VERSION 4.3b

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 19982013 COMSOL

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

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

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Version: May 2013 COMSOL 4.3b

N T E N T S | i

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 2

Why Heat Transfer is Important to Modeling . . . . . . . . . . . . 2

How the Heat Transfer Module Improves Your Modeling. . . . . . . . 2

The Heat Transfer Module Physics Guide. . . . . . . . . . . . . . 3

Where Do I Access the Documentation and Model Library? . . . . . . 11

Overview of the Users Guide 14

C h aC O

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 User Interfaces 18

What is Heat Transfer? . . . . . . . . . . . . . . . . . . . . 18

The Heat Equation . . . . . . . . . . . . . . . . . . . . . . 19

A Note on Heat Flux . . . . . . . . . . . . . . . . . . . . . 21

Heat Flux and Heat Source Variables . . . . . . . . . . . . . . . 23

About the Boundary Conditions for the Heat Transfer User

Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . 34

Radiative Heat Transfer in Transparent Media . . . . . . . . . . . . 36

Consistent and Inconsistent Stabilization Methods for the Heat

Transfer User Interfaces . . . . . . . . . . . . . . . . . . . 38

Moist Air Theory. . . . . . . . . . . . . . . . . . . . . . . 40

About Heat Transfer with Phase Change . . . . . . . . . . . . . . 46

Theory for the Thermal Contact Feature. . . . . . . . . . . . . . 48

About the Heat Transfer Coefficients 53

Heat Transfer Coefficient Theory . . . . . . . . . . . . . . . . 54

Nature of the Flowthe Grashof Number . . . . . . . . . . . . . 55

Heat Transfer Coefficients External Natural Convection . . . . . . . 56

Heat Transfer Coefficients Internal Natural Convection . . . . . . . 58

Heat Transfer Coefficients External Forced Convection . . . . . . . 59

ii | C O N T E N T S

Heat Transfer Coefficients Internal Forced Convection . . . . . . . 59

About Highly Conductive Layers 61

Theory of Out-of-Plane Heat Transfer 63

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

Activating Out-of-Plane Heat Transfer and Thickness . . . . . . . . . 64

Theory for the Bioheat Transfer User Interface 65

Theory for the Heat Transfer in Porous Media User

Interface 66

C h aAbout Handling Frames in Heat Transfer 68

Frame Physics Feature Nodes and Definitions . . . . . . . . . . . . 68

Conversion Between Material and Spatial Frames . . . . . . . . . . 72

References for the Heat Transfer User Interfaces 75

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

About the Heat Transfer Interfaces 78

The Heat Transfer Interface 81

Domain, Boundary, Edge, Point, and Pair Nodes for the Heat

Transfer User Interfaces . . . . . . . . . . . . . . . . . . . 84

Heat Transfer in Solids . . . . . . . . . . . . . . . . . . . . . 86

Translational Motion . . . . . . . . . . . . . . . . . . . . . 88

Heat Transfer in Fluids . . . . . . . . . . . . . . . . . . . . . 89

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . . 93

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

Heat Transfer with Phase Change . . . . . . . . . . . . . . . . 96

Thermal Insulation . . . . . . . . . . . . . . . . . . . . . . 99

Temperature . . . . . . . . . . . . . . . . . . . . . . . . 99

Outflow . . . . . . . . . . . . . . . . . . . . . . . . . 100

Symmetry . . . . . . . . . . . . . . . . . . . . . . . . 101

N T E N T S | iii

Heat Flux. . . . . . . . . . . . . . . . . . . . . . . . . 101

Surface-to-Ambient Radiation . . . . . . . . . . . . . . . . . 103

Periodic Heat Condition . . . . . . . . . . . . . . . . . . . 104

Boundary Heat Source. . . . . . . . . . . . . . . . . . . . 104

Continuity . . . . . . . . . . . . . . . . . . . . . . . . 106

Thin Thermally Resistive Layer. . . . . . . . . . . . . . . . . 106

Thermal Contact . . . . . . . . . . . . . . . . . . . . . . 108

Line Heat Source . . . . . . . . . . . . . . . . . . . . . . 111

Point Heat Source . . . . . . . . . . . . . . . . . . . . . 112

Pressure Work . . . . . . . . . . . . . . . . . . . . . . 112

Viscous Heating . . . . . . . . . . . . . . . . . . . . . . 113

Inflow Heat Flux . . . . . . . . . . . . . . . . . . . . . . 114

Open Boundary . . . . . . . . . . . . . . . . . . . . . . 115C O

Convective Heat Flux . . . . . . . . . . . . . . . . . . . . 116

Highly Conductive Layer Nodes 118

Highly Conductive Layer . . . . . . . . . . . . . . . . . . . 118

Layer Heat Source . . . . . . . . . . . . . . . . . . . . . 120

Edge Heat Flux . . . . . . . . . . . . . . . . . . . . . . 121

Point Heat Flux . . . . . . . . . . . . . . . . . . . . . . 122

Temperature . . . . . . . . . . . . . . . . . . . . . . . 123

Point Temperature . . . . . . . . . . . . . . . . . . . . . 124

Edge Surface-to-Ambient Radiation . . . . . . . . . . . . . . . 125

Point Surface-to-Ambient Radiation . . . . . . . . . . . . . . . 125

Out-of-Plane Heat Transfer Nodes 127

Out-of-Plane Convective Heat Flux . . . . . . . . . . . . . . . 127

Out-of-Plane Radiation . . . . . . . . . . . . . . . . . . . 129

Out-of-Plane Heat Flux . . . . . . . . . . . . . . . . . . . 130

Change Thickness . . . . . . . . . . . . . . . . . . . . . 130

The Bioheat Transfer Interface 132

Biological Tissue . . . . . . . . . . . . . . . . . . . . . . 133

Bioheat . . . . . . . . . . . . . . . . . . . . . . . . . 134

The Heat Transfer in Porous Media Interface 136

Domain, Boundary, Edge, Point, and Pair Nodes for the Heat

Transfer in Porous Media User Interface . . . . . . . . . . . . 137

iv | C O N T E N T S

Heat Transfer in Porous Media. . . . . . . . . . . . . . . . . 137

Thermal Dispersion . . . . . . . . . . . . . . . . . . . . . 143

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 User Interface 146

Boundary, Edge, Point, and Pair Nodes for the Heat Transfer in

Thin Shells User Interface . . . . . . . . . . . . . . . . . . 148

Heat Flux. . . . . . . . . . . . . . . . . . . . . . . . . 149

Thin Conductive Layer. . . . . . . . . . . . . . . . . . . . 150

Heat Source. . . . . . . . . . . . . . . . . . . . . . . . 151

C h aInitial Values. . . . . . . . . . . . . . . . . . . . . . . . 152

Change Thickness . . . . . . . . . . . . . . . . . . . . . 152

Surface-to-Ambient Radiation . . . . . . . . . . . . . . . . . 153

Insulation/Continuity . . . . . . . . . . . . . . . . . . . . 153

Change Effective Thickness . . . . . . . . . . . . . . . . . . 154

Edge Heat Source . . . . . . . . . . . . . . . . . . . . . 154

Point Heat Source . . . . . . . . . . . . . . . . . . . . . 155

Theory for the Heat Transfer in Thin Shells User Interface

156

About Heat Transfer in Thin Shells . . . . . . . . . . . . . . . 156

Heat Transfer Equation in Thin Conductive Shell . . . . . . . . . . 156

Thermal Conductivity Tensor Components . . . . . . . . . . . . 157

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

The Radiation Branch Versions of the Heat Transfer User

Interface 160

The Heat Transfer with Surface-to-Surface Radiation User Interface . . 160

The Heat Transfer with Radiation in Participating Media User

Interface . . . . . . . . . . . . . . . . . . . . . . . . 161

Domain, Boundary, Edge, Point, and Pair Nodes for the Radiation

Branch Versions of the Heat Transfer User Interface . . . . . . . . 161

N T E N T S | v

The Surface-To-Surface Radiation User Interface 164

Domain, Boundary, Edge, Point, and Pair Nodes for the

Surface-to-Surface Radiation User Interface . . . . . . . . . . . 167

Surface-to-Surface Radiation (Boundary Condition) . . . . . . . . . 168

Opaque . . . . . . . . . . . . . . . . . . . . . . . . . 172

Diffuse Mirror . . . . . . . . . . . . . . . . . . . . . . . 173

Prescribed Radiosity . . . . . . . . . . . . . . . . . . . . 174

Radiation Group . . . . . . . . . . . . . . . . . . . . . . 177

External Radiation Source . . . . . . . . . . . . . . . . . . 178

Theory for the Surface-to-Surface Radiation User Interface 182

Wavelength Dependence of Surface Emissivity and Absorptivity . . . . 182

The Radiosity Method for Diffuse-Gray Surfaces . . . . . . . . . . 188C O

The Radiosity Method for Diffuse-Spectral Surfaces . . . . . . . . . 190

View Factor Evaluation . . . . . . . . . . . . . . . . . . . 192

About Surface-to-Surface Radiation . . . . . . . . . . . . . . . 194

Guidelines for Solving Surface-to-Surface Radiation Problems . . . . . 196

Radiation Group Boundaries . . . . . . . . . . . . . . . . . 197

The Radiation in Participating Media User Interface 199

Domain, Boundary, Edge, Point, and Pair Nodes for the Radiation

in Participating Media User Interface . . . . . . . . . . . . . . 201

Radiation in Participating Media . . . . . . . . . . . . . . . . 202

Opaque Surface . . . . . . . . . . . . . . . . . . . . . . 203

Incident Intensity . . . . . . . . . . . . . . . . . . . . . . 205

Continuity on Interior Boundary . . . . . . . . . . . . . . . . 206

Theory for the Radiation in Participating Media User

Interface 207

Radiation and Participating Media Interactions . . . . . . . . . . . 207

Radiative Transfer Equation . . . . . . . . . . . . . . . . . . 208

Boundary Condition for the Transfer Equation. . . . . . . . . . . 209

Heat Transfer Equation in Participating Media . . . . . . . . . . . 210

Discrete Ordinates Method . . . . . . . . . . . . . . . . . . 211

Discrete Ordinates Method Implementation in 2D . . . . . . . . . 212

vi | C O N T E N T S

References for the Radiation User Interfaces 214

C h a p t e 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

The Laminar Flow and Turbulent Flow User Interfaces 216

The Laminar Flow User Interface. . . . . . . . . . . . . . . . 216

The Turbulent Flow, k- User Interface . . . . . . . . . . . . . 219The Turbulent Flow, Low Re k- User Interface . . . . . . . . . . 221Domain, Boundary, Pair, and Point Nodes for Single-Phase Flow . . . . 223

Fluid Properties . . . . . . . . . . . . . . . . . . . . . . 224

Volume Force . . . . . . . . . . . . . . . . . . . . . . . 226Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 227

Wall . . . . . . . . . . . . . . . . . . . . . . . . . . 228

Inlet . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

Outlet . . . . . . . . . . . . . . . . . . . . . . . . . . 233

Symmetry . . . . . . . . . . . . . . . . . . . . . . . . 236

Open Boundary . . . . . . . . . . . . . . . . . . . . . . 237

Boundary Stress . . . . . . . . . . . . . . . . . . . . . . 237

Periodic Flow Condition . . . . . . . . . . . . . . . . . . . 239

Fan . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

Interior Fan . . . . . . . . . . . . . . . . . . . . . . . . 242

Interior Wall . . . . . . . . . . . . . . . . . . . . . . . 244

Grille . . . . . . . . . . . . . . . . . . . . . . . . . . 245

Flow Continuity . . . . . . . . . . . . . . . . . . . . . . 246

Pressure Point Constraint . . . . . . . . . . . . . . . . . . 246

More Boundary Condition Settings for the Turbulent Flow User

Interfaces . . . . . . . . . . . . . . . . . . . . . . . . 247

Theory for the Laminar Flow User Interface 250

Theory for the Inlet Boundary Condition . . . . . . . . . . . . 250

Additional Theory for the Outlet Boundary Condition . . . . . . . 251

Theory for the Fan Defined on an Interior Boundary . . . . . . . . 253

Theory for the Fan and Grille Boundary Conditions . . . . . . . . 254

Non-Newtonian Flow: The Power Law and the Carreau Model . . . . 257

T E N T S | vii

Theory for the Turbulent Flow User Interfaces 260

Turbulence Modeling . . . . . . . . . . . . . . . . . . . . 260

The k-Turbulence Model . . . . . . . . . . . . . . . . . . 264The Low Reynolds Number k- Turbulence Model . . . . . . . . . 270Inlet Values for the Turbulence Length Scale and Turbulent Intensity . . 273

Theory for the Pressure, No Viscous Stress Boundary Condition . . . 274

Solvers for Turbulent Flow . . . . . . . . . . . . . . . . . . 274

Pseudo Time Stepping for Turbulent Flow Models . . . . . . . . . 275

References for the Single-Phase Flow, User Interfaces 276

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

About the Conjugate Heat Transfer User Interfaces 280

Selecting the Right User Interface . . . . . . . . . . . . . . . 280

The Non-Isothermal Flow Options . . . . . . . . . . . . . . . 282

Conjugate Heat Transfer Options . . . . . . . . . . . . . . . 283

The Non-Isothermal Flow and Conjugate Heat Transfer,

Laminar Flow and Turbulent Flow User Interfaces 285

The Non-Isothermal Flow, Laminar Flow User Interface . . . . . . . 285

The Conjugate Heat Transfer, Laminar Flow User Interface . . . . . . 289

The Turbulent Flow, k- and Turbulent Flow Low Re k-User Interfaces . . . . . . . . . . . . . . . . . . . . . . . . 289

Domain, Boundary, Edge, Point, and Pair Nodes Settings for the

NITF User Interfaces. . . . . . . . . . . . . . . . . . . . 292

Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . 293

Wall. . . . . . . . . . . . . . . . . . . . . . . . . . . 300

Interior Wall . . . . . . . . . . . . . . . . . . . . . . . 302

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 303

Open Boundary . . . . . . . . . . . . . . . . . . . . . . 303

Pressure Work . . . . . . . . . . . . . . . . . . . . . . 304

Viscous Heating . . . . . . . . . . . . . . . . . . . . . . 305

Symmetry, Heat . . . . . . . . . . . . . . . . . . . . . . 305

Symmetry, Flow . . . . . . . . . . . . . . . . . . . . . . 306

viii | C O N T E N T S

Theory for the Non-Isothermal Flow and Conjugate Heat

Transfer User Interfaces 308

Turbulent Non-Isothermal Flow Theory . . . . . . . . . . . . . 310

References for the Non-Isothermal Flow and Conjugate Heat

Transfer User Interfaces 315

C h a p t e r 8 : G l o s s a r y

Glossary of Terms 318

1

1I 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 Users Guide

2 | C H A P T E R 1 : I N T 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

The Heat Transfer Module Physics Guide

W

TMorethco

Hlintr

TdCefex

H

TcaO D U C T I O N

Where Do I Access the Documentation and Model Library?

hy Heat Transfer is Important to Modeling

he Heat Transfer Module is an optional package that extends the COMSOL ultiphysics modeling environment with customized user interfaces and functionality

ptimized for the analysis of heat transfer. It is developed for a wide audience including searchers, developers, teachers, and students. To assist users at all levels of expertise, is module comes with a library of ready-to-run example models that appear in the mpanion Heat Transfer Module Model Library.

eat transfer is involved in almost every kind of physical process, and can in fact be the miting factor for many processes. Therefore, its study is of vital importance, and the eed for powerful heat transfer analysis tools is virtually universal. Furthermore, heat ansfer often appears together with, or as a result of, other physical phenomena.

he modeling of heat transfer effects has become increasingly important in product esign including areas such as electronics, automotive, and medical industries. omputer simulation has allowed engineers and researchers to optimize process ficiency and explore new designs, while at the same time reducing costly perimental trials.

ow the Heat Transfer Module Improves Your Modeling

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

Overview of the Physics and Building a COMSOL Model in the COMSOL Multiphysics Reference Manual

L E | 3

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

This book introduces the basic modeling process. The different physics interfaces are described and the modeling strategy for various cases is discussed. These sections cover digucogutrflo

Afuhemm

MtoauapD

T

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

fferent combinations of conductive, convective, and radiative heat transfer. This ide also reviews special modeling techniques for highly conductive layers, thin nductive shells, participating media, and out-of-plane heat transfer. Throughout the ide the topics and examples increase in complexity by combining several heat

ansfer mechanisms and also by coupling these to physics interfaces describing fluid wconjugate heat transfer.

nother source of information is the Heat Transfer Module Model Library, a set of lly-documented models that is divided into broadly defined application areas where at transfer plays an important roleelectronics and power systems, processing and anufacturing, and medical technologyand includes tutorial and verification odels.

ost of the models involve multiple heat transfer mechanisms and are often coupled other physical phenomena, for example, fluid dynamics or electromagnetics. The thors developed several state-of-the art examples by reproducing models that have peared in international scientific journals. See Where Do I Access the ocumentation and Model Library?.

he Heat Transfer Module Physics Guide

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

4 | C H A P T E R 1 : I N T 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 difference is that Fluid is the

PHYSI

F

Lami

T

Tk

LamiO D U C T I O N

default domain node for the Non-Isothermal Flow interfaces.

In the COMSOL Multiphysics Reference Manual:

Studies and the Study Nodes

The Physics User Interfaces

For a list of all the interfaces included with the COMSOL Multiphysics

basic license, see Physics Guide.

CS USER INTERFACE ICON TAG SPACE DIMENSION

AVAILABLE PRESET STUDY TYPE

luid Flow

Single-Phase Flow

nar Flow* spf 3D, 2D, 2D axisymmetric

stationary; time dependent

Turbulent Flow

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

stationary; time dependent

urbulent Flow, Low Re -

spf 3D, 2D, 2D axisymmetric

stationary with initialization; transient with initialization

Non-Isothermal Flow

nar Flow nitf 3D, 2D, 2D axisymmetric

stationary; time dependent

L E | 5

Turbulent Flow

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

stationary; time dependent

Turbulent Flow, Low Re k-

nitf 3D, 2D, 2D axisymmetric

stationary with initialization; transient

Heat

Heat

Heat

Biohe

Heat

C

Lamin

T

T

Tk

R

HSR

PHYSICS USER INTERFACE ICON TAG SPACE DIMENSION

AVAILABLE PRESET STUDY TYPEA B O U T T H E H E A T TR A N S F E R M O D U

with initialization

Heat Transfer

Transfer in Solids* ht all dimensions stationary; time dependent

Transfer in Fluids* ht all dimensions stationary; time dependent

Transfer in Porous Media ht all dimensions stationary; time dependent

at Transfer ht all dimensions stationary; time dependent

Transfer in Thin Shells htsh 3D stationary; time dependent

onjugate Heat Transfer

ar Flow nitf 3D, 2D, 2D axisymmetric

stationary; time dependent

urbulent Flow

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

stationary; time dependent

urbulent Flow, Low Re -

nitf 3D, 2D, 2D axisymmetric

stationary with initialization; transient with initialization

adiation

eat Transfer with urface-to-Surface adiation

ht all dimensions stationary; time dependent

6 | C H A P T E R 1 : I N T R

T

Tm

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 M

rpm 3D, 2D stationary; time

Jo

* Thiadde

PHYSICS USER INTERFACE ICON TAG SPACE DIMENSION

AVAILABLE PRESET STUDY TYPEO D U C T I O N

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

able 1-1 lists the Preset Studies available for the interfaces most relevant to this odule.

edia dependent

Electromagnetic Heating

ule Heating* jh all dimensions stationary; time dependent

s is an enhanced interface, which is included with the base COMSOL package but has d functionality for this module.

Studies and Solvers in the COMSOL Multiphysics Reference Manual

L E | 7

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

PHYSICS INTERFACE TAG DEPENDENT VARIABLES PRESET STUDIES*

ITIA

LIZ

AT

ION

IAL

IZA

TIO

N

F

L

T

T

F

L

T

T

H

H

H

HM

B

H

H

L

T

T

H

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

ST

AT

ION

AR

Y

TIM

E D

EP

EN

DE

NT

ST

AT

ION

AR

Y W

ITH

IN

TR

AN

SIE

NT

WIT

H I

NIT

LUID FLOW>SINGLE-PHASE FLOW

aminar Flow spf u, p urbulent Flow, k- spf u, p, k, ep urbulent Flow, Low Re k- spf u, p, k, ep, G LUID FLOW>NON-ISOTHERMAL FLOW

aminar Flow nitf u, p, T urbulent Flow, k- nitf u, p, k, ep, T urbulent Flow, Low Re k- nitf u, p, k, ep, G, T EAT TRANSFER

eat Transfer in Solids** ht T eat Transfer in Fluids** ht T eat Transfer in Porous edia**

ht T

ioheat Transfer** ht T eat Transfer in Thin Shells htsh T EAT TRANSFER>CONJUGATE HEAT TRANSFER

aminar Flow** nitf u, p, T urbulent Flow, k-** nitf u, p, k, ep, T urbulent Flow, Low Re k-** nitf u, p, k, ep, G, T EAT TRANSFER>RADIATION

eat Transfer with urface-to-Surface Radiation**

ht T, J

8 | C H A P T E R 1 : I N T R

S

Tinto

HP

S

RM

H

Jo

*

*r

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

PHYSICS INTERFACE TAG DEPENDENT VARIABLES PRESET STUDIES*

T ITH

IN

ITIA

LIZ

AT

ION

H I

NIT

IAL

IZA

TIO

NO D U C T I O N

H O W M O R E P H Y S I C S O P T I O N S

here are several general options available for the physics user interfaces and for dividual nodes. This section is a short overview of these options, and includes links additional information when available.

eat Transfer with Radiation in articipating Media**

ht T, I (radiative intensity)

urface-to-Surface Radiation rad J adiation in Participating edia

rpm I (radiative intensity)

EAT TRANSFER>ELECTROMAGNETIC HEATING

ule 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 adiation in participating media. In these cases, J and I are dependent variables.

ST

AT

ION

AR

Y

TIM

E D

EP

EN

DE

N

ST

AT

ION

AR

Y W

TR

AN

SIE

NT

WIT

The links to the features described in the COMSOL Multiphysics Reference Manual (or any external guide) do not work in the PDF, only from within the online help.

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

L E | 9

To display additional options for the physics interfaces and other parts of the model tree, click the Show button ( ) on the Model Builder and then select the applicable option.

After clicking the Show button ( ), additional sections get displayed on the settings window when a node is clicked and additional nodes are available from the context menu when a node is right-clicked. For each, the additional sections that can be displayed include Equation, Advanced Settings, Discretization, Consistent Stabilization, an

Yosodi

FavEq

AthSt

th

O

AopD

S

SE

D

Dc

CIn

C

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

d Inconsistent Stabilization.

u can also click the Expand Sections button ( ) in the Model Builder to always show me sections or click the Show button ( ) and select Reset to Default to reset to splay only the Equation and Override and Contribution sections.

or most nodes, both the Equation and Override and Contribution sections are always ailable. Click the Show button ( ) and then select Equation View to display the uation View node under all nodes in the Model Builder.

vailability of each node, and whether it is described for a particular node, is based on e individual selected. For example, the Discretization, Advanced Settings, Consistent abilization, and Inconsistent Stabilization sections are often described individually roughout the documentation as there are unique settings.

T H E R C O M M O N S E T T I N G S

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

ECTION CROSS REFERENCE

how More Options and xpand Sections

Advanced Physics Sections

The Model Wizard and Model Builder

iscretization Show Discretization

Discretization (Node)

iscretizationSplitting of omplex variables

Compile Equations

onsistent and consistent Stabilization

Show Stabilization

Numerical Stabilization

onstraint Settings Weak Constraints and Constraint Settings

verride and Contribution Physics Exclusive and Contributing Node Types

10 | C H A P T E R 1 : I N T

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

T

Tw

SECTION CROSS REFERENCE

Coordinate System Selection

Coordinate Systems

Da

In

M

M

PR O D U C T I O N

H E L I Q U I D S A N D G A S E S M A T E R I A L S D A T A B A S E

he Heat Transfer Module includes an additional Liquids and Gases material database ith temperature-dependent fluid dynamic and thermal properties.

omain, Boundary, Edge, nd Point Selection

About Geometric Entities

About Selecting Geometric Entities

terface Identifier Predefined Physics Variables

Variable Naming Convention and Scope

Viewing Node Names, Identifiers, Types, and Tags

aterial Type Materials

odel Inputs About Materials and Material Properties

Selecting Physics

Adding Multiphysics Couplings

air Selection Identity and Contact Pairs

Continuity on Interior Boundaries

For detailed information about materials and the Liquids and Gases Material Database, see Materials in the COMSOL Multiphysics Reference Manual.

E | 11

Where Do I Access the Documentation and Model Library?

A number of Internet resources provide more information about COMSOL, including licensing and technical information. The electronic documentation, context help, and the Model Library are all accessed through the COMSOL Desktop.

T

Tfuhael

T

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

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

he COMSOL Multiphysics Reference Manual describes all user interfaces and nctionality included with the basic COMSOL Multiphysics license. This book also s instructions about how to use COMSOL and how to access the documentation

ectronically through the COMSOL Help Desk.

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

Press F1 or select Help>Help ( ) from the main menu for context help.

Press Ctrl+F1 or select Help>Documentation ( ) from the main menu for opening the main documentation window with access to all COMSOL documentation.

Click the corresponding buttons ( or ) on the main toolbar.

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

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

If you have added a node to a model you are working on, click the Help button ( ) in the nodes settings window or press F1 to learn more about it. Under More results in the Help window there is a link with a search string for the nodes name. Click the link to find all occurrences of the nodes name in the documentation, including model documentation and the external COMSOL website. This can help you find more information about the use of the nodes functionality as well as model examples where the node is used.

12 | C H A P T E R 1 : I N T

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.

In most models, SI units are used to describe the relevant properties, parameters, and d

Tthandbb

Tmu

Ifth

C

F

TcosuemR O D U C T I O N

imensions in most examples, but other unit systems are available.

o open the Model Library, select View>Model Library ( ) from the main menu, and en search by model name or browse under a module folder name. Click to highlight y model of interest, and select Open Model and PDF to open both the model and the

ocumentation explaining how to build the model. Alternatively, click the Help utton ( ) or select Help>Documentation in COMSOL to search by name or browse y module.

he model libraries are updated on a regular basis by COMSOL in order to add new odels and to improve existing models. Choose View>Model Library Update ( ) to

pdate your model library to include the latest versions of the model examples.

you have any feedback or suggestions for additional models for the library (including ose developed by you), feel free to contact us at info@comsol.com.

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

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

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

E | 13

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

COMSOL website www.comsol.com

Contact COMSOL www.comsol.com/contact

Support Center www.comsol.com/support

Download COMSOL www.comsol.com/support/download

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

P

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

roduct Updates www.comsol.com/support/updates

OMSOL Community www.comsol.com/community

14 | C H A P T E R 1 : I N T

Ove r v i ew o f t h e U s e r s Gu i d e

The Heat Transfer Module Users Guide gets you started with modeling using COMSOL Multiphysics. The information in this guide is specific to the Chemical Reaction Engineering Module. Instructions how to use COMSOL in general are included with the COMSOL Multiphysics Reference Manual.

T

T

H

Ttrcosein

T

Tptrre

GTfocoHthlaMdbR O D U C T I O N

A 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

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

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

he Heat Transfer Theory chapter starts with the general theory underlying the heat ansfer interfaces used in this module. It then discusses theory about heat transfer efficients, highly conductive layers, and out-of-plane heat transfer. The last three ctions briefly describe the underlying theory for the Bioheat Transfer, Heat Transfer Thin Shells, and Heat Transfer in Porous Media interfaces.

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

he module includes interfaces for the simulation of heat transfer. As with all other hysical descriptions simulated by COMSOL Multiphysics, any description of heat ansfer can be directly coupled to any other physical process. This is particularly levant for systems based on fluid-flow, as well as mass transfer.

eneral Heat Transferhe Heat Transfer Branch chapter details the variety of Heat Transfer interfaces that rm the fundamental interfaces in this module. It covers all the types of heat transfernduction, convection, and radiationfor heat transfer in solids and fluids. About the eat Transfer Interfaces provides a quick summary of each interface, and the rest of e chapter describes these interfaces in details. This includes the highly conductive yer and out-of-plane heat transfer physics features and the Heat Transfer in Porous edia interface. The Heat Transfer with Participating Media (ht) interface is also

escribed as it is a Heat Transfer interface where surface-to-surface radiation is active y default.

As detailed in the section Where Do I Access the Documentation and Model Library? this information can also be searched from the COMSOL Multiphysics software Help menu.

E | 15

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

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

Radiative Heat TransferRTin

T

TLbrdefo

T

TflocoO V E R V I E W O F T H E U S E R S G U I D

adiation Heat Transfer chapter describes the Surface-to-Surface Radiation, the Heat ransfer with Surface-to-Surface Radiation, and the Radiation in Participating Media terfaces.

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

he Conjugate Heat Transfer Branch chapter describes the Non-Isothermal Flow aminar Flow (nitf) and Turbulent Flow (nitf) interfaces found under the Fluid Flow anch, which are identical to the Conjugate Heat Transfer interfaces. Each section scribes the applicable interfaces in detail and concludes with the underlying theory r the interfaces.

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

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

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

17

2

About Handling Frames in Heat TransferH 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 User Interfaces

About the Heat Transfer Coefficients

About Highly Conductive Layers

Theory of Out-of-Plane Heat Transfer

Theory for the Bioheat Transfer User Interface

Theory for the Heat Transfer in Porous Media User Interface

18 | C H A P T E R 2 : H E A

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

The Heat Transfer Interfacetheory is described in this section. This section reviews the theory about the heat transfer equations in COMSOL Multiphysics and heat transfer in1

In

W

HIt

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

general. For more detailed discussions of the fundamentals of heat transfer, see Ref. and Ref. 3.

this section:

What is Heat Transfer?

The Heat Equation

A Note on Heat Flux

Heat Flux and Heat Source Variables

About the Boundary Conditions for the Heat Transfer User Interfaces

Radiative Heat Transfer in Transparent Media

Consistent and Inconsistent Stabilization Methods for the Heat Transfer User Interfaces

Moist Air Theory

About Heat Transfer with Phase Change

Theory for the Thermal Contact Feature

References for the Heat Transfer User Interfaces

hat is Heat Transfer?

eat transfer is defined as the movement of energy due to a difference in temperature. is characterized by the following mechanisms:

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

| 19

ConvectionHeat convection (sometimes called heat advection) takes place through the net displacement of a fluid, which transports the heat content in a fluid through the fluids own velocity. The term convection (especially convective cooling and convective heating) also refers to the heat dissipation from a solid surface to a fluid, typically described by a heat transfer coefficient.

RadiationHeat transfer by radiation takes place through the transport of photons. Participating (or semitransparent) media absorb, emit and scatter photons. Opaque surfaces absorb or reflect them.

T

TcoenTre

w

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

he Heat Equation

he fundamental law governing all heat transfer is the first law of thermodynamics, mmonly referred to as the principle of conservation of energy. However, internal ergy, U, is a rather inconvenient quantity to measure and use in simulations.

herefore, the basic law is usually rewritten in terms of temperature, T. For a fluid, the sulting heat equation is:

(2-1)

here

is the density (SI unit: kg/m3) Cp is the specific heat capacity at constant pressure (SI unit: J/(kgK))

T is absolute temperature (SI unit: K)

u is the velocity vector (SI unit: m/s)

q is the heat flux by conduction (SI unit: W/m2)

p is pressure (SI unit: Pa)

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

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

Cp Tt------- u T+ q :S T----

T------- p

pt------ u p+

Q+ +=

S 12--- u u T+ =

20 | C H A P T E R 2 : H E A

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 eqve

Tco

wcoT

an

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

uation also assumes that mass is always conserved, which means that density and locity must be related through:

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

(2-2)

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

hen k becomes a tensor

d the conductive heat flux is given by

t v + 0=

qi kTxi--------

=

k

kxx kxy kxzkyx kyy kyzkzx kzy kzz

=

qi kijTxj--------

j=

Fouriers law expect that the thermal conductivity tensor is symmetric. Non symmetric tensor leads to unphysical results.

S | 21

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:

The third term represents pressure work and is responsible for the heating of a fluid unfoth

Inhe

Tveob

A

Tisis

TTon

TO

T

A

a:b anmbnmm

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

der adiabatic compression and for some thermoacoustic effects. It is generally small r low Mach number flows. A similar term can be included to account for ermoelastic effects in solids.

serting Equation 2-2 into Equation 2-1, reordering the terms and ignoring viscous ating and pressure work puts the heat equation into a more familiar form:

he Heat Transfer in Fluids physics solves this equation for the temperature, T. If the locity is set to zero, the equation governing pure conductive heat transfer is tained:

Note on Heat Flux

he concept of heat flux is not as simple as it might first appear. The reason is that heat not a conserved property. The conserved property is instead the total energy. There hence heat flux and energy flux that are similar but not identical.

his section briefly describes the theory for the variables for Total Energy Flux and otal Heat Flux. The approximations made do not affect the computational results, ly variables available for results analysis and visualization.

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

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

(2-3)

bove, H0 is the total enthalpy

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

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

u H0 + k T u qr++

22 | C H A P T E R 2 : H E A

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

F

wreore

w

T(Rinso

T

T

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

(2-4)

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

(2-5)

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

Solid materials and ideal gases: HrefCp,refTrefGasliquid: HrefCp,refrefTrefprefref

here the subscript ref indicates that the property is evaluated at the reference state.

he two integrals in Equation 2-5 are sometimes referred to as the sensible enthalpy ef. 7). These are evaluated by numerical integration. The second integral is only cluded for gas/liquid since it is commonly much smaller than the first integral for lids and it is identically zero for ideal gases.

O T A L H E A T F L U X

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

u H 12--- u u + k T u qr++

H Href Cp Td

Tref

T

1--- 1 T---- T------- p

+

pd

pref

p

+ +=

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 depends on the pressure, that dependency must be prescribed either via model input or by using the variable pA, which is the variable for the absolute pressure.

S | 23

(2-6)

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

(2-7)

What is the difference between Equation 2-4 and Equation 2-7? As an example, consider a channel with fully developed incompressible flow with all properties of the fluAlo

Theevishe

Ifth

H

Tanthbe

uU k T qr+

H U p---+=

TABLE

VARIA

tflu

dflu

turb

aflu

trlf

tefl

not aT H E O R Y F O R T H E H E A T TR A N S F E R U S E R I N T E R F A C E

id independent of pressure and temperature. The walls are assumed to be insulated. ssume that the viscous heating is neglected. This is a common approximation for w-speed flows.

here is a pressure drop along the channel that drives the flow. Since there is no viscous ating and the walls are isolated, Equation 2-5 gives that HinHout. Since erything else is constant, Equation 2-4 shows that the energy flux into the channel

higher than the energy flux out of the channel. On the other hand UinUout, so the at flux into the channel is equal to the heat flux going out of the channel.

the viscous heating on the other hand is included, then HinHout (first law of ermodynamics) and UinUout (since work has been converted to heat).

eat Flux and Heat Source Variables

his section lists some predefined variables that are available to compute heat fluxes d sources. All the variable names start with the physics interface prefix. By default e Heat Transfer interface prefix is ht. As an example, the variable named tflux can analyzed using ht.tflux (as long as the physics interface prefix is ht).

2-1: HEAT FLUX VARIABLES

BLE NAME GEOMETRIC ENTITY LEVEL

x Total Heat Flux Domains, boundaries

x Conductive Heat Flux Domains, boundaries

flux Turbulent Heat Flux Domains, boundaries

x Convective Heat Flux Domain, boundaries

lux Translational Heat Flux Domains, boundaries

ux Total Energy Flux Domains, boundaries

pplicable Radiative Heat Flux Domains

24 | C H A P T E R 2 : H E A

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

ntfl

ndfl

nafl

ntrl

ntef

ndfl

ndfl

nafl

nafl

ntrl

ntrl

ntfl

ntfl

ntef

TABLE 2-1: HEAT FLUX VARIABLES

VARIABLE NAME GEOMETRIC ENTITY LEVELT TR A N S F E R T H E O R Y

Out-of-Plane Heat Flux Out-of-plane domains (1D and 2D)

ux Normal Total Heat Flux, Extrapolated

Boundaries

ux Normal Conductive Heat Flux, Extrapolated

Boundaries

ux Normal Convective Heat Flux Boundaries

flux Normal Translational Heat Flux Boundaries

lux Normal Total Energy Flux, Extrapolated

Boundaries

ux_u Internal Normal Conductive Heat Flux, Extrapolated, Upside

Interior boundaries

ux_d Internal Normal Conductive Heat Flux, Extrapolated, Downside

Interior boundaries

ux_u Internal Normal Convective Heat Flux, Extrapolated, Upside

Interior boundaries

ux_d Internal Normal Convective Heat Flux, Downside

Interior boundaries

flux_u Internal Normal Translational Heat Flux, Upside

Interior boundaries

flux_d Internal Normal Translational Heat Flux, Downside

Interior boundaries

ux_u Internal Total Normal Heat Flux, Upside

Interior boundaries

ux_d Internal Total Normal Heat Flux, Downside

Interior boundaries

lux_u Internal Normal Total Energy Flux, Extrapolated, Upside

Interior boundaries

S | 25

D

Oon

nteflux_d Internal Normal Total Energy Flux, Extrapolated, Downside

Interior boundaries

ndflux_acc Normal Conductive Flux, Accurate Exterior boundaries

ntflux_acc Normal Total Heat Flux, Accurate Exterior boundaries

nteflux_acc Normal Total Energy Flux, Accurate Exterior boundaries

ndfl

ndfl

ntfl

ntfl

ntef

ntef

rflu

ccfl

Qtot

Qbto

Ql

Qp

TABLE 2-1: HEAT FLUX VARIABLES

VARIABLE NAME GEOMETRIC ENTITY LEVELT H E O R Y F O R T H E H E A T TR A N S F E R U S E R I N T E R F A C E

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

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

ux_acc_u Internal Normal Conductive Flux, Accurate, Upside

Interior boundaries

ux_acc_d Internal Normal Conductive Flux, Accurate, Downside

Interior boundaries

ux_acc_u Internal Normal Total Heat Flux, Accurate, Upside

Interior boundaries

ux_acc_d Internal Normal Total Heat Flux, Accurate, Downside

Interior boundaries

lux_acc_u Internal Normal Total Energy Flux, Accurate, Upside

Interior boundaries

lux_acc_d Internal Normal Total Energy Flux, Accurate, Downside

Interior boundaries

x Radiative Heat Flux Boundaries

ux Convective Heat Flux Boundaries

Domain Heat Sources Domains

t Boundary Heat Sources Boundaries

Line heat source (Line and Point Heat Sources)

Edges

Point heat source (Line and Point Heat Sources)

Points

26 | C H A P T E R 2 : H E A

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.

Fth

F

CTan

Wis

In

Fth

See Radiative Heat Flux to evaluate the radiative heat flux.T TR A N S F E R T H E O R Y

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

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

onductive Heat Fluxhe conductive heat flux variable, dflux, is evaluated using the temperature gradient d the effective thermal conductivity:

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

In 2D (dz is the domain thickness):

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

the general case keff is the thermal conductivity, k.

or heat transfer in fluids with turbulent flow, keff = k + kT, where kT is the turbulent ermal conductivity.

tflux trlflux dflux+=

tflux aflux dflux+=

dflux keff T=

dflux dzkeff T=

dflux Ackeff T=

S | 27

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

TuTco

CT

Wis

T

w

TrSiva

The Heat Transfer in Porous Media feature requires one of the following products: Batteries & Fuel Cells Module, CFD Module, Chemical Reaction Engineering Module, Corrosion Module, Electrochemistry Module, Electrodeposition Module, Heat Transfer Module, or Subsurface Flow Module.T H E O R Y F O R T H E H E A T TR A N S F E R U S E R I N T E R F A C E

rbulent Heat Fluxhe turbulent heat flux variable, turbflux, enables access to the part of the nductive heat flux that is due to the turbulence.

onvective Heat Fluxhe convective heat flux variable, aflux, is defined using the internal energy, E:

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

In 2D (dz is the domain thickness):

In1D (Ac is the domain thickness):

he internal energy, E, is defined by:

ECpT for solid domainsECpT for ideal gas fluid domainsEHp for other fluid domains

here H is the enthalpy defined by Equation 2-5.

anslational Heat Fluxmilar to convective heat flux but defined for solid domains with translation. The riable name is trlflux.

turbflux kT T=

aflux uE=

aflux dzuE=

aflux AcuE=

28 | C H A P T E R 2 : H E A

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

where the total enthalpy, H0, is defined as

RInd

O

Wflze

CTC

RTR

teflux uH0 dflux u+ +=

H0 Hu u

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

adiative Heat Flux participating media, the radiative heat flux, qr, is not available for analysis on

omains because it is much more accurate to evaluate the radiative heat source:

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

hen the out-of-plane property is activated (1D and 2D only), out-of-plane domain uxes are defined. If there are no out-of-plane physics features, they are evaluated to ro.

onvective Out-of-Plane Heat Fluxhe convective out-of-plane heat flux, ceflux, is generated by the Out-of-Plane onvective Heat Flux feature.

In 2D:

upside:

downside:

In 1D:

adiative Out-of-Plane Heat Fluxhe radiative out-of-plane heat flux, rflux, is generated by the Out-of-Plane adiationfeature.

In 2D:

upside:

Qr qr=

ccflux_u hu Text u T =

ccflux_d hd Text d T =

ccflux_z hz Text z T =

rflux_u u Tamb u4 T4 =

S | 29

downside:

In 1D:

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

B

ATIn

NT

NT

NT

NT

rflux_d d Tamb d4 T4 =

rflux_z z Tamb z4 T4 =T H E O R Y F O R T H E H E A T TR A N S F E R U S E R I N T E R F A C E

ature.

In 2D:

upside:

downside:

In 1D:

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

ll the domain heat fluxes (vector quantity) are also available as boundary heat fluxes. he boundary heat fluxes are then equal to the mean value of the adjacent domains. addition normal boundary heat fluxes (scalar quantity) are available on boundaries.

ormal Total Heat Flux, Extrapolatedhe variable ntflux is defined by:

ormal Conductive Heat Flux, Extrapolatedhe variable ndflux is defined by:

ormal Convective Heat Fluxhe variable naflux is defined by:

ormal Translational Heat Fluxhe variable ntrlflux is defined by:

q0_u hu Text u T =

q0_d hd Text d T =

q0_z hz Text z T =

ntflux mean tflux n=

ndflux mean dflux n=

naflux mean aflux n=

30 | C H A P T E R 2 : H E A

Normal Total Energy Flux, ExtrapolatedThe variable nteflux is defined by:

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

wsu

CCF

Wis

I N

Tbfr

InT

InT

ntrlflux mean trlflux n=

nteflux mean teflux n=T TR A N S F E R T H E O R Y

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

onvective Heat Fluxonvective heat flux, ccflux, is defined as the contribution from the Convective Heat lux boundary condition:

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

In 2D (dz is the domain thickness):

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

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

he internal normal boundary heat fluxes (scalar quantity) are available on interior oundaries. They are calculated using the upside and the downside value of heat fluxes om the adjacent domains.

ternal Normal Conductive Heat Flux, Extrapolated, Upsidehe variable ndflux_u is defined by:

ternal Normal Conductive Heat Flux, Extrapolated, Downsidehe variable ndflux_d is defined by:

rflux Tamb4 T4 G T4 qw+ +=

ccflux h Text T =

ccflux dzh Text T =

ccflux Ach Text T =

ndflux_u up dflux n=

S | 31

Internal Normal Convective Heat Flux, Extrapolated, UpsideThe variable naflux_u is defined by:

Internal Normal Convective Heat Flux, DownsideThe variable naflux_d is defined by:

InT

InT

InT

InT

InT

InT

ndflux_d down dflux n=

naflux_u up aflux n=T H E O R Y F O R T H E H E A T TR A N S F E R U S E R I N T E R F A C E

ternal Normal Translational Heat Flux, Upsidehe variable ntrlflux_u is defined by:

ternal Normal Translational Heat Flux, Downsidehe variable ntrlflux_d is defined by:

ternal Normal Total Energy Flux, Extrapolated, Upsidehe variable nteflux_u is defined by:

ternal Normal Total Energy Flux, Extrapolated, Downsidehe variable nteflux_d is defined by:

ternal Total Normal Heat Flux, Upsidehe variable ntflux_u is defined by:

ternal Total Normal Heat Flux, Downsidehe variable ntlux_d is defined by:

naflux_d down aflux n=

ntrlflux_u up trlflux n=

ntrlflux_d down trlflux n=

nteflux_u up teflux n=

nteflux_d down teflux n=

ntflux_u up tflux n=

ntflux_d down tflux n=

32 | C H A P T E R 2 : H E A

A C C U R A T E F L U X E S

Normal Conductive Flux, AccurateThe variable ndflux_acc is defined by:

Internal Normal Conductive Flux, Accurate, DownsideT

InT

NT

InT

InT

NT

InT

InT

ndflux_acc dep.dflux.T=ndflux_acc dep.uflux.T=

if the adjacent domain is on the downsideif the adjacent domain is on the upsideT TR A N S F E R T H E O R Y

he variable ndflux_acc_d is defined by:

ternal Normal Conductive Flux, Accurate, Upsidehe variable ndflux_acc_u is defined by:

ormal Total Heat Flux, Accuratehe variable ntflux_acc is defined by:

ternal Normal Total Heat Flux, Accurate, Downsidehe variable ntflux_acc_d is defined by:

ternal Normal Total Heat Flux, Accurate, Upsidehe variable ntflux_acc_u is defined by:

ormal Total Energy Flux, Accuratehe variable nteflux_acc is defined by:

ternal Normal Total Energy Flux, Accurate, Downsidehe variable nteflux_acc_d is defined by:

ternal Normal Total Energy Flux, Accurate, Upsidehe variable nteflux_acc_u is defined by:

ndflux_acc_d dep.dflux.T=

ndflux_acc_u dep.uflux.T=

ntflux_acc ndflux_acc naflux ntrlflux+ +=

ntflux_acc_d ndflux_acc_d naflux_d ntrlflux_d+ +=

ntflux_acc_u ndflux_acc_u naflux_u ntrlflux_u+ +=

ntflux_acc nteflux ndflux ndflux_acc+=

nteflux_acc_d nteflux_d ndflux_d ndflux_acc_d+=

S | 33

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 physics features are available in one variable, Qtot (SI unit: W/m

3). This variable Qtot is the sum of:

Q which is the heat source added by Heat Source(described for the Heat Transfer interface and Electromagnetic Heat Source (described for the Joule Heating

B

Tav

L

T

T

nteflux_acc_u nteflux_u ndflux_u ndflux_acc_u+=T H E O R Y F O R T H E H E A T TR A N S F E R U S E R I N T E R F A C E

interface in the COMSOL Multiphysics Reference Manual) feature.

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

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

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

2). This variable Qbtot is the sum of:

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

Qsh which is the boundary heat source added by the Boundary Electromagnetic Heat Source boundary condition (described for the Joule Heating interface in the COMSOL Multiphysics Reference Manual).

Qs: which is the boundary heat source added by a Layer Heat Source subfeature of a highly conductive layer.

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

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

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

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

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

34 | C H A P T E R 2 : H E A

About the Boundary Conditions for the Heat Transfer User 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 specified temperature is of a constraint type and prescribes the temperature at a boundary:

w

w

Tpth

Awcoawrep

O

T T0= on T TR A N S F E R T H E O R Y

hile the latter specifies the inward heat flux

here

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.

he inward heat flux, q0, is often a sum of contributions from different heat transfer rocesses (for example, radiation and convection). The special case q0 0 is called ermal insulation.

common type of heat flux boundary conditions are those where q0hTinfT, here Tinf is the temperature far away from the modeled domain and the heat transfer efficient, h, represents all the physics occurring between the boundary and far ay. It can include almost anything, but the most common situation is that h

presents the effect of an exterior fluid cooling or heating the surface of solid, a henomenon often referred to as convective cooling or heating.

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

n q q0= on

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

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

S | 35

Many boundary conditions are available in heat transfer. Some of them can be associated (for example, Heat Flux and Highly Conductive Layer). Others cannot be associated (for example, Heat Flux and Thermal Insulation).

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

Temperature, Convective Outflow, Open Boundary, Inflow Heat Flux

Thermal Insulation, Symmetry, Periodic Heat Condition

Wtris

TA

A

1

2

3L

4

5

6r

7

8RT H E O R Y F O R T H E H E A T TR A N S F E R U S E R I N T E R F A C E

Highly Conductive Layer

Heat Flux, Convective Heat Flux

Boundary Heat Source, Radiation Group

Surface-to-Surface Radiation, Re-radiating Surface, Prescribed Radiosity, Surface-to-Ambient Radiation

Opaque Surface, Incident Intensity, Continuity on Interior Boundaries

Thin Thermally Resistive Layers, Thermal Contact

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

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

Locate the column that corresponds to the group of B.

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

\B 1 2 3 4 5 6 7 8

-Temperature X X X X

-Thermal Insulation X X X

-Highly Conductive ayer

X X

-Heat Flux X X

-Boundary heat source

-Surface-to-surface adiation

X X

-Opaque Surface X

-Thin Thermally esistive Layer

X X

36 | C H A P T E R 2 : H E A

If the corresponding cell is empty A and B contribute. If it contains an X, B overrides A.

Example 1Cb

ECco

R

Tcoenth

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

onsider a boundary where Temperature is applied. Then a Surface-to-Surface Radiation oundary 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.

xample 2onsider a boundary where Convective Heat Flux is applied. Then a Symmetry boundary ndition is applied on the same boundary afterward.

Convective Heat Flux belongs to group 4.

Symmetry belongs to group 2.

The cell on the line of group 4 and the column of group 2 contains an X so Convective Heat Flux is overridden by Symmetry.

adiative Heat Transfer in Transparent Media

his discussion so far has considered heat transfer by means of conduction and nvection. The third mechanism for heat transfer is radiation. Consider an vironment with fully transparent or fully opaque objects. Thermal radiation denotes e stream of electromagnetic waves emitted from a body at a certain temperature.

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

S | 37

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

Fi

Crembo

Trara

Tir

Uob

Mem

T

G

J =G + T4T H E O R Y F O R T H E H E A T TR A N S F E R U S E R I N T E R F A C E

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

onsider Figure 2-1. A point is located on a surface that has an emissivity , flectivity , absorptivity , and temperature T. Assume the body is opaque, which eans that no radiation is transmitted through the body. This is true for most solid dies.

he total arriving radiative flux at is named the irradiation, G. The total outgoing diative flux is named the radiosity, J. The radiosity is the sum of the reflected diation and the emitted radiation:

(2-8)

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

(2-9)

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

(2-10)

ost opaque bodies also behave as ideal gray bodies, meaning that the absorptivity and issivity are equal, and the reflectivity is therefore given from the following relation:

(2-11)

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

(2-12)

,T ,Tx x

x

xx

J G T4+=

q G J=

q 1 G T4=

1 = =

q G T4 =

38 | C H A P T E R 2 : H E A

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

S

T

Insu

FM

CT

Tthth

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

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

urface-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.

hese assumptions allows the irradiation to be explicitly expressed as

(2-13)

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

(2-14)

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

onsistent and Inconsistent Stabilization Methods for the Heat ransfer User Interfaces

he different versions of the Heat Transfer interface have this advanced option to set e stabilization method parameters. This section provides information pertaining to ese options.

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

G Tamb4=

q Tamb4 T4 =

Theory for the Radiation in Participating Media User Interface

Radiation and Participating Media Interactions

S | 39

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.

StStpeor

CStofofunaddi

I N

TAcodida

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

reamline Diffusionreamline diffusion is active by default and should remain active for optimal rformance for heat transfer in fluids or other applications that include a convective translational term.

rosswind Diffusionreamline diffusion introduces artificial diffusion in the streamline direction. This is ten enough to obtain a smooth numerical solution provided that the exact solution the heat equation does not contain any discontinuities. At sharp gradients, however, dershoots and overshoots can occur in the numerical solution. Crosswind diffusion dresses these spurious oscillations by adding diffusion orthogonal to the streamline rectionthat is, in the crosswind direction.

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

his section contains one inconsistent stabilization method: isotropic diffusion. dding isotropic diffusion is equivalent to adding a term to the physical diffusion efficient. This means that the original problem is not solved, which is why isotropic ffusion is an inconsistent stabilization method. Still, the added diffusion definitely mpens the effects of oscillations, but try to minimize the use of isotropic diffusion.

y default there is no isotropic diffusion. To add isotropic diffusion, select the Isotropic ffusion check box. The field for the tuning parameter id then becomes available. The

Continuous Casting: Model Library path Heat_Transfer_Module/Thermal_Processing/continuous_casting

40 | C H A P T E R 2 : H E A

default value is 0.25; increase or decrease the value of id to increase or decrease the amount of isotropic stabilization.

M

Fcath

H

MTo

wanco

T(S

RT

See Show Stabilization and Stabilization Techniques in the COMSOL Multiphysics Reference Manual.T TR A N S F E R T H E O R Y

oist Air Theory

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

U M I D I T Y

oisture Contenthe moisture content (also called mixing ratio or humidity ratio) is defined as the ratio f water vapor mass mv to dry air mass ma:

(2-15)

here pv is the water vapor partial pressure, pa is the dry air partial pressure, and Ma d Mv are the molar mass of dry air and water vapor, respectively. Without ndensation, the moisture content is not affected by temperature and pressure.

he Moisture content represents a ratio of mass, and it is thus a dimensionless number I unit: 1).

elative Humidityhe relative humidity of an air mixture is expressed as follows:

(2-16)

Heat Transfer in Fluids

xvapmvma-------

pvMvpaMa--------------= =

pvpsat--------=

S | 41

where pv is the water vapor partial pressure and psat is the saturation pressure of water vapor.

According to Daltons law, the total pressure of a mixture of gases is the sum of all the partial pressures of each individual gas; that is, p=pv+pa where pa is the dry air partial pressure.

The relative humidity formulation is often used to quantify humidity. However, for a same quantity of moisture content, the relative humidity changes with temperature ante

Tbere

Twrearm

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

d pressure, so in order to compare different values of , it has to be at the same mperature and pressure conditions.

his quantity is very useful to study the condensation as it defines the boundary tween the liquid phase and the vapor phase. In fact, when the relative humidity aches unity, it means that the vapor is saturated and that water vapor will condense.

he Reference relative humidity (SI unit: 1) is a quantity defined between 0 and 1, here 0 corresponds to dry air and 1 to a water vapor-saturated air. This Reference lative humidity associated to the Reference temperature and the Reference pressure e used to calculate the moisture content. Then the thermodynamical properties of oist air can be deduced through the mixture formula described below.

ecific Humidityhe specific humidity is defined as the ratio of water vapor mv to the total mass

tot=mv+ma:

The Reference relative humidity cannot be greater than one, above which value the water vapor is condensing. If the value is greater than one, the Reference relative humidity value is forced to be one. The condensation area cannot be simulated.

42 | C H A P T E R 2 : H E A

(2-17)

CT

wva

Afo

S

Tthd

Tthco

wis

mvmtot-----------=

As the water vapor only accounts for a few percent in the total mass, the moisture content and the specific humidity are very close: xvap (only for low values). For bigger values of , the two quantities are more precisely related by:T TR A N S F E R T H E O R Y

oncentrationhe concentration is defined by:

(2-18)

here nv is the amount of water vapor in mol and V is the total volume. The water por concentration is defined in this SI unit: mol/m3.

ccording to the ideal gas hypothesis, the saturation concentration is defined as llows:

A T U R A T I O N S T A T E

he saturation state is reached when the relative humidity reaches one. It means that e partial pressure of the water vapor is equal to the saturation pressure (which

epends on the temperature too).

he saturation pressure can be defined using the Clausius-Clapeyron formulation of e vaporization-condensation equilibrium. Under ideal gas hypothesis and nsidering only the gas volume:

(2-19)

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

xvap

1 -------------=

cvnvV------=

csatpsat T

RT------------------=

pdTd

-------hfgMvp

RT2--------------------=

S | 43

By integrating, you can obtain the saturation pressure equation:

(2-20)

where the reference values are: pref101325Pa (1 atm), Tref373.15K (100 C), and hfg=2.26106J/kg.The temperature and saturation pressure can easily be deduced from this formulation.

M

T

Pr

M

X

wamva

R

re

psat prefhfgMv

R---------------- 1

Tref--------- 1

Tsat---------

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

O I S T A I R P R O P E R T I E S

he thermodynamical properties of moist air can be found with some mixture laws.

eliminary Definitions

olar Fraction The molar fraction of dry air Xa and the molar fraction of water vapor

v are defined such as:

(2-21)

(2-22)

here na and nv are respectively the amount of dry air and water vapor, ntot is the total ount of moist air in mol, where pa and pv are the partial pressure of dry air and water

por, p is the pressure, is the relative humidity, and psat is the saturation pressure.

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

(2-23)

Xana

ntot---------

pap-----

p psatp

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

Xvnv

ntot---------

pvp-----

psatp

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

XaXv1

xvapppsat

MvMa-------- xvap+

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

44 | C H A P T E R 2 : H E A

Mixture PropertiesThe thermodynamical properties are built through a mixture formula. The expressions depend on dry air properties and pure steam properties and are balanced by the mass fraction.

Density: According to the ideal gas law, the density mixture m expression is defined as follows:

wrere

S

co

wMo

D

as

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

(2-24)

here Ma and Mv are respectively the molar mass of dry air and water vapor, spectively, and Xa and Xv are the molar fraction of dry air and water vapor, spectively.

pecific heat capacity at constant pressure: According to Ref. 10, the heat capacity at nstant pressure of a mixture is:

(2-25)

here Mm represents the mixture molar fraction and is defined by

m=XaMa+XvMv and where cp,a and cp,v are the heat capacity at constant pressure f dry air and steam, respectively.

ynamic viscosity: According to Ref. 9 and R