube diameter D (SI unit: m) and a Chimney height H (SI unit: m). The defaults are 0.

External Forced ConvectionSelect Plate, averaged transfer coefficient or Plate, local transfer coefficient from the list.

• If Plate, averaged transfer coefficient is selected, enter a Plate length L (SI unit: m) and a Velocity, external fluid Uext (SI unit: m/s). The defaults are 0.

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

Internal Forced ConvectionThe only option is Isothermal tube. Enter a Tube diameter D (SI unit: m) and a Velocity, external fluid Uext (SI unit: m/s). The defaults are 0.


110 | C H A P T E

H i gh l y C ondu c t i v e L a y e r F e a t u r e s

In this section:


• Layer Heat Source

• Edge Heat Flux

• Edge Temperature

• Edge Surface-to-Ambient

Highly Conductive Layer

• 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

Model

Use the Highly Conductive Layer feature 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.

About Highly Conductive Layers

2D

2D Axi

3D

See Also


Right-click to add these additional features:

• Layer Heat Source— to add an 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.

• 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 Edge 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 and Point Surface-to-Ambient Radiation.


From the Selection list, choose the boundaries to model as highly conductive layers.


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.




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

If you also have the Microfluidics Module, the Edge Heat Flux and Point

Heat Flux nodes are not available with the Slip Flow interface.Note

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112 | C H A P T E

Isotropic, Diagonal, Symmetric, or Anisotropic based on the characteristics of the thermal conductivity, and enter another value or expression.


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

Layer Heat Source

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


From the Selection list, choose the boundaries to add the heat source. By default, the selection is the same as for the Highly Conductive Layer feature.

L A Y E R H E A T S O U R C E

Enter a value or expression for the Layer heat source Qs (SI unit: W/m2).

Edge Heat Flux

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

About Highly Conductive Layers

Tip

See Also

Use the Edge Heat Flux feature 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. 3D



From the Selection list, choose the edges to add the heat flux contribution.

H E A T F L U X

Select either the General inward heat flux or Inward heat flux buttons.

• If 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, and its geometry can be omitted.

• If Inward heat flux is selected (in the form q0h·TextT, enter the Heat transfer

coefficient h (SI unit: W/(m2·K)). The default value is 0. Enter an External

temperature Text (SI unit: K). The default value is 293.15 K. The value depends on the geometry and the ambient flow conditions.

Point Heat Flux


From the Selection list, choose the points to add the heat flux contribution.

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

Note

Use the Point Heat Flux feature 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.

2D

2D Axi

Note


114 | C H A P T E

H E A T F L U X

Select either the General inward heat flux or Inward heat flux buttons.

• If 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.

• If Inward heat flux is selected (in the form q0h·TextT, enter the Heat transfer

coefficient h (SI unit: W/(m2·K)). The default value is 0. Enter an External

temperature Text (SI unit: K). The default value is 293.15 K. The value depends on the geometry and the ambient flow conditions.

Edge Temperature


From the Selection list, choose the edges to apply an edge temperature.


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


Enter the value or expression for the Temperature T0 (SI unit: K). The equation for this condition is T = T0 where T0 is the prescribed temperature on the edges.

Use the Edge Temperature feature 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.

3D

Note



To display this section, click the Show button ( ) and select Advanced Physics Options. Select a Constraint type—Bidirectional, symmetric or Unidirectional. If required, select the Use weak constraints check box.

Point Temperature


From the Selection list, choose the points to apply a prescribed temperature.


Enter the value or expression for the Temperature T0 (SI unit: K). The equation for this condition is T = T0 where T0 is the prescribed temperature on the points.


To display this section, click the Show button ( ) and select Advanced Physics Options. Select a Constraint type—Bidirectional, symmetric or Unidirectional. If required, select the Use weak constraints check box.

Edge Surface-to-Ambient

Use the Point Temperature feature 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.

2D

2D Axi

Note

Use the Edge Surface-to-Ambient Radiation feature to add surface-to-ambient radiation to edges representing boundaries of a highly conductive layer.3D


116 | C H A P T E

The net inward heat flux from surface-to-ambient radiation is


Right-click the Highly Conductive Layer node to add this feature.


From the Selection list, choose the edges to add surface-to-ambient radiation contribution.



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.

Point Surface-to-Ambient Radiation

The net inward heat flux from surface-to-ambient radiation is



From the Selection list, choose the points to add surface-to-ambient radiation contribution.

q Tamb4 T4

– =

Use the Point Surface-to-Ambient Radiation feature to add surface-to-ambient radiation to points representing boundaries of a highly conductive layer.

2D

2D Axi

q Tamb4 T4

– =




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.


118 | C H A P T E

Ou t - o f - P l a n e Hea t T r a n s f e r F e a t u r e s

The following features are available for 1D and 2D Heat Transfer models. In this section:

• Out-of-Plane Convective Cooling

• Out-of-Plane Radiation

• Out-of-Plane Heat Flux

• Change Thickness

Out-of-Plane Convective Cooling

Use the Out-of-Plane Convective Cooling node to model upside and downside cooling (or heating) caused by the presence of an ambient fluid. The Out-of-Plane Convective

Cooling feature adds the following contribution to the right-hand side of Equation 2-28 or Equation 2-29:

Surface Resistor: Model Library path Heat_Transfer_Module/

Electronics_and_Power_Systems/surface_resistor

Theory of Out-of-Plane Heat Transfer

Model

See Also

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

1D

2D

hu Text,u T– h+ d Text,d T–



Select the domains where you want to add an out-of-plane convective cooling contribution.

U P S I D E H E A T F L U X

Select a Heat transfer coefficient hu (SI unit: W/(m2·K)) to control the type of convective cooling 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 hu0.

• For all of the options, enter an External temperature, Text,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 Cooling feature, then select an External fluid—Air, Transformer oil, or water. If Air is selected, also enter an Absolute

pressure, pA (SI unit: Pa). The default is 1 atm.

D O W N S I D E H E A T F L U X

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.

Out-of-Plane Radiation

The Out-of-Plane Radiation feature models surface-to-ambient radiation on the upside and downside and adds the following contribution to the right-hand side of Equation 2-28 or Equation 2-29:

About the Heat Transfer CoefficientsSee Also


1D

2D

O U T - O F - P L A N E H E A T TR A N S F E R F E A T U R E S | 119

120 | C H A P T E


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

U P S I D E P A R A M E T E R S

The default Surface emissivity eu, a unitless 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.

Enter an Ambient temperature Tamb,u (SI unit: K). The default is 293.15 K.

D O W N S I D E P A R A M E T E R S

Follow the instructions for the Upside Parameters for the downside parameters ed and Tamb,d.

Out-of-Plane Heat Flux

The Out-of-Plane Heat Flux feature 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 2-28 or Equation 2-29:

u Tamb u4 T4– d Tamb d

4 T4– +

Compare to the equation in the heat theory section about Surface-to-Ambient Radiation.

See Also


1D

2D



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

U P S I D E I N W A R D H E A T F L U X

Select between specifying the upside inward heat flux directly or as a convective term using a heat transfer coefficient.

• Click the General inward heat flux button to specify a value or expression for the inward (or outward, if the quantity is negative) heat flux through the upside (SI unit: W/m2) in the q0,u field.

• 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/m2) as hu·(Text,uT). Enter a value or expression for the heat transfer coefficient in the hu field (SI unit: W/(m2·K) and a value or expression for the external temperature in the Text,u field (SI unit: K). The default value for the external temperature is 293.15 K.

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

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 feature makes it possible model domains with another thickness than the overall thickness that is specified in the Heat Transfer feature’s Physical Model section.


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

dsq0 u dsq0 d+

Select the Out-of-plane heat transfer check box on the Heat Transfer interface to add this feature to 2D models.

2D

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122 | C H A P T E

C H A N G E T H I C K N E S S

Specify a value for the Thickness dz (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.


Th e B i o h e a t T r a n s f e r I n t e r f a c e

When Bioheat Transfer is selected under the Heat Transfer branch ( ) in the Model

Wizard, Biological tissue is automatically selected as the Default model and a Heat

Transfer ( ) interface is added to the Model Builder.

When this version of the interface is added, these default nodes are added to the Model

Builder—Biological Tissue, 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.


Biological tissue is automatically selected as the Default model and. If Heat transfer in

solids or Heat transfer in fluids is selected as the Default model, the Biological Tissue node changes to a Heat Transfer in Solids or Heat Transfer in Fluids node.

This interface is also added when the Heat Transfer in Biological Tissue check box is selected under Advanced Settings on the Heat Transfer interface. To display the section, click the Show button ( ) and select Advanced Physics Options.

Hepatic Tumor Ablation: Model Library path Heat_Transfer_Module/

Medical_Technology/tumor_ablation

Note

Model

The interior and exterior boundary conditions are the same as for the The Heat Transfer Interface.


Note

Tip

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124 | C H A P T E

Biological Tissue

The Biological Tissue feature adds the bioheat equation as the mathematical model for heat transfer in biological tissue. See Equation 2-30.

Right-click the node to add a Bioheat node.




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.




The default uses Thermal conductivity k (SI unit: W/(m·K)) 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.


• Theory for the Bioheat Transfer Interface

• Biological Tissue

• Bioheat

• Working with Geometry in the COMSOL Multiphysics User’s Guide

See Also

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.Tip



The default Density (SI unit: kg/m3) and Heat capacity at constant pressure Cp (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

defined is selected, enter other values or expressions.

Bioheat

A default Bioheat node is added to the Biological Tissue node. This feature provides the source terms that represent blood perfusion and metabolism for modeling heat transfer in biological tissue using the bioheat equation

b Cb b (TbT)

Define the density of blood, specific heat of blood, blood perfusion rate, the arterial blood temperature and the metabolic heat source. Right-click the Biological Tissue node to add more Bioheat subnodes.


From the Selection list, choose the domains to add bioheat source terms. The default selection are the domains in the Biological Tissue feature, but more than one Bioheat feature can be defined with different settings for subsets of the domains where the bioheat transfer occurs.

B I O H E A T

Enter values or expressions for the following properties and source terms in the associated fields:

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

• 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.

• Blood perfusion rate b (SI unit: 1/s, which in this case means (m3/s)/m3), describes the volume of blood per second that flows through a unit volume of tissue.

• Arterial blood temperature Tb (SI unit: K), which is the temperature at which blood 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.

• Metabolic heat source Qmet (SI unit: W/m3), which describes heat generation from metabolism. Enter this quantity as the unit power per unit volume.

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Th e Hea t T r a n s f e r i n Po r ou s Med i a I n t e r f a c e

After selecting Heat Transfer in Porous Media, found under the Heat Transfer

branch ( ) in the Model Wizard, a Heat Transfer ( ) node is added to the Model

Builder with these default nodes: Porous Matrix, Heat Transfer in Fluids, Thermal

Insulation, and Initial Values. Right-click the main node to open a context menu and add as many nodes as required to define the equations, properties and boundary conditions.

The Heat Transfer in Porous Media interface ( ) 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. The ability to define material properties, boundary conditions and more for porous media heat transfer is activated by selecting the Heat transfer in porous media check box on the Heat Transfer settings window (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 the Heat Transfer settings window.

When a different Default model is selected (for example, Heat transfer in

solids), the node in the Model Builder also changes to match. This is a quick way to toggle between the Heat Transfer interfaces.Tip


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

The Heat Transfer in Porous Media Interface has these features described in this section:


• Porous Matrix

• Thermal Dispersion

These domain, boundary, edge, point, and pair conditions are described for The Heat Transfer Interface in the COMSOL Multiphysics User’s Guide (listed in alphabetical order):


• Continuity

• Heat Flux

• Heat Source



• Domain, Boundary, Edge, Point, and Pair Features for the Heat Transfer in Porous Media Interface

• Theory for the Heat Transfer in Porous Media Interface

• Theory for the Heat Transfer Interfaces in the COMSOL Multiphysics User’s Guide

See Also

The links to features described the COMSOL Multiphysics User’s Guide do not work in the PDF, only from within the online help.


Important

Tip

T H E H E A T TR A N S F E R I N P O R O U S M E D I A I N T E R F A C E | 127

128 | C H A P T E


• Outflow




• Symmetry

• Temperature

• Thermal Insulation

• Thin Thermally Resistive Layer and Pair Thin Thermally Resistive Layer

Heat Transfer in Fluids




If user-defined property groups are added, this section displays fields and values that are inputs to expressions that define material properties. There are also two standard model inputs—Absolute pressure and Velocity field.

Enter the Absolute pressure p (SI unit: Pa). The default is atmosphere pressure (101,325 Pa).

From the Velocity field list, select an existing velocity field in the model (for example, Velocity field (spf/fp1) from a Laminar Flow interface) or select User defined to enter values or expressions for the components of the Velocity field (SI unit: m/s).


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

The Heat Transfer in Fluids feature is available with the basic COMSOL Multiphysics license.

Note



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 for the Density (SI unit: kg/m3), Heat capacity at constant pressure CP (SI unit: J/(kg·K)), and the Ratio of specific heats (unitless) take values From material

for a general gas or liquid. Select User defined to enter other values or expressions.

Porous Matrix

The Porous Matrix feature is used to specify the thermal properties of a porous matrix. Right-click to add a Thermal Dispersion subnode.


From the Selection list, choose the domains to define the heat transfer in porous media.


This section contain 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.

I M M O B I L E S O L I D S

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 (unitless) for the solid material.


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 thermal conductivity describes the relationship between the heat flux vector q and the temperature gradient T as in q = kT which is Fourier’s law of heat conduction. Tip


130 | C H A P T E

the characteristics of the thermal conductivity and enter another value or expression in the field or 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 Cp,p (SI unit: J/(kg·K)) uses values From material. If User defined is selected, enter another value or expression.

The equivalent volumetric heat capacity of the solid-liquid system is calculated from

Thermal Dispersion

Right-click the Porous Matrix node to add the Thermal Dispersion feature. This adds an extra term ·kdT to the right-hand side of Equation 2-31 and specifies the values of the longitudinal and transverse dispersivities.


From the Selection list, choose the domains to activate the thermal dispersion.

D I S P E R S I V I T I E S

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

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 = kpT, which is Fourier’s law of heat conduction. Tip

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.

Tip

Cp eq ppCp p LCp+=


where Dij is the dispersion tensor

and ijkl is the fourth order dispersivity tensor

kijd LCp L Dij=

Dij ijklukul

u------------=

ijkl trijkllo tr–

2-------------------- ikjl iljk+ +=


132 | C H A P T E

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

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 Interface

• Theory for the Heat Transfer in Thin Shells Interface

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134 | C H A P T E

Th e Hea t T r a n s f e r i n Th i n S h e l l s I n t e r f a c e

The Thin Conductive Shell (htsh) interface is suitable for solving thermal-conduction 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.

The Thin Conductive Layer is the main feature. It adds the equation for the temperature and provides a setting 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 Thin Conductive Shell node to add other features that implement, for example, edge or point conditions and heat sources.



The default identifier (for the first interface in the model) is htsh.

The Thin Conductive Shell interface ( ) opens after selecting Heat

Transfer in Thin Shells under the Heat Transfer branch ( ) in the Model

Wizard. It is available for 3D models. 3D

Shell Conduction: Model Library path Heat_Transfer_Module/

Tutorial_Models/shell_conductionModel

R 4 : H E A T TR A N S F E R I N T H I N S H E L L S


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.

S H E L L T H I C K N E S S

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


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


To display additional features for the physics interfaces and feature nodes, click the Show button ( ) on the Model Builder and then select Discretization. Select a Frame

type—Spatial (the default) or Material. Select Quadratic (the default), Linear, Cubic, or Quartic for the Temperature. Specify the Value type when using splitting of complex

variables—Real or Complex (the default).

Boundary, Edge, Point, and Pair Conditions for the Thin Conductive Shell Interface

The Heat Transfer in Thin Shells Interface (named Thin Conductive Shell in the Model

Wizard), has the following boundary, edge, point and pair conditions described (listed in alphabetical order):

• Change Effective Thickness


• Boundary, Edge, Point, and Pair Conditions for the Thin Conductive Shell Interface

• Theory for the Heat Transfer in Thin Shells InterfaceSee Also

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

Note

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136 | C H A P T E

• Change Thickness

• Edge Heat Source

• Edge Temperature (only Edge Temperature is available for this interface)

• Initial Values

• Insulation/Continuity

• Heat Flux

• Heat Source

• Out-of-Plane Convective Cooling

• Out-of-Plane Heat Flux

• Out-of-Plane Radiation


• Radiation


• Thin Conductive Layer

Thin Conductive Layer

The Thin Conductive Layer feature adds the heat equation for conductive heat transfer in shells (see Equation 4-1).


From the Selection list, choose the boundaries to define the temperature and the heat transfer equation that defines the temperature field.


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.




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.


Specify 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, and the Density (SI unit: kg/m3). The default settings use values From material. If User

defined is selected, enter other values or expressions.

Heat Source

The Heat Source feature adds a thermal source Q. It adds the following contributions to the right-hand side of Equation 4-1:


From the Selection list, choose the boundaries to define the heat source.

H E A T S O U R C E

Enter a value or expression for the Shell heat source Q (SI unit: W/m3), which describes heat generation within the shell. Express heating and cooling with positive and negative values, respectively.

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.

Thermal Conductivity Tensor ComponentsSee Also

dsQ


138 | C H A P T E


From the Selection list, choose the boundaries to define the initial value.


Enter a value or expression for the initial value of the Temperature T. The default is approximately room temperature, 293.15 K (20º C).

Change Thickness

Use the Change Thickness feature to give parts of the shell a different thickness than that what is specified on the Thin Conductive Shell interface Shell Thickness section.


From the Selection list, choose the boundaries to define the thickness.

C H A N G E T H I C K N E S S

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.

Insulation/Continuity

The Insulation/Continuity feature is the default edge condition. On external edges, this edge condition means that there is no heat flux across the edge:

On internal edges, this edge condition means that the temperature field and its flux is continuous across the edge.


From the Selection list, choose the edges to define the insulation/continuity.


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

Radiation

Use the Radiation feature to add surface-to-ambient radiation to edges. The net inward heat flux from surface-to-ambient radiation is

n kgT 0=


where is the Stefan–Boltzmann constant (a predefined physical constant).


From the Selection list, choose the edges to define the radiation flux.


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


The default Surface emissivity e is used From material. It is a number between 0 and 1. 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. Select User defined to enter another value.


Change Effective Thickness

The Change Effective Thickness feature models edges with another thickness than the overall thickness that is specified in the Thin Conductive Shell interface Shell Thickness section. It defines the height of the part of the edge that is exposed to the ambient surroundings.


From the Selection list, choose the edges to define effective thickness.


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

C H A N G E E F F E C T I V E T H I C K N E S S

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 feature models a linear heat source (or sink).

q de Tamb4 T4

– =


140 | C H A P T E


From the Selection list, choose the edges to define the heat source.


If Edge 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.

E D G E H E A T S O U R C E

Enter a value or expression for the Edge heat source qe (SI unit: W/m). A positive qe means heating while a negative qe means cooling. The added heat source is equal to qe.

Point Heat Source

The Point Heat Source feature models a point heat source (or sink).


From the Selection list, choose the point to define the heat source.

PO I N T H E A T S O U R C E

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.


Th eo r y f o r t h e Hea t T r a n s f e r i n Th i n S h e l l s I n t e r f a c e

The Heat Transfer in Thin Shells Interface theory is described in this section:

• About Thin Conductive Shells

• Heat Transfer Equation in Thin Conductive Shell

• Thermal Conductivity Tensor Components

About Thin Conductive Shells

The Thin Conductive Shell (htsh) interface opens after selecting The Heat Transfer in Thin Shells Interface in the Model Wizard. This 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 setting window for defining the thermal conductivity, the heat capacity and the density:

(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:

(4-2)

Where T is the tangential derivative along the shell and

• is the density (SI unit: kg/m3)

dsCp tT T dskg TT– + 0=

dsCp tT T dskg TT– + dsQ dshu Text u, T– dshd Text d, T– + +=

ds+ u T4amb u, T4

– dsd T4amb d, T4

– dsqu dsqd+ + +

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• ds is the shell thickness (SI unit: m)

• Cp is the heat capacity (SI unit: J/(kg·K)

• kg is the thermal conductivity (SI unit: W/(m·K)

• Q is the heat source (SI unit: W/m3)

• hu and hd are the out-of-plane heat transfer coefficients, upside and downside (SI unit: W/(m2·K))

• Text, u and Text, 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),

• Tamb, u and Tamb, 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 q and the temperature gradient TT as in

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, exl, is the surface tangent vector t1 and the local z direction, ezl, is the normal vector n. Their cross product defines the third orthogonal direction such that:

From this, a transformation matrix between the shell’s local coordinate system and the global coordinate system can be constructed in the following way:

q kg TT–=

exl = t1

eyl = exl ez l n t1=

ez l = n


The thermal conductivity tensor, kg, can be expressed as

A

exlx eylx ezlx

exly eyly ezly

exlz eylz ez lz

=

kg AkAt=

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5

T h e Rad i a t i o n Hea t T r a n s f e r B r an c h

This chapter describes the Heat Transfer Module’s interfaces for modeling radiative heat transfer, including the Theory for the Radiative Heat Transfer Interfaces. The following physics interface for modeling of radiative heat transfer are available in the Heat Transfer>Radiation branch ( ):

• Heat Transfer with Surface-to-Surface Radiation (ht) interface, is a Heat Transfer interface where surface-to-surface radiation is active by default, enabling the Radiation Settings section. All available features are as described in the About the Heat Transfer Interfaces section.

• The Surface-To-Surface Radiation Interface is used for separate radiosity computations.

• The Heat Transfer with Radiation in Participating Media Interface is used to model radiative heat transfer in nontransparent media in combination with the other Heat Transfer physics interfaces for heat transfer through convection, conduction, and surface-to-ambient radiation.

• The Radiation in Participating Media Interface is used for separate computations of radiation in participating media.

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Th e S u r f a c e - T o - S u r f a c e Rad i a t i o n I n t e r f a c e

The Surface-to-Surface Radiation interface ( ), found under the Heat

Transfer>Radiation branch ( ) in the Model Wizard, treats thermal radiation as an energy transfer between boundaries and external heat sources where the medium 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.

When this interface is added, the Initial Values default node is also added to the Model

Builder. Right-click the node to add a Surface-to-Surface Radiation boundary condition or other feature nodes.

For the Surface-to-Surface Radiation 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 User’s Guide.

For this 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 Interface.

For another way to model combinations of conductive, convective, and radiative heat transfer, see the Radiation Settings section described for the Heat Transfer interface.

Important

Note

Tip

R 5 : T H E R A D I A T I O N H E A T TR A N S F E R B R A N C H



The default identifier (for the first interface in the model) is rad.


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.


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.

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.


The dependent variable (field variable) is for the Surface radiosity J. The name can be changed but the names of fields and dependent variables must be unique within a model.

See The Heat Transfer Interface for details about these settings. See Also

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148 | C H A P T E


To display this section, click the Show button ( ) and select Discretization. Select Quadratic (the default), Linear, Cubic, or Quartic for the Surface radiosity. Specify the Value type when using splitting of complex variables—Real or Complex (the default).

Domain, Boundary, Edge, Point, and Pair Conditions for the Surface-to-Surface Radiation Interface

The Surface-To-Surface Radiation Interface has these domain, boundary, edge, point, and pair features available (listed in alphabetical order):

• External Radiation Source

• Initial Values

• Opaque

• Prescribed Radiosity

• Radiation Group

• Reradiating Surface

• Surface-to-Surface Radiation (Boundary Condition)


• Domain, Boundary, Edge, Point, and Pair Conditions for the Surface-to-Surface Radiation Interface

• Radiative Heat Transfer in Transparent Media

• Theory for the Surface-to-Surface Radiation Interface

See Also


• Continuity on Interior Boundaries


• Specifying Boundary Conditions for Identity Pairs

The links to the features described in the COMSOL Multiphysics User’s Guide do not work in the PDF, only from within the online help.

See Also

Important


Surface-to-Surface Radiation (Boundary Condition)

The Surface-to-Surface Radiation boundary condition feature handles radiation with view factor calculation. The feature adds a radiosity dependent variable to its selection and uses it as surface radiosity.


From the Selection list, choose the boundaries to define the surface-to-surface radiation.



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.


Select a Radiation Direction based on the geometric normal (nx, ny, nz):

• Opacity controlled is the default and 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.




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Set Tamb to the far-away temperature in directions where no other boundaries obstruct the view. Inside a closed cavity, the ambient view factor, Famb, is theoretically zero and the value of Tamb therefore should not matter. It is, however, good practice to set Tamb to T in such cases because that minimizes errors introduced by the finite resolution of the view factor evaluation.

When Both sides is selected, define the Ambient temperature Tamb, u and Tamb, d on the up and down side respectively. The geometric normal points from the down side to the up side.

S U R F A C E E M I S S I V I T Y

By default, the Surface emissivity uses values From material, which 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, select an option from the Material on upside and Material on downside lists. The default uses Domain material, and the list contains other options based on the material defined in the model.

When Both sides is selected, also define the Surface emissivity u and d on the up and down side respectively. The geometric normal points from the down side to the up side.

Set 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.

Note

• About the Surface-to-Surface Radiation Boundary Conditions

• Radiation Group Boundaries

• Domain, Boundary, Edge, Point, and Pair Conditions for the Surface-to-Surface Radiation Interface

See Also


Opaque

Use the Opaque feature to assign domains to the surface-to-surface radiation when Opacity controlled is set as the Radiation Direction.


From the Selection list, choose the domains to define the surface-to-surface radiation opacity.

Initial Values

The Initial Values feature adds an initial value for the surface radiosity that can serve as an initial condition for a transient simulation or as an initial guess for a nonlinear solver. Right-click to add more than one set of initial values.


From the Selection list, choose the boundaries to define the initial value of the Surface

radiosity J (SI unit: W/m2).


Enter a value or expression for the initial value.

Reradiating Surface

The Reradiating Surface feature is a variant of the surface-to-surface radiation feature with e = 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. It resembles a mirror that absorbs all irradiation and then radiates it back in all directions: J = G. The feature adds the radiosity dependent variable to its selection and uses it as surface radiosity.

Heat flux on surface-to-surface boundary is zero where q = 0.


From the Selection list, choose the boundaries corresponding to reradiation surfaces.




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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 surface to surface physic interface. It is used in the blackbody radiation intensity expression.


Prescribed Radiosity

Use the Prescribed Radiosity feature to specify radiosity on the boundary. Radiosity can be defined as graybody radiation. The radiosity expressions is then eT4. A user-defined surface radiosity expression can also be defined.


From the Selection list, choose the boundaries to define the prescribed radiosity.



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.

R A D I A T I O N D I R E C T I O N

Select a Radiation Direction based on the geometric normal (nx, ny, nz):

• Opacity controlled is the default and requires that each boundary is adjacent to exactly one opaque domain. Opacity is controlled by the Opaque feature condition.

• Select Negative normal direction to specify that the surface radiates in the negative normal direction.

See Radiation Settings for the Surface-to-Surface Radiation (Boundary Condition).

Note


• Select Positive normal direction if the surface radiates in the positive normal direction.

• Select Both sides if the surface radiates on both sides.

R A D I O S I T Y

Select a Radiosity expression—Graybody radiation, Blackbody radiation, or User defined.

• If Blackbody radiation is selected, it sets the surface radiosity expression J = T4.

• If Graybody radiation is selected, it sets the surface radiosity expression to J = T4. By default, the Surface emissivity is defined From material. In this case, make sure that a material is defined at the boundary level (materials are defined by default at the domain level). If User defined is selected, enter another value for . When Both

sides is selected, define the Surface emissivity u and d on the up and down side respectively. The geometric normal points from the down side to the up side.

• If User defined is selected, it sets the surface radiosity expression to J = J0. which specifies how the radiosity of a boundary is evaluated when that boundary is visible in the calculation of the irradiation onto another boundary in the model. Enter a Surface radiosity expression, J0 (SI unit: W/m2). The default is 0. When Both sides

is selected, define the surface radiosity expression J0, u and J0, d on the up and down side respectively. The geometric normal points from the down side to the up side.

Radiation Group

Radiosity does not directly affect the boundary condition on the boundary where it is specified, but rather how that boundary affects others through radiation.Note

Add a Radiation Group to a Surface-to-Surface Radiation interface or any version of a Heat Transfer 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.

Note


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The Radiation Group feature 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 feature is automatically added to the Model Builder and contains all boundaries selected in the Surface-to-Surface Radiation (Boundary Condition), a Reradiating Surface, or a Prescribed Radiosity) feature.


From the Selection list, choose the boundaries that belong to the same radiation group. This section should contains any boundary that is selected in a Surface-to-Surface

Radiation, a Reradiating Surface, or a Prescribed Radiosity node and that a chance to see one of the boundary that is already selected in the Radiation Group.

External Radiation Source

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, G, on all the boundaries where a Surface-to-surface or Reradiating surface boundary condition is active. The source contribution, Gext, is equal to the product of the view factor of the source by the

Radiation Group BoundariesSee Also

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 interface or in any version of a Heat Transfer interface where the Surface-to-surface radiation check box is selected. Note


source radiosity. For radiation sources located on a point, Gext=Fext Ps. For directional radiative source Gext = Fext q0,s.

S O U R C E

Select a Source position—Point coordinate, Infinite distance, or Solar position. Solar

position is only available in 3D.

• If Point coordinate is selected, define the Source location xs (SI unit: m) and the Source power Ps (SI unit: W, default is 0). The source radiates uniformly in all directions.

• If Infinite distance is selected, define the Incident radiation direction is (unitless) and the Source heat flux q0,s (SI unit: W/m2). The default is 0.

• For 3D models, if Solar position is selected define all the required information to estimate the external radiative heat source due to the sun.

Define the Latitude (decimal value, positive in the northern hemisphere; the default is Greenwich UK latitude, 51.479), the Longitude (decimal value, positive at the East of the Prime Meridian; the default is Greenwich UK longitude, 0.01064), and the Time

zone (number of hours to add to UTC to get local time; the default is Greenwich UK time zone, 0) in the Location table.

The external radiation sources are ignored on the boundaries where neither Surface-to-Surface Radiation nor Reradiating Surface 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 Out-of-plane heat transfer in 2D, External Radiation Source is not available in this case.

Note

2D

xs should not belong to any surface where a Surface-to-surface or Reradiating surface boundary condition is active.

Note


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Enter the Day, (default 1) the Month (default 6, June), and the Year (default 2012) in the date table. The solar position is accurate for a date between the years 2000 and 2199.

Indicate the Hour (default 12), the Minute (default 0) and the Second (0) that defines the local time in the local Time table.

Define the incident radiative intensity coming from the sun, Is (SI unit: W/m2, default value 1000 W/m2), in the Solar energy field. Is 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.

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 (default choice), the time change can be taken into account by adding t to the Second field in the Local time table.Note


Th e Rad i a t i o n i n Pa r t i c i p a t i n g Med i a I n t e r f a c e

The Radiation in Participating Media interface ( ), found under the Heat

Transfer>Radiation branch ( ) in the Model Wizard, enables the modeling of radiative heat transfer inside a participating medium. This interface solves for radiative intensity field.

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.



The default identifier (for the first interface in the model) is rpm.


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.

• 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/Process_and_Manufacturing/boiler

Model

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R E F R A C T I V E I N D E X

Define the Refractive index (a dimensionless number) of the participating media, n. The default value is 1.



Performance IndexSelect 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 0.5. 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.


To display this section, click the Show button ( ) and select Discretization. Select Quadratic, Linear (the default), Cubic, or Quartic for the Radiative intensity. Specify the Value type when using splitting of complex variables—Real (the default) or Complex.


The interface includes a dependent variable for intensity along discrete directions. The number of direction depends on the Discrete ordinates method order selected.


Note


• Domain, Boundary, Edge, Point, and Pair Conditions for the Radiation in Participating Media InterfaceSee Also


3D


If required, edit the Radiative intensities default names I1... In. The name can be changed but the names of fields and dependent variables must be unique within a model.

Domain, Boundary, Edge, Point, and Pair Conditions for the Radiation in Participating Media Interface

The Radiation in Participating Media Interface has these domain, boundary, edge, point, and pair features available:


• Initial Values

• Opaque and Incident Intensity (described for the Heat Transfer interface)



The Radiation in Participating Media feature uses the radiative transfer equation

where

• Is is the radiative intensity at the position s position in the direction


• , , s are absorption, extinction, and scattering coefficients, respectively

• is the blackbody radiation intensity and n is the refractive index


2D


I s Ib T I s –s

4------ I s

0

4

+=

Ib T n2T4

-----------------=


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From the Selection list, choose the domains to define the radiation in participating media.



There is one standard model input—the Temperature T (SI unit: K). The default is 293.15 K and is used in the blackbody radiation intensity expression.

A B S O R P T I O N

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.

S C A T T E R I N G

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 value is 0.

Select the Scattering type—Isotropic, Linear anisotropic, or Nonlinear anisotropic.

• Isotropic (the default) and corresponds to the scattering phase function

• If Linear anisotropic is selected, it defines the scattering phase function as

Enter the Legendre coefficient a1.

• If Nonlinear anisotropic is selected, it defines the scattering phase function

Enter the Legendre coefficients a1, …, a12 as required.

0 1=

0 1 a10+=

0 1 amPm 0

m 1=

12

+=


Initial Values

The Initial Values node defines an initial value for the discrete intensities I1, …, In that can serve as an initial condition for a transient simulation or as an initial guess for a nonlinear solver. Right-click to add more than one set of initial values.


From the Selection list, choose the domains to define an initial value.


Enter a value or expression for the initial value of the Radiative intensities I1, …, In (SI unit: W/m2) in the I1, …, In edit field. The default value is the blackbody temperature, Ib.


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Th e Hea t T r a n s f e r w i t h Rad i a t i o n i n Pa r t i c i p a t i n g Med i a I n t e r f a c e

The Heat Transfer with Radiation in Participating Media 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 node’s settings window. The following default nodes are also added to the Model Builder: Heat Transfer in Solids, Thermal Insulation, Continuity on Interior Boundary, and Initial Values.

Right-click the node to add other boundary conditions and features.



The default identifier (for the first interface in the model) is ht.



Except where described below, most of the settings windows are the same as described for The Radiation in Participating Media Interface and The Heat Transfer Interface.Note


P A R T I C I P A T I N G M E D I A S E T T I N G S

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.

Define the Refractive index of the participating media, n.

Select the Discrete ordinates method order from the list. This order defines the discretization of the radiative intensity direction.

Select Linear (the default), Quadratic, Cubic, Quartic, or Quintic to define the discretization level of the Radiative intensity fields.



Performance IndexSelect 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 0.5. With


Note



3D

2D

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small values, a robust setting for the solver is expected. With large values (up to 1), less memory is needed to solve the model.

Domain, Boundary, Edge, Point, and Pair Conditions for the Heat Transfer with Radiation in Participating Media Interface

The Heat Transfer with Radiation in Participating Media Interface has these domain, boundary, edge, point, and pair features available and described for the Heat Transfer and Radiation in Participating Media interfaces (listed in alphabetical order):




• Continuity

• Heat Flux

• Heat Source





• Incident Intensity

• Infinite Elements

• Initial Values


• Opaque Surface

• Open Boundary

• Outflow




• Theory for the Radiative Heat Transfer Interfaces

• Domain, Boundary, Edge, Point, and Pair Conditions for the Heat Transfer with Radiation in Participating Media Interface

See Also





• Pressure Work



• Symmetry

• Temperature



• Translational Motion


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Th eo r y f o r t h e Rad i a t i v e Hea t T r a n s f e r I n t e r f a c e s

The following sections provide some background information about the radiosity method and view factors and includes these topics:

• The Radiosity Method

• View Factor Evaluation

• Radiation and Participating Media Interactions

• Radiative Transfer Equation

• Boundary Condition for the Transfer Equation

• Heat Transfer Equation in Participating Media

• Discrete Ordinates Method

The Radiosity Method

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 2-12 it is given by

(5-1)

where

• is the surface emissivity, a dimensionless number in the range 01.

• G is the incoming radiative heat flux, or irradiation (SI unit: W/m2).

• is the Stefan-Boltzmann constant (a predefined physical constant equal to 5.670400·108 W/(m2·T4)).

The irradiation, G, at a point can in general be written as a sum according to:

q G T4– =

G Gm Gext FambTamb4

+ +=


where

• Gm is the mutual irradiation, coming from other boundaries in the model (SI unit: W/m2).

• Gext is the irradiation from external radiation sources (SI unit: W/m2). Gext is the sum to 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, G_ext=F_ext(x_s) Ps. For directional radiative source G_ext=F_ext(i_s) q_0,s.

• Famb 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, 0Famb1 must hold at all points.

• Tamb is the assumed far-away temperature in the directions included in Famb.

The Surface-To-Surface Radiation 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:

(5-2)

• Reradiating Surface 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 insulated on one side and for which convection effects can be neglected on the opposite (radiating) side (see Ref. 3). 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 T4. A user-defined surface radiosity expression can also be defined.

The Surface-to-Surface Radiation interface treats the radiosity J as a dependent variable unless J is prescribed.

Gext GextPs Gextq0 s+=

J 1 – G T4+ 1 – Gm J Gext FambTamb

4+ + T4

+= =

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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 Famb in Equation 5-2 are not strictly view factors in the traditional sense. Famb is the view factor of the ambient portion of the field of view, which is considered to be a single boundary with constant radiosity

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 Famb 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.

V I E W F A C T O R F O R E X T E R N A L R A D I A T I O N S O U R C E S

In 3D, the view factor for a point at finite distance is given by

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

Jamb Tamb4

=

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.

Tip

4r2 cos


and the view factor for a source at infinity is cos .

S O L A R P O S I T I O N

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. 5. 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 (isx, isy, isz) 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 (isx, isy, isz) is given by:

R A D I A T I O N I N A X I S Y M M E T R I C G E O M E T R I E S

For an axisymmetric geometry, Gm and Famb 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.

2r cos

isx azi cos zen sin–=

isy azi sin zen sin=

isz zen cos–=


170 | C H A P T E

Select between the hemicube and the direct area integration methods also in axial symmetry. Their settings work the same way as in 3D.

Radiation and Participating Media Interactions

Figure 5-1: 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 Ib, where Ib 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

While Gm and Famb 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.

Note


phase function ij, which gives the probability that a ray coming from one direction i is scattered into the direction j. The phase function (i, j) satisfies

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 sI.

- 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:

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)

where

• I is the radiative intensity at a given position following the direction


• , , s are absorption, extinction, and scattering coefficients, respectively

(5-3)

• Equation 5-3 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

14------ i id

4 1=

s

4------ I i i id

4

I Ib T I s –s4------ I

0

4

+=

Ib T n2T4

-----------------=

'


172 | C H A P T E

scattering phase functions that are defined using the cosine of the scattering angle, 0. The current implementation handles:

• Isotropic phase function:

• Linear anisotropic phase function:

• Nonlinear anisotropic up to the 12th order:

where Pn are nth-order Legendre polynomials.

Legendre polynomials can be defined by the Rodriguez formula:

A quantity of interest is the incident radiation, denoted G and defined by

Boundary Condition for the Transfer Equation

For gray walls, corresponding to opaque surfaces reflecting diffusively and emitting, the radiative intensity Ibnd entering participating media along the direction is

where

(5-4)

• Equation 5-4 is the blackbody radiation intensity and n is the refractive index

• w is the surface emissivity, which is in the range [0, 1]

' 0 1= =

0 1 a10+=

0 1 anPn 0

n 1=

12

+=

Pk x 1

2kk!-----------

xk

k

d

d x2 1– k

=

G I 0

4

=

Ibnd wIb T d------qout for all such that n 0+=

Ib T n2T4

-----------------=


• d1w is the diffusive reflectivity

• n is the outward normal vector

• qout is the heat flux striking the wall:

For black walls w1 and d0. Thus IbndIbT.

Heat Transfer Equation in Participating Media

Heat flux in gray media is defined by

Heat flux divergence can be defined as a function of G and T (see Ref. 1):

In order to couple radiation in participating media, radiative heat flux is taken into account in addition to conductive heat flux: qqcqr. The heat transfer equation reads

and is implemented using following form:

qout wjIjn jn j 0=

qr n I n 0

4

=

Qr q r G 4T4– = =

CpTt------- u T+ qc qr+ – :S T

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

p

pt------ u p+ – Q+ +=

CpTt------- u T+ qc– G 4T4– :S T

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

p

pt------ u p+ – Q+ + +=


174 | C H A P T E

Discrete Ordinates Method

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 nNN2 in 3D (or nNN22 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. 2, 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:

Depending on the value of N, a set of n dependent variables has to be defined and solved for I1, I2, …, In.

Each dependent variable obeys the equation

with the boundary condition

The discrete ordinates method is implemented for 2D and 3D geometries.

2D

3D

I 0

4

wjIj

j 1=

n

i Ii Ib T Ii–s

4------ wjIj j i

j 1=

n

+=


Ii bnd wIb T d

------qout for all i such that n i 0+=


176 | C H A P T E

Th eo r y f o r t h e S u r f a c e - t o - S u r f a c e Rad i a t i o n I n t e r f a c e

The Surface-To-Surface Radiation Interface theory is described in this section:

• About Surface-to-Surface Radiation

• Solving for the Radiosity

• About the Surface-to-Surface Radiation Boundary Conditions

• Guidelines for Solving Surface-to-Surface Radiation Problems

• Radiation Group Boundaries

• References for the Radiation Interfaces

About Surface-to-Surface 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:

(5-5)

where Gm is the mutual irradiation arriving from other surfaces in the modeled geometry and Famb is the ambient view factor. The latter describes the portion of the view from each point that is covered by ambient conditions. Gm on the other hand is determined from the geometry and the local temperatures of the surrounding boundaries. The following sections derive the equations for Gm and Famb for a general 3D case.

Consider a point on a surface as in Figure 5-2. Point 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, Tamb.

Figure 5-2: Example geometry for surface-to-surface radiation.

The mutual irradiation at point is given by the following surface integral:

G Gm FambTamb4

+=

x x

rnn'x'xJ'S'Samb

x


The heat flux that arrives from depends on the local radiosity projected onto . The projection is computed using the normal vectors and along with the vector

, which points from to .

The ambient view factor, Famb, is determined from the integral of the surrounding surfaces S', here denoted as F', determined from the integral below:

The two last equations plug into Equation 5-5 to yield the final equation for irradiative flux.

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

(5-6)

where the integral over 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.

Solving for the Radiosity

The previous section derived equations for the irradiation G at an arbitrary surface point . Now recall the expression for the radiosity leaving from

(5-7)

Gmn'– r n r

r4

------------------------------------J' Sd

S'

=

x' J' xn n'

r x x'

Famb 1 F '– 1 n'– r n r

r4

------------------------------------ Sd

S'

–= =

Gmn'– r n r

2 r3

------------------------------------------J' SdS'=

Famb 1n'– r n r

2 r3

------------------------------------------ SdS'–=

S'

x x

J G T4+=

T H E O R Y F O R T H E S U R F A C E - T O - S U R F A C E R A D I A T I O N I N T E R F A C E | 177

178 | C H A P T E

Inserting the expression for G in Equation 5-7 gives the following equation for the radiosity:

Assuming an ideal graybody, the last equation becomes:

(5-8)

This is the equation used in the Surface-to-Surface Radiation interface to solve for the radiosity, J. It applies to boundaries that participate in surface-to-surface radiation. Equation 5-8 results in a linear equation system in J that is solved in parallel with the equation for the temperature, T.

About the Surface-to-Surface Radiation Boundary Conditions

Heat flux on the Surface-to-Surface Radiation boundary is

where

• e is the surface emissivity, a dimensionless number in the range 01.

• is the Stefan-Boltzmann constant (a predefined physical constant equal to 5.670400·108 W/(m2·T4)).

• G is the incoming radiative heat flux, or irradiation (SI unit: W/m2):

where

• Gm is the mutual irradiation, coming from other boundaries in the model (SI unit: W/m2).

• Famb 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, 0Famb1 must hold at all points.

• Tamb is the assumed far-away temperature in the directions included in Famb.

Surface-to-surface radiation requires accurate evaluation of the mutual irradiation, Gm. The incident radiation at one point on the boundary is a function of the exiting

J Gm FambTamb4

+ T4+=

J 1 – Gm FambTamb4

+ T4+=

q G T4– =

G Gm FambTamb4

+=


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:

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:

• 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 worth 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 consider 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

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, Reradiating Surface, 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

J 1 e– Gm J FambTamb4

+ eT4+=

The Radiation Group feature 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.Note


180 | C H A P T E

added to other radiation groups without being overridden by the manually added radiation groups.

Figure 5-3 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, Reradiating Surface, 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, Reradiating Surface, and Prescribed Radiosity nodes. The numbers on each boundary specify different groups to which the boundary 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 Figure 5-3, case (b) is more efficient than case (d).

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.

Caution

2D Axi


Figure 5-3: Examples of radiation group boundaries.

References for the Radiation Interfaces

1. M.F. Modest, Radiative heat transfer, 2nd ed., Academic Press, San Diego, California, 2003.

2. W.A. Fiveland, “The Selection of Discrete Ordinate Quadrature Sets for Anisotropic Scattering,” Fundamentals of Radiation Transfer, HTD, vol. 160, ASME, 1991.

3. F.P. Incropera and D.P. DeWitt, Fundamentals of Heat and Mass Transfer, 5th ed., John Wiley and Sons, 2002.

4. J.R. Welty, C.E. Wicks, and R.E. Wilson, Fundamentals of Momentum, Heat, and Mass Transfer, 3rd ed., John Wiley and Sons, 1983.

5. http://www.esrl.noaa.gov/gmd/grad/solcalc/

211

1 2 1 2

1 2 321

3 3

2

2

2

2

1

1

1

1

11 1

1

111

inefficient boundary grouping

A B

C D


182 | C H A P T E

6

T h e S i n g l e - P h a s e F l o w B r a n c h

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 Single-Phase Flow branch ( ) in the Model Wizard.

In this chapter:

• The Single-Phase Flow, Laminar Flow Interface

• The Single-Phase Flow, Turbulent Flow Interfaces

• Boundary Conditions for the Single-Phase Flow Interfaces

• Theory for the Laminar Flow Interface

• Theory for the Turbulent Flow Interfaces

183

184 | C H A P T E

Th e S i n g l e - Pha s e F l ow , L am i n a r F l ow I n t e r f a c e

The descriptions in this section are structured based on the order displayed in the Fluid

Flow>Single-Phase Flow branch ( ) of the Model Wizard. Because many of the interfaces are integrated with each other, some features described also cross reference to other interfaces. At the end of this section is a summary of the theory that goes towards deriving each physics interface.

The Heat Transfer Module extends the capabilities of the basic COMSOL Laminar Flow interface.

In this section:

• The Laminar Flow Interface

• Fluid Properties

• Volume Force

• Initial Values

The Laminar Flow Interface

The Laminar Flow 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 feature is Fluid Properties, which adds the



For 2D axisymmetric models, COMSOL Multiphysics takes the axial symmetry boundaries (at r0) into account and automatically adds an Axial Symmetry feature to the model that is valid on the axial symmetry boundaries only.

See Also

2D Axi

R 6 : T H E S I N G L E - P H A S E F L O W B R A N C H

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 features that implement, for example, boundary conditions and volume forces.



The default identifier (for the first interface in the model) is spf.


The default setting is to include All domains in the model to define the fluid pressure and velocity and the Navier-Stokes equations that describe those fields. To choose specific domains, select Manual from the Selection list.


Control the properties of the Laminar Flow interface, which control the overall type of fluid flow model.

• 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

Model

The options available in this section enables switching between other available Single-Phase Flow interfaces. For example,

• This interface changes to The Turbulent Flow, k- Interface when the Turbulence model type selected is RANS (Reynolds-averaged Navier–Stokes).

Tip

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186 | C H A P T E

CompressibilityBy 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


This interface defines these dependent variables (fields). If required, edit the name, but dependent variables must be unique within a model:

• Velocity field u (SI unit: m/s)

• Pressure p (SI unit: Pa)

• Turbulent kinetic energy k (SI unit: m2/s2)

• Turbulent dissipation rate ep (SI unit: m2/s3)

• Reciprocal wall distance G (SI unit: 1/m)



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 calculates the local CFL number (from the Courant–Friedrichs–Lewy condition) from a built-in expression. If Manual is selected, enter a Local CFL

number CFLloc.

By definition no turbulence model is needed when studying laminar flows, and no turbulence model is therefore applied in the interface. The flow state in a fluid flow model, however, is not always known beforehand.Note

The Projection Method for the Navier-Stokes Equations and Pseudo Time Stepping for Laminar Flow Models in the COMSOL Multiphysics User’s GuideSee Also



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 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.

• P2+P1 means second-order elements for the velocity components and linear elements for the pressure field. This is the default for the Creeping Flow interface because second-order elements work well for low flow velocities.

• 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 or Complex (the default).

Fluid Properties

The Fluid Properties feature 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.


From the Selection list, choose the domains to apply the fluid properties.



• Theory for the Laminar Flow InterfaceSee Also

For the Turbulent Flow interfaces, the Fluid Properties feature also adds the equations for the turbulence transport equations.

Note


188 | C H A P T E


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.

F L U I D P R O P E R T I E S

DensityThe default Density (SI unit: kg/m3) uses the value From material. Select User defined to enter a different value or expression.

Dynamic ViscosityThe 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.

D I S T A N C E E Q U A T I O N

Select how the Reference length scale lref (SI unit: m) is defined—Automatic (default) or Manual:

• If Automatic is used, the wall distance is automatically evaluated as half 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.

To define the Absolute Pressure, see the settings for the Heat Transfer in Fluids node as described in the COMSOL Multiphysics User’s Guide.

Tip

This section is only available for Turbulent Flow, Low Reynolds number k- interface since a Wall Distance interface is included.

Important


M I X I N G L E N G T H L I M I T

Select how the Mixing length limit lmix,lim (SI unit: m) is defined—Automatic (default) or Manual:

• If Automatic is used, the wall distance 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.

Volume Force

The Volume Force feature specifies the volume force F on the right-hand side of the incompressible flow equation. Use it, for example, to incorporate the effects of gravity in a model.


From the Selection list, choose the domains where the volume force acts on the fluid.

VO L U M E F O R C E

Enter the components of the Volume force F (SI unit: N/m3).

This section is only available for the Turbulent Flow, k- interface because an upper limit on the mixing length is required.

Note

t

u u u+ pI– u u T+ + F+=

The Boussinesq Approximation in the COMSOL Multiphysics User’s Guide

See Also


190 | C H A P T E

Initial Values

The Initial Values feature 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.


From the Selection list, choose the domains to define initial values.


Enter values or expressions for the initial value of the Velocity field u (SI unit: m/s) and for the Pressure p (SI unit: Pa). The default values are 0.

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.Note


Th e S i n g l e - Pha s e F l ow , T u r bu l e n t F l ow I n t e r f a c e s

The descriptions in this section are structured based on the order displayed in the Fluid Flow branch. Because many of the interfaces are integrated with each other, most of the features cross reference to other interfaces. For example, all nodes that can be added for the Turbulent Flow interfaces are described for The Single-Phase Flow, Laminar Flow Interface

Click the links to go to these section topics:

• The Turbulent Flow, k- Interface

• The Turbulent Flow, Low Re k- Interface

The Turbulent Flow, k- Interface

The Turbulent Flow, k-interface ( ), found under the Single-Phase Flow>Turbulent

Flow branch ( ) in the Model Wizard, 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 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.

Turbulent Flow Over a Backward Facing Step: Model Library path Heat_Transfer_Module/Verification_Models/turbulent_backstep

Model

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Right-click the Turbulent Flow, k- node to add other features that implement, for example, boundary conditions and volume forces.


Turbulence Model Type

Turbulence Model

The Turbulent Flow, Low Re k- Interface

The Turbulent Flow, Low Re k-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.

Except where noted below, see The Laminar Flow Interfacefor all the other settings.

Note

For the Turbulent Flow, k- interface, this defaults to a RANS Turbulence

model type. This enables the Turbulence Model Parameters section.Note

The interface changes to the Turbulent Flow, Low Re k- interface when Low Reynolds number k- is selected.




Tip

See Also


The interface is the same as The Turbulent Flow, k- Interface ( ) except the Turbulence model defaults to Low Reynolds number k-.

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 Guide




Important

See Also

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Bounda r y Cond i t i o n s f o r t h e S i n g l e - Pha s e F l ow I n t e r f a c e s

The boundary features in this section are for all interfaces found under the Fluid

Flow>Single-Phase Flow branch ( ) in the Model Wizard.

In this section:

• Wall

• Inlet

• Outlet

• Symmetry

• Open Boundary

• Boundary Stress

• Periodic Flow Condition

• Flow Continuity

• Pressure Point Constraint

• Interior Fan

• Fan

• Grille

For 2D axisymmetric models, COMSOL Multiphysics takes the axial symmetry boundaries (at r0) into account and automatically adds an Axial Symmetry feature 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.

2D Axi

Note


Wall

The Wall feature includes a set of boundary conditions describing the fluid flow condition at a wall.

• No Slip (the default for laminar flow)

• Slip

• Sliding Wall

• Moving Wall

• Leaking Wall

• Wall Functions (the default for turbulent flow)

• Sliding Wall (Wall Functions)

• Moving Wall (Wall Functions)


From the Selection list, choose the boundaries that represent solid walls.


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

B O U N D A R Y C O N D I T I O N

Select a Boundary condition for the wall. The boundary conditions available vary by interface.

No SlipNo slip is the default boundary condition for a stationary solid wall. The condition prescribes u = 0, that is, that the fluid at the wall is not moving.


• Theory for the Slip Wall Boundary Condition

• Theory for the Sliding Wall Boundary Condition

• The Moving Mesh InterfaceSee Also

B O U N D A R Y C O N D I T I O N S F O R T H E S I N G L E - P H A S E F L O W I N T E R F A C E S | 195

196 | C H A P T E

SlipThe Slip condition 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 WallThe 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.

Moving WallIf the wall moves, so must the fluid. Hence, this boundary condition prescribes u = uw. Enter the components of the Velocity of moving wall uw (SI unit: m/s).

Leaking WallUse 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).

In 2D, the tangential direction is unambiguously defined by the direction of the boundary, 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 Uw (SI unit: m/s). In axial symmetry, if Swirl

flow is selected in the interface properties, also specify a velocity, vw, in the direction.

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 projects it onto the tangential direction. Its magnitude is adjusted to be the same as the magnitude of the vector entered.

2D

3D

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.Important


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

I N T E R F A C E S

Wall FunctionsThe Wall Functions boundary condition applies wall functions to a 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).

Moving Wall (Wall Functions)

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 in the Velocity of moving wall fields (SI unit: m/s).


To display this section, click the Show button ( ) and select Advanced Physics Options. Select a Constraint type—Bidirectional, symmetric or Unidirectional for the following wall boundary conditions: No slip, Moving wall, and Leaking wall. The other types of wall boundary conditions with constraints use unidirectional constraints only. Select the Use weak constraints check box (available for all boundary conditions except Sliding wall, which does not add any constraints) to use weak constraints and create dependent variables for the corresponding Lagrange multipliers.

Boundary Conditions for the Single-Phase Flow InterfacesSee Also

Specifying this boundary condition does not automatically cause the associated wall to move.

Important


198 | C H A P T E

Interior Wall

The Interior Wall boundary condition includes a set of boundary 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 boundary. It allows discontinuities (velocity, pressure, turbulence) across the boundary. You can use the Interior Wall boundary condition to avoid meshing thin structures by instead using no-slip conditions on interior curves and surfaces. You can also prescribe slip conditions and conditions for a moving wall.


From the Selection list, choose the boundaries that represent interior walls.


Select a Boundary condition—No slip (the default), Slip, or Moving wall.

No SlipNo 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.

SlipThe Slip condition 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.

The Interior Wall boundary condition is only available for single-phase flow. It is compatible with laminar and turbulent flows.


• Theory for the Slip Wall Boundary Condition

• The Moving Mesh Interface

Note

See Also


Moving WallIf 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).

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.

• Velocity (the default)

• Pressure, No Viscous Stress

• Normal Stress

• Laminar Inflow

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.Important

In most cases the inlet boundary conditions appear, 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.

Tip

• Theory for the Laminar Inflow Boundary Condition


• Theory for the Pressure, No Viscous Stress Boundary Condition

• Theory for the Normal Stress Boundary ConditionSee Also


200 | C H A P T E


From the Selection list, choose the boundaries that represent inlets. The boundary conditional available is based on interface.


Select a Boundary condition for the inlet.

VE L O C I T Y

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 U0 (SI unit: 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.

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 settings as described in Additional Boundary Condition Settings for Turbulent Flow Interfaces.

See Also

Important

This section displays when Velocity is selected as the Boundary condition. The option is available for the Inlet and Outlet boundary features.

Note


P R E S S U R E , N O V I S C O U S S T R E S S

The Pressure, no viscous stress boundary condition specifies vanishing viscous stress along with a Dirichlet condition on the pressure. Enter the Pressure p0 (SI unit: Pa) at the boundary.

N O R M A L S T R E S S

Enter the magnitude of Normal stress f0 (SI unit: N/m2). Implicitly specifies that .

L A M I N A R I N F L O W

Select a flow quantity to specify for the inlet:

• If Average velocity is selected, enter an Average velocity Uav (SI unit: m/s).

• If Flow rate is selected, enter the Flow rate V0 (SI unit: 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.

Then specify these parameters:

Enter the Entrance length Lentr (SI unit: m) to define the length of the inlet channel outside the model domain. This value must be large enough so that the flow can reach a laminar profile. For a laminar flow, Lentr should be significantly greater than 0.06ReD, 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 4.4Re1/6D. The default is 1 m.

This section displays when Pressure, no viscous stress is selected as the Boundary condition. The option is available for the Inlet and Outlet boundary features. Depending on the pressure field in the rest of the domain, an inlet boundary with this condition can become an outlet boundary.

Note

p f0

This section displays when Normal Stress is selected as the Boundary

condition. The option is available for the Inlet, Outlet, Open Boundary, and Boundary Stress features.Note


202 | C H A P T E

Select the Constrain endpoints to zero 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 boundary with a laminar inflow condition connects to a slip boundary condition, then the laminar profile has a maximum at that end.

A D D I T I O N A L B O U N D A R Y C O N D I T I O N S E T T I N G S F O R TU R B U L E N T F L O W

I N T E R F A C E S

The Turbulent Flow, k- model and Turbulent Flow, low Reynolds number k- models, also require that k and are specified using one of the following:

• Select Specify turbulence length scale and intensity to enter values or expressions for the Turbulent intensity IT (unitless) and Turbulence length scale LT (SI unit: m). For the Turbulent Flow, k- model, also enter a value for the Reference velocity scale Uref (SI unit: m/s).

The Turbulent intensity IT and Turbulence length scale LT values are related to the turbulence variables via

If Specify turbulence variables is selected, enter values or expressions for the Turbulent

kinetic energy k0 (SI unit: m2/s2) and Turbulent dissipation rate, 0 (SI unit: m2/s3).C O N S T R A I N T S E T T I N G S

To display this section, click the Show button ( ) and select Advanced Physics Options. Select a Constraint type—Bidirectional, symmetric or Unidirectional. Select the Use weak

constraints check box as required.

This section displays when Laminar inflow is selected as the Boundary

condition for the Laminar Flow interface. However, it is not available when the Use memory-efficient form check box is selected from Advanced

Settings on the Laminar Flow node’s Settings window.Note

k 32--- U IT 2,= C

3 4 k3 2/

LT-----------=

For recommendations of physically sound values see Inlet Values for the Turbulence Length Scale and Intensity.

Tip


Outlet

The Outlet feature includes a set of boundary 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 happening at an outflow boundary, extend the computational domain to include this phenomenon.

• Pressure, No Viscous Stress (the default)

• Velocity

• Laminar Outflow

• Normal Stress

• Pressure

• No Viscous Stress


From the Selection list, choose the boundaries that represent outlets.

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.

Tip

The Pressure, No Viscous Stress, Velocity, and Normal Stress boundary conditions are described for the Inlet node.

Theory for the Laminar Outflow Boundary Condition

Note

See Also


204 | C H A P T E


Select a Boundary condition for the outlet. Pressure, no viscous stress is the default. The boundary conditional available is based on interface.

P R E S S U R E

This 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 Recuh21 (h is the local mesh element size). It does however work well in most other situations as well.

L A M I N A R O U T F L O W

Select a flow quantity to specify for the inlet:

• If Average velocity is selected, enter an Average velocity Uav (SI unit: m/s).

• If Flow rate is selected, enter the Flow rate V0 (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. This value must be large enough so that the flow can reach a laminar profile. For a laminar flow, Lexit should be significantly greater than 0.06ReD, 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 4.4Re1/6D. The default is 1.

Select the Constrain endpoints to zero 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 boundary with a Laminar inflow condition connects to a Slip boundary condition, then the laminar profile has a maximum at that end.

N O V I S C O U S S T R E S S

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 be combined with pressure constraints on adjacent points.


To display this section, click the Show button ( ) and select Advanced Physics Options. Select a Constraint type—Bidirectional, symmetric or Unidirectional. Select the Use weak

constraints check box to use weak constraints and create dependent variables for the corresponding Lagrange multipliers.

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 Dirichelet condition and a Neumann condition:

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:

This section displays when Laminar Outflow is selected as the Boundary

condition for a Laminar Flow interface. However, it is not available when the Use memory-efficient form check box is selected from Advanced Settings on the Laminar Flow Settings window.Note

u n 0,= pI– u u T+ 23--- u I–

+ n 0=

u n 0,= pI– u u T+ + n 0=


206 | C H A P T E


From the Selection list, choose the boundaries that are symmetry boundaries.

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.


From the Selection list, choose the boundaries that are open boundaries.

B O U N D A R Y C O N D I T I O N S

Select a Boundary condition for the open boundaries—Normal stress (the default) or No

viscous stress. If Normal stress f0 (SI unit: N/m2) is selected, enter a value or expression for the boundary condition.

No Viscous StressIf No viscous stress is selected, which is also available for the Outlet feature, it prescribes vanishing viscous stress:

u n 0,= K K n n– 0=

K u u T+ n=

For 2D axial symmetry, a boundary condition does not need to be defined. For the symmetry axis at r0, the software automatically provides a condition that prescribes ur0 and vanishing stresses in the z direction and adds an Axial Symmetry feature that implements this condition on the axial symmetry boundaries only.

2D Axi

• Inlet

• Outlet

• More Open Boundary Conditions for the Turbulent Flow Interfaces

• Pressure Point ConstraintSee Also


using the compressible and the incompressible formulation respectively.

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

I N T E R F A C E S

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·n0, that is, where flow enters the computational domain.

For the Turbulent Flow, k- and Turbulent Flow, low Reynolds number k-interfaces, these options are available under Exterior turbulence:

• Select Specify turbulent length scale and intensity to enter values or expressions for the Turbulent intensity IT (unitless), Turbulence length scale LT (SI unit: m), and Reference velocity scale Uref (SI unit: m/s).

These values are related to the turbulence variables via

u u T+ 23--- u I–

n 0=

u u T+ n 0=

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.

This boundary condition is not available if the Use memory-efficient form

check box is selected (click the Show button ( ) and select Advanced

Physics Options>Advanced Settings to display the section) on any single-phase flow interface Settings window.

Tip

Note


208 | C H A P T E

• Select Specify turbulence variables to enter values or expressions for the Turbulent

kinetic energy k0 (SI unit: m2/s2), and the Turbulent dissipation rate 0 (SI unit: m2/s3).

Boundary Stress

The Boundary Stress node adds a boundary condition that represents a very general class of conditions also known as traction boundary conditions.


From the Selection list, choose the boundaries to apply boundary stress.


Select a Boundary condition for the boundary stress—General stress (the default), Normal stress, or Normal stress, normal flow.

Normal StressIf Normal stress f0 (SI unit: N/m2) is selected, enter a value or expression.

General StressIf General stress is selected, enter the components of the Stress F (SI unit: N/m2).The total stress on the boundary is set equal to a given stress F:

k 32--- ITUref 2,=

C3 4

LT------------

3 ITUref 2

2----------------------------

32---

=


See Also

InletMore Open Boundary Conditions for the Turbulent Flow InterfacesSee Also

pI– u u T+ 23--- u I–

+ n F=



This boundary condition implicitly sets a constraint on the pressure that for 2D flows is

(6-1)

If unn is small, Equation 6-1 states that pn·F.

Normal Stress, Normal FlowIf Normal stress, normal flow is selected, enter the magnitude of the Normal stress f0 (SI unit: N/m2).

In addition to the stress condition set in the Normal stress condition, this condition also prescribes that there must be no tangential velocities on the boundary:


This boundary condition also implicitly sets a constraint on the pressure that for 2D flows is

(6-2)

If unn is small, Equation 6-2 states that pf0.

pI– u u T+ + n F=

p 2un

n---------- n F–=

pI– u u T+ 23--- u I–

+ n f0n,–= t u 0=

pI– u u T+ + n f0n,–= t u 0=

p 2un

n---------- f0+=



Physics Options>Advanced Settings to display the section) on any single-phase flow interface Settings page.Note


210 | C H A P T E

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

I N T E R F A C E S

Turbulent Boundary TypeSelect a Turbulent boundary type to apply to the turbulence variables—Open boundary, Inlet, or Outlet.

Open BoundaryIf Open boundary is selected, then expect parts of the boundary to be an outlet and parts of the boundary to be an inlet. Under Exterior turbulence, enter values for the turbulence variables, which are used on the parts of the boundary where u · n0, that is, where flow enters the computational domain. These settings are the same as described in More Open Boundary Conditions for the Turbulent Flow Interfaces.

InletSelect Inlet when it is expected that the whole boundary 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 they, for the Inlet option, apply it to the whole boundary. These settings are the same as described in More Open Boundary Conditions for the Turbulent Flow Interfaces.

OutletSelect 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 )


To display this section, click the Show button ( ) and select Advanced Physics Options. Select Use weak constraints 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.

k n 0= n 0=


If the boundaries are not parallel to each other, the velocity vector is automatically transformed.

The Periodic Flow Condition has no input when the interface property Compressibility is set to Compressible flow (Ma<0.3). 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. When Compressibility is set to Incompressible flow, the boundary condition contains an input field for a Pressure difference, psrcpdst. This pressure difference can, for example, drive the flow in a fully developed channel flow.

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.


To display this section, click the Show button ( ) and select Advanced Physics Options. Select Use weak constraints as required.

If the boundaries are curved, it is recommended to only include two boundaries.

Tip



Physics Options>Advanced Settings to display the section) on any single-phase flow interface Settings page.


• Destination Selection

• Using Periodic Boundary Conditions

• Periodic Boundary Condition Example

Note

See Also


212 | C H A P T E

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 subnode is added by default to the Flow Continuity node. The Wall feature 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 boundary condition that applies to exterior boundaries. Right-click the Flow Continuity node to add additional subfeatures.


From the Selection list, choose the boundaries for the selected pairs.


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

Pressure Point Constraint

The Pressure Point Constraint feature adds a pressure constraint at a point. If it is not possible to specify the pressure level using a boundary condition, the pressure must be set in some other way, for example, by specifying a fixed pressure at a point.


From the Selection list, choose the points to use a pressure constraint.

P R E S S U R E C O N S T R A I N T

Enter a point constraint for the Pressure p0 (SI unit: Pa).



• Specifying Boundary Conditions for Identity PairsSee Also



Physics Options>Advanced Settings to display the section) on any single-phase flow interface Settings page.Note



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.

Fan

Use the Fan feature to define the flow direction (inlet or outlet), and the fan parameters on exterior boundaries. Use the Interior Fan node for interior boundaries.


From the Selection list, choose the exterior boundaries to apply the fan.

F L O W D I R E C T I O N

Select a Flow direction—Inlet or Outlet.

P A R A M E T E R S

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.

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.

Select a Static pressure curve to specify a fan curve—Linear (the default), Static pressure

curve data, or User defined.

LinearFor 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) and the Free delivery flow

rate V0,fd (SI unit: m3/s).

The static pressure curve is equal to the static pressure at no flow rate when V00 and equal to 0 when the flow rate is larger than the free delivery flow rate. The default static pressure is 100 Pa and the default free delivery is 0.01 m3/s.

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.

Tip


214 | C H A P T E

User Defined Select User defined to enter different values or expressions. The flow rate across the selection where this boundary 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 DataSelect 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 defined using the Interpolation function type list in the Static Pressure Curve

Interpolation section. Then the units are specified for the flow rate and the static pressure curve in the Units section (described in the next sections).

S T A T I C P R E S S U R E C U R V E D A T A

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).

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


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.

U N I T S


curve. Select Units for the Flow rate (the default SI unit is m3/s) and Static pressure

curve (the default SI unit is Pa).

Theory for the Fan and Grille Boundary ConditionsSee Also


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:

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 feature instead.


From the Selection list, choose the interior boundaries to apply the fan.

I N T E R I O R F A N

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.

u ninlet u n

outlet+ 0=

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 feature. See Linear, Static Pressure Curve Data, and User Defined for details.

Tip

Note


216 | C H A P T E

Grille

The Grille node models the pressure drop caused by having a grille that covers the inlet or outlet.


From the Selection list, choose the boundaries that are covered by the grille.

F L O W D I R E C T I O N

Select a Flow direction—Inlet or Outlet.

P A R A M E T E R S

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.

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.

Select an option from the Static pressure curve list—Linear loss (the default), Quadratic

loss, Static pressure curve data, or User defined.

Linear LossEnter the Linear loss coefficient to define llc. The default is 0 Pas/m3. llc defines the static pressure curve that is a piecewise linear function equal to 0 when flow rate is < 0, equal to V0llc when flow rate is > 0.

Quadratic Loss Enter the Quadratic loss coefficient to define qlc. The default value is 0 Pas2/m6. qlc defines the static pressure curve that is a piecewise quadratic function equal to 0 when flow rate is < 0, equal to V0qlc2 when flow rate is > 0.

Theory for the Fan Defined on an Interior BoundarySee Also

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 Grille node again to update the Graphics window.

Tip


User Defined Select User defined to enter different values or expressions. The flow rate across the selection where this boundary 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 this interface). In order to avoid unexpected behavior, the function used for the grille curve is the maximum between the user-defined function and 0.

Static Pressure Curve DataSelect 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 defined using the Interpolation function type list in the Static Pressure Curve

Interpolation section. Then the units are specified for the flow rate and the static pressure curve in the Units section (described in the next sections).

S T A T I C P R E S S U R E C U R V E D A T A


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).

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


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.

U N I T S


curve. Select Units for the Flow rate (the default SI unit is m3/s) and Static pressure

curve (the default SI unit is Pa).

Theory for the Fan and Grille Boundary Conditions

Note


218 | C H A P T E

Th eo r y f o r t h e L am i n a r F l ow I n t e r f a c e

The Single-Phase Flow, Laminar Flow Interface theory unique to this module is described in this section:

• Theory for the Laminar Inflow Boundary Condition

• Theory for the Laminar Outflow Boundary Condition

• Theory for the Fan Defined on an Interior Boundary

• Theory for the Fan and Grille Boundary Conditions

• Theory for the No Viscous Stress Boundary Condition

Theory for the Laminar Inflow Boundary Condition

In order to prescribe an inlet 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 6-1: 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 a laminar plug flow. If the option is selected that constrains outer edges or endpoints to zero, the assumption is instead that the flow in the fictitious domain is fully developed laminar channel flow (in 2D) or fully developed

For the basic laminar flow theory, see Theory for the Laminar Flow Interface in the COMSOL Multiphysics User’s Guide. This section discusses the theory related to the enhanced features available with this module and for laminar flow.

Also see Theory for the Turbulent Flow Interfaces.

See Also


laminar internal flow (in 3D). This does not affect the boundary condition in the real domain, , where the boundary conditions are always fulfilled.

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.

Theory for the Laminar Outflow Boundary Condition

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 the flow in this fictitious domain is laminar plug flow. If the option is selected that constrains outer edges or endpoints to zero, the assumption is instead that the flow in the fictitious domain is fully developed laminar channel flow (in 2D) or fully developed

Lentr

pentr

Also see Inlet for the node Settings.See Also

T H E O R Y F O R T H E L A M I N A R F L O W I N T E R F A C E | 219

220 | C H A P T E

laminar internal flow (in 3D). This does not affect the boundary condition in the real domain, , where the boundary conditions are always fulfilled.

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.

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 boundary 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.

pexit

Lexit

Also see Outlet for the node Settings.See Also

In both cases, the boundary condition implementation specifies vanishing viscous stress along with a Dirichlet condition on the pressure.

Note


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.

Theory for the Fan and Grille Boundary Conditions

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 Interior Fan for node settings.See Also

See Fan and Grille for node settings.See Also


222 | C H A P T E

D E F I N I N G A D E V I C E A T A N I N L E T

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 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:

(6-3)

(6-4)

The Grille boundary condition sets the following conditions:

(6-5)

(6-6)

where V0 is the flow rate across the boundary, pinput is the pressure at the device’s inlet, and pfanV0) and pgrille(V0) are the static pressure functions of flow rate for the fan and the grille.

Equation 6-3 and Equation 6-5 correspond to the compressible formulation. Equation 6-4 and Equation 6-6 correspond to the incompressible formulation.

u u T+ 23--- u I–

n 0,= p pinput pfan V0 +=

u u T+ n 0,= p pinput pfan V0 +=

u u T+ 23--- u I–

n 0,= p pinput pgrill V0 –=

u u T+ n 0,= p pinput pgrill V0 –=

In 2D the thickness in the third direction, Dz, is used to define the flow rate. Fans are modeled as rectangles in this case.

2D


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.

D E F I N I N G A D E V I C E A T A N O U T L E T

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:

(6-7)

(6-8)

The Grille boundary condition sets the following conditions:

(6-9)

(6-10)

u u T+ 23--- u I–

n 0,= p pext pfan V0 –=

u u T+ n 0,= p pext pfan V0 –=

u u T+ 23--- u I–

n 0,= p pinput pgrill V0 +=

u u T+ n 0,= p pinput pgrill V0 +=


224 | C H A P T E

where V0 is the flow rate across the boundary, pext is the pressure at the device outlet, and pfan(V0), pvacuum pump(V0), and pgrille(V0) are the static pressure function of flow rate for the fan, the vacuum pump, and the grille.

Equation 6-7, Equation 6-8, and Equation 6-9 correspond to the compressible formulation.

Equation 6-8, Equation 6-9, and Equation 6-10 correspond to the incompressible formulation.

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.

Theory for the 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 User’s Guide), the viscous stress condition sets the viscous stress to zero:

In 2D the thickness in the third direction, Dz, is used to define the flow rate. Fans are modeled as rectangles in this case.

2D

u u T+ 23--- u I–

n 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.

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.

u u T+ n 0=

Also see Outlet for the node Settings.See Also


226 | C H A P T E

Th eo r y f o r t h e Tu r bu l e n t F l ow I n t e r f a c e s

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

• Theory for the Pressure, No Viscous Stress Boundary Condition

• Inlet Values for the Turbulence Length Scale and Intensity

• Pseudo Time Stepping for Turbulent Flow Models

• References for the Single-Phase Flow, Turbulent Flow Interfaces

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

(6-11)

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

Theory for the Laminar Flow InterfaceSee Also

Re UL

------------=


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

R E Y N O L D S - A V E R A G E D N AV I E R - S T O K E S ( R A N S ) E Q U A T I O N S

The information below assumes that the flow fluid is incompressible and Newtonian in which case the Navier-Stokes equations take the form:

(6-12)

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 sufficient 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, ui, and an average part, Ui. Because the flow field also varies

t

u u u+ pI– u u T+ + F+=

u 0=

+=

T H E O R Y F O R T H E TU R B U L E N T F L O W I N T E R F A C E S | 227

228 | C H A P T E

on a time scale longer than t2, Ui is still time dependent but is much smoother than the unfiltered velocity ui.

Figure 6-6: The unfiltered velocity component ui, with a time scale t1, and the averaged velocity component, Ui, 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:

(6-13)

where U is the averaged velocity field and is the outer vector product. A comparison with Equation 6-12 indicates that the only difference is the appearance of the last term on the left-hand side of Equation 6-13. 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.

E D D Y V I S C O S I T Y

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

t

U U U u' u' + + P U U T+ F+ +–=

U 0=

u' u' 3---trace u' u' I– T– U U T+ =


where T is the eddy viscosity, also known as the turbulent viscosity. The spherical part can be written

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 importance (in compressible flows, for example) this term must be explicitly included.

TU R B U L E N T C O M P R E S S I B L E F L O W

If the Reynolds average is applied to the compressible form of the Navier-Stokes, terms of the form

appear and need to be modeled. To avoid this, a density-based average, known as the Favre average, is introduced:

(6-14)

It follows from Equation 6-14 that

(6-15)

and a variable, ui, is decomposed in a mass-averaged component, , and a fluctuating component, ui, according to

(6-16)

Using Equation 6-15 and Equation 6-16 along with some modeling assumption for compressible flows (Ref. 7), Equation 6-14 can be written in the form

(6-17)

The Favre-averaged Reynolds stress tensor is modeled using the same argument as for incompressible flows:

3---trace u' u' I 2

3---k=

u

ui˜ 1

--- 1

T----

T lim x ( , )ui x ( , ) d

t

t T+

=

ui ui=

ui

ui ui ui+=

t------

xi------- ui + 0=

uit-------- uj

uixj

--------+pxi

-------–xj

------- uixj

--------ujxi

--------+ 2

3---

ukxk

---------ij– ujui–

Fi++=


230 | C H A P T E

where k is the turbulent kinetic energy. Comparing Equation 6-17 to its incompressible counterpart (Equation 6-13), it can be seen that except for the term

the compressible and incompressible formulations are exactly the same, except that the free variables are instead of

More information about modeling turbulent compressible flows is in Ref. 1 and Ref. 7.

The turbulent transport equations are used in their fully compressible formulations (Ref. 8).

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. 1). 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

(6-18)

where C is a model constant.

The transport equation for k reads:

(6-19)

where the production term is

(6-20)

The transport equation for reads:

ujui– Tuixj

--------ujxi

--------+ 2

3--- T

ukxk

--------- k+

ij–=

2 3 kij–

ui

Ui ui=

T Ck2

------=

kt

------ + u k Tk------+

k Pk –+=

Pk T u: u u T+ 23--- u 2–

23---k u–=


(6-21)

The model constants in Equation 6-18, Equation 6-19, and Equation 6-21 are determined from experimental data (Ref. 1) and the values are listed in Table 6-1.

M I X I N G L E N G T H L I M I T

Equation 6-19 and Equation 6-21 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. 9. The implementation includes an upper limit on the mixing length, :

(6-22)

The mixing length is used to calculated the turbulent viscosity. should not be active in a converged solution but is merely a tool to obtain convergence.

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

The eddy-viscosity model of the Reynolds stress tensor can be written

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 6-18 does not guarantee this. To assert that

the turbulent viscosity is subjected to a realizability constraint. The constraint for 2D and 2D axisymmetry is:

TABLE 6-1: MODEL CONSTANTS

CONSTANT VALUE

C 0.09

C1 1.44

C2 1.92

k 1.0

1.3

t

----- + u T------+

C1

k---P

kC2

2

k-----–+=

lmixlim

lmix max Ck3 2/

----------- lmix

lim =

lmixlim

uiuj 2TSij–23---kij+=

uiui 0 i


232 | C H A P T E

(6-23)

and for 3D and 2D axisymmetry with swirl flow it reads:

(6-24)

Combining equation Equation 6-23 with Equation 6-18 and the definition of the mixing length gives a limit on the mixing length scale:

(6-25)

Equivalently, combining Equation 6-24 with Equation 6-18 and Equation 6-22 gives:

(6-26)

This means there are two limitations on lmix: the realizability constraint and the imposed limit via Equation 6-22.

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. 4, Ref. 5, and Ref. 6.

M O D E L L I M I T A T I O N S

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 underpredicting the spatial extension of recirculation zones (Ref. 1). Furthermore, in the description of rotating flows, the model often shows poor

Tk 2

3 SijSij

-----------------------

Tk

6 SijSij

---------------------------

Swirl flow is not available with the Heat Transfer Module.

Note

lmix23---

kSijSij

-------------------

lmix16

------- kSijSij

-------------------


agreement with experimental data (Ref. 2). In most cases, the limited accuracy is a fair trade-off for the amount of computational resources saved compared to more complicated turbulence models.

WA L L F U N C T I O N S

The flow close to a solid wall is for a turbulent flow and is very 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 are such that the computational domain is assumed to start a distance w from the wall (see Figure 6-7).

Figure 6-7: The computational domain starts a distance w from the wall for wall functions.

The distance w is automatically computed so that

where uC1/4k 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 boundary

Solid wall

w

Mesh cells

w+ uw =


234 | C H A P T E

mesh cell. This means that can become higher than 11.06 if the mesh is relatively coarse.

The boundary conditions for the velocities are a no-penetration condition u n = 0 and a shear stress condition

where

is the viscous stress tensor and

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:

See Ref. 9 and Ref. 10 for further details.


The default initial values for a stationary simulation are (Ref. 9),

w+

Always investigate the solution to check that w is small compared to the dimension of the geometry. Also check that is 11.06 on most of the walls.

If is much higher over a significant part of the walls, the accuracy might become compromised. Both the wall lift-off, w, and the wall lift-off in viscous units, , are available as results and analysis variables.

Tip

w+

w+

w+

n n n n– uuu-------max C

1 4/ k u –=

u u T+ =

uu

1v----- wln B+-----------------------------=

C

3 4/ k3 2/

vw-----------------------=


where is the mixing length limit. For time dependent simulations, the initial value for k is instead

The Low Reynolds Number k- Turbulence Model

In some cases, the accuracy provided by wall functions is not enough. In these cases, 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. 11). The AKN k- model for compressible flows reads (Ref. 8 and Ref. 11):

(6-27)

where

u 0=

p 0=

k 10 0.1 lmix

lim ------------------------------- 2

=

Ckinit

3 2/

0.1 lmixlim

-----------------------=

lmixlim

k 0.1 lmix

lim ------------------------------- 2

=

kt

------ + u k Tk------+

k Pk –+=

t

----- + u T------+

C1

k---P

kfC2

2

k-----–+=


236 | C H A P T E

(6-28)

and

(6-29)

Also, lw is the distance to the closest wall.

Realizability Constraints are applied to the low Reynolds number k- model.

WA L L D I S T A N C E

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 most convenient way to handle the wall distance variable is to solve 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.

WA L L B O U N D A R Y C O N D I T I O N S

The damping terms in the equations for k and allows a no slip condition to be applied to the velocity, that is u0.

Since all velocities must disappear on the wall, so must k. Hence, k0 on the wall.

The correct wall boundary condition for is

Pk T u: u u T+ 23--- u 2–

23---k u–=

T fCk2

------=

f 1 e l* 14–– 2 1 5Rt

3 4/------------e Rt 200 2–+

=

f 1 e l* 3.1–– 2 1 0.3e Rt 6.5 2–– =

l* ulw = Rt k2 = u 1 4/=

C1 1.5= C2 1.9= C 0.09= k 1.4= 1.4=

• The Wall Distance Interface in the COMSOL Multiphysics User’s Guide

• Stationary with Initialization, Transient with Initialization, and Wall Distance Initialization in the COMSOL Multiphysics Reference Guide

See Also

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

(6-30)

is prescribed in those cells. Equation 6-30 can be derived as the first term in a series expansion of

For the expansion to be a valid, it is required that

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

I N L E T V A L U E S F O R T H E TU R B U L E N C E L E N G T H S C A L E A N D I N T E N S I T Y

The guidelines given in Inlet Values for the Turbulence Length Scale and Intensity for selecting turbulence length scale, LT, and the turbulence intensity, IT, apply also to the low-Reynolds number k- model.


The low-Reynolds number k- model has the same default initial guess as the standard k- model (see Initial Values) but with replaced by lref.

The default initial value for the wall distance equations (which solves for the reciprocal wall distance) 2lref.

Inlet Values for the Turbulence Length Scale and Intensity

A value of 0.1% is a low turbulence intensity IT. Good wind tunnels can produce values of as low as 0.05%. Fully turbulent flows usually have intensities between five and ten percent.

2--- k

lw2

-----=

2 k n 2

lc* 1

lc*

lc* 1»

lmixlim


238 | C H A P T E

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 6-2 as a guideline when specifying LT (Ref. 3) where lw is the wall distance, and

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 User’s Guide), the turbulent intensity IT, turbulence length scale LT, and reference velocity scale Uref values are related to the turbulence variables via

is the wall distance in viscous units.

lw+ lw l*=

k 32--- ITUref 2,=

C3 4

LT------------

3 ITUref 2

2----------------------------

32---

=


Also see Inlet and Outlet for the node Settings.See Also

TABLE 6-2: TURBULENT LENGTH SCALES FOR TWO-DIMENSIONAL FLOWS

FLOW CASE LT L

Mixing layer 0.07L Layer width

Plane jet 0.09L Jet half width

Wake 0.08L Wake width

Axisymmetric jet 0.075L Jet half width

Boundary layer (px0)

– Viscous sublayer and log-layer

– Outer layer 0.09L

Boundary layer thickness

Pipes and channels

(fully developed flows)

0.07L Pipe diameter or channel width

lw 1 lw+ 26– exp–


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 as the momentum and continuity equations.

If the automatic expression for CFLloc is, for 2D models:

and for 3D models:

References for the Single-Phase Flow, Turbulent Flow Interfaces

1. D.C. Wilcox, Turbulence Modeling for CFD, 2nd ed., DCW Industries, 1998.

2. 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.

3. H.K. Versteeg and W. Malalasekera, An Introduction to Computational Fluid Dynamics, Prentice Hall, 1995.

4. A. Durbin, “On the k- Stagnation Point Anomality,” International Journal of Heat and Fluid Flow, vol. 17, pp. 89–90, 1986.

5. A, Svenningsson, Turbulence Transport Modeling in Gas Turbine Related Applications,” doctoral dissertation, Department of Applied Mechanics, Chalmers University of Technology, 2006.

6. 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.

7. J. Larsson, Numerical Simulation of Turbulent Flows for Turbine Blade Heat Transfer, doctoral dissertation, Chalmers University of Technology, Sweden, 1998.

t

1.3min niterCMP-1 9 +

if niterCMP 25 9 1.3min niterCMP 25– 9 0 +

if niterCMP 50 90 1.3min niterCMP 50– 9 0

1.3min niterCMP-1 9 +

if niterCMP 30 9 1.3min niterCMP 30– 9 0 +

if niterCMP 60 90 1.3min niterCMP 60– 9 0


240 | C H A P T E

8. 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.

9. 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.

10. H. Grotjans and F.R. Menter, “Wall Functions for General Application CFD Codes,” ECCOMAS 98, Proceedings of the Fourth European Computational Fluid Dynamics Conference, John Wiley & Sons, pp. 1112–1117, 1998.

11. 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.


7

T h e 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 Heat Transfer Module has interfaces for conjugate heat transfer, which are also under the Fluid Flow branch as Non-Isothermal Flow interfaces. 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 Interfaces

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

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

• Shared Feature Settings

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

241

242 | C H A P T E

Abou t t h e Con j u g a t e Hea t T r a n s f e r I n t e r f a c e s

In this section:

• Selecting the Right Interface

• The Non-Isothermal Flow Options

• Conjugate Heat Transfer Options

Selecting the Right 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 interfaces found under the Fluid Flow>Non-Isothermal Flow ( ) and Heat

Transfer>Conjugate Heat Transfer ( ) branches are multiphysics interfaces and contain the physics 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*

INTERFACE (NITF) TURBULENCE MODEL TYPE

TURBULENCE MODEL

HEAT TRANSPORT TURBULENCE MODEL

DEFAULT MODEL

Non-Isothermal Flow, Laminar Flow

None N/A N/A Fluid

Non-Isothermal Flow, Turbulent Flow, k-

RANS k- Kays-Crawford Fluid

R 7 : T H E C O N J U G A T E H E A T TR A N S F E R B R A N C H

Non-Isothermal Flow, Turbulent Flow, Low Re k-

RANS Low Reynolds number k-

Kays-Crawford Fluid

Conjugate Heat Transfer, Laminar Flow

None N/A N/A Heat transfer in solids

Conjugate Heat Transfer, Turbulent Flow, k-

RANS k- Kays-Crawford Heat transfer in solids

Conjugate Heat Transfer, Turbulent Flow, Low Re k-

RANS Low Reynolds number k-

Kays-Crawford Heat transfer in solids

*For all the interfaces, the Neglect initial term (Stokes flow) check box is not selected by default.

TABLE 7-1: THE NON-ISOTHERMAL FLOW AND CONJUGATE HEAT TRANSFER PHYSICAL MODEL DEFAULT SETTINGS*

INTERFACE (NITF) TURBULENCE MODEL TYPE

TURBULENCE MODEL

HEAT TRANSPORT TURBULENCE MODEL

DEFAULT MODEL

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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 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 give you a brief overview of each of the interfaces to help you choose.

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. If a solid is chosen in the Default model list, then the interface is renamed Conjugate Heat Transfer.

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 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.

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- 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.


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In addition to the properties for the different turbulence models mentioned in Theory for the Non-Isothermal Flow and Conjugate Heat Transfer 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. If a liquid regime is chosen in the Default model list, then the interface is renamed Non-Isothermal

Flow, which is the same interface as Conjugate Heat Transfer but with different default settings as in Table 7-1.

C O N J U G A T E H E A T TR A N S F E R , L A M I N A R F L O W

The Conjugate Heat Transfer, Laminar Flow 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 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 Conjugate Heat Transfer, Laminar Flow interface if required. See Table 7-1 for details.

C O N J U G A T E H E A T TR A N S F E R , TU R B U L E N T F L O W

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-e model. The Turbulent Flow, k- and Turbulent Flow Low Re k- 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 BranchSee Also


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Th e Non - I s o t h e rma l F l ow and Con j u g a t e Hea t T r a n s f e r , L am i n a r F l ow I n t e r f a c e s

As discussed in About the Conjugate Heat Transfer Interfaces the two interfaces differ only by where they are selected in the Model Wizard and the default model selected—Heat transfer in solids or Fluids.

In this section:

• The Non-Isothermal Flow, Laminar Flow Interface

• The Conjugate Heat Transfer, Laminar Flow Interface

The Non-Isothermal Flow, Laminar Flow Interface

The Non-Isothermal Flow version of the Laminar Flow 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.




See Also

Viscous Heating in a Fluid Damper: Model Library path Heat_Transfer_Module/Verification_Models/fluid_damper

Model




The default identifier (for the first interface in the model) is nitf.




Define interface properties to control the overall type of model:

Neglect Inertial Term (Stokes Flow)—All InterfacesSelect 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 TypeBy definition, no turbulence model is needed when studying laminar flows. The default Turbulence model type is None.

The flow state in a fluid-flow model is not, however, always known beforehand. Select RANS as the Turbulence model type and select any of k-, k-, Low Re number k- or

This interface changes to a Conjugate Heat Transfer interface when Heat

transfer in solids is selected as the Default model.Tip

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 , L A M I N A R F L O W I N T E R F A C E S | 249

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Spalart-Allmaras as Turbulence model in order to account for turbulence. This changes the interface into the turbulent version. See the Turbulent Flow Interfaces for details.

Default ModelFor the Non-Isothermal Flow interface the Default model is Fluid. For the Conjugate Heat

Transfer interface the Default model is Heat transfer in solids.

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

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



The dependent variables (field variables) are for the Velocity field, Pressure, and 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.


To display this section, click the Show button ( ) and select Discretization. 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. Specify the Value

type when using splitting of complex variables—Real or Complex (the default).

If the default Turbulence model type selected is RANS, the additional turbulence model settings are made available. However, 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. Note

This section is available when the Surface-to-surface radiation check box is selected. See Radiation Settings as described for The Heat Transfer Interface.Note



To display this section, click the Show button ( ) and select Stabilization. Any settings unique to this interface are listed below.

• 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.

The Conjugate Heat Transfer, Laminar Flow Interface

The Conjugate Heat Transfer version of the Laminar Flow 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, 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.



• For Settings window details for the Heat Transfer in Solids feature, see The Heat Transfer Interface

• The Conjugate Heat Transfer, Laminar Flow Interface

See Also

Viscous Heating in a Fluid Damper: Model Library path Heat_Transfer_Module/Verification_Models/fluid_damper


• The Non-Isothermal Flow, Laminar Flow Interface


• The Heat Transfer Interface for Settings window details for the Heat

Transfer in Solids feature.

Model

See Also

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 , L A M I N A R F L O W I N T E R F A C E S | 251

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This interface changes to a Non-Isothermal Flow interface when Fluid is selected as the Default model.

Tip


Th e Non - I s o t h e rma l F l ow and Con j u g a t e Hea t T r a n s f e r , T u r bu l e n t F l ow I n t e r f a c e s

As discussed in About the Conjugate Heat Transfer Interfaces, the Non-Isothermal Flow ( ) and Conjugate Heat Transfer ( ) branches have more than one version of the Turbulent Flow interface. All interfaces use the Reynolds-Averaged Navier-Stokes (RANS) equations as the Turbulence model type, solving for the mean velocity field and pressure. The differences are the Default model is either Heat transfer in solids or Heat transfer in fluids, and the Turbulence model can be k- or the Low Reynolds k- turbulence model.

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

These predefined multiphysics couplings consist of a turbulent flow interface, using a compressible formulation, in combination with a Heat Transfer interface.

• The Turbulent Flow, k- and Turbulent Flow Low Re k- Interfaces




See Also

Turbulent Flow Through a Shell-and-Tube Heat Exchanger: Model Library path Heat_Transfer_Module/Process_and_Manufacturing/

turbulent_heat_exchangerModel

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 , TU R B U L E N T F L O W I N T E R F A C E S | 253

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Most of the setting options are the same as for The Non-Isothermal Flow, Laminar Flow Interface, except where noted below. Right-click any node to add other features that implement, for example, boundary conditions, volume forces, or heat sources.


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.

TU R B U L E N C E M O D E L P A R A M E T E R 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.


The dependent variables (field variables) are for the Velocity field, Pressure, and Temperature. The names can be changed but the names of fields and dependent variables must be unique within a model.

The Neglect inertial term (Stokes flow) check box is only valid for laminar flow.



• Turbulent Non-Isothermal Flow Theory

Note

See Also


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.


To display this section, click the Show button ( ) and select Stabilization. Any settings unique to this interface are listed below.

• 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.

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S h a r e d F e a t u r e S e t t i n g s

All the versions of the Non-Isothermal Flow and Conjugate Heat Transfer interfaces have shared domain, boundary, edge, point, and pair features based on the selections made for the model.

Also because these are all multiphysics interfaces, almost every feature is shared with, and described for, other interfaces. Below are links to the domain, boundary, edge, point, and pair features as indicated.

In this section:

• Fluid

• Initial Values

• Pressure Work

• Viscous Heating

• Wall

These features are described for the Laminar Flow interface (listed in alphabetical order):

• Boundary Stress

• Interior Fan

• Flow Continuity

• Inlet

• Interior Wall

• Open Boundary

• Outlet

• Periodic Flow Condition

• Pressure Point Constraint

• Symmetry

• Volume Force

These features are described for the Heat Transfer interface (listed in alphabetical order):




• Continuity

• Heat Flux

• Heat Source





• Outflow






• Symmetry

• Temperature




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




See Also

Tip

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Fluid

The Fluid feature 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 to the Fluid node.

When the turbulence model type is set to RANS, the Fluid node also adds the equations for k and .


By default, All domains are selected.


Define the model inputs. If no model inputs are required, this section is empty.


The default uses 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 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 = kT 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+TI)T where k is the thermal conductivity tensor, I the identity matrix and T the thermal turbulent conductivity defined by


Select a Fluid type—Gas/Liquid or Ideal gas.

• The heat capacity at constant pressure Cp describes the amount of heat energy required to produce a unit temperature change in a unit mass. For an ideal gas,

To define the Absolute Pressure, see the settings for the Heat Transfer in Fluids node as described in the COMSOL Multiphysics User’s Guide.

Tip

TCpT

PrT--------------=


choose to specify either Cp or the ratio of specific heats, , but not both since they in that case are dependent.

• The ratio of specific heats is the ratio of heat capacity at constant pressure, Cp, to heat capacity at constant volume, Cv. 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.

Gas/LiquidIf Gas/Liquid is selected properties of a non-ideal gas or liquid can be used. By default the Density (SI unit: kg/m3), Heat capacity at constant pressure Cp (SI unit: J/(kg·K)), and Ratio of specific heats (unitless) use data From material. Select User

defined to enter other values or expressions.

Ideal GasIf Ideal gas is selected, the ideal gas law is used to describe the fluid. In this case, specify the thermodynamics properties by selecting a gas constant type and selecting between entering the heat capacity at constant pressure or the ratio of specific heats. For an ideal gas the density is defined as

where pA is the absolute pressure, and T the temperature.

• Select a Gas constant type— Specific gas constant Rs (SI unit: J/(kg·K)) or Mean

molar mass Mn (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.

• From the Specify Cp or list, select Heat capacity at constant pressure Cp (SI unit: J/(kg·K)), and Ratio of specific heats (unitless). The default setting is to use the property value From material. Select User defined to enter another value or expression for either of material property.

D Y N A M I C V I S C O S I T Y

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

MnpA

RT----------------

pARsT-----------= =


260 | C H A P T E

shear-rate dependent viscosity. Examples of non-Newtonian fluids include yoghurt, paper pulp, and polymer suspensions.

Select a Dynamic viscosity (SI unit: Pa·s) from the list—From material (the default), Non-Newtonian power law, Non-Newtonian Carreau model, or User defined. If User defined is selected, use a built-in variable for the shear rate magnitude, spf.sr, which makes it possible to define arbitrary expressions of the dynamic viscosity.

Non-Newtonian Power LawIf Non-Newtonian power law is selected, enter the Power law model parameter m and Model parameter n (both unitless). This selection uses the power law as the viscosity model for a non-Newtonian fluid where the following equation defines dynamic viscosity:

Non-Newtonian Carreau Model 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 (unitless)

This selection uses the Carreau model as the viscosity model for a non-Newtonian fluid where the following equation defines the dynamic viscosity:

M I X I N G L E N G T H L I M I T ( TU R B U L E N C E M O D E L S O N L Y )

This section is only available for the k- and k- models, which need 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:

(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 result that is too large. Then define manually.

• If Manual is selected, enter a value or expression for the Mixing length limit (SI unit: m).

m· n 1–=

0 inf– 1 · 2+ n 1–

2-----------------

+=

lmixlim 0.5lbb=

lmixlim

lmixlim


D I S T A N C E E Q U A T I O N ( TU R B U L E N C E M O D E L S O N L Y )

This section is only available for the low-Reynolds number k- model and the Spalart-Allmaras model, which need 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:

(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 manually.

• If Manual is selected, enter a value or expression for the Reference length scale (SI unit: m).

Wall

For laminar flow, the low Reynolds number k- turbulence model and Spalart-Allmaras turbulence model, the Wall feature 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 when using the k- or k- turbulence model.

About the Thermal Wall FunctionWhenever 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.

lref 0.25lbb=

lref

lref


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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.

Figure 7-2: A simple example that includes both an external wall and an internal wall.

On internal walls, there are two temperatures, one for the solid, Ts, and one for the fluid, Tf. If a temperature is prescribed to an internal wall, the constraint is applied to the temperature for the solid, that is, to Ts.

On external walls, the temperature T is the temperature of the fluid while the wall temperature is represented by the dependent variable Tw. Tw is a variable that is solved for and the equation for Tw is

where qtot is the total heat flux prescribed to the boundary.

If a temperature is prescribed to an external wall, the constraint is applied to the wall temperature Tw.


When using the k- turbulence model or 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).

External wall

OutflowInflow

Solid

Fluid Internal wall

qwf qtot=

Any other heat boundary condition applied to an external wall is wrong in the sense that it acts on the fluid temperature, Ts, instead of the wall temperature, Tw.Note


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).

Initial Values

The Initial Values feature 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. The surface radiosity is only applicable for surface-to-surface radiation.


From the Selection list, choose the domains to define 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 for the velocity and the pressure, and 293.15 K for the temperature.

In a turbulent flow interface, initial values for the turbulence variables are also specified. By default these are specified using the predefined variables defined by the expressions in Initial Guess.

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.

About the Thermal Wall Function

Note

See Also


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The initial value for the surface radiosity J (SI unit: W/m2), for surface-to-surface radiation, has a default value of 0.

Pressure Work

The Pressure Work feature adds the following contribution to the right-hand side of the Heat Transfer in Fluids equation:

(7-3)

The software computes the pressure work using the absolute pressure.


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

P R E S S U R E WO R K F O R M U L A T I O N

select Full formulation or Low mach number formulation. The latter excludes the term u · p from Equation 7-3, which is small for most flows with 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:

(7-4)

Here is the viscous stress tensor and S is the strain rate tensor. Equation 7-4 represents the heating caused by viscous friction within the fluid.


From the Selection list, choose the domains to add pressure work. By default, the selection is the same as for the Fluid feature that it is attached to.

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

p

pt------ u p+ –

:S


Th eo r y f o r t h e Non - I s o t h e rma l F l ow and Con j u g a t e Hea t T r a n s f e r I n t e r f a c e s

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:

(7-5)

where

• is the density (kg/m3)

• u is the velocity vector (m/s)

• p is pressure (Pa)

• is the dynamic viscosity (Pa·s)

• F is the body force vector (N/m3)

It also solves the heat equation, which for a fluid is

where in addition to the quantities above

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

t------ u + 0=

ut------- u u+ p– u u T+ 2

3--- u I–

F++=

CpTt------- u T+ q – :S T

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

p

pt------ u p+ – Q+ +=

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 I N T E R F A C E S | 265

266 | C H A P T E

• T is absolute temperature (SI unit: K)

• q is the heat flux by conduction (SI unit: W/m2)

• 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)

The pressure work term

and the viscous heating term

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:

where E is the elastic contribution to entropy (SI unit: J/(m3·K))

As in the case of fluids, the pressure work term

is not included by default but must be added as a subfeature.

S 12--- u u T+ =

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

p

pt------ u p+

:S

CpTt------- q – T E

t-------– Q+=

T Et

-------

• The Heat Equation

• Turbulent Non-Isothermal Flow Theory

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

See Also


Turbulent Non-Isothermal Flow Theory

Turbulent energy transport is conceptually more complicated than energy transport 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 and is defined by

where the bar denotes the usual Reynolds average. The full field is then decomposed as

With these notations the equation for total internal energy, e, becomes

(7-6)

where h is the enthalpy. The vector

(7-7)

is the laminar conductive heat flux and

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

is model with the Boussinesq approximation:

T˜

T˜ T

-------=

T T˜

T''+=

t

----- euiui

2-----------+

ui''ui''

2-------------------+

xj

------- uj huiui

2-----------+

ujui''ui''

2-------------------+

=+

xj

------- qj– uj''h''– ijui''uj''ui''ui''

2---------------------------–+

xj

------- ui ij ui''uj''– +

qj Txj

-------–=

ij 2Sij23---

ukxk

---------ij–=

ui''u''j–


268 | C H A P T E

(7-8)

where k is the turbulent kinetic energy, which in turned is defined by

(7-9)

The correlation between and in Equation 7-6 is the turbulent transport of heat. It is model analogous to the laminar conductive heat flux

(7-10)

The molecular diffusion,

and turbulent transport term,

are modeled by a generalization of the molecular diffusion and turbulent transport term found in the incompressible k equation

(7-11)

Inserting Equation 7-7, Equation 7-8, Equation 7-9, Equation 7-10 and Equation 7-11 into Equation 7-6 gives

(7-12)

The Favre average can also be applied to the momentum equation, which, using Equation 7-8, can be written

(7-13)

ui''u''j– Tij 2T S

˜ij

13---

ukxk

---------ij– 2

3---kij–= =

k 12---ui''ui''=

uj'' h''

uj''h'' qTj T

T˜

xj

-------–TCpPrT

-------------- T˜

xj

-------–= = =

ijui''

uj''ui''ui'' 2

ijui''uj''ui''ui''

2---------------------------–

T

k------+

kxj

-------=

t

----- euiui

2----------- k+ +

xj------- uj h

uiui2

----------- k+ +

=+

xj

------- qj– qTj–

Tk------+

kxj

-------+

xj------- ui ij T

ij+ +

t

----- ui xj

------- ujui +pxj

-------–xj

------- ij Tij+ +=


Taking the inner product between and Equation 7-13 results in an equation for the resolved kinetic energy, which can be subtracted from Equation 7-12 with the following result:

(7-14)

where the relation

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 7-14 is

(7-15)

Larsson (Ref. 2) suggests to make the split

Since

for all applications of engineering interest, it will follow that

and consequently

(7-16)

where

ui

t

----- e k+ xj

------- uj e k+ pujx

--------j

– +=+

xj

------- qj– qTj–

Tk------+

kxj

-------+

xj------- ui ij T

ij+ +

e h p +=

t

----- e xj

------- uje pujx

--------j

–xj

------- qj– qTj–

xj------- ui ij T

ij+ + +=+

ij ij ij''+=

ij ij''»

ij ij

t

----- e xj

------- uje pujx

--------j

–xj

------- T+ T˜

xj

-------

xj------- uiij

Tot + +=+


270 | C H A P T E

Equation 7-16 is completely analogous to the laminar energy equation and can be expanded using the same theory (see for example Ref. 3):

which is the temperature equation solved in the turbulent Non-Isothermal Flow and Conjugate Heat Transfer interfaces.

TU R B U L E N T C O N D U C T I V I T Y

Kays-CrawfordThis 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

(7-17)

where the Prandtl number at infinity is PrT0.85 and is the conductivity.

Extended Kays-CrawfordWeigand and others (Ref. 5) suggested an extension of Equation 7-17 to liquid metals by introducing

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.

TE M P E R A T U R E WA L L F U N C T I O N S

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 geometry is drawn.

ijTot T+ 2S

˜ij

23---

ukxk

---------ij–

=

CpT˜

t

------- uj˜ T

˜xj

-------+

xj------- T+ T

˜xj

-------

ijS˜

ijT˜

----

T˜

-------

p

pt

------ uj˜ p

xj-------+

–+=

PrT1

2PrT----------------- 0.3

PrT

------------------CpT

-------------- 0.3CpT

--------------

21 e 0.3CpT PrT –– –+

1–

=

PrT 0.85 100CpRe

0.888--------------------------------+=


The heat flux between the fluid with temperature Tf and a wall with temperature Tw, is:

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):

where in turn

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 postprocessing variable on boundaries.

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

qwfCpC

1 4/ k1 2/ Tw Tf–

T+-----------------------------------------------------------=

T+

Prw+ for w

+ w1+

15Pr2 3/ 500w2

+----------–

for w1+ w

+ w2+

Pr-----lnw

+ + for w2+ w

+

=

w+

w C1 2/ k

------------------------------= w1

+ 10Pr1 3/-------------=

w2+ 10 10

PrT---------= Pr

Cp

----------=

15Pr2 3/PrT2--------- 1 ln 1000

PrT---------

+ –=


272 | C H A P T E

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.


8

M a t e r i a l s

The Material Browser contains libraries with an extensive set of mechanical and heat transfer properties for solid materials. In addition, it contains a limited set of fluid properties, which can be used mainly in the physics interfaces for fluid flow and heat transfer.

In this chapter:

• Material Library and Databases

• Liquids and Gases Material Database

273

274 | C H A P T E

Ma t e r i a l L i b r a r y and Da t aba s e s

The Heat Transfer Module includes a Liquids and Gases material database. The materials include temperature-dependent fluid dynamic and thermal properties.

In this section:

• About the Material Databases

• About Using Materials in COMSOL

• Opening the Material Browser

• Using Material Properties

About the Material Databases

All COMSOL modules have predefined material data available to build models. The most extensive material data is contained in the separately purchased Material Library,

For detailed information about all the other materials databases and the separately purchased Material Library, see the section Materials in the COMSOL Multiphysics User’s Guide.Note

Material Browser—select predefined materials in all applications.

Material Library—Purchased separately. Select from over 2500 predefined materials.

Built-In database—Available to all users and contains common materials.

Application specific material databases —Available with specific modules.

User-defined material database library.

Recent Materials—Select from recent materials added to the model.

R 8 : M A T E R I A L S

but all modules contain commonly used or module-specific materials. For example, the Built-In database is available to all users but the MEMS database is included with the MEMS Module and Structural Mechanics Module. Also create custom materials and material libraries by researching and entering material properties.

All the material databases (including the Material Library) are accessed from the Material Browser. These databases are briefly described below.

R E C E N T M A T E R I A L S

From the Recent Materials folder ( ), select from a list of recently used materials, with the most recent at the top. This folder is available after the first time a material is added to a model.

M A T E R I A L L I B R A R Y

An optional add-on database, the Material Library ( ), contains data for over 2500 materials and 20,000 property functions.

B U I L T - I N

Included with COMSOL Multiphysics, the Built-In database ( ) contains common solid materials with electrical, structural, and thermal properties.

A C / D C

Included in the AC/DC Module, the AC/DC database ( ) has electric properties for some magnetic and conductive materials.

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

Included in the Batteries & Fuel Cells Module, the Batteries and Fuel Cells database ( ) includes properties for electrolytes and electrode reactions for certain battery chemistries.

L I Q U I D S A N D G A S E S

Included in the Acoustics Module, CFD Module, Chemical Reaction Engineering Module, Heat Transfer Module, MEMS Module, Pipe Flow Module, and Subsurface

Predefined Built-In Materials for all COMSOL Modules in the COMSOL Multiphysics User’s Guide

See Also

M A T E R I A L L I B R A R Y A N D D A T A B A S E S | 275

276 | C H A P T E

Flow Module, the Liquids and Gases database ( ) includes transport properties and surface tension data for liquid/gas and liquid/liquid interfaces.

M E M S

Included in the MEMS Module and Structural Mechanics Module, the MEMS database ( ) has properties for MEMS materials—metals, semiconductors, insulators, and polymers.

P I E Z O E L E C T R I C

Included in the Acoustics Module, MEMS Module, and Structural Mechanics Module, the Piezoelectric database ( ) has properties for piezoelectric materials.

U S E R - D E F I N E D L I B R A R Y

The User-Defined Library folder ( ) is where user-defined materials databases (libraries) are created. When any new database is created, this also displays in the Material Browser.

About Using Materials in COMSOL

U S I N G T H E M A T E R I A L S I N T H E P H Y S I C S S E T T I N G S

The physics set-up in a model is determined by a combination of settings in the Materials and physics interface nodes. When the first material is added to a model, COMSOL automatically assigns that material to the entire geometry. Different geometric entities can have different materials. The following example uses the

The materials databases shipped with COMSOL Multiphysics are read-only. This includes the Material Library and any materials shipped with the optional modules.

Creating Your Own User-Defined Libraries in the COMSOL Multiphysics User’s Guide

Important

See Also


heat_sink.mph model file contained in the Heat Transfer Module and CFD Module Model Libraries.

Figure 8-1: Assigning materials to a heat sink model. Air is assigned as the material to the box surrounding the heat sink, and aluminum to the heat sink itself.

If a geometry consists of a heat sink in a container, Air can be assigned as the material in the container surrounding the heat sink and Aluminum as the heat sink material itself (see Figure 8-1). The Conjugate Heat Transfer interface, selected during model set-up, has a Fluid flow model, defined in the box surrounding the heat sink, and a Heat

Transfer model, defined in both the aluminum heat sink and in the air box. The Heat

Transfer in Solids 1 settings use the material properties associated to the Aluminum

3003-H18 materials node, and the Fluid 1 settings define the flow using the Air material properties. The other nodes under Conjugate Heat Transfer define the initial and boundary conditions.

All physics interface properties automatically use the correct Materials properties when the default From material setting is used. This means that one node can be used to define the physics across several domains with different materials; COMSOL then uses the material properties from the different materials to define the physics in the domains. If material properties are missing, the Material Contents section on the


278 | C H A P T E

Materials page displays a stop icon ( ) to warn about the missing properties and a warning icon ( ) if the property exists but its value is undefined.

There are also some physics interface properties that by default define a material as the Domain material (that is, the materials defined on the same domains as the physics interface). For such material properties, select any other material that is present in the model, regardless of its selection.

E V A L U A T I N G A N D P L O T T I N G M A T E R I A L P R O P E R T I E S

You can access the material properties for evaluation and plotting like other variables in a model using the variable naming conventions and scoping mechanisms:

• To access a material property throughout the model (across several materials) and not just in a specific material, use the special material container root.material. For example, root.material.rho is the density as defined by the materials in each domain in the geometry. For plotting, you can type the expression material.rho to create a plot that shows the density of all materials.

• To access a material property from a specific material, you need to know the tags for the material and the property group. Typically, for the first material (Material 1) the tag is mat1 and most properties reside in the default Basic property group with the tag def. The variable names appear in the Variable column in the table under Output

properties in the Settings window for the property group; for example, Cp for the heat capacity at constant pressure. The syntax for referencing the heat capacity at constant pressure in Material 1 is then mat1.def.Cp. Some properties are anisotropic tensors, and each of the components can be accessed, such as mat1.def.k11, mat1.def.k12, and so on, for the thermal conductivity. For material properties that are functions, call these with input arguments such as

The Material Page in the COMSOL Multiphysics User’s GuideSee Also

If you use a temperature-dependent material, each material contribution asks for a special model input. For example, rho(T) in a material mat1 asks for root.mat1.def.T, and you need to define this variable (T) manually—if the temperature is not available as a dependent variable—to make the density variable work.

Note


mat1.def.rho(pA,T) where pA and T are numerical values or variables representing the absolute pressure and the temperature, respectively. The functions can be plotted directly from the function nodes’ Settings window by first specifying suitable ranges for the input arguments.

• Many physics interfaces also define variables for the material properties that they use. For example, solid.rho is the density in the Solid Mechanics interface and is equal to the density in a material when it is used in the domains where the Solid Mechanics interface is active. If you define the density in the Solid Mechanics interface using another value, solid.rho represents that value and not the density of the material. If you use the density from the material everywhere in the model, solid.rho and material.rho are identical.

Opening the Material Browser

1 Open or create a model file.

2 From the View menu choose Material Browser or right-click the Materials node and choose Open Material Browser.

The Material Browser opens by default in the same position as the Settings window.

3 Under Material Selection, search or browse for materials.

- Enter a Search term to find a specific material by name, UNS number (Material Library materials only), or DIN number (Material Library materials only). If the search is successful, a list of filtered databases containing that material displays under Material Selection.

When using the Material Browser, the words window and page are interchangeable. For simplicity, the instructions refer only to the Material

Browser.Note

To clear the search field and browse, delete the search term and click Search to reload all the databases.

Tip


280 | C H A P T E

- Click to open each database and browse for a specific material by class (for example, in the Material Library) or physics module (for example, MEMS Materials).

4 When the material is located, right-click to Add Material to Model.

A node with the material name is added to the Model Builder and the Material page opens.

Using Material Properties

Always review the material properties to confirm they are applicable for the model. For example, Air provides temperature-dependent properties that are valid at pressures around 1 atm.Important

For detailed instructions, see Adding Predefined Materials and Material Properties Reference in the COMSOL Multiphysics User’s Guide.

See Also


L i q u i d s and Ga s e s Ma t e r i a l Da t a b a s e

In this section:

• Liquids and Gases Materials

• References for the Liquids and Gases Material Database

Liquids and Gases Materials

The Liquids and Gases materials database contains thermal and fluid dynamic properties for a set of common liquids. All properties are given as functions of temperature and at atmospheric pressure, except the density, which for gases is also a function of the local pressure. The database also contains surface and interface tensions

L I Q U I D S A N D G A S E S M A T E R I A L D A T A B A S E | 281

282 | C H A P T E

for a selected set of liquid/gas and liquid/liquid systems. All functions are based on data collected from scientific publications.

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

TABLE 8-1: LIQUIDS AND GASES MATERIALS

GROUP MATERIAL

Gases References 1, 2, 7, and 8 Air

Nitrogen

Oxygen

Carbon dioxide

Hydrogen

Helium

Steam

Propane

Ethanol vapor

Diethyl ether vapor

Freon12 vapor

SiF4

Liquids References 2, 3, 4, 5, 6, 7, 9, and 10

Engine oil

Ethanol

Diethyl ether

Ethylene glycol

Gasoline

Glycerol

Heptane

Mercury

Toluene

Transformer oil

Water

The MEMS Materials database has data for these materials and a default appearance for 3D models is applied to each material as indicated.

3D


References for the Liquids and Gases Material Database

1. ASHRAE Handbook of Fundamentals, American Society of Heating, Refrigerating and Air Conditioning Engineers, 1993.

2. E. R. G. Eckert and M. Drake, Jr., Analysis of Heat and Mass Transfer, Hemisphere Publishing, 1987.

3. H. Kashiwagi, T. Hashimoto, Y. Tanaka, H. Kubota, and T. Makita, “Thermal Conductivity and Density of Toluene in the Temperature Range 273-373K at Pressures up to 250 MPa,” Int. J. Thermophys., vol. 3, no. 3, pp. 201–215, 1982.

4. C. A. Nieto de Castro, S.F.Y. Li, A. Nagashima, R.D. Trengove, and W.A. Wakeham, “Standard Reference Data for the Thermal Conductivity of Liquids,” J. Phys. Chem. Ref. Data, vol. 15, no. 3, pp. 1073–1086, 1986.

See Working on the Material Page in the COMSOL Multiphysics User’s Guide for more information about customizing the material’s appearance in the Graphics window.

TABLE 8-2: MATERIALS 3D MODEL DEFAULT APPEARANCE SETTINGS

MATERIAL DEFAULT FAMILY

DEFAULT LIGHTING MODEL

CUSTOM DEFAULT SETTINGS

NO

RM

AL

VE

CT

OR

N

OIS

E S

CA

LE

NO

RM

AL

VE

CT

OR

N

OIS

E F

RE

QU

EN

CY

DIF

FU

SE

AN

D A

MB

IEN

T

CO

LO

R O

PA

CIT

Y

SP

EC

UL

AR

EX

PO

NE

NT

RE

FL

EC

TA

NC

E A

T

NO

RM

AL

IN

CID

EN

CE

SU

RF

AC

E R

OU

GH

NE

SS

All gases Air Simple 0.08 3 0.1 - - -

All liquids (except Engine oil, Mercury, and Transformer oil)

Water Cook-Torrance 0.2 0.2 0.2 - 0.7 0.05

Engine oil Custom Cook-Torrance 0.2 0.2 0.2 - 0.7 0.05

Mercury Custom Cook-Torrance 0 1 1 - 0.9 0.1

Transformer oil Plastic Blinn-Phong 0 1 1 64 - -

See Also

L I Q U I D S A N D G A S E S M A T E R I A L D A T A B A S E | 283

284 | C H A P T E

5. B.E. Poling, J.M. Prausnitz, and J.P. O’Connell, The Properties of Gases and Liquids, 5th ed., McGraw-Hill, 2001.

6. C.F. Spencer and B.A. Adler, “A Critical Review of Equations for Predicting Saturated Liquid Density,” J. Chem. Eng. Data, vol. 23, no. 1, pp. 82–88, 1978.

7. N.B.Vargnaftik, Tables of Thermophysical Properties of Liquids and Gases, 2nd ed., Hemisphere Publishing, 1975.

8. R.C.Weast (editor), CRC Handbook of Chemistry and Physics, 69th ed., CRC Press, 1988.

9. M. Zabransky and V. Ruzicka, Jr., “Heat Capacity of Liquid n-Heptane Converted to the International Temperature Scale of 1990,” Phys. Chem. Ref. Data, vol. 23, no. 1, pp. 55–61, 1994.

10. M. Zabransky, V. Ruzicka, Jr., and E.S. Domalski, “Heat Capacity of Liquids: Critical Review and Recommended Values. Supplement I,” J. Phys. Chem. Ref. Data, vol. 30, no. 5, pp. 1199–1397, 2002.


9

G l o s s a r y

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 User’s Guide. For references to more information about a term, see the Index in this or other manuals.

285

286 | C H A P T E

G l o s s a r y o f T e rm sanisotropy The condition of exhibiting properties with different values when measured in different directions.

bioheat equation 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.

blackbody 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.

conduction 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.

advection 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.

convection The term convection is used for the heat dissipation from a solid surface to a fluid, where the heat transfer coefficient and the temperature difference across a fictitious film describes the flux.

emissivity 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.

heat capacity See specific heat.

highly conductive layer 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. The General Heat Transfer physics interface supports heat transfer in highly conductive layers.

irradiation The total radiation that arrives at a surface.

R 9 : G L O S S A R Y

Navier-Stokes equations 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.

opaque material An opaque body does not transmit any radiative heat flux, that is, the surface of an opaque body has a transmissivity equal to 0.

radiation 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.

participating media A media that can absorb, emit, and scatter thermal radiation.

radiosity The total radiation that leaves a surface, that is, both the emitted and the reflected radiation.

specific heat 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 capacity See specific heat.

thin conductive shell An physics interface for modeling heat transfer in a thin shell. “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.

transparent material A transparent body transmits radiative heat flux, that is, the surface of a transparent body has a transmissivity greater than 0.

thermal conductivity 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.

G L O S S A R Y O F TE R M S | 287

288 | C H A P T E
R 9 : G L O S S A R Y

I n d e x

1D and 2D models

out-of-plane heat transfer 64, 74, 131

3D models

infinite elements and 52

thin conductive shells 124

A absorption coefficients 147

acceleration of gravity 55

advanced settings 18

AKN model 219

arterial blood temperature 116

axisymmetric geometries 75, 157, 166

azimuthal sectors 157

B bioheat (node) 116

bioheat transfer interface 114

theory 66

biological tissue 115

black walls 99, 149

blackbody radiation 142

blackbody radiation intensity, definition

158

blood, bioheat properties 115

boundary conditions

bioheat interface 117

heat equation, and 41

heat transfer coefficients, and 53

heat transfer interfaces 88

radiation groups 166

single-phase flow interfaces 180

surface-to-surface radiation, theory

164

thin shells 127

boundary heat source (node) 94

boundary heat source variable 41

boundary stress (node) 198

built-in materials database 253

bulk velocity 55

buoyancy force 55

C Carreau model 238

Cartesian coordinates 49

cell Reynolds number 193

CFL number, pseudo time stepping, and

172

change effective thickness (node) 129

change thickness (node)

out-of-plane heat transfer 112

thin conductive shell interface 127

characteristic length 55

coefficient of volumetric thermal expan-

sion 55

conductive heat flux variable 37

conjugate heat transfer

laminar flow interface 231

turbulent flow interfaces 233

conjugate heat transfer interface

theory 244

consistent stabilization settings 19

constraint settings 19

contacting COMSOL 21

continuity on interior boundary (node)


convection, natural and forced 54

convective cooling (node) 101

convective cooling theory 54

convective heat flux variable 37, 40

convective out-of-plane heat flux varia-

ble 38

crosswind diffusion, consistent stabiliza-

tion method 46

curves, fan 207

D del operators 62

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density, blood 116

dimensionless distance to cell center var-

iable 221

direct area integration, axisymmetric ge-

ometry and 157

direct area integration, radiation settings

74

Dirichlet condition 207

discrete ordinates method (DOM) 160

discretization settings 18

dispersivities, porous media 121

documentation, finding 20

domain heat source variable 40

domain material 256

E eddy viscosity 213

edge heat flux (node) 106

edge heat source (node) 129

edge surface-to-ambient radiation

(node) 107

edge temperature (node) 106

edges

heat flux 106

temperature 106

elastic contribution to entropy 79

emailing COMSOL 21

emission, radiation and 158

equation view 18

evaluating view factors 156

exit length 194

expanding sections 18

External Radiation Source 143

F fan (inlet and outlet boundary condi-

tions) 188

fan (node)


fan curves

fan boundary condition 203

inlet boundary condition 189

theory 207

Favre average 213, 246

flow continuity (node) 201

fluid (node) 236

fluid flow

selecting interfaces 226

turbulent flow theory 211

fluid properties (node) 173

Fourier’s law 32

G Galerkin constraints 89

general stress (boundary stress condi-

tion) 198

geometry, working with 19

Grashof number 55

gravity 55

gray walls 98, 148

graybody radiation 142, 155

grill (inlet and outlet boundary condi-

tions) 189

grouping boundaries 166

guidelines, solving surface-to-surface ra-

diation problems 165

H heat continuity (node) 94

heat equation, highly conductive layers

and 62

heat flux (node) 91

heat flux, theory 33

heat source (node)

heat transfer in porous media 122



heat sources

defining as total power 84, 94

edges, thin shells 129

highly conductive layers 105

line and point 100

point, thin shells 130

heat transfer coefficients 56

out-of-plane heat transfer interfaces

110

theory 54

heat transfer in fluids (node) 80

extended features 120

heat transfer in participating media inter-

face 151

theory 154

heat transfer in porous media interface

118

theory 67

heat transfer in solids (node) 77


selecting 226

theory 30, 154

hemicubes, axisymmetric geometry and

157

hemicubes, radiation settings 74

hide button 18

highly conductive layer (node) 103

highly conductive layers, defined 61

I incident intensity (node)

heat transfer interfaces 99, 149

inconsistent stabilization settings 19

infinite elements (node) 86

inflow heat flux (node) 92

initial values (node)

heat transfer interface 88

non-isothermal flow/conjugate heat

transfer interfaces 242

radiation in participating media inter-

face 150

single-phase, laminar flow interface 176


inlet (boundary stress condition) 200

inlet (node) 186

insulation/continuity (node) 128

Interior wall (node)

single-phase flow, turbulent flow inter-

faces 184

Internet resources 19

isotropic diffusion, inconsistent stabiliza-

tion methods 48

K Kays-Crawford models 248

k-epsilon turbulence model 214

knowledge base, COMSOL 21

L laminar flow

conjugate heat transfer interface 231

non-isothermal flow interface 228


turbulence model 171

turbulent flow k-epsilon 177

turbulent flow, low re k-epsilon 178

laminar inflow (inlet boundary condition)

190

laminar outflow (outlet boundary condi-

tion) 194

layer heat source (node) 105

leaking wall, wall boundary condition 183

Legendre coefficients 147

line heat source (node) 100

line heat source variable 41

liquids and gases materials 259

local CFL number 172, 222

low Reynolds number

k-epsilon turbulence theory 219

neglect inertial term 229

M Mach number

pressure work, and 242

manual scaling (node) 87

mapped infinite elements 49

Material Browser

opening 257

Material Library 253

materials

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databases 253

domain, default 256

liquids and gases 259

properties, evaluating and plotting 256

mean effective thermal conductivity 78

mechanisms of heat transfer 30

metabolic heat source 116

model builder settings 18

Model Library examples


consistent stabilization 47

convective cooling 101

heat transfer in fluids 81

heat transfer in solids 77

heat transfer in thin shells interface

124

heat transfer with surface-to-surface

radiation 74

highly conductive layers 103


non-isothermal flow interface 228, 231

out-of-plane convective cooling 109

radiation in particpating media 145

surface-to-ambient radiation 93

thermodynamics 78

translational motion 79

turbulent flow, k-epsilon interface 177,

233

Model Library, accessing in COMSOL 20

moving wall (wall functions), boundary

condition 184

moving wall, wall boundary condition

183, 185

MPH-files 20

mutual irradiation 162

N nabla operators 62

natural and forced convection 54

Neumann condition 219

no slip, interior wall boundary condition

185

no slip, wall boundary condition 182

no viscous stress (outlet boundary con-

dition) 194

non-isothermal flow interface

laminar flow 228

theory 244

turbulent flow 233

non-Newtonian power law and Carreau

model 238

normal conductive heat flux variable 39

normal convective heat flux variable 39

normal stress (boundary condition) 188

normal stress, normal flow (boundary

stress condition) 199

normal total energy flux variable 40

normal translational heat flux variable 40

Nusselt number 54

O opaque (node)


surface-to-surface radiation interface

140

open boundary (boundary stress condi-

tion) 199

open boundary (node)

heat transfer 93


opening the Model Library 20

outflow (node) 90

outlet (boundary stress condition) 200

outlet (node) 192

out-of-plane convective cooling (node)

109

out-of-plane heat flux (node) 111

out-of-plane heat transfer

change thickness 112

general theory 64

thin shells theory 131

out-of-plane inward heat flux variable 39

out-of-plane radiation (node) 110

override and contribution settings 18

P pair selection 19

pair thin thermally resistive layer (node)

95

parameters, infinite elements 50

participating media, radiative heat trans-

fer 157

Pennes’ approximation 66

perfusion rate, blood 116

periodic flow condition (node) 200

periodic heat condition (node) 94

physics interface settings windows 18

plotting, material properties 256

point heat flux (node) 106

point heat source (node)



point heat source variable 41

point surface-to-ambient radiation

(node) 107

point temperature (node) 106

points

heat flux 106

temperature 106

porous matrix (node) 119

power law, non-Newtonian 238

Prandtl number 54, 248

prescribed radiosity (node) 141

pressure (outlet boundary condition)

193

pressure point constraint (node) 201

pressure work (node)




pressure, no viscous stress (inlet and

outlet boundary conditions) 187

pseudo time stepping

advanced settings 172


pumps, lumped curves and 207

R radiation

axisymmetric geometries, and 75, 157,

166

participating media 157

radiation (node) 128

radiation group (node) 142

radiation groups 166

radiation in participating media (node)



face 147

radiation in participating media interface

145

theory 154

radiation, out-of-plane 110

radiative heat flux variable 40

radiative heat, theory 44

radiative out-of-plane heat flux variable

39

radiative transfer equation 158

radiosity expressions 142

radiosity method 154

raditation intensity, for blackbody 158

RANS

theory, single-phase flow 212

ratio of specific heats 80

Rayleigh number 54

reradiating surface (node) 140

Reynolds number 54–55

extended Kays-Crawford 249

low, turbulence theory 219


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Reynolds stress tensor 213, 216

Reynolds-averaged Navier-Stokes. See

RANS.

Rodriguez formula 159

S scattering coefficients 147

scattering, radiation and 158

sectors, azimuthal 157

selecting

conjugate heat transfer interfaces 226


non-isothermal flow interfaces 226

shell thickness 125

shells, conductive 124, 131

show button 18

single-phase flow


single-phase flow interface

boundary conditions 180

laminar flow 170

turbulent flow low re k-e 178

sliding wall (wall functions), boundary

condition 183

sliding wall, wall boundary condition 182

slip, wall boundary condition 182, 185

solving surface-to-surface radiation

problems 165

source terms, bioheat 116

specific heat capacity, definition 32

specific heat, blood 116

spf.sr variable 174

stabilization settings 19

static pressure curves 189, 203

strain-rate tensors 83

streamline diffusion, consistent stabiliza-

tion methods 46

surface-to-ambient radiation (node) 93

edges and points 107

surface-to-surface radiation (node) 138


136

theory 162

swirl flow theory 216

symmetry (node)



T technical support, COMSOL 21

temperature (node) 89

tensors

Reynolds stress 216

strain-rate 83

theory


conjugate heat transfer interface 244

heat equation definition 31

heat transfer coefficients 54

heat transfer in participating media in-

terface 154

heat transfer in porous media interface

67


non-isothermal flow interface 244

out-of-plane heat transfer 64


face 154

radiative heat transfer interfaces 154


162


turbulent flow k-e interface 211

turbulent flow low re k-e interface 211

thermal conductivity components, thin

shells 132

thermal conductivity, mean effective 78

thermal dispersion (node) 121

thermal expansivity 55

thermal insulation (node) 90

thin conductive layer (node) 125

thin conductive layers, definition 61


theory 131

thin thermally resistive layer (node) 96

total energy flux variable 38

total heat flux 92

total heat flux variable 36

total normal heat flux variable 39

total power 84, 94

traction boundary conditions 198

translational heat flux variable 38

translational motion (node) 78

turbulence models

k-epsilon 214

single-phase flow 171

turbulent compressible flow 213

turbulent conjugate heat transfer inter-

faces

theory 246

turbulent flow k-e interface 177, 233

theory 211

turbulent flow low re k-e interface 178,

233

theory 211

turbulent heat flux variable 37

turbulent kinetic energy theory

k-epsilon model 215

RANS 214

turbulent length scale 222

turbulent non-isothermal flow interfaces

theory 246

turbulent Prandtl number 248

typographical conventions 21

U unbounded domains, modeling 49

user community, COMSOL 21

V variables

dimensionless distance to cell center

221

for material properties 256

shear rate magnitude 174

velocity (inlet and outlet boundary con-

ditions) 187

view factors 156

viscous force 55

viscous heating (node)




volume force (node) 175

W wall (node)

heat transfer interface 98, 148



single-phase flow, laminar flow inter-

faces 181, 184

single-phase flow, turbulent flow inter-

faces 181

wall distance initialization study step 220

wall functions, turbulent flow 217

wall functions, wall boundary condition

183

wall types 148

weak constraint settings 19

web sites, COMSOL 21

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Heat Transfer Module - Montana Tech High Performance Cluster

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