Heat Transfer Module Users Guide with comsol.
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Heat Transfer ModuleUser´s Guide VERSION 4.3b Heat Transfer Module User’s Guide © 1998–2013 COMSOL Protected by U.S. Patents 7,519,518; 7,596,474; and 7,623,991. Patents pending. This Documentation 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. COMSOL, COMSOL Multiphysics, Capture the Concept, COMSOL Desktop, and LiveLink are either registered trademarks or trademarks of COMSOL AB. All other trademarks are the property of their respective owners, and COMSOL AB and its subsidiaries and products are not affiliated with, endorsed by, sponsored by, or supported by those trademark owners. For a list of such trademark owners, see www.comsol.com/tm. Version: May 2013 COMSOL 4.3b Contact Information Visit the Contact Us page at www.comsol.com/contact to submit general inquiries, contact Technical Support, or search for an address and phone number. You can also visit the Worldwide Sales Offices page at www.comsol.com/contact/offices for address and contact information. If you need to contact Support, an online request form is located at the COMSOL Access page at www.comsol.com/support/case. Other useful links include: • Support Center: www.comsol.com/support • Download COMSOL: www.comsol.com/support/download • Product Updates: www.comsol.com/support/updates • COMSOL Community: www.comsol.com/community • Events: www.comsol.com/events • COMSOL Video Center: www.comsol.com/video • Support Knowledge Base: www.comsol.com/support/knowledgebase Part No. CM020801 C o n t e n t s Chapter 1: Introduction 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 User’s Guide 14 C h a p t e r 2 : H e a t Tr a n s f e r T h e o r y Theory for the Heat Transfer 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 Heat Transfer Coefficient Theory 53 . . . . . . . . . . . . . . . . 54 Nature of the Flow—the Grashof Number . . . . . . . . . . . . . 55 Heat Transfer Coefficients — External Natural Convection . . . . . . . 56 Heat Transfer Coefficients — Internal Natural Convection . . . . . . . 58 Heat Transfer Coefficients — External Forced Convection . . . . . . . 59 CONTENTS |i Heat Transfer Coefficients — Internal Forced Convection . . . . . . . 59 About Highly Conductive Layers 61 Theory of Out-of-Plane Heat Transfer 63 Equation Formulation . . . . . . . . . . . . . . . . . . . . . 64 Activating Out-of-Plane Heat Transfer and Thickness . . . . . . . . . 64 Theory for the Bioheat Transfer User Interface 65 Theory for the Heat Transfer in Porous Media User Interface 66 About Handling Frames in Heat Transfer 68 Frame Physics Feature Nodes and Definitions . . . . . . . . . . . . 68 Conversion Between Material and Spatial Frames . . . . . . . . . . 72 References for the Heat Transfer User Interfaces 75 C h a p t e r 3 : T h e H e a t Tr a n s f e r B r a n c h 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 ii | C O N T E N T S Outflow . . . . . . . . . . . . . . . . . . . . . . . . . 100 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . 101 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 . . . . . . . . . . . . . . . . . . . . . . 115 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 CONTENTS | iii Heat Transfer in Porous Media. . . . . . . . . . . . . . . . . 137 Thermal Dispersion . . . . . . . . . . . . . . . . . . . . . 143 C h a p t e 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 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 Initial 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 C h a p t e r 5 : R a d i a t i o n H e a t Tr a n s f e r The Radiation Branch Versions of the Heat Transfer User Interface The Heat Transfer with Surface-to-Surface Radiation User Interface . . 160 160 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 . . . . . . . . iv | C O N T E N T S 161 . . 173 Prescribed Radiosity . . . . . . . . . 194 Guidelines for Solving Surface-to-Surface Radiation Problems . . . . . . . . . . . . . . . . . . . . 177 External Radiation Source . . . . . . . . . . . . . . . 202 Opaque Surface . . 167 Surface-to-Surface Radiation (Boundary Condition) . . . . . . . . 201 Radiation in Participating Media . . . . . . . . . 212 CONTENTS |v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Edge. . . . . . . . . . . . . . . 190 View Factor Evaluation 192 . . . . . . . . . . . .The Surface-To-Surface Radiation User Interface 164 Domain. . . . 196 Radiation Group Boundaries . . 178 Theory for the Surface-to-Surface Radiation User Interface 182 Wavelength Dependence of Surface Emissivity and Absorptivity . . . . . . . . . . . . . . . 168 Opaque . . . and Pair Nodes for the Radiation in Participating Media User Interface . . . . . . . . . . . . . 205 Continuity on Interior Boundary . . . . . . . . . . . . . . . . 203 Incident Intensity . . . . . . . . . . . . 182 The Radiosity Method for Diffuse-Gray Surfaces . . . . . . . . . . . . . . . . . . . Edge. . . . . and Pair Nodes for the Surface-to-Surface Radiation User Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Discrete Ordinates Method Implementation in 2D . . . . . . . . . . . . . 172 Diffuse Mirror . . . . . . . . . . About Surface-to-Surface Radiation . . . Point. 209 Heat Transfer Equation in Participating Media . . . 197 The Radiation in Participating Media User Interface 199 Domain. . . . Boundary. . . . . . 208 Boundary Condition for the Transfer Equation. . . . . . . . . . . 206 Theory for the Radiation in Participating Media User Interface Radiation and Participating Media Interactions . 174 Radiation Group . . . . . . . . . . . . . . 207 207 Radiative Transfer Equation . . . . . . . . . . . Point. . . Boundary. . . . . . . . 188 The Radiosity Method for Diffuse-Spectral Surfaces . . . . . . . . . . 210 Discrete Ordinates Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 The Turbulent Flow. . . . . . . . . . . . . . . . . . . . . . . . Theory for the Laminar Flow User Interface Theory for the Inlet Boundary Condition vi | C O N T E N T S 247 250 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 Grille . . . . 236 Open Boundary . . . . . . . . . 254 Non-Newtonian Flow: The Power Law and the Carreau Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Wall . . . . . . . . . . . 246 Pressure Point Constraint . . . . . . . . . . . . . . . Pair. 251 Theory for the Fan Defined on an Interior Boundary . . 216 216 The Turbulent Flow. . .References for the Radiation User Interfaces 214 Chapter 6: The Single-Phase Flow Branch The Laminar Flow and Turbulent Flow User Interfaces The Laminar Flow User Interface. . . 226 Initial Values. . . . . . . . . . 231 Outlet . . . . . . . . . . 242 Interior Wall . . . 250 Additional Theory for the Outlet Boundary Condition . . . Low Re k- User Interface . . . . . 237 Periodic Flow Condition . . . . . . . 257 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Flow Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Symmetry . . . . . 240 Interior Fan . . . . . . . . . . . . . . . . . . . . . 239 Fan . . . . . . . . . . . . . . . . . . . . . 224 Volume Force . . . . . . 253 Theory for the Fan and Grille Boundary Conditions . . . . . . . . . . . . . . . 237 Boundary Stress . . k- User Interface . . . . . . . . . . . . . . . and Point Nodes for Single-Phase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boundary. . . . . . 246 More Boundary Condition Settings for the Turbulent Flow User Interfaces . . . . . . . . . 219 Domain. . . 228 Inlet . . . 221 Fluid Properties . . . . . . . . 303 Open Boundary . 289 Interfaces . . . . . . . . . . 280 The Non-Isothermal Flow Options . . . . . . . . . . . . . . . . . No Viscous Stress Boundary Condition . . The Low Reynolds Number k- Turbulence Model . . . Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laminar Flow User Interface . . . . . . . . . . . . . . . . . . . . . . . . . .Theory for the Turbulent Flow User Interfaces Turbulence Modeling . . . 273 Theory for the Pressure. . . . . . . . . . . . Point. . . . . 283 The Non-Isothermal Flow and Conjugate Heat Transfer. . . . . . . . . . . . . . . . Edge. . . . . . . . . . . . . . . . 275 References for the Single-Phase Flow. . . . . . . . . . . . Laminar Flow and Turbulent Flow User Interfaces 285 The Non-Isothermal Flow. . . The k-Turbulence Model . . . . . . 282 Conjugate Heat Transfer Options . . . . . . . . 306 CONTENTS | vii . . . . . 304 Viscous Heating . . Flow . 303 Pressure Work . k- and Turbulent Flow Low Re k-User Domain. . . . . . . . . . . . . . and Pair Nodes Settings for the NITF User Interfaces. . . 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 Tr a n s f e r B r a n c h About the Conjugate Heat Transfer User Interfaces 280 Selecting the Right User Interface . . . . 305 Symmetry. . . . . . . . . . . . 285 The Conjugate Heat Transfer. . . . . . . . . . . . . . . 300 Interior Wall . 305 Symmetry. . . . . . 260 260 264 270 Inlet Values for the Turbulence Length Scale and Turbulent Intensity . . . . . . . . . . . . 293 Wall. . . . . . . . . . . . . . . . . . . . . Fluid 292 . . . . . . . . . . . . . . . . . . . . . . 289 The Turbulent Flow. . . . . . . . . . 302 Initial Values. . . . . . . . . 274 Pseudo Time Stepping for Turbulent Flow Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boundary. . . . . . . . . . . . . . . . . . . . 274 Solvers for Turbulent Flow . . . . . . . Laminar Flow User Interface . . . . . . . . . . . . . . . . .Theory for the Non-Isothermal Flow and Conjugate Heat Transfer User Interfaces Turbulent Non-Isothermal Flow Theory . . . . . . . . 308 310 References for the Non-Isothermal Flow and Conjugate Heat Transfer User Interfaces 315 Chapter 8: Glossary Glossary of Terms viii | C O N T E N T S 318 . 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. • About the Heat Transfer Module • Overview of the User’s Guide 1 . The last section is a brief overview with links to each chapter in this guide. an optional package that extends the COMSOL Multiphysics® modeling environment with customized physics interfaces for the analysis of heat transfer.1 Introduction This guide describes the Heat Transfer Module. developers.About the Heat Transfer Module In this section: • Why Heat Transfer is Important to Modeling • How the Heat Transfer Module Improves Your Modeling • The Heat Transfer Module Physics Guide • Where Do I Access the Documentation and Model Library? Overview of the Physics and Building a COMSOL Model in the COMSOL Multiphysics Reference Manual Why Heat Transfer is Important to Modeling The Heat Transfer Module is an optional package that extends the COMSOL Multiphysics modeling environment with customized user interfaces and functionality optimized for the analysis of heat transfer. The modeling of heat transfer effects has become increasingly important in product design including areas such as electronics. Heat transfer is involved in almost every kind of physical process. teachers. It is developed for a wide audience including researchers. while at the same time reducing costly experimental trials. How the Heat Transfer Module Improves Your Modeling The Heat Transfer Module has been developed to greatly expand upon the base capabilities available in COMSOL Multiphysics. and students. and can in fact be the limiting factor for many processes. The module supports all fundamental 2 | CHAPTER 1: INTRODUCTION . Furthermore. heat transfer often appears together with. or as a result of. automotive. and the need for powerful heat transfer analysis tools is virtually universal. Computer simulation has allowed engineers and researchers to optimize process efficiency and explore new designs. its study is of vital importance. other physical phenomena. this module comes with a library of ready-to-run example models that appear in the companion Heat Transfer Module Model Library. To assist users at all levels of expertise. Therefore. and medical industries. 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. The Heat Transfer Module Physics Guide The table below lists all the interfaces available specifically with this module. The different physics interfaces are described and the modeling strategy for various cases is discussed. for example. convective. Using the physics interfaces in this module along with the inherent multiphysics capabilities of COMSOL Multiphysics. and out-of-plane heat transfer.mechanisms including conductive. and radiative heat transfer. fluid dynamics or electromagnetics. Heat Transfer in Solids. participating media. 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 | 3 . These sections cover different combinations of conductive. thin conductive shells. and radiative heat transfer. convective. Laminar interface. and the Single-Phase Flow. Having this module also enhances these COMSOL Multiphysics basic interfaces: Heat Transfer in Fluids. Another source of information is the Heat Transfer Module Model Library. you can model a temperature field in parallel with other features—a versatile combination increasing the accuracy and predicting power of your models. See Where Do I Access the Documentation and Model Library?. Most of the models involve multiple heat transfer mechanisms and are often coupled to other physical phenomena. This book introduces the basic modeling process. Joule Heating. This guide also reviews special modeling techniques for highly conductive layers. The authors developed several state-of-the art examples by reproducing models that have appeared in international scientific journals. and medical technology—and includes tutorial and verification models. 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. 2D axisymmetric stationary. Low Re k- spf 3D. time dependent Turbulent Flow. 2D axisymmetric stationary with initialization. The Non-Isothermal Flow. 2D axisymmetric stationary. 2D. 2D axisymmetric stationary. this also enhances the Heat Transfer in Porous Media interface. 2D. 2D. 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. time dependent Turbulent Flow. 2D. see Physics Guide. time dependent Fluid Flow Single-Phase Flow Laminar Flow* Turbulent Flow Non-Isothermal Flow Laminar Flow 4 | CHAPTER 1: INTRODUCTION . transient with initialization nitf 3D. PHYSICS USER INTERFACE ICON TAG SPACE DIMENSION AVAILABLE PRESET STUDY TYPE spf 3D. The difference is that Fluid is the default domain node for the Non-Isothermal Flow interfaces.If you have an Subsurface Flow Module combined with the Heat Transfer Module. k- spf 3D. 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. 2D axisymmetric stationary with initialization. Low Re k- nitf 3D. time dependent Heat Transfer in Porous Media ht all dimensions stationary. 2D. 2D. 2D axisymmetric stationary with initialization. time dependent Bioheat Transfer ht all dimensions stationary. time dependent Turbulent Flow. time dependent Heat Transfer in Fluids* ht all dimensions stationary. k- nitf 3D.PHYSICS USER INTERFACE ICON TAG SPACE DIMENSION AVAILABLE PRESET STUDY TYPE Turbulent Flow. Low Re k- nitf 3D. time dependent Heat Transfer in Thin Shells htsh 3D stationary. 2D axisymmetric stationary. 2D. 2D axisymmetric stationary. transient with initialization ht all dimensions stationary. 2D. 2D. 2D axisymmetric stationary. time dependent Turbulent Flow. time dependent Turbulent Flow Heat Transfer Conjugate Heat Transfer Laminar Flow Turbulent Flow Radiation Heat Transfer with Surface-to-Surface Radiation 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 | 5 . time dependent Turbulent Flow. transient with initialization Heat Transfer in Solids* ht all dimensions stationary. k- nitf 3D. time dependent nitf 3D. time dependent Surface-to-Surface Radiation rad all dimensions stationary. T 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 Table 1-1 lists the Preset Studies available for the interfaces most relevant to this module. 2D stationary. time dependent Radiation in Participating Media rpm 3D. time dependent Electromagnetic Heating Joule Heating* * This is an enhanced interface.PHYSICS USER INTERFACE ICON TAG SPACE DIMENSION AVAILABLE PRESET STUDY TYPE Heat Transfer with Radiation in Participating Media ht 3D. which is included with the base COMSOL package but has added functionality for this module. time dependent jh all dimensions stationary. 2D stationary. Studies and Solvers in the COMSOL Multiphysics Reference Manual 6 | CHAPTER 1: INTRODUCTION . T Turbulent Flow. ep Turbulent Flow. k. ep. Low Re k- spf u. G. k. k-** nitf u. T Turbulent Flow. k. k- spf u. J HEAT TRANSFER>RADIATION Heat Transfer with Surface-to-Surface Radiation** 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 | 7 . p. p. ep. Low Re k-** nitf u. T Turbulent Flow. p. G TRANSIENT WITH INITIALIZATION DEPENDENT VARIABLES TIME DEPENDENT TAG STATIONARY PHYSICS INTERFACE STATIONARY WITH INITIALIZATION TABLE 1-1: HEAT TRANSFER MODULE DEPENDENT VARIABLES AND PRESET STUDY AVAILABILITY FLUID FLOW>SINGLE-PHASE FLOW FLUID FLOW>NON-ISOTHERMAL FLOW nitf u. k. p. T ht T. ep. G. p. p Turbulent Flow. k- nitf u. ep. p. T 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 Laminar Flow HEAT TRANSFER Heat Transfer in Solids** HEAT TRANSFER>CONJUGATE HEAT TRANSFER Laminar Flow** nitf u. p. ep. k. k.PRESET STUDIES* Laminar Flow spf u. T Turbulent Flow. Low Re k- nitf u. p. only from within the online help. V TRANSIENT WITH INITIALIZATION DEPENDENT VARIABLES TIME DEPENDENT TAG STATIONARY PHYSICS INTERFACE STATIONARY WITH INITIALIZATION TABLE 1-1: HEAT TRANSFER MODULE DEPENDENT VARIABLES AND PRESET STUDY AVAILABILITY HEAT TRANSFER>ELECTROMAGNETIC HEATING Joule Heating** jh * Custom studies are also available based on the interface. In these cases. I (radiative intensity) Surface-to-Surface Radiation rad J Radiation in Participating Media rpm I (radiative intensity) T. in COMSOL Multiphysics. ** For these interfaces. select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree. SHOW MORE PHYSICS OPTIONS There are several general options available for the physics user interfaces and for individual nodes. The links to the features described in the COMSOL Multiphysics Reference Manual (or any external guide) do not work in the PDF. To locate and search all the documentation for this information. 8 | CHAPTER 1: INTRODUCTION . it is possible to enable surface to surface radiation and/or radiation in participating media. J and I are dependent variables.PRESET STUDIES* Heat Transfer with Radiation in Participating Media** ht T. This section is a short overview of these options. and includes links to additional information when available. Consistent Stabilization. the Discretization. and whether it is described for a particular node. and Inconsistent Stabilization. Availability of each node. Consistent Stabilization. For each. You can also click the Expand Sections button ( ) in the Model Builder to always show ) and select Reset to Default to reset to some sections or click the Show button ( display only the Equation and Override and Contribution sections. and Inconsistent Stabilization sections are often described individually throughout the documentation as there are unique settings. Domain. Discretization.To display additional options for the physics interfaces and other parts of the model tree. 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 | 9 . SECTION CROSS REFERENCE Show More Options and Expand Sections Advanced Physics Sections Discretization Show Discretization The Model Wizard and Model Builder Discretization (Node) Discretization—Splitting of complex variables Compile Equations Consistent and Inconsistent Stabilization Show Stabilization Constraint Settings Weak Constraints and Constraint Settings Override and Contribution Physics Exclusive and Contributing Node Types Numerical Stabilization OTHER COMMON SETTINGS At the main level. click the Show button ( ) on the Model Builder and then select the applicable option. or Edge Selection. Boundary. Advanced Settings. Advanced Settings. Click the Show button ( Equation View node under all nodes in the Model Builder. and Dependent Variables. is based on the individual selected. For example. 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. After clicking the Show button ( ). the additional sections that can be displayed include Equation. some of the common settings found (in addition to the Show options) are the Interface Identifier. both the Equation and Override and Contribution sections are always ) and then select Equation View to display the available. For most nodes. see Materials in the COMSOL Multiphysics Reference Manual. SECTION CROSS REFERENCE Coordinate System Selection Coordinate Systems Domain. and Point Selection About Geometric Entities Interface Identifier Predefined Physics Variables About Selecting Geometric Entities Variable Naming Convention and Scope Viewing Node Names. Other sections are common based on application area and are not included here. 10 | CHAPTER 1: INTRODUCTION . Material Type. For detailed information about materials and the Liquids and Gases Material Database. Edge. Boundary. and Tags Material Type Materials Model Inputs About Materials and Material Properties Selecting Physics Adding Multiphysics Couplings Pair Selection Identity and Contact Pairs Continuity on Interior Boundaries THE LIQUIDS AND GASES MATERIALS DATABASE The Heat Transfer Module includes an additional Liquids and Gases material database with temperature-dependent fluid dynamic and thermal properties. Types. Boundary. Coordinate System Selection.At the nodes’ level. Identifiers. some of the common settings found (in addition to the Show options) are Domain. or Point Selection. Edge. and Model Inputs. • 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. This book also has instructions about how to use COMSOL and how to access the documentation electronically through the COMSOL Help Desk. including model documentation and the external COMSOL website. and then either enter a search term or look under a specific module in the documentation tree. This can help you find more information about the use of the node’s functionality as well as model examples where the node is used. the blue links do not work to open a model or content referenced in a different guide. and documentation sets. Under More results in the Help window there is a link with a search string for the node’s name. THE DOCUMENTATION The COMSOL Multiphysics Reference Manual describes all user interfaces and functionality included with the basic COMSOL Multiphysics license. To locate and search all the documentation. click the Help button ( ) in the node’s settings window or press F1 to learn more about it. Click the link to find all occurrences of the node’s name in the documentation. The electronic documentation. if you are using the online help in COMSOL Multiphysics.Where Do I Access the Documentation and Model Library? A number of Internet resources provide more information about COMSOL. However. including licensing and technical information. in COMSOL Multiphysics: • Press F1 or select Help>Help ( ) from the main menu for context help. model examples. these links work to other modules. and the Model Library are all accessed through the COMSOL Desktop. If you have added a node to a model you are working on. context help. If you are reading the documentation as a PDF file on your computer. A B O U T T H E H E A T TR A N S F E R M O D U L E | 11 . parameters. and select Open Model and PDF to open both the model and the documentation explaining how to build the model. If you have any feedback or suggestions for additional models for the library (including those developed by you).THE MODEL LIBRARY Each model comes with documentation that includes a theoretical background and step-by-step instructions to create the model. contact COMSOL at info@comsol. You can use the step-by-step instructions and the actual models as a template for your own modeling and applications. Click to highlight any model of interest. Alternatively. CONT ACT ING COMSOL BY EMAIL For general product information. select View>Model Library ( then search by model name or browse under a module folder name. To receive technical support from COMSOL for the COMSOL products. and dimensions in most examples. An automatic notification and case number is sent to you by email.com.com. In most models. The model libraries are updated on a regular basis by COMSOL in order to add new models and to improve existing models. but other unit systems are available. Choose View>Model Library Update ( ) to update your model library to include the latest versions of the model examples. 12 | CHAPTER 1: INTRODUCTION . and To open the Model Library. please contact your local COMSOL representative or send your questions to support@comsol. feel free to contact us at info@comsol. SI units are used to describe the relevant properties. The models are available in COMSOL as MPH-files that you can open for further investigation. click the Help button ( ) or select Help>Documentation in COMSOL to search by name or browse by module. ) from the main menu.com. com/support/updates COMSOL Community www.com/contact Support Center www.comsol.comsol.com/support/knowledgebase Product Updates www.comsol.COMSOL WEBSITES COMSOL website www.com/support Download COMSOL www.comsol.comsol.com/support/download Support Knowledge Base www.comsol.com/community A B O U T T H E H E A T TR A N S F E R M O D U L E | 13 .comsol.com Contact COMSOL www. G L O S S A R Y. and Heat Transfer in Porous Media interfaces. see the Contents. Heat Transfer in Thin Shells. 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 The module includes interfaces for the simulation of heat transfer. This includes the highly conductive layer and out-of-plane heat transfer physics features and the Heat Transfer in Porous Media interface. and Index. as well as mass transfer. 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. General Heat Transfer The Heat Transfer Branch chapter details the variety of Heat Transfer interfaces that form the fundamental interfaces in this module. The last three sections briefly describe the underlying theory for the Bioheat Transfer. 14 | CHAPTER 1: INTRODUCTION . and the rest of the chapter describes these interfaces in details. any description of heat transfer can be directly coupled to any other physical process. About the Heat Transfer Interfaces provides a quick summary of each interface. It covers all the types of heat transfer— conduction. 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 heat transfer coefficients. A N D I N D E X To help you navigate through this guide. This is particularly relevant for systems based on fluid-flow.Overview of the User’s Guide The Heat Transfer Module User’s Guide gets you started with modeling using COMSOL Multiphysics®. The information in this guide is specific to the Chemical Reaction Engineering Module. TA B L E O F C O N T E N T S . and radiation—for heat transfer in solids and fluids. and out-of-plane heat transfer. Glossary. 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. highly conductive layers. Instructions how to use COMSOL in general are included with the COMSOL Multiphysics Reference Manual. As with all other physical descriptions simulated by COMSOL Multiphysics. convection. T 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 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. THE FLUID FLOW USER INTERFACES The Single-Phase Flow Branch chapter describe the single-phase laminar and turbulent flow interfaces in detail. and the Radiation in Participating Media interfaces. Heat Transfer in Thin Shells The Heat Transfer in Thin Shells chapter describes the interface. Radiative Heat Transfer Radiation Heat Transfer chapter describes the Surface-to-Surface Radiation.Bioheat Transfer The Bioheat Transfer Interface section discusses modeling heat transfer within biological tissue using the Bioheat Transfer interface. O V E R V I E W O F T H E U S E R ’S G U I D E | 15 . which are identical to the Conjugate Heat Transfer interfaces. the Heat Transfer with Surface-to-Surface Radiation. Each section describes the applicable interfaces in detail and concludes with the underlying theory for the interfaces. Each section describes the applicable interfaces in detail and concludes with the underlying theory for the interfaces. which is suitable for solving thermal-conduction problems in thin structures. 16 | CHAPTER 1: INTRODUCTION . 2 Heat Transfer Theory This chapter discusses some fundamental heat transfer theory. Theory related to individual interfaces is discussed in those chapters. In this chapter: • Theory for the Heat Transfer User Interfaces • About the Heat Transfer Coefficients • About Highly Conductive Layers • Theory of Out-of-Plane Heat Transfer • Theory for the Bioheat Transfer User Interface • Theory for the Heat Transfer in Porous Media User Interface • About Handling Frames in Heat Transfer 17 . 18 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y . 3. It is characterized by the following mechanisms: • Conduction—Heat conduction takes place through different mechanisms in different media. 1 and Ref. Typical for heat conduction is that the heat flux is proportional to the temperature gradient. Theoretically it takes place in a gas through collisions of the molecules. see Ref. in a fluid through oscillations of each molecule in a “cage” formed by its nearest neighbors. For more detailed discussions of the fundamentals of heat transfer.Theory for the Heat Transfer User Interfaces The Heat Transfer Interfacetheory is described in this section. 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. In this section: • What is Heat Transfer? • The Heat Equation • A Note on Heat Flux • Heat Flux and Heat Source Variables • About the Boundary Conditions for the Heat Transfer User Interfaces • Radiative Heat Transfer in Transparent Media • Consistent and Inconsistent Stabilization Methods for the Heat Transfer User Interfaces • Moist Air Theory • About Heat Transfer with Phase Change • Theory for the Thermal Contact Feature • References for the Heat Transfer User Interfaces What is Heat Transfer? Heat transfer is defined as the movement of energy due to a difference in temperature. This section reviews the theory about the heat transfer equations in COMSOL Multiphysics® and heat transfer in general. • Convection—Heat convection (sometimes called heat advection) takes place through the net displacement of a fluid. the resulting heat equation is: T p T C p ------. typically described by a heat transfer coefficient. is a rather inconvenient quantity to measure and use in simulations. For a fluid. internal energy. Participating (or semitransparent) media absorb. commonly referred to as the principle of conservation of energy.+ u p + Q t T p t (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): 1 S = --. Therefore.------. -----. which transports the heat content in a fluid through the fluid’s own velocity. U. However. The Heat Equation The fundamental law governing all heat transfer is the first law of thermodynamics. u + u T 2 • Q contains heat sources other than viscous heating (SI unit: W/m3) 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 | 19 . emit and scatter photons. • Radiation—Heat transfer by radiation takes place through the transport of photons.+ u T = – q + :S – ---. T. Opaque surfaces absorb or reflect them. the basic law is usually rewritten in terms of temperature. The term convection (especially convective cooling and convective heating) also refers to the heat dissipation from a solid surface to a fluid. 1. Specific heat capacity at constant pressure is the amount of energy required to raise one unit of mass of a substance by one degree while maintained at constant pressure. This quantity is also commonly referred to as specific heat or specific heat capacity. which states that the conductive heat flux. Non symmetric tensor leads to unphysical results. is proportional to the temperature gradient: T q i = – k -------x i (2-2) where k is the thermal conductivity (SI unit: W/(m·K)). In deriving Equation 2-1.For a detailed discussion of the fundamentals of heat transfer. 20 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y . The equation also assumes that mass is always conserved. q. a number of thermodynamic relations have been used. see Ref. it has different values in different directions). the thermal conductivity can be anisotropic (that is. In a solid. which means that density and velocity must be related through: + v = 0 t The heat transfer interfaces use Fourier’s law of heat conduction. Then k becomes a tensor k xx k xy k xz k = k yx k yy k yz k zx k zy k zz and the conductive heat flux is given by qi = – T kij ------x j j Fourier’s law expect that the thermal conductivity tensor is symmetric. 5) u H 0 + – k T + u + q r (2-3) Above. If the velocity is set to zero. This section briefly describes the theory for the variables for Total Energy Flux and Total Heat Flux. H0 is the total enthalpy 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 | 21 . reordering the terms and ignoring viscous heating and pressure work puts the heat equation into a more familiar form: T C p ------. An analogous term arises from the internal viscous damping of a solid. There is hence heat flux and energy flux that are similar but not identical. 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. the equation governing pure conductive heat transfer is obtained: T C p ------.The second term on the right of Equation 2-1 represents viscous heating of a fluid. It is generally small for low Mach number flows.+ – k T = Q t 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. only variables available for results analysis and visualization. The conserved property is instead the total energy.+ C p u T = kT + Q t The Heat Transfer in Fluids physics solves this equation for the temperature. Inserting Equation 2-2 into Equation 2-1. chapter 3. A similar term can be included to account for thermoelastic effects in solids. T. 4. The approximations made do not affect the computational results. The operation “:” is a contraction and can in this case be written on the following form: a:b = anm bnm n m The third term represents pressure work and is responsible for the heating of a fluid under adiabatic compression and for some thermoacoustic effects. it is important that the dependence of Cp. Potential energy is therefore often excluded and the total energy flux is approximated by 1 u H + --. H. the enthalpy. Tref. For the evaluation of H to work. 6): 22 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y .1 H 0 = H + --. for example. In Equation 2-3 is the viscous stress tensor and qr is the radiative heat flux. and on the temperature are prescribed either via model input or as a function of the temperature variable. any value can be assigned to Href (Ref. These are evaluated by numerical integration. but it is in general rather difficult to derive. 7). It can be formulated in some special cases. In theory. which is the variable for the absolute pressure. is the enthalpy at reference temperature.15 K and pref is one atmosphere. and reference pressure. . 4). that dependency must be prescribed either via model input or by using the variable pA. 1 + ---. u u 2 where in turn H is the enthalpy. but for practical reasons. has the form (Ref. 5) T H = H ref + p 1 T - Cp dT + --. Tref is 298.refrefTrefprefref where the subscript “ref” indicates that the property is evaluated at the reference state. If Cp. . 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. The reference enthalpy. or depends on the pressure. -----T T ref p ref dp p (2-5) where p is the absolute pressure.refTref • Gasliquid: HrefCp. it is given a positive value according to the following approximations • Solid materials and ideal gases: HrefCp. u u – k T + u + q r 2 (2-4) For a simple compressible fluid. Href. in Equation 2-3 is the force potential.4 in Ref. 7). The two integrals in Equation 2-5 are sometimes referred to as the sensible enthalpy (Ref. for gravitational effects (Chapter 1. TO T A L H E A T F L U X The total heat flux vector is defined as (Ref. pref. Since everything else is constant. the variable named tflux can be analyzed using ht. Heat Flux and Heat Source Variables This section lists some predefined variables that are available to compute heat fluxes and sources. If the viscous heating on the other hand is included. There is a pressure drop along the channel that drives the flow. boundaries teflux Total Energy Flux Domains. Since there is no viscous heating and the walls are isolated. boundaries dflux Conductive Heat Flux Domains. boundaries turbflux Turbulent Heat Flux Domains. boundaries not applicable Radiative Heat Flux Domains 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 | 23 . This is a common approximation for low-speed flows. boundaries trlflux Translational Heat Flux Domains. Assume that the viscous heating is neglected. On the other hand UinUout. then HinHout (first law of thermodynamics) and UinUout (since work has been converted to heat). The walls are assumed to be insulated.tflux (as long as the physics interface prefix is ht). It is related to the enthalpy via p H = U + -- (2-7) What is the difference between Equation 2-4 and Equation 2-7? As an example. consider a channel with fully developed incompressible flow with all properties of the fluid independent of pressure and temperature. By default the Heat Transfer interface prefix is ht. TABLE 2-1: HEAT FLUX VARIABLES VARIABLE NAME GEOMETRIC ENTITY LEVEL tflux Total Heat Flux Domains. Equation 2-4 shows that the energy flux into the channel is higher than the energy flux out of the channel. so the heat flux into the channel is equal to the heat flux going out of the channel.uU – k T + q r (2-6) where U is the internal energy. All the variable names start with the physics interface prefix. As an example. boundaries aflux Convective Heat Flux Domain. Equation 2-5 gives that HinHout. Extrapolated Boundaries ndflux Normal Conductive Heat Flux. Downside Interior boundaries ntrlflux_u Internal Normal Translational Heat Flux. Extrapolated. Downside Interior boundaries ntflux_u Internal Total Normal Heat Flux. Downside Interior boundaries nteflux_u Internal Normal Total Energy Flux. Extrapolated Boundaries ndflux_u Internal Normal Conductive Heat Flux. Upside Interior boundaries ndflux_d Internal Normal Conductive Heat Flux. Extrapolated. Upside Interior boundaries ntflux_d Internal Total Normal Heat Flux. Upside Interior boundaries naflux_d Internal Normal Convective Heat Flux. Upside Interior boundaries ccflux_d ccflux_z rflux_u rflux_d rflux_z q0_u q0_d q0_z 24 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y . Extrapolated Boundaries naflux Normal Convective Heat Flux Boundaries ntrlflux Normal Translational Heat Flux Boundaries nteflux Normal Total Energy Flux. boundaries Out-of-Plane Heat Flux Out-of-plane domains (1D and 2D) ntflux Normal Total Heat Flux. Upside Interior boundaries ntrlflux_d Internal Normal Translational Heat Flux.TABLE 2-1: HEAT FLUX VARIABLES VARIABLE NAME GEOMETRIC ENTITY LEVEL ccflux_u Convective Out-of-Plane Heat Flux Out-of-plane domains (1D and 2D) Radiative Out-of-Plane Heat Flux Out-of-plane domains (1D and 2D). Downside Interior boundaries naflux_u Internal Normal Convective Heat Flux. Extrapolated. Extrapolated. Their definition can vary depending on the active physics nodes and selected properties. Accurate Exterior boundaries ntflux_acc Normal Total Heat Flux. Accurate. Downside Interior boundaries nteflux_acc_u Internal Normal Total Energy Flux. Downside Interior boundaries ntflux_acc_u Internal Normal Total Heat Flux. Accurate Exterior boundaries nteflux_acc Normal Total Energy Flux. Extrapolated.TABLE 2-1: HEAT FLUX VARIABLES VARIABLE NAME GEOMETRIC ENTITY LEVEL nteflux_d Internal Normal Total Energy Flux. Upside Interior boundaries ndflux_acc_d Internal Normal Conductive Flux. Accurate. Accurate. Upside Interior boundaries nteflux_acc_d Internal Normal Total Energy Flux. Accurate. Downside Interior boundaries ndflux_acc Normal Conductive Flux. Downside Interior boundaries rflux Radiative Heat Flux Boundaries ccflux Convective Heat Flux Boundaries Qtot Domain Heat Sources Domains Qbtot Boundary Heat Sources Boundaries Ql Line heat source (Line and Point Heat Sources) Edges Qp Point heat source (Line and Point Heat Sources) Points DOMAIN HEAT FLUXES On domains the heat fluxes are vector quantities. Accurate. Accurate Exterior boundaries ndflux_acc_u Internal Normal Conductive Flux. 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 | 25 . Upside Interior boundaries ntflux_acc_d Internal Normal Total Heat Flux. Accurate. keff = k + kT. the total heat flux is defined by: tflux = aflux + dflux Conductive Heat Flux The conductive heat flux variable. heat transfer in fluids). for example heat transfer in solids and biological tissue domains. is evaluated using the temperature gradient and the effective thermal conductivity: dflux = – k eff T When the out-of-plane property is activated (1D and 2D only) the conductive heat flux is defined as follows: • In 2D (dz is the domain thickness): dflux = – d z k eff T • In 1D (Ac is the cross-section area): dflux = – A c k eff T In the general case keff is the thermal conductivity. For heat transfer in fluids with turbulent flow. See Radiative Heat Flux to evaluate the radiative heat flux. the total heat flux is defined by: tflux = trlflux + dflux For fluid domains (for example.Total Heat Flux On domains the total heat flux. where kT is the turbulent thermal conductivity. dflux. For accuracy reasons the radiative heat flux is not included. tflux. For solid domains. 26 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y . k. corresponds to the conductive and convective heat flux. For heat transfer in porous media. E. Turbulent Heat Flux The turbulent heat flux variable. E: aflux = uE When the out-of-plane property is activated (1D and 2D only) the convective heat flux is defined as follows: • In 2D (dz is the domain thickness): aflux = d z uE • In1D (Ac is the domain thickness): aflux = A c uE The internal energy. turbflux. turbflux = – k T T Convective Heat Flux The convective heat flux variable. The variable name is trlflux. CFD Module. Corrosion Module. keff = keq. Electrochemistry Module. or Subsurface Flow Module. Translational Heat Flux Similar to convective heat flux but defined for solid domains with translation. The Heat Transfer in Porous Media feature requires one of the following products: Batteries & Fuel Cells Module. Electrodeposition Module. is defined by: • ECpT for solid domains • ECpT for ideal gas fluid domains • EHp for other fluid domains where H is the enthalpy defined by Equation 2-5. Chemical Reaction Engineering Module. is defined using the internal energy. Heat Transfer Module. enables access to the part of the conductive heat flux that is due to the turbulence. where keq is the equivalent conductivity defined in the Heat Transfer in Porous Media feature. aflux. 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 | 27 . is defined as uu H 0 = H + -----------2 Radiative Heat Flux In participating media. is not available for analysis on domains because it is much more accurate to evaluate the radiative heat source: Qr = qr OUT-OF-PLANE DOMAIN FLUXES When the out-of-plane property is activated (1D and 2D only). is defined when viscous heating is enabled: teflux = uH 0 + dflux + u where the total enthalpy. • In 2D: 4 4 upside: rflux_u = u T amb u – T 28 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y . qr. is generated by the Out-of-Plane Convective Heat Flux feature. H0. • In 2D: upside: ccflux_u = h u T ext u – T downside: ccflux_d = h d T ext d – T • In 1D: ccflux_z = h z T ext z – T Radiative Out-of-Plane Heat Flux The radiative out-of-plane heat flux. teflux. ceflux. out-of-plane domain fluxes are defined. If there are no out-of-plane physics features. rflux.Total Energy Flux The total energy flux. they are evaluated to zero. is generated by the Out-of-Plane Radiationfeature. the radiative heat flux. Convective Out-of-Plane Heat Flux The convective out-of-plane heat flux. is generated by the Out-of-Plane Heat Flux feature. Extrapolated The variable ntflux is defined by: ntflux = mean tflux n Normal Conductive Heat Flux. • In 2D: upside: q0_u = h u T ext u – T downside: q0_d = h d T ext d – T • In 1D: q0_z = h z T ext z – T BOUNDARY HEAT FLUXES All the domain heat fluxes (vector quantity) are also available as boundary heat fluxes. The boundary heat fluxes are then equal to the mean value of the adjacent domains.4 4 downside: rflux_d = d T amb d – T • In 1D: 4 4 rflux_z = z T amb z – T Out-of-Plane Heat Flux The convective out-of-plane heat flux. q0. In addition normal boundary heat fluxes (scalar quantity) are available on boundaries. Normal Total Heat Flux. Extrapolated The variable ndflux is defined by: ndflux = mean dflux n Normal Convective Heat Flux The variable naflux is defined by: naflux = mean aflux n Normal Translational Heat Flux The variable ntrlflux is defined by: 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 | 29 . is a scalar quantity defined as: 4 4 4 rflux = T amb – T + G – T + q w where the terms respectively account for surface-to-ambient radiative flux. rflux. Downside The variable ndflux_d is defined by: 30 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y . is defined as the contribution from the Convective Heat Flux boundary condition: ccflux = h T ext – T When the out-of-plane property is activated (1D and 2D only) the convective heat flux is defined as follows: • In 2D (dz is the domain thickness): ccflux = d z h T ext – T • In 1D (Ac is the cross section area): ccflux = A c h T ext – T INTERNAL BOUNDARY HEAT FLUXES The internal normal boundary heat fluxes (scalar quantity) are available on interior boundaries. Upside The variable ndflux_u is defined by: ndflux_u = up dflux n Internal Normal Conductive Heat Flux. Extrapolated. Internal Normal Conductive Heat Flux. and radiation in participating net flux. They are calculated using the upside and the downside value of heat fluxes from the adjacent domains. surface-to-surface radiative flux.ntrlflux = mean trlflux n Normal Total Energy Flux. Convective Heat Flux Convective heat flux. Extrapolated. Extrapolated The variable nteflux is defined by: nteflux = mean teflux n Radiative Heat Flux On boundaries the radiative heat flux. ccflux. Upside The variable naflux_u is defined by: naflux_u = up aflux n Internal Normal Convective Heat Flux. Extrapolated. Downside The variable nteflux_d is defined by: nteflux_d = down teflux n Internal Total Normal Heat Flux. Downside The variable ntlux_d is defined by: ntflux_d = down tflux n T H E O R Y F O R T H E H E A T TR A N S F E R U S E R I N T E R F A C E S | 31 . Downside The variable ntrlflux_d is defined by: ntrlflux_d = down trlflux n Internal Normal Total Energy Flux. Upside The variable ntflux_u is defined by: ntflux_u = up tflux n Internal Total Normal Heat Flux. Downside The variable naflux_d is defined by: naflux_d = down aflux n Internal Normal Translational Heat Flux. Extrapolated. Upside The variable nteflux_u is defined by: nteflux_u = up teflux n Internal Normal Total Energy Flux. Upside The variable ntrlflux_u is defined by: ntrlflux_u = up trlflux n Internal Normal Translational Heat Flux. Extrapolated.ndflux_d = down dflux n Internal Normal Convective Heat Flux. ACCURATE FLUXES Normal Conductive Flux. Upside The variable ntflux_acc_u is defined by: ntflux_acc_u = ndflux_acc_u + naflux_u + ntrlflux_u Normal Total Energy Flux.T Internal Normal Conductive Flux. Upside The variable nteflux_acc_u is defined by: 32 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y . Accurate The variable nteflux_acc is defined by: ntflux_acc = nteflux – ndflux + ndflux_acc Internal Normal Total Energy Flux. Accurate.T Normal Total Heat Flux. Accurate The variable ndflux_acc is defined by: ndflux_acc = – dep.T if the adjacent domain is on the downside if the adjacent domain is on the upside Internal Normal Conductive Flux. Accurate The variable ntflux_acc is defined by: ntflux_acc = ndflux_acc + naflux + ntrlflux Internal Normal Total Heat Flux.uflux.dflux. Accurate. Accurate. Accurate.dflux. Downside The variable nteflux_acc_d is defined by: nteflux_acc_d = nteflux_d – ndflux_d + ndflux_acc_d Internal Normal Total Energy Flux. Accurate.T ndflux_acc = – dep. Downside The variable ndflux_acc_d is defined by: ndflux_acc_d = dep.uflux. Downside The variable ntflux_acc_d is defined by: ntflux_acc_d = ndflux_acc_d + naflux_d + ntrlflux_d Internal Normal Total Heat Flux. Upside The variable ndflux_acc_u is defined by: ndflux_acc_u = dep. Accurate. The out-of-plane contributions (convective heat flux. Qtot (SI unit: W/m3).tot (SI unit: W/m2). • 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). BOUNDARY HEAT SOURCES The sum of the boundary heat sources added by different boundary conditions is available in one variable. LINE AND POINT HEAT SOURCES The sum of the line heat sources is available in a variable called Ql (SI unit: W/m). Qb.nteflux_acc_u = nteflux_u – ndflux_u + ndflux_acc_u DOMAIN HEAT SOURCES The sum of the domain heat sources added by different physics features are available in one variable. • Qs: which is the boundary heat source added by a Layer Heat Source subfeature of a highly conductive layer. heat flux. This variable Qbtot is the sum of: • Qb which is the boundary heat source added by the Boundary Heat Source boundary condition. the SI unit for the variable Qp is W/m. and the blood contribution in Bioheat are considered flux so that they are not added to Qtot. In 2D axisymmetric models. 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 | 33 . This variable Qtot is the sum of: • Q which is the heat source added by Heat Source(described for the Heat Transfer interface and Electromagnetic Heat Source (described for the Joule Heating interface in the COMSOL Multiphysics Reference Manual) feature. • Qmet which is the heat source added by the Bioheat feature. and radiation). The sum of the point heat sources is available in a variable called Qp (SI unit: W). O VE R R I D I N G M E C H A N I S M F O R H E A T TR A N S F E R B O U N D A R Y C O N D I T I O N S This section includes information for features that may require additional modules. q0. but the most common situation is that h represents the effect of an exterior fluid cooling or heating the surface of solid. normal to the boundary. The inward heat flux. A common type of heat flux boundary conditions are those where q0h·TinfT. a phenomenon often referred to as convective cooling or heating. represents all the physics occurring between the boundary and “far away. 34 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y . The Heat Transfer Module contains a set of correlations for convective heat flux and heating. • q0 is inward heat flux (SI unit: W/m2). is often a sum of contributions from different heat transfer processes (for example. • n is the normal vector of the boundary. The specified temperature is of a constraint type and prescribes the temperature at a boundary: T = T0 on while the latter specifies the inward heat flux –n q = q0 on where • q is the conductive heat flux vector (SI unit: W/m2) where q = kT. The special case q0 0 is called thermal insulation. where Tinf is the temperature far away from the modeled domain and the heat transfer coefficient. See About the Heat Transfer Coefficients.About the Boundary Conditions for the Heat Transfer User Interfaces TE M P E R A T U RE 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.” It can include almost anything. h. radiation and convection). Heat Flux and Thermal Insulation). Incident Intensity. 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 | 35 . • Locate the column that corresponds to the group of B. Continuity on Interior Boundaries • Thin Thermally Resistive Layers. Heat Flux and Highly Conductive Layer). Thermal Contact TABLE 2-2: OVERRIDING RULES FOR HEAT TRANSFER BOUNDARY CONDITIONS A\B 1 2 3 1-Temperature X X X 2-Thermal Insulation X X 3-Highly Conductive Layer X 4-Heat Flux X 4 5 6 7 8 X X X X 5-Boundary heat source 6-Surface-to-surface radiation X X 7-Opaque Surface 8-Thin Thermally Resistive Layer X X X When there is a boundary condition A above a boundary condition B in the model tree and both conditions apply to the same boundary.Many boundary conditions are available in heat transfer. Surface-to-Ambient Radiation • Opaque Surface. Prescribed Radiosity. Some of them can be associated (for example. Radiation Group • Surface-to-Surface Radiation. Table 2-2 gives the overriding rules for these groups. Inflow Heat Flux • Thermal Insulation. Convective Heat Flux • Boundary Heat Source. Others cannot be associated (for example. Symmetry. Re-radiating Surface. Several categories of boundary condition exist in heat transfer. Periodic Heat Condition • Highly Conductive Layer • Heat Flux. Convective Outflow. • Temperature. Open Boundary. use Table 2-2 to determine if A is overridden by B or not: • Locate the line that corresponds to the A group (see above the definition of the groups). In the table above only the first member of the group is displayed. Radiative Heat Transfer in Transparent Media This discussion so far has considered heat transfer by means of conduction and convection. Then a Surface-to-Surface Radiation boundary condition is applied on the same boundary afterward. Consider an environment with fully transparent or fully opaque objects. If it contains an X. But they might be overridden. • 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. the boundary conditions contribute. That means that they never override any other boundary condition. • Convective Heat Flux belongs to group 4. Example 1 Consider a boundary where Temperature is applied. Then a Symmetry boundary condition is applied on the same boundary afterward. • Surface-to-surface radiation belongs to group 6. • Temperature belongs to group 1. Example 2 Consider a boundary where Convective Heat Flux is applied. • Symmetry belongs to group 2.• If the corresponding cell is empty A and B contribute. 36 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y . if Symmetry followed by Convective Heat Flux is added. The third mechanism for heat transfer is radiation. B overrides A. Thermal radiation denotes the stream of electromagnetic waves emitted from a body at a certain temperature. In Example 2 above. Group 4 and group 5 boundary conditions are always contributing. • The cell on the line of group 1 and the column of group 6 is empty so Temperature and Surface-to-Surface radiation contribute. This is true for most solid bodies. for ideal gray bodies. q. and temperature T. q = 1 – G – T 4 (2-10) Most opaque bodies also behave as ideal gray bodies. reflectivity . A point x is located on a surface that has an emissivity . J. absorptivity . q is given by: 4 q = G – T (2-12) 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 | 37 . is then given the difference between the irradiation and the radiosity: q = G–J (2-9) Using Equation 2-8 and Equation 2-9 J can be eliminated and a general expression is obtained for the net inward heat flux into the opaque body based on G and T. and the reflectivity is therefore given from the following relation: = = 1– (2-11) Thus. which means that no radiation is transmitted through the body. G.T . The total outgoing radiative flux x is named the radiosity.T Figure 2-1: Arriving irradiation (left). The radiosity is the sum of the reflected radiation and the emitted radiation: J = G + T 4 (2-8) The net inward radiative heat flux. The total arriving radiative flux at x is named the irradiation. leaving radiosity (right).DERIVING THE RADIATIVE HEAT FLUX J =G + T4 G x x . Assume the body is opaque. meaning that the absorptivity and emissivity are equal. Consider Figure 2-1. and zero reflectivity. • The ambient surroundings behave as a blackbody. These assumptions allows the irradiation to be explicitly expressed as 4 G = T amb (2-13) Inserting Equation 2-13 into Equation 2-12 results in the net inward heat flux for surface-to-ambient radiation 4 4 q = T amb – T (2-14) For boundaries where a surface-to-ambient radiation is specified. COMSOL Multiphysics adds this term to the right-hand side of Equation 2-14. SURFACE-TO-AMBIENT RADIATION Surface-to-ambient radiation assumes the following: • The ambient surroundings in view of the surface have a constant temperature. . click the Show button ( 38 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y ) and select Stabilization. • Theory for the Radiation in Participating Media User Interface • Radiation and Participating Media Interactions Consistent and Inconsistent Stabilization Methods for the Heat Transfer User Interfaces The different versions of the Heat Transfer interface have this advanced option to set the stabilization method parameters. The Heat Transfer interface supports both types of radiation. is different for each of them. This section provides information pertaining to these options. This means that the emissivity and absorptivity are equal to 1. 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. Tamb. To display this section. G. but the irradiation term.This is the equation used as a radiation boundary condition. This is often enough to obtain a smooth numerical solution provided that the exact solution of the heat equation does not contain any discontinuities. These are consistent stabilization methods. The field for the tuning parameter id then becomes available. the added diffusion definitely dampens the effects of oscillations. which is why isotropic diffusion is an inconsistent stabilization method. Continuous Casting: Model Library path Heat_Transfer_Module/ Thermal_Processing/continuous_casting Streamline Diffusion Streamline diffusion is active by default and should remain active for optimal performance for heat transfer in fluids or other applications that include a convective or translational term. Still. Crosswind Diffusion Streamline diffusion introduces artificial diffusion in the streamline direction. By default there is no isotropic diffusion.CONSISTENT STABILIZATION This section contains two consistent stabilization methods: streamline diffusion and crosswind diffusion. however. select the Isotropic diffusion check box. The 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 | 39 . A stabilization method is active when the corresponding check box is selected. in the crosswind direction. Crosswind diffusion addresses these spurious oscillations by adding diffusion orthogonal to the streamline direction—that is. INCONSISTENT STABILIZATION This section contains one inconsistent stabilization method: isotropic diffusion. The consistent stabilization methods take effect for fluids and for solids with Translational Motion. To add isotropic diffusion. but try to minimize the use of isotropic diffusion. At sharp gradients. which means that they do not perturb the original transport equation. Adding isotropic diffusion is equivalent to adding a term to the physical diffusion coefficient. undershoots and overshoots can occur in the numerical solution. This means that the original problem is not solved. Without condensation.= -------------ma pa M a (2-15) where pv is the water vapor partial pressure.25.default value is 0. The Moisture content represents a ratio of mass. The following theory assumes that the moist air is an ideal gas. respectively. pa is the dry air partial pressure. and Ma and Mv are the molar mass of dry air and water vapor. Moist Air Theory For the Heat Transfer in Fluids physics. increase or decrease the value of id to increase or decrease the amount of isotropic stabilization. Heat Transfer in Fluids HUMIDITY Moisture Content The moisture content (also called mixing ratio or humidity ratio) is defined as the ratio of water vapor mass mv to dry air mass ma: pv Mv mv x vap = ------. and it is thus a dimensionless number (SI unit: 1). See Show Stabilization and Stabilization Techniques in the COMSOL Multiphysics Reference Manual. the moisture content is not affected by temperature and pressure. the moist air functionality is provided to calculate the relative humidity and to deduce if there is condensation. Relative Humidity The relative humidity of an air mixture is expressed as follows: pv = -------p sat 40 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y (2-16) . Then the thermodynamical properties of moist air can be deduced through the mixture formula described below. the total pressure of a mixture of gases is the sum of all the partial pressures of each individual gas. The Reference relative humidity cannot be greater than one. it means that the vapor is saturated and that water vapor will condense. This quantity is very useful to study the condensation as it defines the boundary between the liquid phase and the vapor phase. above which value the water vapor is condensing. However. the Reference relative humidity value is forced to be one. If the value is greater than one. for a same quantity of moisture content. p=pv+pa where pa is the dry air partial pressure. the relative humidity changes with temperature and pressure. it has to be at the same temperature and pressure conditions. The relative humidity formulation is often used to quantify humidity. This Reference relative humidity associated to the Reference temperature and the Reference pressure are used to calculate the moisture content. In fact. when the relative humidity reaches unity. where 0 corresponds to dry air and 1 to a water vapor-saturated air. According to Dalton’s law. so in order to compare different values of . The Reference relative humidity (SI unit: 1) is a quantity defined between 0 and 1.where pv is the water vapor partial pressure and psat is the saturation pressure of water vapor. The condensation area cannot be simulated. Specific Humidity The specific humidity is defined as the ratio of water vapor mv to the total mass mtot=mv+ma: 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 | 41 . that is. mv = ----------m tot (2-17) As the water vapor only accounts for a few percent in the total mass. the moisture content and the specific humidity are very close: xvap (only for low values). Mv is the molar mass of water vapor. The water vapor concentration is defined in this SI unit: mol/m3. T is the temperature. The saturation pressure can be defined using the Clausius-Clapeyron formulation of the vaporization-condensation equilibrium. the saturation concentration is defined as follows: p sat T c sat = -----------------RT SATURATION STATE The saturation state is reached when the relative humidity reaches one. Under ideal gas hypothesis and considering only the gas volume: h fg M v p dp ------= ------------------2 dT RT (2-19) where p is the pressure. According to the ideal gas hypothesis. hfg is the latent heat of vaporization. For bigger values of . the two quantities are more precisely related by: x vap = ------------1– Concentration The concentration is defined by: nv c v = -----V (2-18) where nv is the amount of water vapor in mol and V is the total volume. It means that the partial pressure of the water vapor is equal to the saturation pressure (which depends on the temperature too). and R is the universal gas constant. 42 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y . By integrating.– --------- R T T ref sat (2-20) where the reference values are: pref101325Pa (1 atm).= --------------------p n tot p (2-21) pv p sat nv X v = --------.= ----. Tref373.26106J/kg. is the relative humidity.= -----------n tot p p (2-22) where na and nv are respectively the amount of dry air and water vapor.= ----. where pa and pv are the partial pressure of dry air and water vapor. and hfg=2. MOIST AIR PROPERTIES The thermodynamical properties of moist air can be found with some mixture laws.+ x vap Ma (2-23) 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 | 43 . p is the pressure. Preliminary Definitions Molar Fraction The molar fraction of dry air Xa and the molar fraction of water vapor Xv are defined such as: p – p sat pa na X a = --------. you can obtain the saturation pressure equation: h fg M v 1 1 p sat = p ref exp ---------------. XaXv1 Relation Between Relative Humidity And Moisture Content Moisture content and relative humidity can be linked with the following expression: x vap p = --------------------------------------M v p sat -------. and psat is the saturation pressure.15K (100 °C). The temperature and saturation pressure can easily be deduced from this formulation. ntot is the total amount of moist air in mol. --------. Dynamic viscosity: According to Ref.v Mm Mm (2-25) where Mm represents the mixture molar fraction and is defined by Mm=XaMa+XvMv and where cp. the heat capacity at constant pressure of a mixture is: Ma Mv c p.v are the heat capacity at constant pressure of dry air and steam. Density: According to the ideal gas law.X a c p. the density mixture m expression is defined as follows: p m = -------. respectively. M a X a + M v X v RT (2-24) where Ma and Mv are respectively the molar mass of dry air and water vapor.v where ij is given by 44 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y (2-26) . 10.X v c p. respectively. Specific heat capacity at constant pressure: According to Ref.a + --------.v j = a. and Xa and Xv are the molar fraction of dry air and water vapor. the accuracy lost by this assumption is small as the pure steam represents a small fraction.Mixture Properties The thermodynamical properties are built through a mixture formula. 9 and Ref. 10. In fact.m = --------. The expressions depend on dry air properties and pure steam properties and are balanced by the mass fraction. respectively.a and cp. The ideal gas assumption sets the compressibility factor and the enhancement factor to the unity. the dynamic viscosity is defined as follows: m = Xi i -------------------------X j ij i = a. 15K for steam properties. The steam properties are based on the Industrial Formulation IAPWS-IF97.v (2-27) j = a.1 --- ij 1 2 --- i 2 Mj 4 1 + ----- ------- j M i = ----------------------------------------------1 Mi 8 1 + ------- M j --2 Here. • Condensation indicator condInd. The polynomials have been computed according to Ref. The valid temperature range is 200KT1200K for dry air properties and 273. 10. • Relative humidity phi. 1 for dry air properties and Ref. 9 and Ref. respectively. 8 for pure steam properties.15KT873. 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 | 45 . • Concentration of water vapor c. Thermal conductivity: According to Ref. • Vapor mass fraction omega_moist. the thermal conductivity of the mixture is defined similarly: km = Xi ki -------------------------X j ij i = a.v where ka and kv are the thermal conductivity of dry air and steam. a and v are the dynamic viscosity of dry air and steam. respectively. this indicator is set to 1 if condensation has been detected (and 0 if not. Pure Component Properties The dry air and steam properties used to define the mixture properties are temperature-dependent high-order polynomials. This variable corresponds to the calculated with the system temperature and pressure). Results and Analysis Variables The following variables are provided to display the related quantities: • Moisture content xvap. Functions Three functions are defined and can be used as feature parameters as well as in post processing. T. . pA).fpsat(T).fluid1.fluid1. The enthalpy H is expressed by: H = 1 – H phase1 + H phase2 where Hphase1 and Hphase2 are the enthalpies when the material is in phase 1 or in phase 2. representing the fraction of phase after transition.fluid1. where T is the temperature (SI unit: K). Instead of adding a latent heat L in the energy balance equation when the material reaches its phase change temperature Tpc. T is the temperature (SI units: K) and pA is the pressure (SI units Pa). pA). It returns the corresponding water vapor concentration (SI unit: mol/m^3). which is equal to 0 before Tpc T2 and to 1 after Tpc T 2. where RH is the relative humidity 0 1 . T is the temperature (SI unit: K) and pA is the pressure (SI unit: Pa). Differentiating with respect to temperature. • ht. It returns the saturation pressure (SI unit: Pa). it is assumed that the transformation occurs in a temperature interval between Tpc T2 and Tpc T2.fc(RH. It returns the moisture content (SI unit: 1). the material phase is modeled by a smoothed function.fxvap(RH. About Heat Transfer with Phase Change The Heat Transfer with Phase Change node is used to solve the heat equation after specifying the properties of a phase change material according to the apparent heat capacity formulation. this equality provides the following formula for the specific heat capacity: d C p = 1 – d H p phase1 + d H p phase2 + H phase2 – H phase1 dT dT dT 46 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y . respectively. • ht. The concentration computation assumes that the ideal gas assumption is valid • ht.T. where RH is the relative humidity 0 1 . In this interval. respectively. when is the Heaviside function (equal to 0 before Tpc d and to 1 after Tpc). is the Dirac pulse. the apparent heat capacity Cp. It should not depend on the temperature. used in the heat equation. The specific heat capacity is the sum of an equivalent heat capacity Ceq: C eq = 1 C p phase1 + 2 C p phase2 and the distribution of latent heat CL: C L T = H phase2 – H phase1 d dT In the ideal case.which can be rewritten as: C p = 1 C p phase1 + 2 C p phase2 + H phase2 – H phase1 d dT Here. 1 and 2 are equal to 1 and . The latent heat distribution CL is approximated by CL T = L d dT so that the total heat per unit mass released during the phase transformation coincides with the latent heat: T T pc + -------2 C L T dT T T pc – -------2 = L T T pc + -------2 d dT T T pc – -------. CL is the enthalpy jump L at temperature Tpc that is added when you have a pure substance. dT Therefore. Finally. is given by: C p = 1 C p phase1 + 2 C p phase2 + C L The equivalent heat conductivity and volumetric heat capacity reduce to: 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 | 47 .d T 2 = L The latent heat L may depend on the absolute pressure. dependent on the temperature. When the material density is not constant over time.k = 1 k phase1 + 2 k phase2 and C p = 1 phase1 C p phase1 + 2 phase1 C p phase2 The density is thus given by: 1 phase1 C p phase1 + 2 phase1 C p phase2 = --------------------------------------------------------------------------------------------------- 1 C p phase1 + 2 C p phase2 To satisfy energy and mass conservation in phase change models. particular attention should be paid to the density in time simulations. if Moving Mesh is not used. it is recommended to set material density to a constant value. The heat fluxes at the upside and downside boundaries depend on the temperature difference according to the relations: – n d – k d T d = – h T u – T d + rQ fric – n u – k u T u = – h T d – T u + 1 – r Q fric 48 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y . Theory for the Thermal Contact Feature The Thermal Contact feature has correlations to evaluate the joint conductance at two contacting surfaces. Moving Mesh User Interface (described in the COMSOL Multiphysics Reference Manual) has the tools to deform the geometry accordingly. for example. However. volume change is expected. u Figure 2-2: Contacting surfaces at the microscopic level. ------- asp H c Here. hc. p is the contact pressure and kcontact is the harmonic mean of the contacting surface conductivities: 2k u k d k contact = -----------------ku + kd 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 | 49 . the gap conductance. Hc is the microhardness of the softer material. from the contact spots.95 h c = 1. The joint conductance h has three contributions: the constriction conductance. contact is made at a finite number of spots as in Figure 2-2. due to the fluid at the interstitial space. masp.25k contact ----------.u Y asp.16 in Ref. 11): asp = u2 asp + d2 asp m asp = m u2 asp + m d2 asp CONSTRICTION CONDUCTANCE Cooper-Mikic-Yovanovich (CMY) Correlation The Cooper-Mikic Yovanovich (CMY) correlation is valid for isotropic rough surfaces and assumes plastic deformation of the surface asperities. hg. It relates hc to the asperities and pressure load at the contact interface: m asp p 0.At a microscopic level. The RMS values asp and masp are (4. and the radiative conductance. hr: h = hc + hg + hr SURFACE ASPERITIES The microscopic surface asperities are characterized by the average height uasp and dasp and the average slope muasp and mdasp. 071c2 p p ------.0 -------- – 0.94 m asp 2p h c = 1.+ --------------Eu Ed E contact where Eu and Ed are the Young’s moduli of the two contacting surfaces and u and d are the Poisson’s ratios.= -------------------------------------------------- c2 Hc asp . Econtact is an effective Young’s modulus for the contact interface.60 GPa.16.m asp c 1 1.16.442 -------c1 The Brinell hardness is denoted by HB and H0 is equal to 3.178 GPa. 50 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y . satisfying (4. kd) is not isotropic.0 – 5. the Vickers correlation coefficient and size index: 1 ---------------------------------- 1 + 0.1 in Ref. It gives hc by the following relation: 0.54k contact ----------. 11): c1 HB 3 HB HB 2 ------= 4.+ 4.62 -------- 0 Here 0 is equal to 1 µm.30 and 7. ------------------------ asp mE contact Here.61 -------- H 0 H0 H0 H0 HB c 2 = – 0. c1 and c2 are given by the correlations below (4.16.37 + 0. 11): 1 – u2 1 – d2 1 -----------------= --------------. 11) for the relative pressure using c1 and c2. it is replaced by its normal conductivity nTkun (resp. The relative pressure pHc can be evaluated by specifying Hc directly or using the following relation (4. For materials with Brinell hardness between 1. Mikic Elastic Correlation The Mikic correlation is valid for isotropic rough surfaces and assumes elastic deformations of surface asperities.3 in Ref.When ku (resp. nTkdn).1 in Ref.77 -------. and Tg is the gap temperature equal to: Tu + Td T g = -------------------2 RADIATIVE CONDUCTANCE At high temperatures. Qfric. If the two bodies are identical. is a gas property parameter (equal to 1. Y denotes the mean separation thickness (see Figure 2-2).GAP CONDUCTANCE The gap conductance due to interstitial fluid cannot be neglected for high fluid thermal conductivity or high contact pressure. D is the average gas particle diameter. kB is the Boltzmann constant. T u3 + T u2 T d + T u T d2 + T d3 u + d – u d which implies that: u d h r T u – T d = ---------------------------------.5 so that half of the friction heat goes to each surface. T d4 – T u4 u + d – u d THERMAL FRICTION The friction heat. is partitioned into rQfric and 1rQfric at the contact interface. above 600 °C. pg is the gas pressure (often the atmospheric pressure). The parallel-plate gap gas correlation assumes that the interstitial fluid is a gas and defines hg by: kg h g = -----------------Y + Mg Here kg is the gas conductivity. However.7 for air). radiative conductance needs to be considered. and Mg is the gas parameter equal to: M g = kB Tg = -----------------------2 2D p g In these relations. T u4 – T d4 u + d – u d u d h r T d – T u = ---------------------------------. r and 1r would be 0. in the general case where the two bodies 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 | 51 . is the contact thermal accommodation parameter. is the gas mean free path. The gray-diffuse parallel plate model provides the following formula for hr: u d h r = ---------------------------------. The Charron’s relation (Ref.are made of different materials. Thermal Contact Contact Switch: Model Library path Heat_Transfer_Module/ Thermal_Contact_and_Friction/contact_switch 52 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y . 1r is: 1 1 – r = --------------1 + u u = d C p d k d ------------------------ u C p u k u For anisotropic conductivities. 12) defines r as: 1 r = --------------1 + d d = u C p u k u ------------------------ d C p d k d and symmetrically. the partition rate might not be 0.5. ku). nTkun) replaces kd (resp. nTkdn (resp. About the Heat Transfer Coefficients One of the most common boundary conditions when modeling heat transfer is convective cooling or heating whereby a fluid cools or heats a surface by natural or forced convection. and the surface temperature—and. The first method is simple. The main difficulty in using heat transfer coefficients is in calculating or specifying the appropriate value of the h coefficient. In this section: • Heat Transfer Coefficient Theory • Nature of the Flow—the Grashof Number • Heat Transfer Coefficients — External Natural Convection • Heat Transfer Coefficients — Internal Natural Convection A B O U T T H E H E A T TR A N S F E R C O E F F I C I E N T S | 53 . for forced-convection. it is possible to model this process in two ways: • Use a heat transfer coefficient on the surfaces • Extend the model to describe the flow and heat transfer in the surrounding fluid The second approach is the correct approach if the geometry or the external flow is complicated. In addition. the geometrical configuration affects the coefficient. However. The Heat Transfer interface provides built-in functions for heat transfer coefficients. also on the fluid’s flow rate. That coefficient depends on the fluid’s material properties. such a simulations can become costly. both in terms of computational time and memory requirement. the heat flux is described by the equation – n – k T = h T inf – T where h is a heat transfer coefficient and Tinf the temperature of the external fluid far from the boundary. In principle. For most engineering purposes. The Heat Transfer Module includes the Conjugate Heat Transfer interface for this purpose. yet powerful and efficient. Convective heat flux is then modeled by specifying the heat flux on the boundaries that interface with the fluid as being proportional to the temperature difference across a fictitious thermal boundary layer. Mathematically. the use of these coefficients is an accurate and numerically efficient modeling approach. buoyancy forces induced by temperature differences and the thermal expansion of the fluid drive the flow. In natural convection. resulting in a total of eight types of convection. The difference between natural and forced convection is that in the latter case an external force such as a fan creates the flow.• Heat Transfer Coefficients — External Forced Convection • Heat Transfer Coefficients — Internal Forced Convection • The Heat Transfer Interface • Theory for the Heat Transfer User Interfaces Heat Transfer Coefficient Theory It is possible to divide convection heat flux into four main categories depending on the type of convection conditions (natural or forced) and on the type of geometry (internal or external convection flow). these four cases can all experience either laminar or turbulent flow conditions. Natural Forced External Internal Laminar Flow Turbulent Flow Figure 2-3: The eight possible categories of convective heat flux. 54 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y . In addition. as in Figure 2-3. Re. Gr. • U is the bulk velocity (SI unit: m/s). which is defined as the ratio between the buoyancy force and the viscous force. and the temperature scale (temperature difference).Heat transfer handbooks generally contain a large set of empirical and theoretical correlations for h coefficients. the velocity is largely unknown for internally driven flows such as natural convection. Pr. This module includes a subset of them. • T is the temperature difference between surface and external fluid bulk (SI unit: K). such as forced convection. The expressions are based on the following set of dimensionless numbers: • The Nusselt number. ReLU L/ • The Prandtl number. the fluid’s physical properties. Nature of the Flow—the Grashof Number In cases of externally driven flow. RaGr Pr 2 gCp T L3/(k) where • h is the heat transfer coefficient (SI unit: W/(m2·K)). Similar to the Reynolds number it requires the definition of a length scale. Ra)hL/k • The Reynolds number. NuL(Re. PrCp/k • The Rayleigh number. characterizes the flow. The Grashof number is defined as: 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 . In such cases the Grashof number. • Cp equals the heat capacity of the fluid (SI unit: J/(kg·K)). • is the thermal expansivity (SI unit: 1/K) Further. • is the fluid density (SI unit: kg/m3). • L is the characteristic length (SI unit: m). which describes the ratio of the inertial to viscous forces. • g is the acceleration of gravity (SI unit: m/s2). • is the dynamic viscosity (SI unit: Pa·s). • k is the thermal conductivity of the fluid (SI unit: W/(m·K)). However. Gr refers to the Grashof number. the flow’s nature is characterized by the Reynolds number. It describes the ratio of the internal driving force (buoyancy force) to a viscous force acting on the fluid. Heat Transfer Coefficients — External Natural Convection VE R T I C A L WA L L The correlations are equations 9.492k 1 + ------------------- C p (2-28) where L.825 + ------------------------------------------------------------- Ra L 10 9 9 / 16 8 / 27 L 0. represents the fluid’s dynamic viscosity. ------- T p which for an ideal gas reduces to = 1T The transition from laminar to turbulent flow occurs at a Gr value of 109. the height of the wall.492k 1 + ------------------ C p h = 2 k 0. 1: 0. ----------------------------------------------------------0. In general.27 in Ref. the coefficient of volumetric thermal expansion is given by 1 = – --. is the fluid’s coefficient of volumetric thermal expansion. is a correlation input and g T p C p T – T ext L 3 Ra L = --------------------------------------------------------------------------k 56 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y (2-29) . T0 equals the temperature of the surrounding air.387Ra L1 / 6 ---.67Ra L1 / 4 k.3 g T s – T 0 L GrL = ------------------------------------2 where g is the acceleration of gravity. 0. L is the length scale.68 + Ra L 10 9 L 9 / 16 4 / 9 0. and is the density. the flow is turbulent for larger values.26 and 9. Ts denotes the temperature of the hot surface. All material properties are evaluated at TText2.81 m/s2. RaL. These correlations are valid for 60° 60°.387Ra L1 / 6 ---.where in turn g is the acceleration of gravity equal to 9. The definition of Raleigh number..825 + ------------------------------------------------------------- Ra L 10 9 9 / 16 8 / 27 L 1 + 0. correlations for inclined walls are only satisfactory for the top side of a cold plate or the down face of a hot plate. There is some data available in Ref. INCLINED WALL The correlations are equations 9. Hence. 1 (same as for vertical wall): 0. --- Ra L 10 9 L 0. 1 is used instead of cos in the expression for h. According to Ref. = 0 for vertical walls). 3 but this data gives only approximations of this transition. 1.81 m/s2. the height of the wall. Unfortunately. few data is available about this transition. 0. For this reasons.26 and 9. is a correlation input and is the tilt angle (angle between the wall and the vertical direction. The laminar-turbulent transition depends on (see Ref. In addition.27 in Ref.492k ------------------- C p h = 2 k 0. independently of value. For turbulent flow. these correlations are not recommended for the bottom side of a cold face and for the top side of a hot plate. equal to 9. 3).68 + --------------------------------------------------------9 / 16 4 / 9 1 + 0. data is only provided for water (Pr around 6).492k ------------------- C p (2-30) where L. according to the authors. 3). is analog to these for vertical walls and is given by the following: g T p C p T – T ext L 3 Ra L = --------------------------------------------------------------------------k (2-31) where in turn g denotes the gravitational acceleration. when 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 | 57 .67 cos Ra L 1 / 4 k. we define a flow as turbulent. because this gives better accuracy (see Ref. RaL is given by Equation 2-29. 3. HORIZONTAL PLATE. Heat Transfer Coefficients — Internal Natural Convection N A R R O W C H I M N E Y. then k 1 h = ----. are correlation inputs (equation 7.30–9. then RaL is given by Equation 2-29. 3). 1 but can also be found as equations 7. upside. and H.Ra L H 24 (2-34) where L.g T p C p T – T ext L 3 -------------------------------------------------------------------------.0.77 and 7. The material data are evaluated at TText2.32 in Ref. UPSIDE The correlations are equations 9.-----. If TText. see Ref. 58 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y . The material data are evaluated at TText2.15Ra L1 / 3 Ra L 10 7 L (2-32) k h = ---. and L. Otherwise it is the same implementation as for Horizontal plate. DOWNSIDE Equation 2-32 is used when T Text and Equation 2-33 is used when T Text. the plate diameter (defined as area/perimeter. the plate distance.78 in Ref. 3) is a correlation input. 10 9 k All material properties are evaluated at TText2.54Ra L1 / 4 Ra L 10 7 L k--0.27Ra L1 / 4 L (2-33) while if T Text. the chimney height.96 in Ref. P A R A L L E L P L A T E S If RaL HL and T Text. HORIZONTAL PLATE. then h = k--0. 1: 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 | 59 . L O C A L TR A N S F E R C O E F F I C I E N T This correlation corresponds to equations 5. A VE R A G E D TR A N S F E R C O E F F I C I E N T This correlation is an assembly of equations 7.0. Circular Tube If RaDHD. 3: h = k --------------------------------.61 in Ref.41 in Ref.Narrow Chimney. and Uext.-------------------------------------------------------------- 2 --Re L 5 10 5 h = L 1 + 0. are correlation inputs (table 7. 1: 0. Heat Transfer Coefficients — Internal Forced Convection I S O T H E R M A L TU B E This correlation corresponds to equations 8.79b and 5. The material data are evaluated at TText2. the tube diameter.332Pr 1 / 3 Re x1 / 2 Re x 5 10 5 max x eps k --------------------------------.2 in Ref. the exterior velocity are correlation inputs. The material data are evaluated at TText2.Ra D H 128 where D.0468 Pr 2 / 3 1 / 4 k 1/3 2 --4/5 5 L.34 and 7.037Re L – 871 Re L 5 10 (2-35) where Prcpk and ReLUextL. L. the position along the plate.3387Pr 1 / 3 Re L1 / 2 k.0. 3 with DhD). P L A T E . then k 1 h = ----.---------. RaD is given by Equation 2-29 with L replaced by D.Pr 0.55 and 8.0296Pr 1 / 3 Re x4 / 5 Re x 5 10 5 max x eps (2-36) where Prcpk and RexUextx. the plate length and Uext. Heat Transfer Coefficients — External Forced Convection P L A T E . The material data are evaluated at TText2. and H.131 in Ref. x. the chimney height. the exterior velocity are correlation inputs. All material data are evaluated at Text except T which is evaluated at the wall temperature. the tube diameter and Uext.4 if T Text.3 if TText and n0.027Re D Re D 2500 T D (2-37) where Prcpk. are correlation inputs. ReDUextD and n0. the exterior velocity.66 Re D 2500 D h = k 0.0. 60 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y . k--3.14 4 / 5 Pr n ----------- ---. D. T. More generally. 2D. The material in the thin layer must be a good thermal conductor. A good example is a copper trace on a printed circuit board.About Highly Conductive Layers This module supports heat transfer in highly conductive layers in 3D. which simplifies the geometry and reduces the required number of mesh elements. 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. where the traces are good thermal conductors compared to the board’s substrate material. 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. Also see Highly Conductive Layer Nodes. Figure 2-4 shows an example where a highly conductive layer reduces the mesh density significantly. you can assume that no variations in temperature and in-plane heat flux exist along the layer’s thickness. ABOUT HIGHLY CONDUCTIVE LAYERS | 61 . The highly conductive layer feature is efficient for modeling heat transfer in thin layers without the need to create a fine mesh for them. and 2D axisymmetry. Copper wire modeled with a mesh Copper wire represented as a highly conductive layer Figure 2-4: Modeling a copper wire as a domain (top) requires a denser mesh compared to modeling it as a boundary with a highly conductive layer (bottom). the Highly Conductive Layer feature uses a variant of the heat equation that describes the in-plane heat flux in the layer: T d s s C s ------.+ t – d s k s tT = q – q + d s Q S = – q s t (2-38) Here the operator t denotes the del or nabla operator projected onto the plane of the highly conductive layer. the general heat flux boundary condition becomes T – n q = – d s s C p s ------.– t – d s k s tT on t 62 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y . 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. To describe heat transfer in highly conductive layers. Out-of-Plane Heat Transfer Nodes T H E O R Y O F O U T . 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. Figure 2-5 shows examples of likely situations where this type of geometry reduction can be applied.O F . it is efficient to reduce the model geometry to 2D or even 1D and use the out-of-plane heat transfer mechanism. q qup qdown Figure 2-5: Geometry reduction from 3D to 1D (top) and from 3D to 2D (bottom). For such objects. Instead.P L A N E H E A T TR A N S F E R | 63 . the reduced geometry does not represent the upside and downside surfaces of the plate in Figure 2-5 as boundaries. For example. there is usually only a small variation in temperature along the object’s thickness or cross section.Theory of Out-of-Plane Heat Transfer Out-of-Plane Heat Transfer Nodes When the object to model in COMSOL Multiphysics® is thin or slender enough along one of its geometry dimensions. The reduced geometry does not include all the boundaries of the original 3D geometry. Equation Formulation When out-of-plane heat transfer is enabled. dz. 64 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y .+ u T = d z kT + d z Q t (2-40) The Pressure Work attribute on Solids and Fluids and the Viscous Heating attribute on Fluids are not available when out-of-plane heat transfer is activated. is replaced by T C p d z ------.– d z kT = d z Q t (2-39) where dz is the thickness of the domain in the out-of-plane direction. Equation 3-1 is replaced by T d z C p ------. Activating Out-of-Plane Heat Transfer and Thickness Using a 1D or 2D model. Equation 3-2. The equation for heat transfer in fluids. activate the physics features for out-of-plane heat transfer and the thickness property by clicking the main Heat Transfer node and selecting the Out-of-plane heat transfer check box under Physical Model. Heat Source nodes that are added to a model with out-of-plane heat transfer enabled are multiplied by the thickness. Boundary conditions are also adjusted. the equation for heat transfer in solids. To model Equation 2-41 add the Biological Tissue model equation. This feature uses Pennes’ approximation to represent heat sources from metabolism and blood perfusion. This is used to model heat transfer within biological tissue. 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.T he o r y f o r t he Bi oh eat Tran sfer U ser Interface The Bioheat Transfer Interface uses the bioheat equation and the corresponding physics nodes in the Heat Transfer interface. heat capacity Cp. The Biological Tissue model provides the left-hand side of Equation 2-41 while the Bioheat node provides the two source terms on the right-hand side of Equation 2-41. T H E O R Y F O R T H E B I O H E A T TR A N S F E R U S E R I N T E R F A C E | 65 . with a Bioheat feature. The equation for conductive heat transfer using this approximation: Cp T + – k T = b C b b T b – T + Q met t (2-41) The density . + C p u T = k eq T + Q t (2-42) with the following material properties: • is the fluid density. k by k eq = p k p + L k The equivalent volumetric heat capacity of the solid-fluid system is calculated by C p eq = p p C p p + L C p Here p denotes the solid material’s volume fraction. either an analytic expression or a velocity field from a fluid-flow interface. where L is the fluid’s volume fraction.Theory for the Heat Transfer in Porous Media User Interface The Heat Transfer in Porous Media Interface uses the following version of the heat equation as the mathematical model for heat transfer in porous media (Ref. or equivalently the porosity. • Q is the heat source (or sink). • u is the fluid velocity field. 14): T C p eq ------. • keq is the equivalent thermal conductivity (a scalar or a tensor if the thermal conductivities are anisotropic). which is related to the volume fraction of the liquid L (or porosity) by L + p = 1 66 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y . • Cp is the fluid heat capacity at constant pressure. The average linear velocity (the velocity within the pores) can be calculated as uLuL. keq. the volume flow rate per unit cross-sectional area. that is. Add one or several heat sources as separate physics features. • (Cp)eq is the equivalent volumetric heat capacity at constant pressure. u should be interpreted as the Darcy velocity. is related to the conductivity of the solid kp and to the conductive of the fluid. The equivalent thermal conductivity of the solid-fluid system. For a steady-state problem the temperature does not change with time. and the first term in the left-hand side of Equation 2-42 disappears. T H E O R Y F O R T H E H E A T TR A N S F E R I N P O R O U S M E D I A U S E R I N T E R F A C E | 67 . About Handling Frames in Heat Transfer This section discusses heat transfer analysis with moving frames. the user inputs for certain features are defined on the material and are converted so that all the corresponding variables contain the value on the spatial frame. Edge. Table 2-4. When a moving mesh is detected. Because the heat transfer variables and equations are defined on the spatial frame. the inputs are internally converted to the spatial frame. all heat transfer physics account for deformation effects on heat transfer properties. Point. and Pair Nodes for the Heat Transfer User Interfaces • Theory for the Heat Transfer User Interfaces Frame Physics Feature Nodes and Definitions This subsection contains the list of all heat transfer nodes and the corresponding definition frame. a Heat Transfer Module or a CFD Module. for example. Boundary. When the Enable conversions between material and spatial frame check box is selected. • The Heat Transfer Interface • Domain. when spatial and material frames do not coincide. The following explains the different values listed in the definition frame column in Table 2-3. 68 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y . The entire physics (equations and variables) are defined on the spatial frame. Some of the physics require additional licenses. and Table 2-5: Material: •The inputs are entered by the user and defined on the material frame. Material/(Spatial): •For these physics nodes. (Material)/Spatial: •For these physics nodes. Domain Nodes TABLE 2-3: DOMAIN PHYSICS NODES FOR FRAMES NODE NAME DEFINITION FRAME Heat Transfer in Solids Material Translational Motion Material Heat Transfer in Fluids Spatial Biological Tissue Material Heat Transfer with Phase Change Spatial Heat Transfer in Porous Media Material (Solid part) Spatial (Fluid part) Thermal Dispersion Spatial Immobile Fluids Spatial Geothermal Heating Material Infinite Elements Spatial Pressure Work Spatial Viscous Heating Spatial Heat Source Material/(Spatial) Bioheat Material Opaque N/A Out-of-Plane Convective Heat Flux Spatial Out-of-Plane Radiation Spatial Out-of-Plane Heat Flux Spatial A B O U T H A N D L I N G F R A M E S I N H E A T TR A N S F E R | 69 . N/A: There is no definition frame for this physics node. The default definition frame is the spatial frame.Spatial: •The inputs are entered by the user are defined on the spatial frame. select from a menu to decide if the inputs are defined on the material or spatial frame. No conversion is done. select from a menu to decide if the inputs are defined on the material or spatial frame. The default definition frame is the material frame. TABLE 2-3: DOMAIN PHYSICS NODES FOR FRAMES 70 | NODE NAME DEFINITION FRAME Radiation in Participating Media Spatial Change Thickness Spatial Initial Values Spatial C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y . Boundary Nodes TABLE 2-4: BOUNDARY PHYSICS NODES FOR FRAMES NODE NAME DEFINITION FRAME Temperature Spatial Thermal Insulation N/A Outflow N/A Symmetry N/A Heat Flux (Material)/Spatial Inflow Heat Flux Spatial Open Boundary Spatial Thin Thermally Resistive Layer Material Thermal Contact Material Surface-to-Ambient Radiation Spatial Surface-to-Surface Radiation Spatial Prescribed Radiosity Spatial Reradiating Surface Spatial Radiation Group N/A Periodic Heat Condition Spatial Boundary Heat Source Material/(Spatial) Heat Continuity Spatial Pair Thin Thermally Resistive Layer Material Pair Thermal Contact Material Pair Boundary Heat Source Material/(Spatial) Convective Heat Flux Spatial Highly Conductive Layer Material Layer Heat Source Material Opaque Surface Spatial Continuity on Interior Boundary Spatial Edge and Point Nodes TABLE 2-5: EDGE AND POINT NODES FOR FRAMES NODE NAME DEFINITION FRAME Line Heat Source Material/(Spatial) Point Heat Source Material A B O U T H A N D L I N G F R A M E S I N H E A T TR A N S F E R | 71 . or tensors) and on their density order. the following variables are relative scalars of weight one (also called scalar densities): the mass density . D E N S I T Y. H E A T S O U R C E . H E A T F L U X Scalar density variables do not have the same value in the material and in the spatial frame. the heat flux q0. The conversion depends on the dimension of the variables (scalars. the heat transfer coefficient h. vectors. In heat transfer physics. and the production/absorption coefficient qs. 72 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y .TABLE 2-5: EDGE AND POINT NODES FOR FRAMES NODE NAME DEFINITION FRAME Edge Heat Flux (Material)/Spatial Point Heat Flux Spatial Temperature Spatial Point Temperature Spatial Edge Surface-to-Ambient Spatial Point Surface-to-Ambient Spatial TABLE 2-6: HEAT TRANSFER IN THIN SHELLS NODES NODE NAME DEFINITION FRAME Thin Conductive Layer Material Heat Source Material/(Spatial) Change Thickness Spatial Initial Values Spatial Out-of-Plane Convective Heat Flux Spatial Out-of-Plane Radiation Spatial Out-of-Plane Heat Flux Spatial Heat Flux Spatial/(Material) Surface-to-Ambient Radiation Spatial Temperature Spatial Change Effective Thickness Spatial Edge Heat Source Material/(Spatial) Conversion Between Material and Spatial Frames This subsection explains how the user inputs are converted. the heat source Q. On domains it corresponds to the local volume change from the material to the spatial frame while it corresponds to local surface or length change on boundaries and edges. The relationship between the value on the spatial frame and the material frame is 1 T k x y z = ----------------. As a consequence.detInvF is the inverse of spatial. no transformation is done because the user input is defined on the spatial frame.detInvF*500[kg/m^3] (on the spatial frame). THERMAL CONDUCTIVITY Thermal conductivity is a tensor density. to evaluate or integrate the mass density on the material frame. if =500[kg/m^3] is defined in the Heat Transfer in Fluids (definition frame = spatial frame) the variable ht.detInvF to get the corresponding value on the spatial frame. F is the coordinate transform matrix from the material frame to the spatial frame defined in the paragraph above.rho has to be multiplied by spatial. spatial.When a feature has its definition frame on the spatial frame. spatial. the user input is defined on the material frame so it has to be multiplied by spatial.detF. if =500[kg/m^3] is defined in the Heat Transfer in Solids (definition frame = material frame) the variable ht. For example. A B O U T H A N D L I N G F R A M E S I N H E A T TR A N S F E R | 73 . When a feature has its definition frame on the material frame.detF has different definitions based on the dimension of the geometric entity where it is evaluated. For example.detF. the value of ht.rho is equal to 500[kg/m^3] (on the spatial frame).rho is equal to spatial. VE L O C I T Y VE C T O R The relationship between u x y z and u X Y Z is T u x y z = F u X Y Z where F is the coordinate transform matrix from the material to the spatial frame: xX yX zX F = xY yY zY xZ yZ zZ with xX corresponding to the derivative of x with respect to X.F k X Y Z F det F where k x y z is the thermal conductivity tensor in the spatial frame and k X Y Z is the thermal conductivity tensor in the material frame. detInvF r 74 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y . heat flux. the conversion leads to: R Q =500[W/m^3]* ---. and so forth) are multiplied by R ---r which corresponds to the ratio of the material first cylindrical coordinate over the spatial one. and so on. The density variables (density. The transformation matrix uses tangential derivatives and is defined as xT X yT X zT X Ftang = xT Y yT Y zT Y xT Z yT Z zT Z where xTX corresponds to the tangential derivative x with respect to X. AXISYMMETRIC GEOMETRIES In 1D axisymmetric and 2D axisymmetric models an additional conversion is done between the material frame and the spatial frame. if you enter a heat source Q =500[W/m^3]in the material frame in axisymmetric cases. For example.THERMAL CONDUCTIVITY OF HIGHLY CONDUCTIVE LAYER The same transformations are applied to thermal conductivity but with different transformation matrices.spatial. heat source. K. 13. R. 1943. Charron. John Wiley & Sons. DeWitt. 185–210. Zhang. Heat Transfer. Veynante. vol. Batchelor. T.. 7. 1996. no. “Effect of Relative Humidity on the Prediction of Natural Convection Heat Transfer Coefficients. “Comparison of Some Finite Element Methods for Solving the Diffusion-Convection-Reaction Equation.. 1998. Incompressible Flow. Cambridge University Press. Codina. pp. 2002. no. M. R E F E R E N C E S F O R T H E H E A T TR A N S F E R U S E R I N T E R F A C E S | 75 .References for the Heat Transfer User Interfaces 1. no. 12. An Introduction to Fluid Dynamics. Tsilingiris. F. 2007.P. and J. Meth.. 2nd ed. Principles of Convective Heat Transfer. Panton. Wiley. F. 29. 1098–1110. and H-J Kretzschmar. Gupta. 1993. International Steam Tables.. 4. 28. W. 335–342. T. Bakera. G. Bergman and A.” Energy Conversion and Management. 6. Springer. F. Publication Scientifique et Technique du Ministère de l'Air. Mech. Bejan et al. pp. 2000.” Comp. 2006. Lavine. Fundamentals of Heat and Mass Transfer. 2001.P. 182.S. 2. 8. vol. A. 2008. 2003. Poinsot and D. Wagner. J. Incropera and D. 156. 2nd ed. John Wiley & Sons. DeWitt.L. 4. 9. Partage de la chaleur entre deux corps frottants [Heat Partition Between Two Rubbing Bodies]. “Thermophysical and Transport Properties of Humid Air at Temperature Range Between 0 and 100 °C. pp. Incropera. 3. 10. Edwards. R. 5. Springer. 2008. fifth ed. Engrg. 2007. Heat Transfer Handbook.Appl. 2nd ed. Fundamentals of Heat and Mass Transfer. D. John Wiley & Sons. John Wiley & Sons. vol.” Heat Transfer Engineering. A. Bejan. 2005. Theoretical and Numerical Combustion.P. Kaviany.L. A.T. 11.P. Second Edition. P. Sixth edition. 14. J. Bear and Y. Bachmat, Introduction to Modeling of Transport Phenomena in Porous Media, Kluwer Academic Publisher, 1990. 76 | C H A P T E R 2 : H E A T TR A N S F E R T H E O R Y 3 The Heat Transfer Branch This chapter details the variety of interfaces found under the Heat Transfer branch ( ) in the Model Wizard and these form the fundamental interfaces in the Heat Transfer Module. It covers all the types of heat transfer—conduction, convection, and radiation—for heat transfer in solids and fluids. For information about surface-to-surface radiation see Radiation Heat Transfer. In this chapter: • About the Heat Transfer Interfaces • The Heat Transfer Interface • Highly Conductive Layer Nodes • Out-of-Plane Heat Transfer Nodes • The Bioheat Transfer Interface • The Heat Transfer in Porous Media Interface 77 About the Heat Transfer Interfaces The Heat Transfer interfaces model heat transfer by conduction and convection. Surface-to-ambient radiation effects around edges and boundaries can also be included. The interfaces are available in 1D, 2D, and 3D and for axisymmetric models with cylindrical coordinates in 1D and 2D. The default dependent variable is the temperature, T. After selecting a version of the Heat Transfer interface in the Model Wizard, default nodes are added under the main node. For example: • If Heat Transfer in Solids ( ) is selected, a Heat Transfer (ht) node is added with a default Heat Transfer in Solids model. • If Heat Transfer in Fluids ( added. ) is selected, a default Heat Transfer in Fluids model is The benefit of the different versions of the standard Heat Transfer interface, all with ht as the identifier (see Table 3-1), is that it is easy to add the default settings when selecting the interface from the Model Wizard. At any time, right-click the parent node to add a Heat Transfer in Fluids or Heat Transfer in Solids node—the functionality is always available. The interface options are also available from the Heat Transfer interface by selecting a specific check box under the Physical Model section (for surface-to-surface radiation, biological tissue, radiation in participating media, or porous media). See Table 3-1 and Table 3-2. TABLE 3-1: THE HEAT TRANSFER (HT) PHYSICS INTERFACE OPTIONS ICON 78 | NAME DEFAULT PHYSICAL MODEL Heat Transfer in Solids not applicable Heat Transfer in Fluids not applicable Heat Transfer in Porous Media The Heat transfer in porous media check box is selected. Heat Transfer with Surface-to-Surface Radiation (under the Radiation branch) The Surface-to-surface radiation check box is selected (which enables the Radiation Settings section). C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H TABLE 3-1: THE HEAT TRANSFER (HT) PHYSICS INTERFACE OPTIONS ICON NAME DEFAULT PHYSICAL MODEL Heat Transfer with Radiation in Participating Media (under the Radiation branch) The Radiation in participating media check box is selected (which enables the Participating Media Settings section). Bioheat Transfer The Heat transfer in biological tissue check box is selected. TABLE 3-2: ADDITIONAL HEAT TRANSFER PHYSICS OPTIONS ICON NAME ID DEFAULT PHYSICAL MODEL Laminar Flow (under the nitf See Table 7-1 for details. Turbulent Flow k-and Turbulent Flow, Low Re k-(under the Conjugate Heat Transfer branch) nitf See Table 7-1 for details. Heat Transfer in Thin Shells htsh No Physical Model section, but the Surface-to-Surface Radiation check box is available to activate the Radiation Settings section. Surface-to-Surface Radiation (under the Radiation branch) rad No Physical Model section, but the Radiation Settings section is automatically available by default. Radiation in Participating Media (under the Radiation branch) rpm not applicable Joule Heating (under the jh No check boxes are selected under Physical Model. Conjugate Heat Transfer branch) Electromagnetic Heating branch) A B O U T T H E H E A T TR A N S F E R I N T E R F A C E S | 79 • The Heat Transfer Interface • The Bioheat Transfer Interface • The Heat Transfer in Porous Media Interface • The Heat Transfer in Thin Shells User Interface • The Conjugate Heat Transfer Branch • Radiation Heat Transfer • The Joule Heating User Interface in the COMSOL Multiphysics Reference Manual 80 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H The Heat Transfer Interface The Heat Transfer user interfaces model heat transfer by conduction and convection. Surface-to-ambient radiation effects around edges and boundaries can also be included. The interfaces are available in 1D, 2D, and 3D and for axisymmetric models with cylindrical coordinates in 1D and 2D. The default dependent variable is the temperature, T. The Heat Transfer user interfaces include the equations, boundary conditions, and sources for modeling conductive and convective heat transfer and solving for the temperature. After selecting a version of the physics user interface in the Model Wizard (as described in About the Heat Transfer Interfaces), default nodes are added under the main node. For example: ) is selected, a Heat Transfer in Solids (ht) node is added • If Heat Transfer in Solids ( with a default Heat Transfer in Solids model as a subnode. • If Heat Transfer in Fluids ( ) is selected, a Heat Transfer in Fluids (ht) node is added with a default Heat Transfer in Fluids model as a subnode. The benefit of the different versions of the Heat Transfer user interfaces, with ht as the common default identifier (see Table 3-1), is that it is easy to add the default settings when selecting the interface from the Model Wizard. At any time, right-click the parent node to add a Heat Transfer in Fluids or Heat Transfer in Solids node—the functionality is always available. 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. Depending on the version of the interface selected, the default nodes vary. T H E H E A T TR A N S F E R I N T E R F A C E | 81 INTERFACE IDENTIFIER The interface identifier is used primarily as a scope prefix for variables defined by the physics user interface. the identifier string must be unique. The default is 1 m and applies to the entire geometry. 82 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H . no check boxes are selected for the standard version of the Heat Transfer interface. click the Show button ( ) and select Stabilization. It provides extra diffusion in the region of sharp gradients. select Manual from the Selection list. To choose specific domains. Refer to such interface variables in expressions using the pattern <identifier>. In order to distinguish between variables belonging to different physics user interfaces. numbers and underscores (_) are permitted in the Identifier field. The first character must be a letter. The Crosswind diffusion check box is also selected by default. select the Out-of-plane heat transfer check box and then enter the Thickness of the plane (dz). so streamline diffusion and crosswind diffusion can be used simultaneously. DOMAIN SELECTION The default setting is to include All domains in the model to define heat transfer and a temperature field. The default identifier (for the first interface in the model) is ht. Click to select any of the available check boxes to activate the other versions of the ht interface as detailed in Table 3-1 and Table 3-2. 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. About the Heat Transfer Interfaces CONSISTENT STABILIZATION To display this section. The added diffusion is orthogonal to the streamline diffusion. use the Change Thickness node. PHYSICAL MODEL By default.<variable_name>. If another thickness is specified for some of the domains. Only letters. If required for 2D or 1D models. The smoothing can provide a more well-behaved flux value close to singularities.INCONSISTENT STABILIZATION To display this section. specify the Value type when using splitting of complex variables—Real (the default) or Complex. • The Compute boundary fluxes check box is selected by default so that COMSOL computes predefined accurate boundary flux variables (with the suffix _acc such as ht. or Quintic. ADVANCED SETTINGS Add both a Heat Transfer (ht) and Moving Mesh (ale) interface (found under the Mathematics>Deformed Mesh branch in the Model Wizard) then click the Show button ( ) and select Advanced Physics Options to display this section. the heat transfer features automatically account for deformation effects on heat transfer properties. Quartic. • In the table. The Isotopic diffusion check box is not selected by default. Editing the name of a scalar dependent variable changes both its field name and the dependent variable name. Linear. With moving mesh.ndflux_acc for the accurate—as opposed to the standard extrapolated— normal conductive heat flux). Rotation effects on thermal conductivity of an anisotropic material T H E H E A T TR A N S F E R I N T E R F A C E | 83 . or with a component name belonging to some other field. In particular the effects for volume changes on the density are considered. This option has no effect when the model does not contain a moving frame since the material and spatial frames are identical in this case. If a new field name coincides with the name of another field of the same type. A new field name must not coincide with the name of a field of another type. click the Show button ( ) and select Stabilization. and when this option is active. DEPENDENT VA RIA BLES The Heat Transfer user interfaces have a dependent variable for the Temperature T. DISCRETIZATION To display this section. click the Show button ( ) and select Discretization. the Enable conversions between material and spatial frame check box is selected by default. When the model contains moving mesh. The dependent variable names can be changed. • The Apply smoothing to boundary fluxes check box is selected by default. • Select an element order (shape function order) for the Temperature—Quadratic (the default). Cubic. the fields will share degrees of freedom and dependent variable names. Edge. Edge. the feature inputs (for example. Point. • About Handling Frames in Heat Transfer • Show More Physics Options • Domain. Point. edge. Heat Flux. Boundary. more generally. boundary. and Pair Nodes for the Heat Transfer User Interfaces The Heat Transfer Interface has these domain. 84 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H . and pair nodes and subnodes available (listed in alphabetical order including out-of-plane and highly conductive layer features). are also covered. point. To locate and search all the documentation. Boundary. in COMSOL Multiphysics. deformation effects on arbitrary thermal conductivity. When the Enable conversions between material and spatial frame check box is not selected. Heat Source. and Pair Nodes for the Heat Transfer User Interfaces • Consistent and Inconsistent Stabilization Methods for the Heat Transfer User Interfaces • Theory for the Heat Transfer User Interfaces • Show Stabilization in the COMSOL Multiphysics Reference Manual Domain. Boundary Heat Source. select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree.and. and Line Heat Source) are not converted and all are defined on the Spatial frame. The following nodes are also available for some versions of the Heat Transfer interface and described in Radiation Heat Transfer chapter. 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. • Incident Intensity • Opaque T H E H E A T TR A N S F E R I N T E R F A C E | 85 .• Boundary Heat Source • Out-of-Plane Convective Heat Flux • Change Thickness • Out-of-Plane Heat Flux • Convective Heat Flux • Out-of-Plane Radiation • Continuity • Periodic Heat Condition • Edge Heat Flux • Point Heat Flux • Edge Surface-to-Ambient Radiation • Point Heat Source • Heat Flux • Point Temperature • Heat Source • Heat Transfer in Fluids • Point Surface-to-Ambient Radiation • Heat Transfer in Solids • Pressure Work • Heat Transfer with Phase Change • Surface-to-Ambient Radiation • Highly Conductive Layer • Symmetry • Initial Values • Temperature • Inflow Heat Flux • Thermal Contact • Layer Heat Source • Thermal Insulation (the default boundary condition) • Line Heat Source • Open Boundary • Outflow • Thin Thermally Resistive Layer • Translational Motion • Viscous Heating For axisymmetric models. DOMAIN SELECTION For a default node. the heat capacity Cp. you can select Manual from the Selection list to choose specific domains or select All domains as required.– kT = Q t (3-1) For a steady-state problem the temperature does not change with time and the first term disappears. also right-click to add Pressure Work or Opaque nodes. the model inputs appear here. and the thermal conductivity k (a scalar or a tensor if the thermal conductivity is anisotropic). 86 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H . When nodes are added from the context menu. and a heat source (or sink) Q—one or more heat sources can be added separately. If you have the Heat Transfer Module. the selection is automatically selected and is the same as for the interface. MODEL INPUTS This section contains fields and values that are inputs to expressions that define material properties. and cannot be edited.• Opaque Surface • Radiation in Participating Media Heat Transfer in Solids The Heat Transfer in Solids node uses the heat equation version in Equation 3-1 as the mathematical model for heat transfer in solids: T C p ------. When parts of the model are moving in the material frame. also right-click to add a Pressure Work node. The Opaque subnode is automatically added to the entire selection when Surface-to-surface radiation is activated. the setting inherits the selection from the parent node. right-click the Heat Transfer in Solids node to add a Translational Motion node to take this into account. Initially. The selection can be edited. that is. this section is empty. The equation includes the following material properties: the density . If such user-defined materials are added. If you have the CFD Module. Enter this quantity as power per length and temperature.kxx. and so on (using the default interface identifier ht). THERMODYNAMICS. HEAT CONDUCTION. The components of a thermal diffusivity in the case that it is a tensor (xx. which is Fourier’s law of heat conduction. Fourier’s law expect that the thermal conductivity tensor is symmetric. The heat capacity at constant pressure describes the amount of heat energy required to produce a unit temperature change in a unit mass. Symmetric. ht.kyy. The thermal diffusivity can be interpreted as a measure of thermal inertia (heat propagates slowly where the thermal diffusivity is low. Select User defined to enter other values or expressions. The thermal conductivity describes the relationship between the heat flux vector q and the temperature gradient T as in q = kT. Diagonal. and so on. kyy. The coordinate system is used for interpreting directions of orthotropic and anisotropic thermal conductivity. If User defined is selected. representing an anisotropic thermal conductivity) are available as ht. Thermal Diffusivity In addition. and kzz. and so on. A non symmetric tensor can lead to unphysical results. SOLID The default Density (SI unit: kg/m3) and Heat capacity at constant pressure Cp (SI unit: J/(kg·K)) use values From material. T H E H E A T TR A N S F E R I N T E R F A C E | 87 . choose Isotropic. yy. The single scalar mean effective thermal conductivity ht.COORDINATE SYSTEM SELECTION The Global coordinate system is selected by default. or Anisotropic based on the characteristics of the thermal conductivity. SOLID The default setting is to use the Thermal conductivity k (SI unit: W/(m·K)) From material. kyy. the thermal diffusivity is defined as k/(Cp) (SI unit: m2/s) is also a predefined quantity. for example). The Coordinate system list contains any additional coordinate systems that the model includes (except boundary coordinate systems).kmean is the mean value of the diagonal elements kxx. The components of a thermal conductivity k in the case that it is a tensor (kxx. and enter another value or expression. alphaTdMean is the mean value of the diagonal elements xx. The Heat Flux boundary condition does not. The single scalar mean thermal diffusivity ht. which provides movement by translation to model heat transfer in solids. work at boundaries where n·u0. 88 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H .representing an anisotropic thermal diffusivity) are available as ht. for example. The denominator Cp is the effective volumetric heat capacity. each covering a subset of the Heat Transfer in Solids node’s selection. the selection is the same as for the Heat Transfer in Solids node that it is attached to. a moving heat source. but it is possible to use more than one Heat Translation subnode. choose the domains to define. • Axisymmetric Transient Heat Transfer: Model Library path COMSOL_Multiphysics/Heat_Transfer/heat_transient_axi • 2D Heat Transfer Benchmark with Convective Cooling: Model Library path COMSOL_Multiphysics/Heat_Transfer/heat_convection_2d Translational Motion Right-click the Heat Transfer in Solids node to add the Translational Motion node. DOMAIN SELECTION From the Selection list. It adds the following contribution to the right-hand side of Equation 3-1. for example. By default.C_eff.alphaTdxx. and zz. and so on (using the default interface identifier ht). defined in the parent node: – C p u T The contribution describes the effect of a moving coordinate system that is required to model.alphaTdyy. yy. ht. ht. Special care must be taken at boundaries where n·u0. and is also available as a predefined quantity. Cp. • Velocity field u (SI unit: m/s)—either an analytic expression or a velocity field from a fluid-flow interface. • Thermal conductivity k (SI unit: W/(m·K))—a scalar or a tensor if the thermal conductivity is anisotropic. • The Ratio of specific heats (dimensionless)— the ratio of heat capacity at constant pressure. • The heat source (or sink) Q—one or more heat sources can be added separately. y. 1. T H E H E A T TR A N S F E R I N T E R F A C E | 89 . fields. See Thermodynamics.TR A N S L A T I O N A L M O T I O N Enter component values for x. For common diatomic gases such as air. Heat Generation in a Disc Brake: Model Library path Heat_Transfer_Module/Thermal_Contact_and_Friction/brake_disc 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: T C p ------. Most liquids have 1. It is also used if the ideal gas law is applied. Solid. Cv. is used in the streamline stabilization and in the variables for heat fluxes and total energy fluxes. and z (in 3D) for the Velocity field utrans (SI unit: m/ s). When using the ideal gas law to describe a fluid.0. specifying is enough to evaluate Cp. to heat capacity at constant volume.+ C p u T = kT + Q t (3-2) For a steady-state problem the temperature does not change with time and the first term disappears.1 while water has 1. and sources: • Density (SI unit: kg/m3) • Heat capacity at constant pressure Cp (SI unit: J/(kg·K))—describes the amount of heat energy required to produce a unit temperature change in a unit mass. This equation includes the following material properties.4 is the standard value. for example). or Pressure Work nodes to the Heat Transfer in Fluids feature. Solid. MODEL INPUTS This section has fields and values that are inputs to expressions that define material properties. Opaque. If such user-defined property groups are added. the selection is automatically selected and is the same as for the interface. and cannot be edited. Heat Transfer by Free Convection: Model Library path COMSOL_Multiphysics/Multiphysics/free_convection DOMAIN SELECTION For a default node. Solve for the absolute 90 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H . you can select Manual from the Selection list to choose specific domains or select All domains as required.Right-click to add Viscous Heating (for heat generated by viscous friction). that is. Absolute Pressure This section controls both the variable as well as any property value (reference pressures) used when solving for pressure. See Thermodynamics. Absolute pressure is also used if the ideal gas law is applied. the setting inherits the selection from the parent node. When nodes are added from the context menu. There are also two standard model inputs—Absolute pressure and Velocity field. the model inputs appear here. The absolute pressure is used in some predefined quantities that include the enthalpy (the energy flux. There are usually two ways to calculate the pressure when describing fluid flow and mass and heat transfer. COORDINATE SYSTEM SELECTION The Global coordinate system is selected by default. 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. it may not with physics interfaces that it is being coupled to. The defaults are 0 m/s. Velocity Field The default Velocity field u (SI unit: m/s) is User defined. For example. Or select an existing velocity field in the model (for example. the pressure variables solved can also be selected from the list. When User defined is selected. For example. Velocity field (spf/fp1) from a Laminar Flow interface). reduces the chances for stability and convergence during the solving process for this variable. While this check box maintains control over the pressure variable and instances where absolute pressure is required within this respective physics interface. if a fluid-flow interface is added you can select Pressure (spf/fp) from the list. When a Pressure variable is selected.325 Pa). which. In such models. The Coordinate system list contains any additional coordinate systems that the model includes (except boundary T H E H E A T TR A N S F E R I N T E R F A C E | 91 . Using one or the other option usually depends on the system and the equations being solved for. The default Absolute pressure pA (SI unit: Pa) is User defined and is 1 atm (101. 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. if included. In other cases. the pressure drop over the modeled domain is probably many orders of magnitude less than atmospheric pressure. When additional physics interfaces are added to the model. in a straight incompressible flow problem. enter values or expressions for the components based on space dimension. the Reference pressure check box is selected by default and the default value of pref is 1[atm] (1 atmosphere).pressure or a pressure (often denoted gauge pressure) that relates back to the absolute pressure through a reference pressure. check the coupling between any interfaces using the same variable. Select User defined to enter another value for either of these material properties. Enter this quantity as power per length and temperature. which is a built-in physical constant. 92 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H . Gas/Liquid Select Gas/Liquid to specify the Density. Heat capacity at constant pressure Cp (SI unit: J/(kg·K)). FLUID The default Thermal conductivity k (SI unit: W/(m·K)) is taken From material. heat capacity. The default settings are to use data From material. For both properties. and the Ratio of specific heats for a general gas or liquid. choose Isotropic. THERMODYNAMICS. or ratio of specific heats. 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.314 J/(mol·K). The coordinate system is used for interpreting directions of orthotropic and anisotropic thermal conductivity. and Ratio of specific heats (dimensionless) for a general gas or liquid use values From material.coordinate systems). If Mean molar mass is selected. Then: • Select a Gas constant type—Specific gas constant Rs (SI unit: J/(kg·K)) or Mean molar mass Mn (SI unit: kg/mol). Select a Fluid type—Gas/Liquid. Select User defined to enter other values or expressions. Symmetric. FLUID The default Density (SI unit: kg/m3). the software uses the universal gas constant R 8. Select User defined to enter another value for the density. the default setting is to use the property value from the material. HEAT CONDUCTION. If User defined is selected. or Anisotropic based on the characteristics of the thermal conductivity. and enter another value or expression. Moist air. the Heat capacity at constant pressure. Diagonal. Ideal Gas Select Ideal gas to use the ideal gas law to describe the fluid. or Ideal gas. These three reference values are used to estimate the mass fraction of vapor. Right-click to add additional Initial Values nodes. . For both properties. Four different options are available from the Input quantity list to define the amount of vapor in the moist air: • Select Vapor mass fraction (the default) to define the vapor mass fraction (SI unit: kg/kg). specify either Cp or the ratio of specific heats. a Reference temperature (SI unit: K). the default setting is to use the property value From material. but not both since these. Once this option is selected a Concentration model input is automatically added in the Models Inputs section. in that case.• From the list under Specify Cp or . • Select Concentration to define the concentration of vapor (SI unit: mol/m3). which is used to define the thermodynamic properties of the moist air. Moist Air Theory Initial Values The Initial Values node adds an initial value for the temperature that can serve as an initial condition for a transient simulation or as an initial guess for a nonlinear solver. • Select Moisture content to define the moisture content of the moist air (SI unit: kg/ kg). • Select Relative humidity to define the quantity of vapor from a Reference relative humidity (SI unit: 1). Select User defined to define another value for either of these material properties. T H E H E A T TR A N S F E R I N T E R F A C E | 93 . are dependent. Moist Air If Moist air is selected. and a Reference pressure (SI unit: Pa). For an ideal gas. the thermodynamics properties are defined as a function of the quantity of vapor in the moist air. select Heat capacity at constant pressure Cp (SI unit: J/(kg·K)) or Ratio of specific heats (dimensionless). Specify the heat source as the heat per volume in the domain. See also Additional General Source Options. or as a total heat source (power). enter the Production/absorption coefficient qs (SI unit: W/(m3·K)). choose the domains to add the heat source to. Linear source. respectively. DOMAIN SELECTION From the Selection list. The default is 0 W/m3(that is. enter a value for the distributed heat source Q (SI unit: W/m3) when the default option. You express heating and cooling with positive and negative values. as a linear heat source. • If Total power is selected. The default value is approximately room temperature. enter the total heat source. or Total power button. V = 94 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H 1 . • If Linear source (Qqs·T) is selected. 293. the selection is automatically selected and is the same as for the interface.15 K (20 ºC). you can select Manual from the Selection list to choose specific domains or select All domains as required. . no heat source). When nodes are added from the context menu. The default is 0 W. In this case Q = Ptot/V. is selected. Heat Source The Heat Source describes heat generation within the domain. (SI unit: W). INITIAL VALUES Enter a value or expression for the initial value of the Temperature T (SI unit: K). the setting inherits the selection from the parent node. Ptot. The default is 0 W/(m3·K).DOMAIN SELECTION For a default node. User defined. that is. and cannot be edited. HEAT SOURCE Click the General source (the default). where V is the total volume of the selected domains. In 3D and 2D axial symmetry. • If General source is selected. Add one or more nodes as required—all heat sources within a domain contribute to the total heat source. a text field is available to define Ac. The advantage of writing the source in this second form is that it can be stabilized by the streamline diffusion. The theory covers qs that is independent of the temperature. there are predefined heat sources available (in addition to a User defined heat source) when simulating heat transfer together with electrical or electromagnetic physics user interfaces. a text field is available to define dz. for example. If the out-of-plane property is not active. ohmic heating and induction heating. If the out-of-plane property is not active. • With the addition of an Electric Currents physics interface. • With the addition of any version of the Electromagnetic Waves user interface (which requires the RF Module). the Total power dissipation density (emw/wee1) and Electromagnetic power loss density (emw/wee1) heat sources are available from the General source list. but some stability can be gained as long as qs is only weakly dependent on the temperature. the Total power dissipation density (ec/cucn1) heat source is available from the General source list. T H E H E A T TR A N S F E R I N T E R F A C E | 95 . In 1D: V = Ac 1 where Ac is the cross-sectional area. Additional General Source Options For the general heat source Q. Such sources represent.In 2D and 1D axial symmetry: V = dz 1 where dz is the out-of-plane thickness. The following options are also available from the General source list above but require additional interfaces and/or licenses as indicated. If Material (the default) is selected. Opaque. • About Handling Frames in Heat Transfer • The Heat Transfer Interface • Stabilization Techniques in the COMSOL Multiphysics Reference Manual Heat Transfer with Phase Change The Heat Transfer with Phase Change node is used to solve the heat equation after specifying the properties of a phase change material according to the apparent heat capacity formulation. add both a Heat Transfer (ht) and a Moving Mesh (ale) user interface (found under the Mathematics>Deformed Mesh branch in the Model Wizard). • For the Heat Transfer in Porous Media user interface. Right-click to add Viscous Heating (for heat generated by viscous friction). heat sources from the electrochemical current distribution interfaces are available. the Electromagnetic heating (mef/alc1) heat source is available from the General source list. or Pressure Work nodes to the Heat Transfer with Phase Change node. Then click the Show button ( ) and select Advanced Physics Options. FRAME SELECTION To display this section. When the model contains a moving mesh. Corrosion Module. or Electrodeposition Module. the variables take their values from the edit fields. a conversion from the material to the spatial frame is applied to the edit field values. which in turn enables this section. Use Frame Selection to select the frame where the input variables are defined. with the addition of physics interfaces from the Batteries & Fuel Cells Module. the Electromagnetic heating (mf/al1) heat source is available from the General source list. the Enable conversions between material and spatial frame check box is selected by default on the Heat Transfer interface. • With the addition of a Magnetic and Electric Fields user interface (which requires the AC/DC Module). If Spatial is selected.• With the addition of a Magnetic Fields user interface (a 3D model requires the AC/ DC Module). 96 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H . Enter any additional latent heat values as per the Number of phase transitions. The default is 10 K. Enter any additional transition intervals as per the Number of phase transitions. choose 2 in the Number of phase transitions list. or any couple of successive phase transformations. and -iron are allotropes of solid iron that can be considered as phases with distinct phase change temperatures. About the Phases The different phases are ordered according to the temperatures of fusion. or evaporation. or allotropic varieties of a substance. only one phase transition is needed to simulate solidification. Enter a Latent heat from phase 1 and phase 2 L12 (SI unit: J/kg). NUMBER OF TRANSITIONS To display this section.12 (SI unit: K).DOMAIN SELECTION From the Selection list. melting. T H E H E A T TR A N S F E R I N T E R F A C E | 97 . click the Show button ( ) and select Advanced Physics Options. For example. In most cases. If you want to model successive melting and evaporation. . choose the domains to define where heat transfer with phase change occurs. Enter any additional phase change temperatures as per the Number of phase transitions. Enter a Transition interval between phase 1 and phase 2 T12 (SI unit: K). PHASE CHANGE Enter a Phase change temperature between phase 1 and phase 2 Tpc.12. composite materials. The value of T12 must be strictly positive.12 while the material properties of phase 2 hold for TTpc.15 K. . Choose the Number of phase transitions to model. metal alloys. It is useful to choose 3 or more transitions to handle extra changes of material properties such as in mixtures of compounds. The default is 273. The default value is 1 and the maximum value is 5. The default is 333 kJ/kg. A value near 0K corresponds to a behavior close to a pure substance. MODEL INPUTS This section is the same as for Heat Transfer in Fluids. the material properties of phase 1 are valid when TTpc. Hence. Diagonal. • Ratio of specific heats phase[1.When more than one transition is modeled.. Tpc. which can point to any material in the model.. The default is 1 W/(m·K). T23 or L23). or Anisotropic based on the characteristics of the thermal conductivity. If User defined is selected.] (SI unit: W/(m·K)) uses the material values for phase i. the values of Tjj1 are chosen so that the ranges between Tpc.] (SI unit: J/(kg·K))... • Heat capacity at constant pressure Cp. Or select User defined to enter a different value or expression for each: • The default Thermal conductivity kphase[1.2.. the number of phases exceeds 2 and new variables are created (for example.jj1 where the material properties of phase j only apply..] (dimensionless)... The values of Tjj1 must all be strictly positive.2.].2. PHASE For each Phase section (based on the Number of phase transitions) select or enter the following: Select a Material. The phase change temperatures Tpc. and enter another value or expression. The default is 1.phase[1.. choose Isotropic. • Density phase[1.jj1 are increasing and satisfy T pc 1 2 T pc 2 3 This defines distinct domains of temperature bounded by Tpc.23.. The default uses the Domain material. If this condition is not satisfied. The default is 4200 J/(kg·K). In addition.] (SI unit: kg/m3).. The defaults for the following use values From material. The default is 1000 kg/m3.. unexpected behaviors may occur because some phases would never form completely.j1j and Tpc.2.1 About Heat Transfer with Phase Change 98 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H .jj1Tjj12 do not overlap..jj1Tjj12 and Tpc.2.. phase [1. Symmetric. An interesting numerical check for convergence is the numerical evaluation of the thermal insulation condition along the boundary. for example. For this to be true. BOUNDARY SELECTION From the Selection list. and cannot be edited. Intuitively this equation says that the temperature gradient across the boundary must be zero.15 K. 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. For The Heat Transfer in Thin Shells User Interface. the setting inherits the selection from the parent node. on boundaries. heat cannot transfer across it. When nodes are added from the context menu. the temperature on one side of the boundary must equal the temperature on the other side. Ideally the contour lines are perpendicular to any insulated boundary. The default is 293. choose the boundaries to define. Because there is no temperature difference across the boundary. Another check is to plot the temperature field as a contour plot. Temperature Use the Temperature node to specify the temperature somewhere in the geometry.Thermal Insulation The Thermal Insulation node is the default boundary condition for all Heat Transfer interfaces. T H E H E A T TR A N S F E R I N T E R F A C E | 99 . that is. this condition can also be applied to edges and pairs. you can select Manual from the Selection list to choose specific boundaries or select All boundaries as required. This boundary condition means that there is no heat flux across the boundary: n kT = 0 This condition specifies where the domain is well insulated. Enter the value or expression for the Temperature T0 (SI unit: K). BOUNDARY SELECTION For a default node. the selection is automatically selected and is the same as for the interface. 100 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H . • Select the Discontinuous Galerkin constraints button when Classic constraints do not work satisfactorily. BOUNDARY SELECTION In most cases. • By default Classic constraints is selected. this condition states that the only heat transfer over a boundary is by convection.CONSTRAINT SETTINGS To display this section. • Select the Use weak constraints check box to replace the standard constraints with a weak implementation. the Outflow node does not require any user input. The Discontinuous Galerkin constraints option is especially useful to prevent oscillations on inlet boundaries where convection dominates. click the Show button ( ) and select Advanced Physics Options. and there is no radiation. If required. Unlike the Classic constraints. Otherwise. In a model with convective heat transfer. This is relevant on fluid inlets where the temperature may not be enforced on the walls at the inlet extremities. select Current physics (internally symmetric) or Individual dependent variables to restrict the reaction terms as required. This is usually a good approximation of the conditions at an outlet boundary in a heat transfer model with fluid flow. Show More Physics Options Outflow The Outflow node provides a suitable boundary condition for convection-dominated heat transfer at outlet boundaries. The temperature gradient in the normal direction is zero. select All physics (symmetric). these constraints do not enforce the temperature on the boundary extremities. select the boundaries that are convection-dominated outlet boundaries. To Apply reaction terms on all dependent variables. PAIR SELECTION If this node is selected from the Pairs menu. Ctrl-click to deselect. and it means that there is no heat flux across the boundary. choose the pair to define. define the symmetry boundaries. For inlet boundaries. HEAT FLUX Click to select the General inward heat flux (the default). Inward heat flux. This boundary condition is similar to a Thermal Insulation condition.Symmetry The Symmetry node provides a boundary condition for symmetry boundaries. choose the boundaries or edges to define. If required. Heat Flux Use the Heat Flux node to add heat flux across boundaries and edges. the node does not require any user input. use the Inflow Heat Flux condition instead. BOUNDARY SELECTION In most cases. The symmetry condition only applies to the temperature field. It has no effect on the radiosity (surface-to-surface radiation) and on the radiative intensity (radiation in participating media). BOUNDARY OR EDGE SELECTION From the Selection list. A positive heat flux adds heat to the domain. or Total heat flux button. T H E H E A T TR A N S F E R I N T E R F A C E | 101 . An identity pair has to be created first. This feature is not applicable to inlet boundaries. For example. any electric heater is well represented by this condition. Enter a value for q0 to represent a heat flux that enters the domain. The default is 0 W. enter the total heat flux qtot (SI unit: W) for the total heat flux across the boundaries where the Heat Flux node is active.General Inward Heat Flux If General inward heat flux q0 (SI unit: W/m2) is selected. 102 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H . it adds to the total flux across the selected boundaries. The default is 0 W/m2. enter the Heat transfer coefficient h (SI unit: W/(m2·K)). In this case q0 = qtot/A. Inward heat flux is defined by q0 hText T. A = 1 . see Incropera and DeWitt in Ref. and its geometry can be omitted. The default is 0 W/(m2·K).15 K. a text field is available to define dz. Inward Heat Flux If Inward heat flux is selected. Total Heat Flux If Total heat flux is selected. where A is the total area of the selected boundaries. Also enter an External temperature Text (SI unit: K). In 3D and 2D axial symmetry. For a thorough introduction about how to calculate heat transfer coefficients. If the out-of-plane property is not active. The default is 293. In 2D and 1D axial symmetry: A = dz 1 where dz is the out-of-plane thickness. The value depends on the geometry and the ambient flow conditions. 1. FRAME SELECTION The settings are the same for the Heat Source node and described under the Frame Selection section. choose the boundaries to define. this section is empty. and Tamb is the ambient temperature. 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. a text field is available to define Ac. • About Handling Frames in Heat Transfer • The Heat Transfer Interface Surface-to-Ambient Radiation Use the Surface-to-Ambient Radiation condition to add surface-to-ambient radiation to boundaries. The net inward heat flux from surface-to-ambient radiation is 4 4 q = T amb – T where is the surface emissivity. Initially. T H E H E A T TR A N S F E R I N T E R F A C E | 103 . is the Stefan-Boltzmann constant (a predefined physical constant). MODEL INPUTS This section contains fields and values that are inputs to expressions that define material properties. If such user-defined materials are added. the model inputs appear here. If the out-of-plane property is not active. BOUNDARY SELECTION From the Selection list. SURFACE-TO-AMBIENT RADIATION The default Surface emissivity (a dimensionless number between 0 and 1) is taken From material.In 1D: A = Ac 1 where Ac is the cross-sectional area. choose the boundaries to define. User defined.15 K. choose the pair to define. When selected as a Pair Boundary Heat Source. • If General source is selected. enter a value for the boundary heat source Qb (SI unit: W/m2) when the default option. The default is 293. is selected. An identity pair has to be created first. Right-click to add a Destination Selection node. Ctrl-click to deselect. The default is 0 W/m2. BOUNDARY HEAT SOURCE Click the General source (the default) or Total boundary power button. it also prescribes that the temperature field is continuous across the pair. choose the boundaries to define. For the general boundary heat 104 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H . A positive Qb is heating and a negative Qb is cooling. In the COMSOL Multiphysics Reference Manual: • Periodic Condition and Destination Selection • Periodic Boundary Conditions Boundary Heat Source The Boundary Heat Source models a heat source (or heat sink) that is embedded in the boundary.Enter an Ambient temperature Tamb (SI unit: K). BOUNDARY SELECTION From the Selection list. PAIR SELECTION When Pair Boundary Heat Source is selected from the Pairs menu. Continuous Casting: Model Library path Heat_Transfer_Module/ Thermal_Processing/continuous_casting Periodic Heat Condition Use the Periodic Heat Condition to add a periodic heat condition to boundaries. BOUNDARY SELECTION From the Selection list. where A is the total area of the selected boundaries. Such sources represent. In this case Qb = Pb. a text field is available to define Ac. a text field is available to define dz. A = 1 . FRAME SELECTION The settings are the same for the Heat Source node and described under the Frame Selection section. tot (SI unit: W). there are predefined heat sources available when simulating heat transfer together with electrical or electromagnetic physics user interfaces. • If Total boundary power is selected. tot/A. • About Handling Frames in Heat Transfer • The Heat Transfer Interface T H E H E A T TR A N S F E R I N T E R F A C E | 105 . The default is 0 W. In 1D: A = Ac 1 where Ac is the cross-sectional area. If the out-of-plane property is not active. If the out-of-plane property is not active. ohmic heating and induction heating. In 2D and 1D axial symmetry: A = dz 1 where dz is the out-of-plane thickness. In 3D and 2D axial symmetry.source Qb. for example. enter the total power (total heat source) Pb. In the COMSOL Multiphysics Reference Manual: • Continuity on Interior Boundaries • Identity and Contact Pairs Thin Thermally Resistive Layer Use the Thin Thermally Resistive Layer node to define the thickness and thermal conductivity of a resistive material located on boundaries. It prescribes that the temperature field is continuous across the pair. An identity pair has to be created first. Ctrl-click to deselect. PAIR SELECTION When this node is selected from the Pairs menu. It can be added to pairs by selecting Pair Thin Thermally Resistive Layer from the Pairs menu.Continuity The Continuity node can be added to pairs. This material can be formed of one or more layers. Continuity is only suitable for pairs where the boundaries match. choose the pair to define. BOUNDARY SELECTION The selection list in this section shows the boundaries for the selected pairs. The resistive material can also be defined through the Thermal Resistance: ds R s = ----ks The heat flux across the Thin Thermally Resistive Layer is defined by Tu – Td – n d – k d T d = – k s -------------------ds Td – Tu – n u – k u T u = – k s -------------------ds 106 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H . In order to model a thin resistive layer made of several materials. Like any pair feature. Initially. If such user-defined materials are added. use the Multiple layers option. When using the Pair Thin Thermally Resistive Layer node. MODEL INPUTS This section contains fields and values that are inputs to expressions that define material properties. However. choose the pair to define. respectively. the model inputs appear here. then the u and d subscripts refer to the upside and the downside of the pair. which is available with the Heat Transfer Module. When the material has a multilayer structure ks and ds in the expressions above are replaced by dtot and ktot. respectively. do not use two conditions on the same pair. T H E H E A T TR A N S F E R I N T E R F A C E | 107 . the Pair Thin Thermally Resistive Layer condition contributes with any other pair feature. PAIR SELECTION If this node is selected from the Pairs menu.where the u and d subscripts refer to the upside and the downside of the slit. this section is empty. instead of the slit. An identity pair has to be created first. which are defined according to Equation 3-3 and Equation 3-4: nl d tot = dsj (3-3) j=1 d tot k tot = ----------------nl d sj ------k sj (3-4) j=1 where nl is the number of layers. choose the boundaries to define. Ctrl-click to deselect. BOUNDARY SELECTION From the Selection list. 2.THIN THERMALLY RESISTIVE LAYER By default the Multiple layers check box is not selected. • For each layer. The default is 0. 108 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H .. MODEL INPUTS This section contains fields and values that are inputs to expressions that define material properties. Select User defined to enter another value or expression. • The default Thermal conductivity ks (SI unit: W/(m·K)) is taken From material. PAIR SELECTION When Pair Thermal Contact is selected from the Pairs menu. The default is 0. Thermal Contact The Thermal Contact node defines correlations for the conductance h at the interface of two bodies in contact. and so on) list to assign a material to each layer. enter the Layer thickness ds (SI unit: m). click to select the check box. Ctrl-click to deselect. the model inputs appear here. An identity pair has to be created first.01 W/(m·K). If such user-defined materials are added. this section is empty. choose the pair to define. The conductance h is involved in the heat flux across the surfaces in contact according to: – n d – k d T d = – h T u – T d + rQ fric – n u – k u T u = – h T d – T u + 1 – r Q fric where u and d subscripts refer respectively to the upside and downside of the slit. It can be added to pairs by selecting Pair Thermal Contact from the Pairs menu.. To define multiple sandwiched thin layers with different thermal conductivities.0050 m. Solid material 2. • Select an option from the Solid material (Solid material 1. choose the boundaries to add a thermal contact condition. Boundary material. . takes the material from the boundary. BOUNDARY SELECTION From the Selection list.). Then select the Number of layers to define (1 to 5) and set the properties for each layer selected. which is then taken from the material selected in Solid material (1. Initially. The default setting. CONTACT Constriction Conductance Correlation Choose the Constriction conductance correlation—Cooper-Mikic-Yovanovich correlation (the default). enter a value or expression for the Constriction conductance hc (SI unit: W/(m2·K)). The default is 0 W/(m2·K). • Contact pressure p (SI unit: Pa). or Brinell hardness. • If User defined is selected. • Asperities average slope masp (dimensionless). • If Microhardness is selected. choose a Hardness definition— Microhardness (the default). The default is 0. Radiative Conductance Correlation When the Surface-to-surface radiation check box is selected under the Physical Model section on a physics interface. Gap Conductance Correlation Choose the Gap conductance correlation—User defined or Parallel-plate gap gas conductance (the second option is only available if Cooper-Mikic-Yovanovich correlation or Mikic elastic correlation is chosen as the Constriction conductance correlation). T H E H E A T TR A N S F E R I N T E R F A C E | 109 . The default is 0 Pa. enter a value for the Gap conductance hg (SI unit: W/ (m2·K)). enter a value for Hc (SI unit: Pa). • If User defined is selected. choose the Radiative conductance correlation—User defined or Gray-diffuse parallel surfaces (the default). If User defined is selected. The default is 0 W/(m2·K). The default is 1 µm. Vickers hardness. or User defined. enter a value for the Radiative conductance hT (SI unit: W/ (m2·K)). The default is 3 GPa. The default is 0 W/(m2·K). CONTACT SURFACE PROPERTIES This section displays if Cooper-Mikic-Yovanovich correlation or Mikic elastic correlation are chosen under Contact. Mikic elastic correlation. When Cooper-Mikic-Yovanovich correlation is selected. Enter values for the: • Asperities average height asp (SI unit: m).4. RADIATIVE CONDUCTANCE This section is available when Gray-diffuse parallel surfaces is selected as the Radiative conductance correlation under Contact. The default is 1. enter a value for the Heat partition coefficient r (dimensionless). The default is 3 GPa. The default is 1 GPa. enter values or expressions for the Young’s modulus E (SI unit Pa) and Poisson’s ratio (dimensionless). enter a value for HB (SI unit: Pa). or Anisotropic based on the characteristics of the gas thermal conductivity. • If Weighted harmonic mean is selected.5. The default is 1. THERMAL FRICTION Select a Heat partition coefficient definition—Charron’s relation (the default) or User defined. If User defined is selected. Symmetric. The default value is 1 atm. The default value is 0.60 GPa. • Gas thermal accommodation parameter (dimensionless). The default Gas thermal conductivity kgap (SI unit: W/(m·K)) is taken From material.7.• If Vickers hardness is selected. Select User defined to enter another value or expression. When Mikic elastic correlation is selected. Also enter the following: • Gas pressure pgap (SI unit: Pa). enter a value for the Vickers correlation coefficient c1 (SI unit: Pa) and Vickers size index c2 (dimensionless). By default the Surface emissivity (dimensionless) is taken From material. and enter another value or expression. • If User defined enter another value or expression for the Contact interface Young’s modulus Econtact (SI unit: Pa). If User defined is selected. also choose Isotropic. • Gas fluid parameter and (dimensionless).025 W/(m·K). • If Brinell hardness is selected. The default is 0.37 nm. respectively. The defaults are 5 GPa and 0.1. HB should be between 1.30 and 7. choose the Contact interface Young’s modulus definition—Weighted harmonic mean (the default) or User defined. • Gas particles diameter D (SI unit: m). The defaults are 0. Diagonal. GAP PROPERTIES This section is available when Parallel-plate gap gas conductance is selected as the Gap conductance correlation under Contact.7. The default is 0. The default is 1. 110 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H . Positive Ql is heating while a negative Ql is cooling. The default is 0 W/m. Theory for the Thermal Contact Feature Line Heat Source The Line Heat Source node models a heat source (or sink) that is so thin that it has no thickness in the model geometry. T H E H E A T TR A N S F E R I N T E R F A C E | 111 .tot (SI unit: W). • When General source is selected. Ql (SI unit: W/m) in unit power per unit length. The default is 0 W/m2. This is because in 2D it is a boundary and in 1D it is a domain. LINE HEAT SOURCE Click the General source (the default) or Total line power button. • If Total line power is selected. enter a value for the distributed heat source. EDGE SELECTION From the Selection list.Enter a Frictional heat source Qfric (SI unit: W/m2). but that distribution must be interpreted in a weak sense. The finite element discretization used in COMSOL Multiphysics returns a finite temperature distribution along the line. In theory. Select this node from the Edges submenu. 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 Line Heat Source node is only available in 3D. The default is 0 W. enter the total power (total heat source) Pl. choose the edges to define. the Pressure Work node adds the following term to the right-hand side of the Heat Transfer in Solids equation: 112 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H . The Point Heat Source is available in 3D and 2D. Pressure Work Right-click the Heat Transfer in Solidsor Heat Transfer in Fluids node to add the Pressure Work subnode. When added under Heat Transfer in Solidsnode. POINT SELECTION From the Selection list. POINT HEAT SOURCE Enter the Point heat source Qp (SI unit: W) in unit power. Select this node from the Points menu. The finite element discretization used in COMSOL Multiphysics returns a finite value.FRAME SELECTION The settings are the same for the Heat Source node and described under the Frame Selection section. choose the points to define. The default is 0 W. In 1D it is not available because points are boundaries there (possibly interior boundaries). 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). • About Handling Frames in Heat Transfer • The Heat Transfer Interface 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. but that value must be interpreted in a weak sense. Positive Qp is heating while a negative Qp is cooling. In theory. choose the domains to define. T H E H E A T TR A N S F E R I N T E R F A C E | 113 . By default. The low Mach number formulation excludes the term u · p from Equation 3-6. which is small for most flows with a low Mach number. DOMAIN SELECTION From the Selection list. P RE S S U RE WO R K For the Heat Transfer in Solids model. – T ----. When added under Heat Transfer in Fluids node. the Pressure Work feature adds the following contribution to the right-hand side of the Heat Transfer in Fluids equation: T p – ---. which adds the following term to the right-hand side of the heat transfer in fluids equation: :S (3-7) where is the viscous stress tensor and S is the strain-rate tensor. Viscous Heating Right-click the Heat Transfer in Fluids to add the Viscous Heating node.------. The default is 0 Jm3·K).+ u p T p t (3-6) The software computes the pressure work using the absolute pressure. the selection is the same as for the parent node (Heat Transfer in Solidsor Heat Transfer in Fluids) it is attached to. choose the domains to define. select a Pressure work formulation—Full formulation (the default) or Low Mach number formulation. the selection is the same as for the Heat Transfer in Fluids feature that it is attached to. -----. Equation 3-7 represents the heating caused by viscous friction within the fluid. For the Heat Transfer in Fluids model. enter a value or expression for the Elastic contribution to entropy Ent (SI unit: Jm3·K)). By default. DOMAIN SELECTION From the Selection list.S el t (3-5) where Sel is the elastic contribution to entropy. choose the boundaries to define. Inflow Heat Flux Use the Inflow Heat Flux node to model inflow of heat through a virtual domain with a heat source. This boundary condition estimates the heat flux through the system boundary 1 – n – kT = q 0 – u n ------------------. 1 + T ---. The second integral in Equation 3-5 is neglected if the feature is applied to the boundary of a solid domain. . The temperature at the outer boundary of the virtual domain is known. The default is 0 W. define q0 (SI unit: W/m2) to add to the total flux across the selected boundaries. INFLOW HEAT FLUX Select the Inward heat flux (the default) or Total heat flux buttons.+ h in – h ext u n un (3-8) where h in – h ext = T in T ext 1 - --. This feature is applicable to inlet boundaries. Select User defined to enter another value or expression. COMSOL uses the dynamic viscosity together with the velocity expressions to compute the viscous stress tensor. The default value is 0 W/m2. The default is 0 Pa·s. In this case q0qtot/A.DYNAMIC VISCOSITY The Dynamic viscosity (SI unit: Pa·s) uses the value of the viscosity From material. 114 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H . BOUNDARY SELECTION From the Selection list. ----- dp T ext p pA C p dT + p (3-9) A positive heat flux adds heat to the domain. • When Total heat flux is selected. where A is the total area of the selected boundaries. • When Inward heat flux is selected. define qtot. In 2D and 1D axial symmetry A = dz 1 where dz is the out-of-plane thickness. the heat can flow out of the domain or into the domain with a specified exterior temperature. a text field is available to define dz or Ac.15 K) and the External absolute pressure pext (SI unit: Pa) (the default is 1 atm). a text field is available to define dz or Ac.For either selection. Open Boundary The Open Boundary node adds a boundary condition for modeling heat flux across an open boundary. BOUNDARY SELECTION From the Selection list. If the out-of-plane property is not active. In 3D and 2D axial symmetry. T H E H E A T TR A N S F E R I N T E R F A C E | 115 . Use this node to limit a modeling domain that extends in an open fashion. A = 1 . In 1D A = Ac 1 where Ac is the cross-sectional area. OPEN BOUNDARY Enter the exterior Temperature T0 (SI unit: K) outside of the open boundary. enter a value or expression for the External temperature Text (SI unit: K) (the default is 273. choose the boundaries to define. If the out-of-plane property is not active. pA (SI unit: Pa). The Convective Heat Flux node adds the following heat flux contribution to its boundaries: h T ext – T where the heat transfer coefficient. HEAT FLUX Select a Heat transfer coefficient h (SI unit: W/(m2·K)) to control the type of convective heat flux to model—User defined (the default). • If Inclined wall is selected. Horizontal plate. enter a Wall height L (SI unit: m) and the Tilt angle (SI unit: rad). follow the individual instructions below and select an External fluid—Air (the default). L is approximated by the ratio between the surface area and its perimeter.15 K. The default is 0 m. External natural convection. The default is 1 atm. BOUNDARY SELECTION From the Selection list. or Horizontal plate. Transformer oil. can be user defined or by using a library of predefined coefficients described in About the Heat Transfer Coefficients. The tilt angle is the angle between the wall and the vertical direction. choose Vertical wall. enter an External temperature. or water. define the Plate diameter (area/perimeter) L (SI unit: m). Inclined wall. Text (SI unit: K).Convective Heat Flux This feature was previously called Convective Cooling. downside from the list under Heat transfer coefficient. External Natural Convection If External natural convection is selected. upside or Horizontal plate. If Air is selected. h. • For all options. Internal natural convection. choose the boundaries to define. • For all options (except User defined). The default is 0 m. The default is 293. also enter an Absolute pressure. External forced convection. or Internal forced convection. • If Horizontal plate. upside. The default is 0 rad. • If Vertical wall is selected. enter a Wall height L (SI unit: m). 116 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H . downside is selected. for vertical walls. circular tube is selected. enter a Plate distance L (SI unit: m) and a Chimney height H (SI unit: m). local transfer coefficient is selected. Internal Forced Convection If Internal forced convection is selected. The defaults are each 0 m. parallel plates is selected. • If Plate. averaged transfer coefficient or Plate. averaged transfer coefficient is selected. choose Plate. choose Narrow chimney. enter a Tube diameter D (SI unit: m) and a Chimney height H (SI unit: m). Enter a Tube diameter D (SI unit: m). external fluid Uext (SI unit: m/s). enter a Position along the plate xpl (SI unit: m) and a Velocity. parallel plates or Narrow chimney. • If Narrow chimney. circular tube from the list under Heat transfer coefficient. • If Plate. the default is 0 m/s.Internal Natural Convection If Internal natural convection is selected. • Power Transistor: Model Library path Heat_Transfer_Module/ Power_Electronics_and_Electronic_Cooling/ power_transistor • Free Convection in a Water Glass: Model Library path Heat_Transfer_Module/Tutorial_Models_Forced_and_Natural_Convection/ cold_water_glass T H E H E A T TR A N S F E R I N T E R F A C E | 117 . The defaults are each 0 m. external fluid Uext (SI unit: m/s). the default is 0 m. The defaults are each 0 m. local transfer coefficient from the list under Heat transfer coefficient. the only option is Isothermal tube. External Forced Convection If External forced convection is selected. and a Velocity. The defaults are each 0 m. enter a Plate length L (SI unit: m) and a Velocity. external fluid Uext (SI unit: m/s). • If Narrow chimney. This feature can also be added to 2D axisymmetric models. 118 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H .Highly Conductive Layer Nodes In this section: • Highly Conductive Layer • Layer Heat Source • Edge Heat Flux • Point Heat Flux • Temperature • Point Temperature • Edge Surface-to-Ambient Radiation • Point Surface-to-Ambient Radiation About Highly Conductive Layers • Heat Transfer in a Surface-Mount Package for a Silicon Chip: Model Library path Heat_Transfer_Module/Electronics_and_Power_Systems/ surface_mount_package • Copper Layer on Silica Glass: Model Library path Heat_Transfer_Module/Tutorial_Models/copper_layer Highly Conductive Layer Use the Highly Conductive Layer node to model heat transfer in thin highly conductive layers on boundaries in 2D and 3D. If such user-defined property groups are added.About Highly Conductive Layers Right-click the Highly Conductive Layer node to add these additional features: • Layer Heat Source—to add a layer internal heat source. MODEL INPUTS This section contains fields and values that are inputs to expressions that define material properties. See Temperature and Point Temperature. See Edge Heat Flux and Point Heat Flux. • Edge (3D) or Point (2D and 2D axisymmetric) Surface-to-Ambient Radiation—adds a surface-to-ambient radiation for the highly conductive layer. • 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. choose the boundaries to define. BOUNDARY SELECTION From the Selection list. within the highly conductive layer. the model inputs appear here. Qs. The Coordinate system list contains any additional coordinate systems that the model includes (except boundary coordinate systems). See Edge Surface-to-Ambient Radiation and Point Surface-to-Ambient Radiation. COORDINATE SYSTEM SELECTION The Global coordinate system is selected by default. the Edge Heat Flux and Point Heat Flux nodes are not available with the Slip Flow interface. HIGHLY CONDUCTIVE LAYER NODES | 119 . The Layer thickness ds. The coordinate system is used for interpreting directions of orthotropic and anisotropic thermal conductivity. displays in this section.01 m. • 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. If you also have the Microfluidics Module. The default value is 0. Add one or more heat sources. If the thickness is zero. By default. and enter another value or expression. choose the boundaries to define. Right-click the Highly Conductive Layer node to add this feature. If User defined is selected. or Anisotropic based on the characteristics of the thermal conductivity. choose Isotropic. respectively.HEAT CONDUCTION The default Layer thermal conductivity ks (SI unit: W/(m·K)) is taken From material and describes the layer’s ability to conduct heat. BOUNDARY SELECTION From the Selection list. 120 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H . the selection is the same as for the Highly Conductive Layer node. About Highly Conductive Layers Layer Heat Source Use a Layer Heat Source node to add an internal heat source. Diagonal. THERMODYNAMICS The default Layer density s (SI unit: kg/m3) and Layer heat capacity Cs (SI unit: J/ (kg·K)) are taken From material.01 m. Select User defined to enter other values or expressions. within the highly conductive layer. the highly conductive layer does not take effect. Qs. Enter a value or expression for the Layer thickness ds (SI unit: m). The defaults are 0 kg/m3 and 0 J/ (kg·K). Symmetric. The default is 0. EDGE SELECTION From the Selection list. enter a value or expression for Qs (SI unit: W/m3) as a heat source per volume. define the total power Ps. Edge Heat Flux Use the Edge Heat Flux node for 3D models to add heat flux across boundaries of a highly conductive layer.tot/V where V=A·ds with A equal to the area of the boundary selection. choose the edges to define. In 3D and 2D axial symmetry. The default is 0 W. The default is 0 W/m3. • When the General source button is selected. HIGHLY CONDUCTIVE LAYER NODES | 121 . If the out-of-plane property is not active. a text field is available to define dz. A = 1 .tot (SI unit: W). A positive heat flux adds heat to the layer.LAYER HEAT SOURCE Select the General source (the default) or Total power button to define Qs. A = dz 1 where dz is the out-of-plane thickness. the Edge Heat Flux node is not available with the Slip Flow interface. In 2D. Right-click the Highly Conductive Layer node to add this feature. If you also have the Microfluidics Module. In this case Qs = Ps. • If the Total power button is selected. Enter an External temperature Text (SI unit: K). 122 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H . In this case q0 = qtot/A where A=L·ds with L equal to the length of the edge selection. it adds to the total flux across the selected edges. The default value is 293.EDGE HEAT FLUX Select either the General inward heat flux (the default). any electric heater is well represented by this condition. enter the total heat flux qtot (SI unit: W). • If Inward heat flux is selected (in the form q0h·TextT. and its geometry can be omitted. Inward heat flux. FRAME SELECTION The settings are the same for the Heat Source node and described under the Frame Selection section. the Point Heat Flux node is not available with the Slip Flow interface. or Total heat flux buttons. The default value is 0 W/(m2·K). If you also have the Microfluidics Module. The default is 0 W/m2. The value depends on the geometry and the ambient flow conditions. • If Total heat flux is selected. Enter a value for q0 to represent a heat flux that enters the layer. Right-click the Highly Conductive Layer node to add this feature. A positive heat flux adds heat to the layer.15 K. • When General inward heat flux q0 (SI unit: W/m2) is selected. The default is 0 W. About Handling Frames in Heat Transfer Point Heat Flux Use the Point Heat Flux node for 2D and 2D axisymmetric models to add heat flux across boundaries of a highly conductive layer. enter the Heat transfer coefficient h (SI unit: W/(m2·K)). For example. Right-click the Highly Conductive Layer node to add this feature. The value depends on the geometry and the ambient flow conditions. • If Inward heat flux is selected (in the form q0h·TextT. The default value is 0 W/(m2·K). choose the edges to define. it adds to the total flux across the selected points. • When General inward heat flux q0 (SI unit: W/m2) is selected. For example. The equation for this condition is T = T0 where T0 is the prescribed temperature on the edges. Enter an External temperature Text (SI unit: K). Only edges adjacent to the boundaries can be selected in the parent node.POINT S EL EC TION From the Selection list. and its geometry can be omitted. The default value is 293.15 K. The default is 0 W/m2. any electric heater is well represented by this condition. PAIR SELECTION If this node is selected from the Pairs menu. choose the pair to define. HIGHLY CONDUCTIVE LAYER NODES | 123 . An identity pair has to be created first. EDGE SELECTION From the Selection list. HEAT FLUX Select either the General inward heat flux (the default) or Inward heat flux buttons. enter the Heat transfer coefficient h (SI unit: W/(m2·K)). choose the points to define. Temperature Use the Temperature node to specify the temperature on a set of edges that represent thin boundary surfaces of the highly conductive layer. Ctrl-click to deselect. TE M P E R A T U R E Enter the value or expression for the Temperature T0 (SI unit: K). The default is 293.15 K. Enter a value for q0 to represent a heat flux that enters the layer. choose the points to define. The equation for this condition is T = T0 where T0 is the prescribed temperature on the points. click the Show button ( ) and select Advanced Physics Options. select All physics (symmetric). To Apply reaction terms on all dependent variables. Select the Use weak constraints check box to replace the standard constraints with a weak implementation. select Current physics (internally symmetric) or Individual dependent variables to restrict the reaction terms as required. 124 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H . select All physics (symmetric). Right-click the Highly Conductive Layer node to add this feature. select Current physics (internally symmetric) or Individual dependent variables to restrict the reaction terms as required. POINT SELECTION From the Selection list.CONSTRAINT SETTINGS To display this section. PO I N T TE M P E R A T U RE Enter the value or expression for the Temperature T0 (SI unit: K). Select the Use weak constraints check box to replace the standard constraints with a weak implementation.15 K. Point Temperature Use the Point Temperature node to specify the temperature on a set of points that represent thin boundary surfaces of the highly conductive layer. Otherwise. Otherwise. click the Show button ( ) and select Advanced Physics Options. The default is 293. Only points adjacent to the boundaries can be selected in the parent node. CONSTRAINT SETTINGS To display this section. To Apply reaction terms on all dependent variables. and Tamb is the ambient temperature. Right-click the Highly Conductive Layernode to add this feature. Right-click the Highly Conductive Layernode to add this feature.15 K. HIGHLY CONDUCTIVE LAYER NODES | 125 . choose the edges to define. The default is 0. 4 4 The net inward heat flux from surface-to-ambient radiation is q = T amb – T where is the surface emissivity. is the Stefan-Boltzmann constant (a predefined physical constant). The default is 293. choose the points to define. The net inward heat flux from surface-to-ambient radiation is 4 4 q = T amb – T where is the surface emissivity. POINT S EL EC TION From the Selection list. Point Surface-to-Ambient Radiation Use the Point Surface-to-Ambient Radiation node to add surface-to-ambient radiation to points representing boundaries of a highly conductive layer. is the Stefan-Boltzmann constant (a predefined physical constant). and Tamb is the ambient temperature. SURFACE-TO-AMBIENT RADIATION Enter an Ambient temperature Tamb (SI unit: K).Edge Surface-to-Ambient Radiation Use the Edge Surface-to-Ambient Radiation node to add surface-to-ambient radiation to edges representing boundaries of a highly conductive layer. 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. EDGE SELECTION From the Selection list. The default Surface emissivity (a dimensionless number between 0 and 1) is taken From material. 126 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H . 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.15 K. The default is 0.SURFACE-TO-AMBIENT RADIATION Enter an Ambient temperature Tamb (SI unit: K). The default Surface emissivity (a dimensionless number between 0 and 1) is taken From material. The default is 293. O U T . these features are available for 3D models.Out-of-Plane Heat Transfer Nodes The following nodes are available for 1D and 2D Heat Transfer models and in 3D for the Heat Transfer in Thin Shells interface.P L A N E H E A T TR A N S F E R N O D E S | 127 .O F . In this section: • Out-of-Plane Convective Heat Flux • Out-of-Plane Radiation • Out-of-Plane Heat Flux • Change Thickness Theory of Out-of-Plane Heat Transfer Select the Out-of-plane heat transfer check box on a Heat Transfer interface to add these nodes to 2D and 1D models. For the Heat Transfer in Thin Shells interface. Surface Resistor: Model Library path Heat_Transfer_Module/ Thermal_Stress/surface_resistor Out-of-Plane Convective Heat Flux Use the Out-of-Plane Convective Heat Flux node to model upside and downside cooling or heating caused by the presence of an ambient fluid. use the default. 128 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H . UPSIDE HEAT FLUX About the Heat Transfer Coefficients Select a Heat transfer coefficient hu (SI unit: W/(m2·K)) to control the type of convective heat flux to model—User defined (the default). or Internal forced convection. then select an External fluid—Air.This feature adds the following contribution h u T ext.– d z kT = d z Q t (3-10) T C p d z ------. Transformer oil. External natural convection. Text. The default is 1 atm.d – T to the right-hand side of Equation 3-10 or Equation 3-11 T d z C p ------. pA (SI unit: Pa). If Air is selected. • For all of the options (except User defined). If only convective flux is required on the downside. enter an External temperature. For the Heat Transfer in Thin Shells interface these features are available in 3D. which sets hu0.+ u T = d z kT + d z Q t (3-11) Select the Out-of-plane heat transfer check box on a Heat Transfer interface to add these nodes to 2D and 1D models. or water. follow the individual instructions in the Heat Flux section described for the Convective Heat Flux feature.u – T + h d T ext. also enter an Absolute pressure. Internal natural convection. DOMAIN SELECTION Select the domains where you want to add an out-of-plane convective heart flux contribution. • For all of the options.u (SI unit: K). External forced convection. DOWNSIDE PARAMETERS Follow the instructions for the Upside Parameters section to define the downside parameters d and Tamb. The feature adds the following contribution to the right-hand side of Equation 3-10 or Equation 3-11: 4 4 4 4 u T amb u – T + d T amb d – T Compare to the equation in the section Surface-to-Ambient Radiation. The default is 293.O F .P L A N E H E A T TR A N S F E R N O D E S | 129 . For the Heat Transfer in Thin Shells interface these features are available in 3D. the Heat transfer coefficient is hd. Select the Out-of-plane heat transfer check box on a Heat Transfer interface to add these nodes to 2D and 1D models. 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. DOMAIN SELECTION Select the domains where you want to add an out-of-plane surface-to-ambient heat transfer contribution. The default is 0. Select User defined to enter another value. UPSIDE PARAMETERS The default Surface emissivity u (a dimensionless number between 0 and 1) is taken From material. Enter an Ambient temperature Tamb.u (SI unit: K). O U T . for example. Out-of-Plane Radiation The Out-of-Plane Radiation node models surface-to-ambient radiation on the upside and downside.15 K.d.DOWNSIDE HEAT FLUX The controls in the Downside Heat Flux section are the same as those in the Upside Heat Flux section except that it is applied to the downside instead of the upside. The default value for the external temperature is 293. Click the Inward heat flux button to specify an inward (or outward.u field (SI unit: K).uT).Out-of-Plane Heat Flux The Out-of-Plane Heat Flux node adds a heat flux q0.15 K. if the quantity is negative) heat flux through the upside (SI unit: W/m2) as hu·(Text. DOMAIN SELECTION Select the domains where you want to add an out-of-plane heat flux.u field.d as a downward heat flux to the right-hand side of Equation 3-10 or Equation 3-11: d s q 0 u + d s q 0 d Select the Out-of-plane heat transfer check box on a Heat Transfer interface to add these nodes to 2D and 1D models. if the quantity is negative) heat flux through the upside (SI unit: W/m2) in the q0. The default is 0 W/m2. 130 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H . For the Heat Transfer in Thin Shells interface these features are available in 3D. The General inward heat flux button is selected by default. UPSIDE INWARD HEAT FLUX Select between specifying the upside inward heat flux directly or as a convective term using a heat transfer coefficient.u as an upside heat flux and a heat flux q0. Enter a value or expression for the inward (or outward. 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. DOWNSIDE INWARD HEAT FLUX The controls in the Downside Inward Heat Flux section are identical to those in the Upside Inward Heat Flux section except that they apply to the downside instead of the upside. Change Thickness The Change Thickness node makes it possible model domains with another thickness than the overall thickness that is specified in the Heat Transfer interface Physical Model section. CHANGE THICKNESS Specify a value for the Thickness dz (SI unit: m). O U T .O F . This value replaces the overall thickness in the domains that are selected in the Domain Selection section. The default value is 1 m.P L A N E H E A T TR A N S F E R N O D E S | 131 .DOMAIN SELECTION Select the domains where you want to use a different thickness. and Pair Nodes for the Heat Transfer User Interfaces Hepatic Tumor Ablation: Model Library path Heat_Transfer_Module/ Medical_Technology/tumor_ablation 132 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H . Thermal Insulation (the default boundary condition). PHYSICAL MODEL The Heat transfer in biological tissue check box is automatically selected. Right-click the Heat Transfer node to add other features that implement boundary conditions and sources. Edge. Boundary. • The Heat Transfer Interface • Theory for the Bioheat Transfer User Interface • Biological Tissue • Bioheat • Domain. these default nodes are added to the Model Builder—Biological Tissue (with a default Bioheat node). and Initial Values. Point. Biological tissue is automatically selected as the default physical model and a Heat Transfer (ht) ( ) interface is added to the Model Builder. All functionality to include both solid and fluid domains is also available. When this version of the interface is added. The rest of the settings as well as the interior and exterior boundary conditions are the same as for The Heat Transfer Interface.T h e B i o h e a t T r a ns 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. See Equation 2-41. the setting inherits the selection from the parent node. or Anisotropic based on the characteristics of the thermal conductivity and enter another value or expression in the field or matrix. which includes the effect of the movement by translation that requires a moving coordinate system. also right-click to add a Translational Motion node. If such user-defined property groups are added. The coordinate system is used for interpreting directions of orthotropic and anisotropic thermal conductivity. and cannot be edited. MODEL INPUTS This section has fields and values that are inputs to expressions that define material properties. Symmetric. COORDINATE SYSTEM SELECTION The Global coordinate system is selected by default. T H E B I O H E A T TR A N S F E R I N T E R F A C E | 133 . The Opaque subnode is automatically added to the entire selection when the Surface-to-surface radiation check box is selected on the Bioheat Transfer interface settings window. When parts of the model (for example. The Coordinate system list contains any additional coordinate systems that the model includes (except boundary coordinate systems). When nodes are added from the context menu. DOMAIN SELECTION For a default node. choose Isotropic. The selection can be edited. Right-click the node to add a Bioheat node. Diagonal. you can select Manual from the Selection list to choose specific domains or select All domains as required. The defaults are 0 W/(m·K). Initially. the selection is automatically selected and is the same as for the interface. If User defined is selected. HEAT CONDUCTION The default Thermal conductivity k (SI unit: W/(m·K)) uses values From material. the model inputs appear here. that is. a heat source) are moving.Biological Tissue The Biological Tissue node adds the bioheat equation as the mathematical model for heat transfer in biological tissue. this section is empty. Bioheat A default Bioheat node is added to the Biological Tissue node. you can select Manual from the Selection list to choose specific domains or select All domains as required. respectively. which is the dependent variable that is solved for and not a material property. the setting inherits the selection from the parent node. The defaults are 0 kg/m3 and 0 J/(kg·K).THERMODYNAMICS 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.15 K. If User defined is selected. BIOHEAT Enter values or expressions for these properties and source terms: • Arterial blood temperature Tb (SI unit: K). 134 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H . The default is 0 J/(kg·k). which is the temperature at which blood leaves the arterial blood veins and enters the capillaries. blood Cb (SI unit: J/(kg·k)). This feature provides the source terms that represent blood perfusion and metabolism to model heat transfer in biological tissue using the bioheat equation: b Cb b (TbT) Right-click the Biological Tissue node to add more Bioheat subnodes. and cannot be edited. describes the volume of blood per second that flows through a unit volume of tissue. the selection is automatically selected and is the same as for the interface. that is. which in this case means (m3/s)/m3). T is the temperature in the tissue. • Blood perfusion rate b (SI unit: 1/s. The default tis 310. The default is 0 1/s. enter other values or expressions. which describes the amount of heat energy required to produce a unit temperature change in a unit mass of blood. • Specific heat. When nodes are added from the context menu. DOMAIN SELECTION For a default node. which describes heat generation from metabolism. • Metabolic heat source Qmet (SI unit: W/m3). The default is 0 W/m3. which is the mass per volume of blood. blood b (SI unit: kg/m3).• Density. T H E B I O H E A T TR A N S F E R I N T E R F A C E | 135 . The default is 0 kg/m3. Enter this quantity as the unit power per unit volume. When this interface is added. is an extension of the generic Heat Transfer interface that includes modeling heat transfer through convection. and more for porous media heat transfer is activated by selecting the Heat transfer in porous media check box. The rest of the settings for this interface are the same as for the Heat Transfer interface. and non-isothermal flow. Right-click the main node to open a context menu and add as many physics features as required to define the equations. and Initial Values. default nodes are added to the Model Builder—Heat Transfer in Porous Media. boundary conditions. Phase Change: Model Library path Heat_Transfer_Module/Phase_Change/ phase_change 136 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H . Figure 3-1: The ability to model porous media heat transfer is activated by selecting the Heat transfer in porous media check box in any Heat Transfer (ht) settings window under Physical Model. The ability to define material properties.T he He a t T r a n sfer i n Porou s Med i a Interface The Heat Transfer in Porous Media user interface ( ). Thermal Insulation (the default boundary condition). properties and boundary conditions. (Figure 3-1). found under the Heat Transfer branch ( ) in the Model Wizard. conjugate heat transfer. conduction and radiation. select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree. • Boundary Heat Source • Periodic Heat Condition • Continuity • Point Heat Source • Heat Flux • Surface-to-Ambient Radiation • Heat Source • Symmetry • Heat Transfer with Phase Change • Temperature • Heat Transfer in Solids • Thermal Contact • Line Heat Source • Thermal Insulation • Outflow • Thin Thermally Resistive Layer Heat Transfer in Porous Media The Heat Transfer in Porous Media node is used to specify the thermal properties of a porous matrix. and Pair Nodes for the Heat Transfer in Porous Media User Interface The Heat Transfer in Porous Media Interface has these nodes described in this section: • Heat Transfer in Porous Media • Thermal Dispersion These domain. Edge. and pair nodes are described forThe Heat Transfer Interface (listed in alphabetical order): To locate and search all the documentation. Boundary. The Heat Transfer 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 | 137 . point. Point.• Show More Physics Options • Domain. and Pair Nodes for the Heat Transfer in Porous Media User Interface • Theory for the Heat Transfer in Porous Media User Interface • Theory for the Heat Transfer User Interfaces Domain. Right-click to add a Thermal Dispersion subnode. Boundary. in COMSOL Multiphysics. edge. Point. boundary. Edge. The selection can be edited.in Porous Media model uses the following version of the heat equation as the mathematical model for heat transfer in fluids: T C p ------. See Thermodynamics. that is. the setting inherits the selection from the parent node. • Velocity field u (SI unit: m/s): Either an analytic expression or a velocity field from a fluid-flow interface. is used in the streamline stabilization and in the variables for heat fluxes and total energy fluxes. • The heat source (or sink) Q: One or more heat sources can be added separately. If you have the Heat Transfer Module. It is also used if the ideal gas law is applied. • Thermal conductivity k (SI unit: W/(m·K)): A scalar or a tensor if the thermal conductivity is anisotropic. When using the ideal gas law to describe a fluid. It has these material properties: • Density (SI unit: kg/m3) • Heat capacity at constant pressure Cp (SI unit: J/(kg·K)): This describes the amount of heat energy required to produce a unit temperature change in a unit mass. • The Ratio of specific heats (dimensionless): The ratio of heat capacity at constant pressure.4 is the standard value. Cv. to heat capacity at constant volume. 138 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H . the selection is automatically selected and is the same as for the interface. When nodes are added from the context menu. specifying is enough to evaluate Cp. Most liquids have 1. Porous Matrix. For common diatomic gases such as air. Cp.+ Cp u T = kT + Q t (3-12) For a steady-state problem the temperature does not change with time and the first term disappears. DOMAIN SELECTION For a default node. 1. you can select Manual from the Selection list to choose specific domains or select All domains as required.1 while water has 1. the Opaque subnode is automatically added to the entire selection when Surface-to-surface radiation is activated.0. and cannot be edited. 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 | 139 . Porous Matrix. the pressure drop over the modeled domain is probably many orders of magnitude less than atmospheric pressure. if included. Solve for the absolute pressure or a pressure (often denoted gauge pressure) that relates back to the absolute pressure through a reference pressure. the absolute pressure may be required to be solved for. In other cases. in a straight incompressible flow problem. Using one or the other option usually depends on the system and the equations being solved for. such as where pressure is a part of an expression for gas volume or diffusion coefficients. There are usually two ways to calculate the pressure when describing fluid flow and mass and heat transfer. the pressure variables solved can also be selected from the list. See Thermodynamics. If such user-defined property groups are added. There are also two standard model inputs—Absolute pressure and Velocity field. When additional physics interfaces are added to the model. The absolute pressure is used in some predefined quantities that include the enthalpy (the energy flux. which. Absolute Pressure This section controls both the variable as well as any property value (reference pressures) used when solving for pressure.325 Pa). reduces the chances for stability and convergence during the solving process for this variable. For example. for example). For example. the model inputs appear here. if a fluid-flow interface is added you can select Pressure (spf/fp) from the list. The default Absolute pressure pA (SI unit: Pa) is User defined and is 1 atm (101.MODEL INPUTS This section has fields and values that are inputs to expressions that define material properties. Absolute pressure is also used if the ideal gas law is applied. and the Ratio of specific heats for a general gas or liquid. The default settings are to use data 140 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H . it may not with physics interfaces that it is being coupled to. Heat capacity at constant pressure Cp (SI unit: J/(kg·K)). or Ideal gas. 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. Diagonal. While this check box maintains control over the pressure variable and instances where absolute pressure is required within this respective physics interface. Moist air. If User defined is selected. and Ratio of specific heats (dimensionless) for a general gas or liquid use values From material. THERMODYNAMICS. When User defined is selected. the Reference pressure check box is selected by default and the default value of pref is 1[atm] (1 atmosphere). Velocity Field The default Velocity field u (SI unit: m/s) is User defined. HEAT CONDUCTION. and enter another value or expression. Enter this quantity as power per length and temperature. In such models. Select a Fluid type—Gas/Liquid. Velocity field (spf/fp1) from a Laminar Flow interface). or Anisotropic based on the characteristics of the thermal conductivity. FLUID The default Thermal conductivity k (SI unit: W/(m·K)) is taken From material. FLUID The default Density (SI unit: kg/m3). such as for gas flow governed by the gas law. Or select an existing velocity field in the model (for example. Select User defined to enter other values or expressions. Gas/Liquid Select Gas/Liquid to specify the Density. choose Isotropic.When a Pressure variable is selected. 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. The defaults are 0 m/s. enter values or expressions for the components based on space dimension. the Heat capacity at constant pressure. Symmetric. check the coupling between any interfaces using the same variable. If Mean molar mass is selected. • From the list under Specify Cp or . Four different options are available from the Input quantity list to define the amount of vapor in the moist air: • Select Vapor mass fraction (the default) to define the vapor mass fraction (SI unit: kg/kg). or ratio of specific heats. which is a built-in physical constant. in that case. For both properties. select Heat capacity at constant pressure Cp (SI unit: J/(kg·K)) or Ratio of specific heats (dimensionless). Select User defined to define another value for either of these material properties. heat capacity. For both properties. the software uses the universal gas constant R 8. the thermodynamics properties are defined as a function of the quantity of vapor in the moist air. and a Reference pressure 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 | 141 . • Select Relative humidity to define the quantity of vapor from a Reference relative humidity (SI unit: 1). • Select Moisture content to define the moisture content of the moist air (SI unit: kg/ kg). For an ideal gas.314 J/(mol·K).From material. Moist Air If Moist air is selected. Once this option is selected a Concentration model input is automatically added in the Models Inputs section. • Select Concentration to define the concentration of vapor (SI unit: mol/m3). the default setting is to use the property value from the material. . are dependent. Ideal Gas Select Ideal gas to use the ideal gas law to describe the fluid. but not both since these. the default setting is to use the property value From material. a Reference temperature (SI unit: K). Then: • Select a Gas constant type—Specific gas constant Rs (SI unit: J/(kg·K)) or Mean molar mass Mn (SI unit: kg/mol). Select User defined to enter another value for either of these material properties. specify either Cp or the ratio of specific heats. Select User defined to enter another value for the density. or Anisotropic based on the characteristics of the thermal conductivity and enter another value or expression in the field or matrix. POROUS MATRIX 3 The default Density p (SI unit: kg/m ) uses values From material. POROUS MATRIX The default Thermal conductivity kp (SI unit: W/(m·K)) uses values From material. enter another value or expression. These three reference values are used to estimate the mass fraction of vapor. which is used to define the thermodynamic properties of the moist air. Diagonal. Enter a Volume fraction p (dimensionless) for the solid material. If User defined is selected. THERMODYNAMICS. The Solid material list can point to any material in the model. The equivalent volumetric heat capacity of the solid-liquid system is calculated from C p eq = p p C p p + L C p 142 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H .(SI unit: Pa). If User defined is selected. The default Specific heat capacity Cp. Symmetric. choose Isotropic. which is Fourier’s law of heat conduction. 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. enter another value or expression. If User defined is selected. 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. HEAT CONDUCTION. Moist Air Theory IMMOBILE SOLIDS This section contains fields and values that are inputs to expressions that define material properties.p (SI unit: J/(kg·K)) uses values From material. Thermal Dispersion Right-click the Heat Transfer in Porous Media node to add the Thermal Dispersion node. DOMAIN SELECTION From the Selection list.+ C p u T = k eq T + Q t and specifies the values of the longitudinal and transverse dispersivities. DISPERSIVITIES Define the Longitudinal dispersivity lo (SI unit: m) and Transverse dispersivity tr (SI unit: m). choose the domains to define. For the Transverse vertical dispersivity the Thermal Dispersion node defines the tensor of dispersive thermal conductivity d = C k ij L p L D ij where Dij is the dispersion tensor uk ul D ij = ijkl -----------u and ijkl is the fourth order dispersivity tensor lo – tr ijkl = tr ij kl + -------------------. This adds an extra term ·kdT to the right-hand side of T C p eq ------. ik jl + il jk 2 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 | 143 . 144 | C H A P T E R 3 : T H E H E A T TR A N S F E R B R A N C H . 4 Heat Transfer in Thin Shells This chapter describes the Heat Transfer in Thin Shells interface found under the Heat Transfer branch ( ) in the Model Wizard. In this chapter: • The Heat Transfer in Thin Shells User Interface • Theory for the Heat Transfer in Thin Shells User Interface 145 . It adds the equation for the temperature and provides a settings window for defining the thermal conductivity. the heat capacity and the density (see Equation 4-1). select Manual from the Selection list. 146 | C H A P T E 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 . numbers and underscores (_) are permitted in the Identifier field. solving for the temperature. SURFACE-TO-SURFACE RADIATION Select the Surface-to-surface radiation check box to add a Radiation Settings section. the identifier string must be unique. edge and point conditions. Only letters. BOUNDARY SELECTION The default setting is to include All boundaries in the model to define the dependent variables and the equations. these default nodes are also added to the Model Builder— Thin Conductive Layer. and Initial Values. found under the Heat ) in the Model Wizard. The default identifier (for the first interface in the model) is htsh. Insulation/Continuity (a default boundary condition). To choose specific boundaries. is suitable for solving thermal-conduction problems in thin structures and has the equations. The default is 0. SHELL THICKNESS Define the Shell thickness ds (SI unit: m) (see Equation 4-1). In order to distinguish between variables belonging to different physics user interfaces.T he H e a t T r a nsfer i n Th i n Sh el l s U ser Interface The Heat Transfer in Thin Shells (htsh) user interface ( ). edge or point conditions. When this interface is added. for example. INTERFACE IDENTIFIER The interface identifier is used primarily as a scope prefix for variables defined by the physics user interface.01 m. and heat sources for modeling heat transfer in thin conductive shell.<variable_name>. and heat sources. Right-click the Heat Transfer in Thin Shells node to add other features that implement. Refer to such interface variables in expressions using the pattern <identifier>. It is available for 3D models Transfer branch ( The Thin Conductive Layer is the main node. The first character must be a letter. Select the Number of wavelength intervals. The variable name can be changed. and when this option is active. The default is 256. DEPENDENT VA RIA BLES The dependent variable (field variable) is for the Temperature T. Select Linear (the default). It is possible to define up to 5 wavelength intervals. select the Radiation integration order. Select the Use radiation groups check box to enable the ability of defining radiation groups. select the Radiation resolution. This option has no effect when the model does not contain a moving frame since the material and spatial frames are identical in this case. 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. It is then possible to define the surface emissivity per spectral band. When the model contains moving mesh. Default is one which correspond to a diffuse gray radiation model. Select the Surface-to-surface radiation method—Hemicube or Direct area integration. Cubic. the heat transfer physics automatically account for deformation effects T H E H E A T TR A N S F E R I N T H I N S H E L L S U S E R I N T E R F A C E | 147 . When the Number of wavelength intervals is greater than one. This can speed up the radiation calculations in many cases. With moving mesh. a diffuse spectral model is used. the Enable conversions between material and spatial frame check box is selected by default. ADVANCED SETTINGS Add both a Heat Transfer in Thin Shells (htsh) and Moving Mesh (ale) interface (found under the Mathematics>Deformed Mesh branch in the Model Wizard) then click the Show button ( ) and select Advanced Physics Options to display this section. Quadratic. • If Direct area integration is selected.RADIATION SETTINGS See The Heat Transfer Interface for details about the Surface-to-surface radiation method and Radiation resolution settings. Quartic or Quintic to define the Discretization level used for the surface radiosity shape function. Modify the Transparent media refractive index it is different from 1 that corresponds to vacuum refractive index and that is a good approximation for air refractive index. but the names of dependent variables must be unique within a model. When the Enable conversions between material and spatial frame check box is not selected. are also covered. Specify the Value type when using splitting of complex variables—Real (the default) or Complex for each of the variables in the table. Heat Flux. click the Show button ( ) on the Model Builder and then select Discretization. Select Quadratic (the default). edge. Edge. or Quartic for the Temperature. and Line Heat Source) are not converted and all are defined on the Spatial frame. point and pair nodes described (listed in alphabetical order): 148 | C H A P T E 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 . In particular the effects for volume changes on the density are considered. and Pair Nodes for the Heat Transfer in Thin Shells User Interface • Theory for the Heat Transfer in Thin Shells User Interface Shell Conduction: Model Library path Heat_Transfer_Module/ Tutorial_Models_Thin_Structure/shell_conduction Boundary. Boundary Heat Source. deformation effects on arbitrary thermal conductivity.on heat transfer properties. and Pair Nodes for the Heat Transfer in Thin Shells User Interface The Heat Transfer in Thin Shells User Interface has the following boundary. Linear. Point. more generally. the feature inputs (for example. Heat Source. Cubic. Point. Edge. • About Handling Frames in Heat Transfer • Show More Physics Options • Boundary. DISCRETIZATION To display this section. Rotation effects on thermal conductivity of an anisotropic material and. BOUNDARY OR EDGE SELECTION From the Selection list. Heat Flux Use the Heat Flux node to add heat flux across boundaries. select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree. To locate and search all the documentation.The following are described in this section: • Change Effective Thickness • Heat Flux • Change Thickness • Heat Source • Edge Heat Source • Point Heat Source • Initial Values • Surface-to-Ambient Radiation • Insulation/Continuity • Thin Conductive Layer These nodes are described for the Heat Transfer interfaces: • External Radiation Source • Prescribed Radiosity • Opaque • Radiation Group • Out-of-Plane Heat Flux • Out-of-Plane Convective Heat Flux • Surface-to-Surface Radiation (Boundary Condition) • Out-of-Plane Radiation • Temperature When nodes are described for other interfaces. It adds a heat flux qdeq0. A positive heat flux adds heat to the domain. The Heat Flux feature adds a heat source (or sink) to edges. in COMSOL Multiphysics. choose the boundaries or edges to define. T H E H E A T TR A N S F E R I N T H I N S H E L L S U S E R I N T E R F A C E | 149 . the difference for this interface is that Boundaries are selected instead of Domains. Then click the Show button ( ) and select Advanced Physics Options. MODEL INPUTS This section has fields and values that are inputs to expressions that define material properties. and cannot be edited. choose the pair to define. When nodes are added from the context menu.PAIR SELECTION If this node is selected from the Pairs menu. The rest of the settings are the same for the Heat Source node as described under the Frame Selection section. that is. which are the same for this interface. you can select Manual from the Selection list to choose specific boundaries or select All boundaries as required. Heat Flux settings section for The Heat Transfer Interface. BOUNDARY SELECTION For a default node. If such user-defined property groups have been added. 150 | C H A P T E 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 model inputs are included here. An identity pair has to be created first. Ctrl-click to deselect. HEAT FLUX See the Heat Flux node. the setting inherits the selection from the parent node. FRAME SELECTION To display this section add both a Heat Transfer in Thin Shells (htsh) and Moving Mesh (ale) interface (found under the Mathematics>Deformed Mesh branch in the Model Wizard). About Handling Frames in Heat Transfer Thin Conductive Layer The Thin Conductive Layer node adds the heat equation for conductive heat transfer in shells (see Equation 4-1). the selection is automatically selected and is the same as for the interface. The Heat Source describes heat generation within the shell. choose Isotropic. Diagonal. as linear heat source. respectively. HEAT SOURCE See the Heat Source node. or Anisotropic based on the characteristics of the thermal conductivity and enter other values or expressions in the field or matrix. choose the boundaries to define. enter other values or expressions. all heat sources within a boundary contribute to the total heat source. Symmetric. THERMODYNAMICS Specify the Density (SI unit: kg/m3) and the Heat capacity at constant pressure Cp (SI unit: J/(kg·K)) to describe the amount of heat energy required to produce a unit temperature change in a unit mass. If User defined is selected. T H E H E A T TR A N S F E R I N T H I N S H E L L S U S E R I N T E R F A C E | 151 . or as a total heat source (power). The default settings use values From material for both. Specify the heat source as the heat per volume in the domain. which are the same for this interface. the Thermal conductivity k (SI unit: W/(m·K)) uses values From material. If User defined is selected. Express heating and cooling with positive and negative values. BOUNDARY SELECTION From the Selection list.HEAT CONDUCTION Thermal Conductivity Tensor Components By default. Heat Source The Heat Source node adds a thermal source Q. It adds the following contributions to the right-hand side of Equation 4-1: d s Q . Add one or more nodes as required. Heat Source settings section for The Heat Transfer Interface. CHANGE THICKNESS Specify a value for the Shell thickness ds (SI unit: m). right-click the Initial Values node.01 m.Jinit W/m2. and cannot be edited. The default is approximately room temperature. BOUNDARY SELECTION From the Selection list. Also enter a Surface radiosity J (SI unit: W/m2). the setting inherits the selection from the parent node. If more than one set of initial values is needed. the selection is automatically selected and is the same as for the interface. Surface-to-Surface Radiation boundary condition as r T 04 – 1 – r T amb Change Thickness Use the Change Thickness node to give parts of the shell a different thickness than that what is specified on the Heat Transfer in Thin Shells interface Shell Thickness section. choose the boundaries to define. This variable is defined under the 4 . that is. When nodes are added from the context menu. INITIAL VALUES Enter a value or expression for the initial value of the Temperature T. 152 | C H A P T E 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 . 293.15 K (20º C). BOUNDARY SELECTION For a default node.FRAME SELECTION The settings are the same for the Heat Flux node Frame Selection section About Handling Frames in Heat Transfer 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. you can select Manual from the Selection list to choose specific boundaries or select All boundaries as required. This value replaces the overall thickness for the boundaries that are selected. The default is htsh. The default value is 0. choose the pair to define.Surface-to-Ambient Radiation Use the Surface-to-Ambient Radiation condition to add surface-to-ambient radiation to edges. this edge condition means that the temperature field and its flux is continuous across the edge. Ctrl-click to deselect. Enter an Ambient temperature Tamb (SI unit: K). An emissivity of 0 means that the surface emits no radiation at all and an emissivity of 1 means that it is a perfect blackbody. The default is 293. MODEL INPUTS This section contains fields and values that are inputs to expressions that define material properties. choose the edges to define. On external edges. is the Stefan-Boltzmann constant (a predefined physical constant). T H E H E A T TR A N S F E R I N T H I N S H E L L S U S E R I N T E R F A C E | 153 .15 K. PAIR SELECTION If this node is selected from the Pairs menu. this section is empty. Insulation/Continuity The Insulation/Continuity node is the default edge condition. If such user-defined materials are added. and Tamb is the ambient temperature. the model inputs appear here. An identity pair has to be created first. The net inward heat flux from surface-to-ambient radiation is 4 4 n – d s k T T = T amb – T where is the surface emissivity. Initially. this edge condition means that there is no heat flux across the edge: n k g T = 0 On internal edges. EDGE SELECTION From the Selection list. SURFACE-TO-AMBIENT RADIATION The default Surface emissivity (a dimensionless number between 0 and 1) is taken From material. An identity pair has to be created first. A positive q means heating while a negative q means cooling. EDGE SELECTION From the Selection list. When nodes are added from the context menu. choose the pair to define. the selection is automatically selected and is the same as for the interface. PAIR SELECTION If this node is selected from the Pairs menu. that is. Ctrl-click to deselect. EDGE SELECTION From the Selection list.01 m. It adds a heat source qQl or qdeQb. Edge Heat Source The Edge Heat Source node models a linear heat source (or sink). PAIR SELECTION If this node is selected from the Pairs menu. Change Effective Thickness The Change Effective Thickness node models edges with another thickness than the overall thickness that is specified in the Heat Transfer in Thin Shells interface Shell Thickness section. The default is 0. CHANGE EFFECTIVE THICKNESS Enter a value for the Effective thickness de (SI unit: m). and cannot be edited. 154 | C H A P T E 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 setting inherits the selection from the parent node. choose the edges to define. Ctrl-click to deselect. An identity pair has to be created first. Ctrl-click to deselect. It defines the height of the part of the edge that is exposed to the ambient surroundings. This value replaces the overall thickness in the edges selected in the Edges section. choose the pair to define.EDGE SELECTION For a default node. An identity pair has to be created first. choose the pair to define. you can select Manual from the Selection list to choose specific edges or select All edges as required. PAIR SELECTION If this node is selected from the Pairs menu. choose the edges to define. If Heat source defined per unit of length is selected. • If Total boundary power is selected. The default is 0 W. • When General source is selected.tot (SI unit: W). enter the total power (total heat source) Pl. POINT HEAT SOURC E Enter a value or expression for the Point heat source Qp (SI unit: W). click the General source (the default) or Total line power button. The default is 0 W. The default is 0 W/m2. tot (SI unit: W). FRAME SELECTION The settings are the same for the Heat Flux node Frame Selection section About Handling Frames in Heat Transfer Point Heat Source The Point Heat Source node models a point heat source (or sink). The default is 0 W/m. Ql (SI unit: W/m) in unit power per unit length. A positive Qb is heating and a negative Qb is cooling.EDGE HEAT SOURCE From the Edge heat source type list. enter the boundary heat source Qb (SI unit: W/m2). • If General source is selected. The added heat source is equal to Qp. The default is 0 W. POINT S EL EC TION From the Selection list. T H E H E A T TR A N S F E R I N T H I N S H E L L S U S E R I N T E R F A C E | 155 . click the General source (the default) or Total boundary power button. Positive Ql is heating while a negative Ql is cooling. choose the points to define. If Heat source defined per unit of area is selected. • If Total line power is selected. A positive Qp means heating while a negative Qp means cooling. select Heat source defined per unit of length (the default) or Heat source defined per unit of area. enter the total power (total heat source) Pb. enter a value for the distributed heat source. u – T + d s h d T ext. d – T t (4-2) 4 4 + d s u T 4 amb. the heat capacity and the density: d s C p T + T – d s k g T T = 0 t (4-1) Heat Transfer Equation in Thin Conductive Shell The dependent variable is the temperature T. u – T + d s d T 4 amb. assume constant temperature through the shell thickness. The governing equation for heat transfer in thin shells is: d s C p T + T – d s k g TT = d s Q + d s h u T ext. It adds the equation for the temperature and provides a settings window for defining the thermal conductivity.T he o r y f o r the H eat Tran sfer i n Th i n Shells User Interface The Heat Transfer in Thin Shells User Interface theory is described in this section: • About Heat Transfer in Thin Shells • Heat Transfer Equation in Thin Conductive Shell • Thermal Conductivity Tensor Components About Heat Transfer in Thin Shells The Heat Transfer in Thin Shells User Interface supports two types of heat transfer: conduction and out-of-plane heat transfer and is suitable for solving thermal-conduction problems in thin structures. The interface is defined on 3D faces. Because the thermal conductivity across the shell thickness is very large or the shell is so thin. The Thin Conductive Layer node is the main feature. d – T + d s q u + d s q d Where T is the tangential derivative along the shell and • is the density (SI unit: kg/m3) • ds is the shell thickness (SI unit: m) 156 | C H A P T E 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 . exl.• 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. The local x direction. upside and downside (SI unit: K) • qu and qd are the out-of-plane inward heat fluxes. Their cross product defines the third orthogonal direction such that: e xl = t1 e yl = e xl e zl e zl = n = n t1 From this. is the normal vector n. upside and downside (SI unit: 1). is the surface tangent vector t1 and the local z direction. • Tamb. d are the out-of-plane ambient temperatures. u and Tamb. ezl. 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 q = – k g TT which is Fourier’s law of heat conduction (see also The Heat Equation). upside and downside (SI unit: K) • u and d are the out-of-plane surface emissivities. upside and downside (SI unit: W/(m2·K)) • Text. a transformation matrix between the shell’s local coordinate system and the global coordinate system can be constructed in the following way: e xlx e ylx e zlx A = e xly e yly e zly e xlz e ylz e zlz 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 H I N S H E L L S U S E R I N T E R F A C E | 157 . d are the out-of-plane external temperatures. The tensor components are specified in the shell local coordinate system. u and Text. which is defined from the geometric tangent and normal vectors. can be expressed as k g = AkA t 158 | C H A P T E 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 thermal conductivity tensor. kg. In this chapter: • The Radiation Branch Versions of the Heat Transfer User Interface • The Surface-To-Surface Radiation User Interface • Theory for the Surface-to-Surface Radiation User Interface • The Radiation in Participating Media User Interface • Theory for the Radiation in Participating Media User Interface • References for the Radiation User Interfaces 159 . The physics interface for modeling radiative heat transfer are available under the Heat Transfer>Radiation branch ( ) and described in the About the Heat Transfer Interfaces section.5 R a d i a t i o n H eat Transfer This chapter describes the interfaces for modeling radiative heat transfer. RADIATION SETTINGS See the Radiation Settings section for The Surface-To-Surface Radiation User Interface. Thermal Insulation. to model heat transfer that includes surface-to-surface radiation. It is a Heat Transfer interface with the Surface-to-surface radiation check box selected under Physical Model. the rest of the settings are the same as for The Heat Transfer Interface. The following default nodes are also added to the Model Builder—Heat Transfer in Solids (with a default Opaque node). and Initial Values. 160 | C H A P T E R 5 : R A D I A T I O N H E A T TR A N S F E R . found under the Radiation branch ( ). Right-click the node to add other boundary conditions and features. which enables the Radiation Settings section.The Radiation Branch Versions of the Heat Transfer User Interface • The Heat Transfer with Surface-to-Surface Radiation User Interface • The Heat Transfer with Radiation in Participating Media User Interface • About the Heat Transfer Interfaces • The Heat Transfer Interface The Heat Transfer with Surface-to-Surface Radiation User Interface Use the Heat Transfer with Surface-to-Surface Radiation (ht) user interface ( ). Except for the Radiation Settings section. The Heat Transfer with Radiation in Participating Media User Interface The Heat Transfer with Radiation in Participating Media (ht) user interface ( ). This interface solves for radiative intensity and temperature fields. Thermal Insulation. the Radiation in participating media check box is selected in the Physical Model section of the main Heat Transfer settings window. and Initial Values. found under the Heat Transfer>Radiation branch ( ) in the Model Wizard. Domain. Edge. The following default nodes are also added to the Model Builder—Heat Transfer in Solids. combines features from the Radiation in Participating Media and Heat Transfer interfaces. and Pair Nodes for the Radiation Branch Versions of the Heat Transfer User Interface Both The Heat Transfer with Surface-to-Surface Radiation User Interface and The Heat Transfer with Radiation in Participating Media User Interface are versions of the Heat Transfer interface. and described for. Boundary. This means all the nodes are shared with. When this interface is added. the rest of the settings are the same as for The Heat Transfer Interface. Except for the Participating Media Settings section (described for the The Radiation in Participating Media User Interface). Continuity on Interior Boundary. T H E R A D I A T I O N B R A N C H VE R S I O N S O F 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 | 161 . Point. Right-click the node to add other boundary conditions and features. Opaque Surface. This enables the modeling of radiative heat transfer inside a participating medium combined with heat transfer in solids and fluids. The Heat Transfer with Surface-to-Surface Radiation (ht) interface also has the following nodes available and described for the Surface-to-Surface Radiation interface: • External Radiation Source • Opaque • Prescribed Radiosity • Radiation Group • Diffuse Mirror • Surface-to-Surface Radiation (Boundary Condition) The Heat Transfer with Radiation in Participating Media (ht) interface also has the following nodes available: • Continuity on Interior Boundary • Incident Intensity • Opaque Surface • Radiation in Participating Media To locate and search all the documentation.The Heat Transfer Interface. select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree. 162 | C H A P T E R 5 : R A D I A T I O N H E A T TR A N S F E R . in COMSOL Multiphysics. including nodes described for the Highly Conductive Layer Nodes and Out-of-Plane Heat Transfer Nodes. point. and pair nodes and subnodes (listed in alphabetical order): • Boundary Heat Source • Outflow • Continuity • Periodic Heat Condition • Convective Heat Flux • Point Heat Flux • Edge Heat Flux • Point Heat Source • Edge Surface-to-Ambient Radiation • Point Surface-to-Ambient Radiation • Temperature • Point Temperature • Heat Flux • Pressure Work • Heat Source • Surface-to-Ambient Radiation • Heat Transfer in Fluids • Symmetry • Heat Transfer in Solids • Temperature • Heat Transfer with Phase Change • Thermal Contact • Highly Conductive Layer • Thermal Insulation • Inflow Heat Flux • Thin Thermally Resistive Layer • Initial Values • Translational Motion • Line Heat Source • Viscous Heating • Open Boundary T H E R A D I A T I O N B R A N C H VE R S I O N S O F 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 | 163 . boundary. edge.The following are links to the domain. use The Heat Transfer with Surface-to-Surface Radiation User Interface instead. Refer to such interface variables in expressions using the pattern <identifier>. but the interface is compatible with all standard study types. Right-click the node to add a Surface-to-Surface Radiation boundary condition or other nodes. found under the Heat ) in the Model Wizard. Absolute (thermodynamical) temperature units must be used. then select The Radiation in Participating Media User Interface. This physics user interface solves only for the surface radiosity. The surface-to-surface radiation is always stationary (that is. the radiation time scale is assumed to be shorter than any other time scale). See Specifying Model Equation Settings in the COMSOL Multiphysics Reference Manual. To solve for surface radiosity and temperature.<variable_name>. The radiation therefore contributes to the boundary conditions rather than to the heat equation itself. If the media participate in the radiation.The Surface-To-Surface Radiation User Interface The Surface-to-Surface Radiation (rad) user interface ( ). INTERFACE IDENTIFIER The interface identifier is used primarily as a scope prefix for variables defined by the physics user interface. COMSOL Multiphysics works under the assumption that the domain medium does not participate in the radiation process. For the Surface-to-Surface Radiation user interface. For this user interface. The process transfers energy directly between boundaries and external radiation sources. In order to distinguish between variables 164 | C H A P T E R 5 : R A D I A T I O N H E A T TR A N S F E R . select a Stationary or Time Dependent study as a preset study type. treats thermal radiation as an Transfer>Radiation branch ( energy transfer between boundaries and external heat sources where the medium does not participate in the radiation (radiation in transparent media). Define the Wavelength dependence of emissivity. 2. the surface emissivity has the same definition for all wavelength.S U R F A C E R A D I A T I O N U S E R I N T E R F A C E | 165 . select the Surface-to-surface radiation check box under Physical Model settings window. to define a diffuse gray radiation model. • Select Solar and ambient to define a diffuse spectral radiation model with two spectral bands. select Manual from the Selection list. Still the surface emissivity can depend on other quantities. (solar radiation) and one for large wavelengths. to define a diffuse spectral radiation model.belonging to different physics user interfaces. to adjust the wavelength intervals corresponding to the solar and ambient radiation. . numbers and underscores (_) are permitted in the Identifier field. • Keep the default value. Update Intervals endpoint (SI unit m). To choose specific boundaries. Constant. (ambient radiation). 1.. • Choose Multiple Spectral Bands and set the Number of wavelength intervals value (2 to 5). 1. 0 and N (with N equal to the value selected to define of Number of wavelength intervals). Modify the Transparent media refractive index if it is different from 1 that corresponds to vacuum refractive index and that is usually a good approximation for air refractive index. It is then possible to provide a definition of the surface emissivity for each spectral band. Only letters. [. one for short wavelengths. The first character must be a letter. to define the wavelength intervals [i-1.[. It is then possible to define the Intervals endpoint (SI unit m). To display this section for any version of the Heat Transfer interface. The default identifier (for the first user interface in the model) is rad.1]. T H E S U R F A C E . In this case. The surface properties can then be defined for each spectral band. [1.. in particular it can be temperature dependent.TO . In particular it is possible to define the solar absorptivity for short wavelengths and the surface emissivity for large wavelengths. RADIATION SETTINGS This section is alway visible in the Surface-to-Surface Radiation interface. the identifier string must be unique. BOUNDARY SELECTION The default setting is to include All boundaries in the model to define the dependent variables and the equations.i[ for i from 1 to Number of wavelength intervals. are predefined and equal to 0 and respectively. Note that the first and the last endpoints. Try to balance this number against the mesh resolution in the rz-plane. Quartic or Quintic to define the Discretization level used for the surface radiosity shape function. in many cases. Select a Surface-to-surface radiation method—Hemicube (the default) or Direct area integration. The number of z-buffer pixels on each side of the 3D hemicube equals the specified resolution squared. speed up the radiation calculations. For an axisymmetric geometry. Thus the time required to evaluate the irradiation increases quadratically with resolution. 166 | C H A P T E R 5 : R A D I A T I O N H E A T TR A N S F E R . 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. Cubic. excluded from the integrals. • If Hemicube is selected. select a Radiation integration order—4 is the default. Elements facing away from each other are. Hemicube Hemicube is the default method for the heat transfer interfaces. Select Linear (the default). Its accuracy can be influenced by setting the Radiation resolution of the virtual snapshots. Gm and Famb must be evaluated in a corresponding 3D geometry obtained by revolving the 2D boundaries about the axis. Quadratic. however. as well. • If Direct area integration is selected. which is the same as the number of elements to a full revolution. Direct Area Integration COMSOL Multiphysics evaluates the mutual irradiation between surface directly. In 2D. and thus the time is. See below for descriptions of each method. The resolution can be controlled in the azimuthal direction by setting the number of azimuthal sectors. without considering which face elements are obstructed by others. select a Radiation resolution—256 is the default. This means that shadowing effects (that is. in 2D only three directions are needed). Think of it as rendering digital images of the geometry in five different directions (in 3D. COMSOL Multiphysics creates this virtual 3D geometry by revolving the 2D boundary mesh into a 3D mesh.Also select the Use radiation groups check box to enable the ability to define radiation groups. which can. the number of z-buffer pixels is proportional to the resolution property. and counting the pixels in each mesh element to evaluate its view factor. surface elements being obstructed in nonconvex cases) are not taken into account. This section is empty for Surface-to-Surface Radiation (rad) interface which define the shape functions discretization in Radiation Settings. Edge.Direct area integration is fast and accurate for simple geometries with no shadowing. and Pair Nodes for the Radiation Branch Versions of the Heat Transfer User Interface • The Surface-To-Surface Radiation User Interface • Theory for the Heat Transfer User Interfaces • Thermo-Photo-Voltaic Cell: Model Library path Heat_Transfer_Module/Thermal_Radiation/tpv_cell • Free Convection in a Light Bulb: Model Library path Heat_Transfer_Module/Thermal_Radiation/light_bulb Domain.S U R F A C E R A D I A T I O N U S E R I N T E R F A C E | 167 . Point. and Pair Nodes for the Surface-to-Surface Radiation User Interface The Surface-To-Surface Radiation User Interface has these domain. edge. or where the shadowing can be handled by manually assigning boundaries to different groups. click the Show button ( ) and select Discretization. Control the accuracy by specifying a Radiation integration order. Point. Boundary. • About the Heat Transfer Interfaces • The Heat Transfer Interface • Domain. point. DISCRETIZATION To display this section.TO . Sharp angles and small gaps between surfaces may require a higher integration order for accuracy but also more time to evaluate the irradiation. Boundary. Edge. and pair nodes available (listed in alphabetical order): • External Radiation Source • Diffuse Mirror • Opaque T H E S U R F A C E . boundary. If shadowing is ignored. global energy is not conserved. Where the radiation is defined on both sides the radiative heat source is defined on both sides too. choose the boundaries where the boundary condition is applied. ny. the radiative heat source is ignored. RADIATION SETTINGS When Wavelength dependence of emissivity is set to Constant in the interface settings. MODEL INPUTS This section has fields and values that are inputs to expressions that define material properties. The default is the temperature variable in the Heat Transfer interface or 293. There is one standard model input—the Temperature T (SI unit: K). When the model input does not contain a dependent variable. for the fractional emissive power. The feature adds one radiosity shape function per spectral interval to its selection and uses it as surface radiosity.15 K in the Surface-to-Surface Radiation interface. the model inputs are included here.• Prescribed Radiosity • Radiation Group • Surface-to-Surface Radiation (Boundary Condition) Surface-to-Surface Radiation (Boundary Condition) The Surface-to-Surface Radiation boundary condition feature handles radiation with view factor calculation. This model input is used in the expression for the blackbody radiation intensity and. The temperature model input is also used to determine the variable that receives the radiative heat source. when multiple wavelength intervals are used. BOUNDARY SELECTION From the Selection list. Surface-to-Surface Radiation boundary condition adds radiative heat source contribution q = G – e b T on the side of the boundary where the radiation is defined. If such user-defined property groups have been added. select a Radiation direction based on the geometric normal (nx. nz)—Opacity 168 | C H A P T E R 5 : R A D I A T I O N H E A T TR A N S F E R . • Select Negative normal direction to specify that the surface radiates in the negative normal direction. Positive normal direction. Opacity is controlled by the Opaque boundary condition.TO . Enter an Ambient temperature Tamb (SI unit: K). By default Define ambient temperature on each side is not selected. select a Radiation direction for each spectral band—Opacity controlled (the default).S U R F A C E R A D I A T I O N U S E R I N T E R F A C E | 169 .3-2. or Both sides. The default is 293. The additional choice.controlled (the default). When Wavelength dependence of emissivity is set to Solar and ambient or Multiple spectral bands in the interface settings. This is useful for example to represent that glass is opaque to radiation outside of the 0. When Define ambient temperature on each side is selected. The Radiation direction defines the radiation direction for each spectral band similarly as when Wavelength dependence of emissivity is Constant. fire) and to a cold temperature on the other side (for example.5 µm wavelength range. Positive normal direction. Negative normal direction. in the Radiation Settings section. Defining a radiation direction for each spectral band make it possible to build models where the transparency/opacity properties defer between spectral bands.15 K. • Select Positive normal direction if the surface radiates in the positive normal direction. This is needed to define ambient temperature for a surface that radiates on both side and that is exposed to a hot temperature on one side (for example. or Both sides. Negative normal direction. The Wavelength dependence of emissivity is defined in the interface settings. AMBIENT Select Define ambient temperature on each side when the ambient temperature differs between the sides of a boundary. define the Ambient temperature Tamb. u T H E S U R F A C E . None is used when adjacent domains are either both transparent or both opaque for a given spectral band. or None. external temperature). • Select Both sides if the surface radiates on both sides. • Opacity controlled requires that each boundary is adjacent to exactly one opaque domain. SURFACE EMISSIVITY In diffuse gray and diffuse spectral radiation models. is theoretically zero and the value of Tamb therefore should not matter. the surface emissivity and the absorptivity must be equal. Famb. d on the up and down side respectively. Inside a closed cavity. in the Radiation Settings section. FEPBi for each spectral band. All fractional emissive powers are expected to be in [0. good practice to set Tamb to T or to a typical temperature value for the cavity surfaces in such cases because that minimizes errors introduced by the finite resolution of the view factor evaluation.1] and their sum is expected to be equal to 1. For this reason it is equivalent to define the surface emissivity or the absorptivity. The surface emissivity settings are defined per spectral interval. the ambient view factor. however. The geometric normal points from the down side to the up side. Set Tamb to the far-away temperature in directions where no other boundaries obstruct the view.and Tamb. It is. When the Fractional emissive power is User defined. When the Fractional emissive power is Blackbody/Graybody. define the Fractional emissive power. SURF ACE FRA CTIONAL EMISS IVE POWER This section is available only when Wavelength dependence of emissivity is Solar and ambient or Multiple spectral bands. the fractional emissive power is automatically computer for each spectral band as a function of the band endpoints and surface temperature. 170 | C H A P T E R 5 : R A D I A T I O N H E A T TR A N S F E R . The Wavelength dependence of emissivity is defined in the interface settings. The default Surface radiosity is ht. When the Radiation direction is set to Both sides for a spectral band.init (SI unit: W/m2) is defined as J Bi init = Bi e b T init + 1 – Bi e b T amb When Both sides is selected as the Radiation direction. Set the surface emissivity to a number between 0 and 1.TO . INIT IA L VA LUES The surface radiosity initial values are defined per spectral interval. The geometric normal points from the down side to the up side. init. The list contains other options based on the materials defined in the model. or Positive normal direction for a spectral band Bi. init. d (SI unit: W/m2). JBi. When the Radiation direction is Opacity controlled. or Positive normal direction for a spectral band. no information is needed for this spectral band in the Surface Emissivity section. define the Material on upside and Material on downside: • The default for both Material on upside and Material on downside use Boundary material. This is a property of the material surface that depends both on the material itself and the structure of the surface.JBiinitU and ht.When the Radiation direction is Opacity controlled.S U R F A C E R A D I A T I O N U S E R I N T E R F A C E | 171 . where 0 represents diffuse mirror and 1 is appropriate for a perfect blackbody. Negative normal direction. u and JBi. respectively. • Define the Surface emissivity on the upside and downside. the Surface emissivity (dimensionless) uses values From material. the default Surface radiosity.JBiinitD (SI unit: W/m2). The proper value for a physical material lies somewhere in-between and can be found from tables or measurements. The Wavelength dependence of emissivity is defined in the interface settings. Enter initial values for the Surface radiosity JBi. J Bi init u = Bi u e b T init + 1 – Bi u e b T amb u T H E S U R F A C E . Negative normal direction. When the Radiation direction is set to None for a spectral band. Make sure that a material is defined at the boundary level (by default materials are defined at the domain level). in the Radiation Settings section. by default. d are named. init.. surface-to-surface radiation propagates in non-opaque domains. in the Radiation Settings section. respectively. When the Radiation direction is defined by Opacity controlled in surface-to-surface boundary features. up to the maximum number of spectral intervals. u and JBi. Jinit. JBi.. Alternatively the Radiation direction can be defined using the normal orientation or on both sides of boundaries. init. u and Jinit. init. DOMAIN SELECTION From the Selection list. no surface radiosity is defined hence no initial value is needed. In this case the Opaque node is ignored. choose the domains to define as opaque. Boundary.. Jinit. The Opaque node enables to define the surface-to-surface radiation direction on boundaries surrounding the domains where the Opaque node is defined. right-click the Surface-to-Surface Radiation (rad) node to add an Opaque node.J Bi init d = Bi d e b T init + 1 – Bi d e b T amb d When None is selected as the Radiation direction. • In the notation used here. Bi stands for B1. 172 | C H A P T E R 5 : R A D I A T I O N H E A T TR A N S F E R . Edge. Point. d • Guidelines for Solving Surface-to-Surface Radiation Problems • Radiation Group Boundaries • Domain. B2. • When the model contains one spectral interval. and Pair Nodes for the Surface-to-Surface Radiation User Interface Opaque In the Surface-to-Surface Radiation interface. • The Wavelength dependence of emissivity is defined in the interface settings. JBi. The node adds radiosity shape function for each spectral band to its selection and uses it as surface radiosity. If such user-defined property groups have been added. When Wavelength dependence of emissivity is set to Solar and ambient or Multiple spectral bands in the interface settings. this section is empty.15 K in the T H E S U R F A C E .S U R F A C E R A D I A T I O N U S E R I N T E R F A C E | 173 . There is one standard model input—the Temperature T (SI unit: K). Diffuse mirror surfaces are common as approximations of a surface that is well insulated on one side and for which convection effects can be neglected on the opposite (radiating) side. Diffuse Mirror The Diffuse Mirror node is a variant of the surface-to-surface radiation node with a surface emissivity equal to zero. in the Radiation Settings section. select the spectral bands for which the opacity is defined by selecting corresponding Opaque on spectral band i check box. MODEL INPUTS This section has fields and values that are inputs to expressions that define material properties. The Wavelength dependence of emissivity is defined in the interface settings. The opacity is then defined for all wavelengths. By default the Opaque feature is active for all spectral bands. the model inputs are included here. It resembles a mirror that absorbs all irradiation and then radiates it back in all directions. BOUNDARY SELECTION From the Selection list. The radiative heat flux on a diffuse mirror boundary is zero. choose the boundaries where this boundary condition is applied.TO . The default is the temperature variable in the heat transfer physic interface or 293.OPAQUE When Wavelength dependence of emissivity is set to Constant in the user interface settings. • Opacity controlled requires that each boundary is adjacent to exactly one opaque domain. • Select Negative normal direction to specify that the surface radiates in the negative normal direction. ny. Positive normal direction. and Initial Values sections are the same as for the Surface-to-Surface Radiation (Boundary Condition). It is used in the blackbody radiation intensity expression. Ambient. The Radiation Settings. BOUNDARY SELECTION From the Selection list. select a Radiation direction based on the geometric normal (nx.surface-to-surface physic interface. It is used in the blackbody radiation intensity expression. MODEL INPUTS This section has fields and values that are inputs to expressions that define material properties. choose the boundaries where this boundary condition is applied.15 K in the Surface-to-Surface Radiation interface. the model inputs are included here. Negative normal direction. nz)—Opacity controlled (the default). Opacity is controlled by the Opaque boundary condition. Radiosity can be defined as blackbody or graybody radiation. Prescribed Radiosity Use the Prescribed Radiosity node to specify radiosity on the boundary for each spectral band. There is one standard model input—the Temperature T (SI unit: K). RADIATION DIRECTION When Wavelength dependence of emissivity is set to Constant in the interface settings. 174 | C H A P T E R 5 : R A D I A T I O N H E A T TR A N S F E R . The default is the temperature variable in the Heat Transfer interface or 293. A user-defined surface radiosity expression can also be defined. or Both sides. If such user-defined property groups have been added. • Select Positive normal direction if the surface radiates in the positive normal direction. • Select Both sides if the surface radiates on both sides. select a Radiation direction for each spectral band—Opacity controlled (the default). • When Wavelength dependence of emissivity is set to Constant in the interface settings. or Both sides. When Wavelength dependence of emissivity is set to Solar and ambient or Multiple spectral bands in the interface settings. it defines for each spectral band JBi = FEPBi(T)eb(T) when radiation is defined on one side or T H E S U R F A C E . but rather how that boundary affects others through radiation.S U R F A C E R A D I A T I O N U S E R I N T E R F A C E | 175 . Select a Radiosity expression—Graybody radiation (the default). Positive normal direction. The Radiation direction defines the radiation direction for each spectral band similarly as when Wavelength dependence of emissivity is Constant. or User defined. in the Radiation Settings section. The Wavelength dependence of emissivity is defined in the interface settings. Negative normal direction. or None. Blackbody radiation.TO . RADIOSITY Radiosity does not directly affect the boundary condition on the boundary where it is specified. Blackbody Radiation When Blackbody radiation is selected it sets the surface radiosity expression corresponding to a blackbody. • When Wavelength dependence of emissivity is set to Solar and ambient or Multiple spectral bands. it defines J = eb(T) when radiation is defined on one side or Ju = eb(Tu) and Jd = eb(Td) when radiation is defined on both sides. for all spectral bands. • When the temperature varies across a pair (for example with a Thin Thermally Resistive Layer condition is active on the same boundary). If Wavelength dependence of emissivity is set to Solar and ambient or Multiple spectral bands. Material on downside and the Surface emissivity Bi.JBi. If User defined is selected for the Surface emissivity. the temperature used to define the radiosity is evaluated on the side were the surface radiation is defined. If Wavelength dependence of emissivity is set to Constant in the interface settings. make sure that a material is defined at the boundary level (materials are defined by default at the domain level). define the Material on upside. the Surface emissivity Bi. or • when radiation is defined on both sides for Bi spectral band. • The blackbody hemispherical total emissive power is defined by ebTn2T4 Graybody Radiation When Graybody radiation is selected it sets the surface radiosity expression corresponding to a graybody. the Surface emissivity (dimensionless) is defined From material. define the Surface emissivity Bito set JBi = FEPBiBieb(T).u. enter another value for . respectively. define the Material on upside. In this case. The surface radiosity on upside and downside is then defined by Ju = ueb(Tu) and Jd = deb(Td) respectively. or • when radiation is defined on both sides.d(T)eb(Tu) and Jd = FEPBi. the Surface emissivity u. define the Surface emissivity to set J = eb(T). Material on downside and the Surface emissivity d on the upside and downside. respectively. • when radiation is defined on one side.d(T)eb(Td) when radiation is defined on both sides. • when radiation is defined on one side for Bi spectral band. By default. The surface radiosity on upside and 176 | C H A P T E R 5 : R A D I A T I O N H E A T TR A N S F E R . d on the upside and downside.d = FEPBi. respectively.ueb(Tu) and Jd = FEPBi(T)Bi. which means that all radiative boundaries belong to the same radiation group. T H E S U R F A C E . 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. where 0 represents diffuse mirror and 1 is appropriate for a perfect blackbody.TO . Radiation Group Add a Radiation Group to a Surface-to-Surface Radiation (rad) interface or any version of a Heat Transfer (ht) interface where the Surface-to-surface radiation check box is selected. or a Prescribed Radiosity feature. The Radiation Group node enables you to specify radiation groups to speed up the radiation calculations and gather boundaries in a radiation problem that have a chance to see one another. When the Use radiation groups check box is selected. it sets the surface radiosity expression to J = J0. J0 (SI unit: W/m2).S U R F A C E R A D I A T I O N U S E R I N T E R F A C E | 177 .downside is then defined by Ju = FEPBi(T)Bi. The surface emissivity to a number between 0 and 1. If Wavelength dependence of emissivity is Solar and ambient or Multiple spectral bands.deb(Td) respectively. Enter a Surface radiosity expression. similar settings are available for each spectral band. Select the Use radiation groups check box under Radiation Settings.d on the upside and downside. When the Radiation direction is set to Both sides (under Radiation Settings) also define the surface Radiosity expression J0. the node is automatically added to the Model Builder and contains all boundaries selected in the Surface-to-Surface Radiation (Boundary Condition). By default the check box is not selected.u and J0. The proper value for a physical material lies somewhere in-between and can be found from tables or measurements. The geometric normal points from the downside to the upside. User Defined If Wavelength dependence of emissivity is set to Constant in the interface settings and Radiosity expression is set to User defined. The default is 0 W/m2. When the Wavelength dependence of emissivity is Solar and ambient or Multiple spectral bands, the radiation groups are defined per spectral band. BOUNDARY SELECTION From the Selection list, choose the boundaries that belong to the same radiation group. This selection should contain any boundary that is selected in a Surface-to-Surface Radiation, a Diffuse Mirror, or a Prescribed Radiosity node and that has a chance to see one of the boundary that is already selected in the Radiation Group. RADIATION GROUP When the Wavelength dependence of emissivity is Constant, the radiation group is valid for all wavelengths, this is empty. When the Wavelength dependence of emissivity is set to Solar and ambient or Multiple spectral bands, the radiation group is define for all spectral bands by default. Clear Radiation group defined on spectral band i check boxes to remove Bi spectral band from this radiation group. Radiation Group Boundaries External Radiation Source The External Radiation Source node is selected from the Global submenu and is available for 2D and 3D models in the Surface-to-Surface Radiation (rad) interface or in any version of a Heat Transfer (ht) interface where the Surface-to-surface radiation check box is selected. Add an External Radiation Source to define an external radiation source as a point or directional radiation source with view factor calculation. Each External Radiation Source feature contributes to the incident radiative heat flux on all spectral bands, GBi, on all the boundaries where a Surface-to-surface or Diffuse mirror boundary condition is active. The source contribution, Gext,Bi, is equal to the product of the view factor of 178 | C H A P T E R 5 : R A D I A T I O N H E A T TR A N S F E R the source by the source radiosity. For radiation sources located on a point, Gext,Bi=Fext,Bi Ps,Bi. For directional radiative source Gext,Bi = Fext,Bi q0,s. The number of spectral bands is defined in the interface settings, in the Radiation Settings. When only one spectral band is defined, the Bi subscript in variable names is removed. The external radiation sources are ignored on the boundaries when neither Surface-to-Surface Radiation nor Diffuse Mirror is 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. SOURCE Select a Source position—Point coordinate (the default) or Infinite distance. In 3D, Solar position is also available as an option Point Coordinate If Point coordinate is selected, define the Source location xs (SI unit: m) and the Source power Ps (SI unit: W, default is 0 W). The source radiates uniformly in all directions. If Wavelength dependence of emissivity is Solar and ambient or Multiple spectral bands, set the Source power definition to Blackbody or User defined. When Blackbody is selected, enter the Source temperature, Ts (SI unit: K, default is 5780K), to define the source power on the spectral band Bi as Ps,Bi= FEPBi(Ts)Ps where FEPBi(Ts) is the fractional black body emissive power over Bi interval at Ts. When User defined is selected, enter an expression to define the source power on each spectral band Bi, Ps,Bi (SI unit: W, default is 0 W). xs should not belong to any surface where a Surface-to-surface or Diffuse Mirror boundary condition is active. T H E S U R F A C E - TO - S U R F A C E R A D I A T I O N U S E R I N T E R F A C E | 179 The Wavelength dependence of emissivity is defined in the interface settings, in the Radiation Settings section. Infinite Distance If Infinite distance is selected, define the Incident radiation direction is (dimensionless) and the Source heat flux q0,s (SI unit: W/m2). The default is 0 W/m2. If Wavelength dependence of emissivity is Solar and ambient or Multiple spectral bands, set the Source heat flux definition to Blackbody or User defined. When Blackbody is selected, enter the Source temperature, Ts (SI unit: K, default is 5780K), to define the source heat flux on the spectral band Bi as q0,s,Bi= q0,sFEPBi(Ts) where FEPBi(Ts) is the fractional black body emissive power over Bi interval at Ts. When User defined is selected, enter an expression to define the source heat flux on each spectral band Bi, q0,s,Bi (SI unit: W, default is 0 W). Solar Position Solar position is available for 3D models. When this option is selected use it to estimate the external radiative heat source due to the sun. North, West, and the up directions correspond to the x, y, and z directions, respectively. In the Location table define the: • Latitude, a decimal value, positive in the northern hemisphere; the default is Greenwich UK latitude, 51.479; as the value is expected to represent degrees and as the model’s unit for angles maybe different (SI unit for angle is radian), the value should be enter without any unit to avoid double conversion; • Longitude, a decimal value, positive at the East of the Prime Meridian; the default is Greenwich UK longitude, 0.01064; as the value is expected to represent degrees and as the model’s unit for angles maybe different (SI unit for angle is radian), the value should be enter without any unit to avoid double conversion; and • Time zone, the number of hours to add to UTC to get local time; the default is Greenwich UK time zone, 0. 180 | C H A P T E R 5 : R A D I A T I O N H E A T TR A N S F E R In the Date table, enter the: • Day, the default is 01; as the value is expected to represent days and as the model’s unit for time maybe different (SI unit for time is second), the value should be enter without any unit to avoid double conversion; • Month, the default is 6, which is June; as the value is expected to represent months and as the model’s unit for time maybe different (SI unit for time is second), the value should be enter without any unit to avoid double conversion; and • Year, the default is 2012. As the value is expected to represent years and as the model’s unit for time maybe different (SI unit for time is second), the value should be enter without any unit to avoid double conversion. The solar position is accurate for a date between 2000 and 2199. In the Local time table, enter the: • Hour, the default is 12; as the value is expected to represent hours and as the model’s unit for time maybe different (SI unit for time is second), the value should be enter without any unit to avoid double conversion; • Minute, the default is 0; as the value is expected to represent minutes and as the model’s unit for time maybe different (SI unit for time is second), the value should be enter without any unit to avoid double conversion; and • Second, the default is 0. The sun position is updated if the location, date, or local time changes during a simulation. In particular for transient analysis, if the unit system for the time is in seconds (the default), the time change can be taken into account by adding t to the Second field in the Local time table. In the Solar irradiance field Is (SI unit: W/m2) define the incident radiative intensity coming from the sun. The default is 1000 W/m2. 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. If Wavelength dependence of emissivity is Solar and ambient or Multiple spectral bands, the solar irradiance is divided among all spectral bands Bi as q0,s,Bi= q0,sFEPBi(Tsun) where FEPBi(Tsun) is the fractional black body emissive power over Bi interval at Tsun=5780K. T H E S U R F A C E - TO - S U R F A C E R A D I A T I O N U S E R I N T E R F A C E | 181 Theory for the Surface-to-Surface Radiation User Interface The Surface-To-Surface Radiation User Interface theory is described in this section: • Wavelength Dependence of Surface Emissivity and Absorptivity • The Radiosity Method for Diffuse-Gray Surfaces • The Radiosity Method for Diffuse-Spectral Surfaces • View Factor Evaluation • About Surface-to-Surface Radiation • Guidelines for Solving Surface-to-Surface Radiation Problems • Radiation Group Boundaries • References for the Radiation User Interfaces Wavelength Dependence of Surface Emissivity and Absorptivity The surface properties for radiation, the emissivity and absorptivity can be dependent on the angle of emission/absorption, surface temperature or radiation wavelength. The surface-to-surface radiation feature in the Heat Transfer module implements the radiosity method that enable arbitrary temperature dependence and assumes that the emissivity and absorptivity is independent of the angle of emission.absorption. It is also possible to account for wavelength dependency of the surface emissivity and absorptivity. PLANK SPECTRAL DISTRIBUTION The Planck’s distribution of emissive power for a blackbody in vacuum is given as a function of the surface temperature and of wavelength. The blackbody hemispherical emissive power is noted eb,(,T) and defined as 2n 2 C 1 e b T = -----------------------------C2 ------ 5 T e – 1 182 | C H A P T E R 5 : R A D I A T I O N H E A T TR A N S F E R (5-1) were • is the wavelength in vacuum 2 • C 1 = hc0 • C 2 = hc0 k • h is the Planck's constant • k is the Boltzmann constant • c0 is the speed of the light in vacuum The graphs below show the hemispherical spectral emissive power for a blackbody at 5780K (sun black body temperature) and for a black body at 300K. The dotted vertical lines delimit the visible wavelength (0.4 to 0.7µm). Figure 5-1: Planck distribution of a blackbody at 5780K THEORY FOR THE SURFACE-TO-SURFACE RADIATION USER INTERFACE | 183 (.Figure 5-2: Planck distribution of a blackbody at 300K The integral of eb. 2 F 1 T 2 T = e T d --------------------------------------- b e T d 0 b 1 One can notice that and that F 1 T 2 T = F 0 2 T – F 0 1 T and F 0 = 1 184 | C H A P T E R 5 : R A D I A T I O N H E A T TR A N S F E R .T) over a spectral band represents the power radiated on the spectral band and is defined by 2 eb T d = F T T 0 eb T d = F T T n2 T4 1 1 2 1 2 where F 1 T 2 T is the fractional black body emissive power. THEORY FOR THE SURFACE-TO-SURFACE RADIATION USER INTERFACE | 185 . This setting is rarely applicable if there is a solar radiation. DIFFUSE-GRAY SURFACES Diffuse-gray surfaces correspond to the hypothesis that surface properties are independent of the radiation wavelength and of the angle between the surface normal and the radiation direction. This is likely the case when the radiation is emitted by surface at temperatures in limited range. The assumption that the surface emissivity is independent of the radiation wavelength is often valid when most of the radiative power is concentrated on a relatively narrow spectral band.Note also that eb T = 0 eb T d = n 2 T 4 as defined for a black surface by the Stefan-Boltzmann law. properties are then described in terms of a solar absorptivity and an emissivity.5µm or shorter.5µm). it is appropriate to use a two-band approach with a solar band (for wavelengths shorter than 2.SOLAR AND AMBIENT SPECTRAL BANDS When solar radiation is part of the model. It is interesting to notice that about 97% of the radiated power • from a blackbody at 5800K is at wavelengths of 2. Figure 5-3: Normalized Planck distribution of blackbodies at 700K and 5800K Many problems have a solar load.5µm) and a ambient band (for wavelengths above 2. • from a blackbody at 700K is at wavelengths of 2. 186 | C H A P T E R 5 : R A D I A T I O N H E A T TR A N S F E R . For each surface. In such cases.5µm or longer. but the peak temperatures are below 700K. it is possible to enhance diffuse-gray surface model by considering two spectral bands: one for short wavelengths and one for large wavelengths. > 2.5um.5 µm Figure 5-4: Absorption of solar radiation and emission to surroundings By splitting the bands at the default of 2.5 µm Re-radiation to surroundings. the fraction of absorbed solar radiation on each surface is defined primarily by the solar absorptivity. The re-radiation at longer wavelengths (objects below ~700K) and the re-absorption of this radiation is defined primarily via the emissivity Emissivity Wavelength Figure 5-5: Solar and ambient spectral band approximation of the surface emissivity by a constant per band emissivity THEORY FOR THE SURFACE-TO-SURFACE RADIATION USER INTERFACE | 187 . < 2.Solar irradiation. According to Equation 2-9 it is given by q = G–J where • G is the incoming radiative heat flux. • J is the total outgoing radiative flux. or radiosity (SI unit: W/m2). G. By definition. The Radiosity Method for Diffuse-Gray Surfaces The radiation interacts with convective and conductive heat transfer through the source term in the Heat Flux and Boundary Heat Source boundary conditions. or irradiation (SI unit: W/m2). The irradiation. at a point can in general be written as a sum according to: G = G m + G ext + G amb where • Gm is the mutual irradiation. 188 | C H A P T E R 5 : R A D I A T I O N H E A T TR A N S F E R (5-2) .GENERAL DIFFUSE-SPECTRAL SURFACES Diffuse-spectral surfaces correspond to the hypothesis that surface properties are wavelength dependent but independent of the angle between the surface normal and the radiation direction. Emissivity 1 2 3 Wavelength The multiple spectral bands approach is used in cases when the surface emissivity varies significantly over the bands of interest. coming from other boundaries in the model (SI unit: W/m2). The heat transfer module enables to define constant surface properties per spectral bands (with up to 5 spectral bands) and to adjust spectral intervals endpoints. this source must be the difference between incident radiation and radiation leaving the surface. It requires accurate evaluation of the mutual irradiation. Diffuse-gray surface hypothesis corresponds to surfaces where is independent on the radiation wavelength. The incident radiation at one point on the boundary is a function of the exiting radiation. at every other point in view. • is the Stefan-Boltzmann constant (a predefined physical constant equal to 5. which leads to an implicit radiation balance: J = 1 – G + e b T = 1 – G m J + G ext + G amb + e b T (5-3) • Diffuse mirror is a variant of the Surface-to-Surface Radiation radiation type with = 0. G ext = Gext Ps + Gext q0 s 4 • G amb = n 2 F amb T amb is the ambient irradiation. J is defined by: J = G + e b T where • is the surface reflectivity which is equal to 1- for diffuse-gray surfaces. Tamb is the assumed far-away temperature in the directions included in Famb. The radiosity. • n is the transparent media refractive index. The radiosity. Gext is the sum of the products. a dimensionless number in the range 01. 4 • e b T = n 2 T is the blackbody hemispherical total emissive power. For diffuse-gray surface. in turn. is a function of Gm. of the external heat sources view factor by the corresponding source radiosity. Gm. Reradiation surfaces are common as an approximation of a surface that is well THEORY FOR THE SURFACE-TO-SURFACE RADIATION USER INTERFACE | 189 . for each external source.• Gext is the irradiation from external radiation sources (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. J. Therefore. For radiation sources located on a point. For directional radiative source Gext=Fext(is) q0. J (W/m2). • T is the surface temperature (SI unit: K) The Surface-To-Surface Radiation User Interface includes these radiation types: • Surface-to-Surface Radiation is the default radiation type. • is the surface emissivity. or radiosity.670400·108 W/(m2·T4)).s. Gext=Fext(xs) Ps. is the sum of the reflected irradiation and the emitted irradiation. 0Famb1 must hold at all points. by definition. (.. The radiosity expressions is then eb(T). The Surface-to-Surface Radiation interface treats the radiosity J as a shape function unless J is prescribed.N : 190 | C H A P T E R 5 : R A D I A T I O N H E A T TR A N S F E R .N with 0 = 0 and N = The surface properties can then be defined per spectral band: • Surface emissivity on Bi: i T = T i – 1 i • Ambient irradiation on Bi. The Surface-To-Surface Radiation User Interface assumes that the surface emissivity and opacity properties are constant per spectral band. • Prescribed radiosity makes it possible to specify graybody radiation. 4).insulated on one side and for which convection effects can be neglected on the opposite (radiating) side (see Ref. assuming that the ambient fractional emissive power corresponds to these of a blackbody at Tamb: G amb i = i = i–1 G amb d = F i – 1 T i T F amb e b T amb • External radiation sources on Bi with q 0 s i and P s i the external radiation source heat flux and source power over Bi i 1. a dimensionless number in the range 01.T) is the hemispherical spectral surface emissivity. Diffuse-spectral surface corresponds to a surface where is dependent on the radiation wavelength and surface temperature. It defines N spectral band (N=2 when solar and ambient radiation model is used).. • T is the surface temperature (SI unit: K) • eb. It resembles a mirror that absorbs all irradiation and then radiates it back in all directions.T) is the black body hemispherical emissive power defined by Equation 5-1. B i = i – 1 i i 1. A user-defined surface radiosity expression can also be defined. The Radiosity Method for Diffuse-Spectral Surfaces For a general diffuse spectral surface: J = =0 T eb T d where • (. The incident radiation over Bi spectral band at one point on the boundary is a function of the existing radiation. A user-defined surface radiosity expression can also be defined. 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. the radiosity is then F 1 T 2 T e b T .G ext i = i = i–1 G ext d = F ext i i s q 0 s i or G ext i = i = i–1 G ext d = F ext i i s P s i When the external source fractional emissive power corresponds to these of a blackbody at Text. in turn. 4). or radiosity. • Prescribed radiosity makes it possible to specify the surface radiation for each spectral band. THEORY FOR THE SURFACE-TO-SURFACE RADIATION USER INTERFACE | 191 . is a function of Gm. which leads to an implicit radiation balance: J i = 1 – i G i + i e b T = 1 – i G m i J i + G ext i + G amb i + i e b T (5-4) • Diffuse mirror is a variant of the Surface-to-Surface Radiation radiation type with i = 0. external radiation sources can be defined on Bi from the external radiation source heat flux and source power over all wavelengths. q 0 s and P s : G ext i = F ext i F i – 1 T i T i s q 0 s or G ext i = F ext i F i – 1 T i T i s P s The Surface-To-Surface Radiation User Interface includes these radiation types: • Surface-to-Surface Radiation is the default radiation type. Ji (W/m2).i. The Surface-to-Surface Radiation interface treats the radiosity Ji as a shape function unless Ji is prescribed. The radiosity. Using graybody radiation definition. at every other point in view. It resembles a mirror that absorbs all irradiation and then radiates it back in all directions. The quantities Gm and Famb in Equation 5-3 are not strictly view factors in the traditional sense. Famb is the view factor of the ambient portion of the field of view. which are selected in the Radiation Settings section from the Heat Transfer interface. flat-faceted representation of the geometry. A separate evaluation is performed for each unique point where Gm or Famb is requested. View factors are always calculated directly from the mesh. To improve the accuracy of the radiative heat transfer simulation. think of it as a product of a view factor matrix and a radiosity vector. and r is the distance from the source. In 2D the view factor for a point at finite distance is given by cos 2r 192 | C H A P T E R 5 : R A D I A T I O N H E A T TR A N S F E R . however. is the integral over all visible points of a differential view factor times the radiosity of the corresponding source point. the first time they are needed. on the other hand. VIEW FACTOR FOR EXTERNAL RADIATION SOURCES In 3D. In the discrete model. typically for each quadrature point during solution. This is. not necessarily the way the calculation is performed. Differential view factors are normally computed only once. 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 Heat Transfer Module supports two surface-to-surface radiation methods. which is considered to be a single boundary with constant radiosity J amb = e b T amb Gm. For a source at infinity. the view factor for a point at finite distance is given by 2 cos 4r where is the angle between the normal to the irradiated surface and the direction of the source. which is a polygonal. Loosely speaking. and then stored in memory until next time the model definition or the mesh is changed. the view factor is given by cos .View Factor Evaluation The strategy for evaluating view factors is central to any radiation simulation. the mesh must be refined rather than raising the element order. y. The relation between azi. The estimated solar position is accurate for a date between year 2000 and 2199. and z directions. is accessible from the Radiation Settings section in physics interfaces for heat transfer. Their settings work the same way as in 3D. and the up directions correspond to the x. COMSOL Multiphysics creates this virtual 3D geometry by revolving the 2D boundary mesh into a 3D mesh. isy.and the view factor for a source at infinity is cos . 6. date. the West. This number. The savings compared to a full 3D simulation are therefore substantial despite the full 3D view factor code being used. which is the same as the number of elements to a full revolution. in the model. While Gm and Famb are in fact evaluated in a full 3D. respectively. The position of the sun is necessary to determine the direction of the corresponding external radiation source. longitude. time zone. isy. Select between the hemicube and the direct area integration methods also in axial symmetry. the number of points where they are requested is limited to the quadrature points on the boundary of a 2D geometry. zen and (isx. The resolution can be controlled in the azimuthal direction by setting the number of azimuthal sectors. and time using similar method as described in Ref. Azimuthal sectors. isz) is given by: is x = – cos azi sin zen is y = sin azi sin zen is z = – cos zen RADIATION IN AXISYMMETRIC GEOMETRIES For an axisymmetric geometry. THEORY FOR THE SURFACE-TO-SURFACE RADIATION USER INTERFACE | 193 . isz) in Cartesian coordinates assuming that the North. due to an approximation used in the Julian Day calculation. Try to balance this number against the mesh resolution in the rz-plane. Gm and Famb must be evaluated in a corresponding 3D geometry obtained by revolving the 2D boundaries about the axis. SOLA R POSITION The sun is the most common example of an external radiation source. The zenith angle (zen) and the azimuth (azi) angles of the sun are converted into a direction vector (isx. The direction of sunlight (zenith angle and the solar elevation) is automatically computed from the latitude. • G amb = F amb e b T amb is the ambient irradiation (SI unit: W/m2). Famb is the ambient view factor and Tamb is the assumed far-away temperature in the directions included in Famb. A generalized equation for the irradiative flux is: G = G m + G ext + G amb (5-5) where • Gm is the mutual irradiation (SI unit: W/m2) arriving from other surfaces in the modeled geometry. It includes radiation from both the ambient surroundings and from other surfaces. Point x can see points on other surfaces as well as the ambient surrounding. while the ambient surrounding has a constant temperature. Tamb. The following sections derive the equations for Gm. J'. 194 | C H A P T E R 5 : R A D I A T I O N H E A T TR A N S F E R . Gext and Gamb for a general 3D case. Consider a point x on a surface as in Figure 5-6. Famb describes the portion of the view from each point that is covered by ambient conditions. Assume that the points on the other surfaces have a local radiosity. Gm on the other hand is determined from the geometry and the local temperatures of the surrounding boundaries. • Gext is irradiation from external sources (SI unit: W/m2). n is the transparent media refractive index.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. Famb. The mutual irradiation at point x is given by the following surface integral: Gm = – n' r n r . The projection is computed using the normal vectors n and n' along with the vector r . The irradiation from external radiation sources is the sum of the ### The ambient view factor.J' dS ----------------------------------4 r S' The heat flux that arrives from x' depends on the local radiosity J' projected onto x . THEORY FOR THE SURFACE-TO-SURFACE RADIATION USER INTERFACE | 195 . here denoted as F'.dS ----------------------------------4 r S' The two last equations plug into Equation 5-5 to yield the final equation for irradiative flux. determined from the integral below: F amb = 1 – F ' = 1 – – n' r n r . which points from x to x' .Figure 5-6: Example geometry for surface-to-surface radiation. is determined from the integral of the surrounding surfaces S'. 2D geometries results in simpler integrals. By default. 196 | C H A P T E R 5 : R A D I A T I O N H E A T TR A N S F E R . In axisymmetric geometries. a virtual 3D geometry must be constructed. Using a smaller block size also leads to more frequent updates of the progress bar. The additional nonzero elements in the matrix appear in the rows and columns corresponding to the radiosity degrees of freedom.The equations used so far apply to the general 3D case. It is therefore common practice to keep the element order of the radiosity variable. 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. When you need to increase the resolution of your temperature field. It may be useful to consider a block size as small as 100. For 2D the resulting equations for the mutual irradiation and ambient view factor are Gm = – n' r n r . low. • 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.dS ----------------------------------------3 2r S ' where the integral over S ' denotes the line integral along the boundaries of the 2D geometry. the irradiation and ambient view factor cannot be computed directly from a closed-form expression. Instead. linear Lagrange elements are used irrespective of the shape-function order specified for the temperature. J.J' dS ----------------------------------------3 2r F amb = 1 – (5-6) S ' – n' r n r . it might be worth considering raising the order of the temperature elements instead of refining the mesh. and the view factors evaluated according to Equation 5-6. A default group contains all boundaries selected in a Surface-to-Surface Radiation. and Prescribed Radiosity nodes. in THEORY FOR THE SURFACE-TO-SURFACE RADIATION USER INTERFACE | 197 . when defining groups. These boundaries do not irradiate other boundaries. On boundaries that have no number. the user has set a node among the Surface-to-Surface Radiation. Figure 5-7 shows four examples of possible boundary groupings. but to different groups in a planar model using the same 2D geometry. or Prescribed Radiosity node. the user has NOT set a node among the Surface-to-Surface Radiation. For example.Radiation Group Boundaries The Radiation Group node is only available when the Use radiation groups check box is selected under Radiation Settings. On boundaries that belong to one or more radiation group. and Prescribed Radiosity nodes. consider the full 3D geometry. it is good practice to specify as many groups as possible as opposed to specifying few but large groups. The grouping cannot be based on which boundaries have a free view toward each other in the 2D geometry. The numbers on each boundary specify different groups to which the boundary belongs. neither do other boundaries irradiate them. obtained by revolving the model geometry about the z axis. For example. a boundary grouping can be applied to save computational time. Be careful when grouping boundaries in axisymmetric geometries. A radiation group can be defined using a Radiation Group node. Diffuse Mirror. Instead. see Radiation Group for details. it is overridden in the default group. To obtain optimal computational performance. Diffuse Mirror. parallel vertical boundaries must typically belong to the same group in 2D axisymmetric models. Diffuse Mirror. For radiation problems. By default this check box is not selected. Then this boundary can be added to other radiation groups without being overridden by the manually added radiation groups. When a node is added to another radiation group. which means that all radiative boundaries belong to the same radiation group. Figure 5-7. A B 1 1 1 2 1 12 C 2 123 2 3 3 1 D 1 inefficient boundary grouping 1 1 1 1 1 2 2 2 2 1 Figure 5-7: Examples of radiation group boundaries. 198 | C H A P T E R 5 : R A D I A T I O N H E A T TR A N S F E R 1 1 1 . case (b) is more efficient than case (d). THE RADIATION IN PARTICIPATING MEDIA USER INTERFACE | 199 . In order to distinguish between variables belonging to different physics user interfaces. Transfer>Radiation branch ( When the interface is added. these default nodes are also added to the Model Builder— Radiation in Participating Media. DOMAIN SELECTION The default setting is to include All domains in the model to define the dependent variables and the equations. Solves only for radiation variables.<variable_name>. use a Heat Transfer interface. Right-click the main node to add boundary conditions or other features. The first character must be a letter. Only letters. In order to solve for radiation and temperature. This interface solves for radiative intensity field. Refer to such interface variables in expressions using the pattern <identifier>.T he R a di a t i o n i n Part i ci p at i n g Med i a U s e r Inte r f a c e The Radiation in Participating Media (rpm) user interface ( ). and Initial Values. To choose specific domains. numbers and underscores (_) are permitted in the Identifier field. INTERFACE IDENTIFIER The interface identifier is used primarily as a scope prefix for variables defined by the physics user interface. 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. select Manual from the Selection list. enables the modeling of radiative heat transfer inside a participating medium. Continuity on Interior Boundary. The default identifier (for the first interface in the model) is rpm. found under the Heat ) in the Model Wizard. the identifier string must be unique. Wall. With small values. S6. With large values (up to 1). 24. Cubic. respectively. Discretization Level Select Linear (the default). S4. DISCRETIZATION To display this section. Quartic. S6. respectively. and 80 directions.PARTICIPATING MEDIA SETTINGS Refractive Index Define the Refractive index nr (dimensionless) of the participating media. The default is 0. . In 2D. and S8 generate 8. 12. The default is S4. Quadratic. less memory is needed to solve the model. S2. The same refractive index is used for the whole model. Discrete Ordinates Method Select the Discrete ordinates method order from the list.4. a robust setting for the solver is expected. 24. S2. This order defines the discretization of the radiative intensity direction. S4. and 40 directions. Performance Index Select a Performance index Pindex from the list. and S8 generate 4. or Quintic to define the Discretization level. Select a value between 0 and 1 that modifies the strategy used to define automatic solver settings. click the Show button ( 200 | C H A P T E R 5 : R A D I A T I O N H E A T TR A N S F E R ) and select Discretization. In 3D. 48. and Pair Nodes for the Radiation in Participating Media User Interface • Radiative Heat Transfer in Finite Cylindrical Media: Model Library path Heat_Transfer_Module/Tutorial_Models/cylinder_participating_media • Radiative Heat Transfer in a Utility Boiler: Model Library path Heat_Transfer_Module/Thermal_Radiation/boiler Domain. Boundary. point. and pair nodes available and described here: • Radiation in Participating Media • Continuity on Interior Boundary • Incident Intensity • Opaque Surface These nodes are described for other interfaces: • Opaque To locate and search all the documentation. and Pair Nodes for the Radiation in Participating Media User Interface The Radiation in Participating Media User Interface has these domain. select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree.This section is empty for Radiation in Participating Media (rpm) interface which define the shape functions discretization in Discretization Level. Point. • About the Heat Transfer Interfaces • The Heat Transfer Interface • About Handling Frames in Heat Transfer • Theory for the Heat Transfer User Interfaces • Theory for the Radiation in Participating Media User Interface • Show More Physics Options • Domain. Edge.. Edge. Point. edge. boundary. THE RADIATION IN PARTICIPATING MEDIA USER INTERFACE | 201 . in COMSOL Multiphysics. Boundary. The absorption coefficient defines the amount of radiation.Radiation in Participating Media The Radiation in Participating Media node uses the radiative transfer equation s I = I b T – I + -----4 4 0 I d where • I) is the radiative intensity in the direction • T is the temperature • . If such user-defined materials are added. the model inputs appear here. ABSORPTION The default Absorption coefficient (SI unit: 1/m) uses the value From material. choose the domains to define. and scattering coefficients • Ib is the blackbody radiative intensity • is the scattering phase function The following equation is the blackbody radiation intensity and n is the refractive index: 2 4 n T I b T = ---------------- It also adds the radiative heat source term in the heat transfer equation: Q r = q r = G – 4T 4 DOMAIN SELECTION From the Selection list. extinction. s and s are absorption. If User defined is selected. MODELS INPUTS This section contains fields and values that are inputs to expressions that define material properties. enter another value or expression. I. 202 | C H A P T E R 5 : R A D I A T I O N H E A T TR A N S F E R . that is absorbed by the medium. The default is to use the heat transfer dependent variable. There is one standard model input—the Temperature T (SI unit: K). INIT IA L VA LUES For The Heat Transfer with Radiation in Participating Media User Interface. The default is 0. It is also available for The Radiation in Participating Media User Interface. THE RADIATION IN PARTICIPATING MEDIA USER INTERFACE | 203 . ….Ibinit (SI unit: W/m2). Enter the Legendre coefficient a1. Select the Scattering type—Isotropic. • If Linear anisotropic is selected. a12 as required. that is absorbed by the surface. The Opaque Surface node prescribes incident intensities on a boundary and accounts for the net radiative heat flux. 0: • Isotropic (the default) corresponds to the scattering phase function 0 = 1 . If User defined is selected. it defines the scattering phase function 12 0 = 1 + am Pm 0 m=1 Enter the Legendre coefficients a1. an Initial Values section is added. enter another value or expression. Opaque Surface The Opaque Surface node is available for The Heat Transfer with Radiation in Participating Media User Interface version of the Heat Transfer interface. Linear anisotropic. This provides three options to define the scattering phase function using the cosine of the scattering angle.SCATTERING The default Scattering coefficient s (SI unit: 1/m) uses the value From material. • If Polynomial anisotropic is selected. qw. it defines the scattering phase function as 0 = 1 + a 1 0 . The Opaque Surface node defines a boundary opaque to radiation. which has a default Radiative intensity of ht. or Polynomial anisotropic. Both are dimensionless numbers between 0 and 1 that satisfy the relation dw1. If d1w. When nodes are added from the context menu. An identity pair has to be created first. you can select Manual from the Selection list to choose specific boundaries or select All boundaries as required. an emissivity of 0 means that the surface emits no radiation at all and that all outgoing radiation is diffusely reflected by this boundary. Enter a Diffusive reflectivity d.15 K and is used in the black-body radiative intensity expression. Gray Wall If Gray wall is selected the default Surface emissivity e value is taken From material (a material defined on the boundaries).BOUNDARY SELECTION For a default node. The boundary temperature definition can differ from the that of the temperature in the adjacent domain. choose the pair to define. By default d1w. WA LL SE TTIN GS Select a Wall type to define the behavior of the wall—Gray wall or Black wall. An emissivity of 1 means that the surface is a perfect black body. outgoing radiation is fully absorbed on this boundary. Radiative intensity (W/m2 in SI units) along incoming discrete directions on this boundary is defined by 204 | C H A P T E R 5 : R A D I A T I O N H E A T TR A N S F E R . The default is 293. and cannot be edited. There is one standard model input—the Temperature T (SI unit: K). 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. MODELS INPUTS This section contains fields and values that are inputs to expressions that define material properties. that is. Select User defined to enter another value or expression. PAIR SELECTION If this node is selected from the Pairs menu. Ctrl-click to deselect. In this case. If such user-defined materials are added. the setting inherits the selection from the parent node. the selection is automatically selected and is the same as for the interface. the model inputs appear here. d I i bnd = w I b T + -----. no user input is required and the radiative intensity along the incoming discrete directions on this boundary is defined by I i bnd = I b T Values of radiative intensity along outgoing discrete directions are not prescribed. Ctrl-click to deselect. choose the pair to define. Incident Intensity The Incident Intensity node is available for The Heat Transfer with Radiation in Participating Media User Interface version of the Heat Transfer interface. BOUNDARY SELECTION From the Selection list. Use an Incident Intensity node to specify the radiative intensity along incident directions on a boundary. An identity pair has to be created first.q out Black Wall If Black wall is selected. choose the boundaries to define. PAIR SELECTION If this node is selected from the Pairs menu. THE RADIATION IN PARTICIPATING MEDIA USER INTERFACE | 205 . It is also available for The Radiation in Participating Media User Interface. sz correspond to the components of discrete ordinate vectors. and cannot be edited. PAIR SELECTION When this node is selected from the Pairs menu. This represents the value of radiative intensity along incoming discrete directions. 206 | C H A P T E R 5 : R A D I A T I O N H E A T TR A N S F E R .sz where interfaceIdentifier is the physics interface identifier.sx. the selection is automatically selected and is the same as for the interface. interfaceIdentifier. ht. the setting inherits the selection from the parent node. 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.sy. Ctrl-click to deselect. choose the pair to define. The components of each discrete ordinate vector can be used in this expression. An identity pair has to be created first.INCIDENT INTENSITY Enter a Boundary radiation intensity Iwall (SI unit: W/m2). that is. interfaceIdentifier. and ht. the Heat Transfer interface identifier is ht so ht.sy. BOUNDARY SELECTION For a default node.sx. The syntax is interfaceIdentifier. By default. Values of radiative intensity on outgoing discrete directions are not prescribed. Let I denote the radiative intensity traveling in a given direction. The phase function (i. where Ib is the blackbody radiation intensity. Different kinds of interactions are observed: • Absorption: The medium absorbs a fraction of the incident radiation. The amount of emitted radiative intensity is equal to Ib. • Scattering: A part of the radiation coming from a given direction is scattered in other directions. • Emission: The medium emits radiation in all directions. The scattering properties of the medium are described by the scattering phase function ij. . The amount of absorbed radiation is I where is the absorption coefficient. which gives the probability that a ray coming from one direction i is scattered into the direction j. j) satisfies THEORY FOR THE RADIATION IN PARTICIPATING MEDIA USER INTERFACE | 207 . In some applications the medium is not completely transparent and the radiation rays interact with the medium.Theory for the Radiation in Participating Media User Interface Radiation and Participating Media Interactions Figure 5-8: Example of interactions between participating media and radiation. s are absorption. The general radiative transfer equation can be written as (see Ref. 1) s I = I b T – I + -----4 4 0 I d where • I is the radiative intensity at a given position following the direction • T is the temperature • .1 -----4 i d i = 1 4 Radiative intensity in a given direction is attenuated and augmented by scattering: .It is augmented because a part of radiative intensity coming from other directions is scattered in all direction. emission. The phase function’s definition is material dependent and its definition can be complicated.It is attenuated because a part of incident radiation in this direction is scattered into other directions. extinction. and scattering) can now be formulated. It is common to use approximate 208 | C H A P T E R 5 : R A D I A T I O N H E A T TR A N S F E R . . The amount of radiation augmented by scattering is obtained by integrating scattering coming form all directions i: ------s 4 I i i di 4 Radiative Transfer Equation The balance of the radiative intensity including all contributions (propagation. 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. and scattering coefficients. respectively 2 4 n T I b T = ---------------- (5-7) • Equation 5-7 is the blackbody radiation intensity. s. The amount of radiation attenuated by scattering is sI. including the direction we are looking at. absorption. The current implementation handles: • Isotropic phase function: ' = 0 = 1 • Linear anisotropic phase function: 0 = 1 + a1 0 • Polynomial anisotropic up to the 12th order: 12 0 = 1 + an Pn 0 n=1 where Pn are nth-order Legendre polynomials. the radiative intensity Ibnd entering participating media along the direction is d I bnd = w I b T + -----. 0.scattering phase functions that are defined using the cosine of the scattering angle. Legendre polynomials can be defined by the Rodriguez formula: k 1 dk 2 P k x = ----------x – 1 k k 2 k! d x A quantity of interest is the incident radiation. which is in the range [0. denoted G and defined by 4 G = 0 I Boundary Condition for the Transfer Equation For gray walls.q out for all such that n 0 where 2 4 n T I b T = ---------------- (5-8) • Equation 5-8 is the blackbody radiation intensity and n is the refractive index • w is the surface emissivity. corresponding to opaque surfaces reflecting diffusively and emitting. 1] THEORY FOR THE RADIATION IN PARTICIPATING MEDIA USER INTERFACE | 209 . ------. Thus IbndIbT. Heat Transfer Equation in Participating Media Heat flux in gray media is defined by qr n = 4 0 I n Heat flux divergence can be defined as a function of G and T (see Ref.+ u T = – q c + G – 4nT 4 + :S – ---. -----. radiative heat flux is taken into account in addition to conductive heat flux: qqcqr.• d1w is the diffusive reflectivity • n is the outward normal vector • qout is the heat flux striking the wall: q out = wj Ij n j n j 0 For black walls w1 and d0. 1): Q r = q r = G – 4nT 4 In order to couple radiation in participating media. The heat transfer equation reads T p T C p ------.+ u p + Q t T p t 210 | C H A P T E R 5 : R A D I A T I O N H E A T TR A N S F E R .+ u p + Q t T p t and is implemented using following form: T p T C p ------. -----.+ u T = – q c + q r + :S – ---.------. S6. Radiative intensity is defined for any direction . LSE symmetric quadrature fulfills the half. Several sets are available in the literature. the angular space is discretized. Since it is not possible to fulfill exactly all these conditions. 1). it is also recommended that the quadrature fulfills the half moment for vectors of Cartesian basis. S4. accuracy should be improved when N increases. first.Discrete Ordinates Method The discrete ordinates method is implemented for 3D and 2D geometries. integrals over directions are replaced by numerical quadratures of discrete directions: 4 0 n I d wj Ij j=1 Depending on the value of N. and third moments (see Ref. the implementation uses LSE symmetric quadrature for S2. I2. Thanks to angular space discretization. Following the conclusion of Ref. A set should satisfy first. …. second. The SN approximation provides a discretization of angular space into nNN2 in 3D (or nNN22 in 2D) discrete directions. and third moments. a set of n dependent variables has to be defined and solved for I1. Each dependent variable obeys the equation s i I i = I b T – I i + -----4 n w j I j j i j=1 with the boundary condition THEORY FOR THE RADIATION IN PARTICIPATING MEDIA USER INTERFACE | 211 . because the angular space is continuous. In. In order to treat radiative intensity equation numerically. second. It consists of a set of directions and quadrature weights. 3. and S8. In addition I j. = 2I˜j j- i+ 212 | C H A P T E R 5 : R A D I A T I O N H E A T TR A N S F E R . as it is in COMSOL Multiphysics implementation.have opposite component in z direction. and j- i+ = j+ i.have the same components in the (x.and I i + (which are equal in 2D) we get: n s 2 i I˜i = 2I b T – 2I˜i + -----4 w j I j j i + j i + - j=1 which can be rewritten as: s i I˜i = I b T – I˜i + -----8 n w j I j j i + j i + - j=1 In addition if i j can be rewritten as a function of i j . then j+ i+ = j- i. let’s define 2 indexes.q out for all i such that n i 0 Discrete Ordinates Method Implementation in 2D For a given index i.+ -----4 n w j I j j i - j=1 By summing the two above equations and introducing I˜i which is equal to I i. Assuming that a model is invariant in the z direction. y) plane and • i+ and i. i+ and i-. j- i+ + I j+ j+ i. I i. = 2I˜j j+ i. i+ and i-: i + I i + s = I b T – I i+ + -----4 n wj Ij j i + j=1 s i. we can write the DOM form of the radiative transfer equation in two directions.= I b T – I i. so that • i+ and i.d I i bnd = w I b T + -----. In other expressions than the scattering term. multiplied by 2. the z component of the radiative intensities Ii and of the discrete directions i can by ignored (or set to zero) and the weight wi. on half of the 3D DOM directions. except for the scattering term. One can also notice that 4 0 n2 n I d wj Ij j=1 = n2 wj Ij + wj Ij - j=1 - + + = wi Ii ˜ ˜ (5-10) j=1 with w˜ i = 2w i . ˜ i .so we can simplify above equation: n s ˜ i I˜i = I b T – I˜i + -----4 wj Ii j i ˜ (5-9) j=1 with 1 ˜ i = i 1 0 since the z component of I˜i is null in 2D. I˜i . THEORY FOR THE RADIATION IN PARTICIPATING MEDIA USER INTERFACE | 213 . Using results from Equation 5-9 and Equation 5-10 we can formulate DOM in 2D using only radiative intensities. References for the Radiation User Interfaces 1. and R. “The Selection of Discrete Ordinate Quadrature Sets for Anisotropic Scattering. 3rd ed. 6. Fundamentals of Momentum. 1983. John Wiley and Sons.A. http://www. John Wiley and Sons. Wicks. Heat. Academic Press. 2nd ed.P.F. Taylor & Francis. 2003. Modest. Wilson.gov/gmd/grad/solcalc 214 | C H A P T E R 5 : R A D I A T I O N H E A T TR A N S F E R .. J. California. 5. 160. ASME. HTD.” Fundamentals of Radiation Transfer. Sieger. Thermal Radiation Heat Transfer. 2002.R. J. DeWitt.. New York.esrl. Welty. 2. 5th ed.P. Incropera and D. Fundamentals of Heat and Mass Transfer.noaa. F. M.E. and Mass Transfer. Fiveland.. 1991. Radiative Heat Transfer.E. vol. Howell. C. San Diego. W. 3. 4th ed. R.. 2002. 4. In this chapter: • The Laminar Flow and Turbulent Flow User Interfaces • Theory for the Laminar Flow User Interface • Theory for the Turbulent Flow User Interfaces • References for the Single-Phase Flow. This chapter describes ) in the the fluid flow groups under the Fluid Flow>Single-Phase Flow branch ( Model Wizard. This enables modeling of forced or temperature gradient-driven flows in both laminar and turbulent regimes. User Interfaces 215 .6 The Single-Phase Flow Branch The Heat Transfer Module extends the CFD capability of COMSOL Multiphysics® by adding turbulence modeling and support for low Mach number compressible flows. Refer to such interface variables in expressions using the pattern 216 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH . has the equations. solving for the velocity field and the pressure. • Domain. found under the Single-Phase Flow branch ( ) in the Model Wizard. boundary conditions. which adds the Navier-Stokes equations and provides an interface for defining the fluid material and its properties. and Point Nodes for Single-Phase Flow • Theory for the Laminar Flow User Interface The Laminar Flow User Interface The Laminar Flow (spf) user interface ( ). INTERFACE IDENTIFIER The interface identifier is used primarily as a scope prefix for variables defined by the physics user interface. When this interface is added. COMSOL Multiphysics takes the axial symmetry boundaries (at r0) into account and automatically adds an Axial Symmetry node to the model that is valid on the axial symmetry boundaries only. these default nodes are also added to the Model Builder— Fluid Properties. Low Re k- User Interface For 2D axisymmetric models.The Laminar Flow and Turbulent Flow User Interfaces In this section: • The Laminar Flow User Interface • The Turbulent Flow. The main node is Fluid Properties. and volume forces for modeling freely moving fluids using the Navier-Stokes equations. Pair. and Initial Values. k- User Interface • The Turbulent Flow. boundary conditions and volume forces. Right-click the Laminar Flow node to add other nodes that implement. Boundary. for example. Wall (the default boundary condition is No slip). The flow state in a fluid flow model.<identifier>. the identifier string must be unique. Turbulence Model Type By default. Only letters. A new field name must not coincide with the name of a field of another type. click the Show button ( ) and select Stabilization. DEPENDENT VA RIA BLES These dependent variables (fields) are defined for this interface—Velocity field u (SI unit: m/s) and its components. Observe that P1+P1 elements require Streamline diffusion to be active. k- User Interface when the Turbulence model type selected is RANS (Reynolds-averaged Navier– Stokes). For example. Component names must be unique within a model except when two fields share a common field name. or with a component name belonging to some other field. If a new field name coincides with the name of another field of the same type. CONSISTENT STABILIZATION To display this section. The first character must be a letter. If T H E L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R I N T E R F A C E S | 217 . and Pressure p (SI unit: Pa). Editing the name of a scalar dependent variable changes both its field name and the dependent variable name. PHYSICAL MODEL By default the interface uses the Compressible flow (Ma<0. numbers and underscores (_) are permitted in the Identifier field.<variable_name>. The consistent stabilization methods are applicable to the Navier-Stokes equations— Streamline diffusion and Crosswind diffusion.3) formulation of the Navier-Stokes equations. click to clear one or both of the Streamline diffusion and Crosswind diffusion check boxes. If required. and dependent variable names. These check boxes are selected by default. In order to distinguish between variables belonging to different physics user interfaces. is not always known beforehand. the fields will share degrees of freedom and dependent variable names. however. component. edit the field. Selecting an option in this section switches between available Single-Phase Flow (spf) interfaces. None is selected as the Turbulence model type. Select Incompressible flow to use the incompressible (constant density) formulation. If required. The default identifier (for the first interface in the model) is spf. this interface changes to The Turbulent Flow. P2+P1. click the Show button ( ) and select Stabilization. click the Show button ( ) and select Advanced Physics Options. From the Discretization of fluids list select the element order for the velocity components and the pressure: P1+P1 (the default). Click to select as required. Normally these settings do not need to be changed. thereby improving the numerical robustness. If Manual is selected. It controls the discretization (the element types used in the finite element formulation). When selected. also choose a CFL number expression—Automatic (the default) or Manual. or P3+P2. INCONSISTENT STABILIZATION To display this section. Pseudo Time Stepping for Laminar Flow Models and About Pseudo Time Stepping in the COMSOL Multiphysics Reference Manual DISCRETIZATION To display this section. Linear elements are computationally cheaper than higher-order elements and are also less prone to introducing spurious oscillations.you deactivate Streamline diffusion. make sure that your model uses P2+P1 elements or higher. Make sure that Streamline Diffusion in the Consistent Stabilization section is selection when using P1+P1 elements. the Isotropic diffusion check box is not selected for the Navier-Stokes equations. This is the default element order for the Laminar Flow and Turbulent Flow interfaces. 218 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH . By default. • P1+P1 (the default) means linear elements for both the velocity components and the pressure field. 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. P1+P1 elements require streamline diffusion to be a numerically valid discretization. click the Show button ( ) and select Discretization. Automatic sets the local CFL number (from the Courant–Friedrichs–Lewy condition) to the built-in variable CFLCMP which in turns trigger a PID regulator for the CFL number. ADVANCED SETTINGS To display this section. enter a Local CFL number CFLloc (dimensionless). k-(spf)user interface ( ). Wall (the default boundary condition is Wall functions). the pressure. solving for the turbulent kinetic energy k and the rate of dissipation of turbulent kinetic energy . which adds the Navier-Stokes equations and the transport equations for k and . has the equations. • P3+P2 means third-order elements for the velocity components and second-order elements for the pressure field. Specify the Value type when using splitting of complex variables—Real (the default) or Complex for each of the variables in the table. solving for the mean velocity field. • Show More Physics Options • Domain. boundary Flow>Turbulent Flow branch ( conditions. Pair. and Initial Values. found under the Single-Phase ) in the Model Wizard. and provides an interface for defining the fluid material and its properties. T H E L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R I N T E R F A C E S | 219 . and volume forces for modeling turbulent flow using the Reynolds averaged Navier-Stokes (RANS) equations. This can add additional accuracy but it also adds additional degrees of freedom compared to P2+P1 elements. k- User Interface The Turbulent Flow. Boundary. Except where included below. When this interface is added. see The Laminar Flow User Interface for all the other settings.• P2+P1 means second-order elements for the velocity components and linear elements for the pressure field. and the standard k- model. The main feature is Fluid Properties. these default nodes are also added to the Model Builder—Fluid Properties. and Point Nodes for Single-Phase Flow • Theory for the Laminar Flow User Interface • Flow Past a Cylinder: Model Library path COMSOL_Multiphysics/ Fluid_Dynamics/cylinder_flow • Terminal Falling Velocity of a Sand Grain: Model Library path COMSOL_Multiphysics/Fluid_Dynamics/falling_sand The Turbulent Flow. 035) under the Turbulence variables scale parameters subsection. e. C. Ce2. DEPENDENT VARIABLES These dependent variables (fields) are defined for this interface: • Velocity field u (SI unit: m/s) and its components • Pressure p (SI unit: Pa) • Turbulent kinetic energy k (SI unit: m2/s2) • Turbulent dissipation rate ep (SI unit: m2/s3) ADVANCED SETTINGS To display this section. The parameters are used when a new default solver for a transient study step 220 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH . the Turbulence model type defaults to RANS and the Turbulence model defaults to k-. The scaling parameters must only contain numerical values. This enables the Turbulence Model Parameters section. enter a value for Uscale (SI unit: m/s) (the default is 1 m/s) and Lfact (dimensionless) (the default is 0. click the Show button ( ) and select Advanced Physics Options. v. In addition to the settings described for the Laminar Flow interface.PHYSICAL MODEL For this interface. The Turbulence variables scale parameters subsection is available when the Turbulence model type is set to RANS. but for some special cases. The scaling parameters cannot contain variables. k. The Uscale and Lfact parameters are used to calculate absolute tolerances for the turbulence variables. and B. units or parameters defined under Global Definitions. For this interface the parameters are Ce1. 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. better performance can be obtained by tuning the model parameters. solving for the mean velocity field. Transient with Initialization. If you change the parameters. found under the Single-Phase Flow>Turbulent Flow branch. PHYSICAL MODEL For this interface. For study information. boundary conditions. Low Re k-(spf) user interface ( ).is generated. The interface also includes a wall distance equation that solves for the reciprocal wall distance. has the equations. the pressure. see Stationary with Initialization. This enables the Turbulence Model Parameters section. Low Re k- User Interface The Turbulent Flow. and Wall Distance Initialization in the COMSOL Multiphysics Reference Manual. and the AKN low-Reynolds number k- model. the new values take effect the next time you generate a new default solver • The Laminar Flow User Interface • About Pseudo Time Stepping in the COMSOL Multiphysics Reference Manual • Theory for the Turbulent Flow User Interfaces • Show More Physics Options Turbulent Flow Over a Backward Facing Step: Model Library path Heat_Transfer_Module/Verification_Models/turbulent_backstep The Turbulent Flow. solving for the turbulent kinetic energy k and the rate of dissipation of turbulent kinetic energy . 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. the Turbulence model type defaults to RANS and the Turbulence model defaults to Low Reynolds number k-. T H E L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R I N T E R F A C E S | 221 . and volume forces for modeling turbulent flow using the Reynolds averaged Navier-Stokes (RANS) equations. k- User Interface for all the other settings. better performance can be obtained by tuning the model parameters. k. and v. Ce2.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. e. • The Laminar Flow User Interface • About Pseudo Time Stepping in the COMSOL Multiphysics Reference Manual • Theory for the Turbulent Flow User Interfaces • Show More Physics Options 222 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH . C. For this interface the parameters are Ce1. but for some special cases. DEPENDENT VARIABLES These dependent variables (fields) are defined for this interface: • Velocity field u (SI unit: m/s) and its components • 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) See The Laminar Flow User Interface and The Turbulent Flow. John Wiley & Sons.L. and point nodes (listed in alphabetical order): • Boundary Stress • Interior Wall • Fan • Open Boundary • Flow Continuity • Outlet • Fluid Properties • Periodic Flow Condition • Grille • Pressure Point Constraint • Initial Values • Symmetry • Inlet • Volume Force • Interior Fan • Wall For 2D axisymmetric models. Sani. Other interfaces also share many of these domain. pair. To locate and search all the documentation. Incompressible Flow and the Finite Element Method. Boundary. select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree. Gresho and R. Pair. T H E L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R I N T E R F A C E S | 223 . The theory about most boundary conditions is found in P. in COMSOL. boundary. 2000. COMSOL Multiphysics takes the axial symmetry boundaries (at r0) into account and adds an Axial Symmetry node to the model that is valid on the axial symmetry boundaries only. Volume 2: Isothermal Laminar Flow. and Point Nodes for Single-Phase Flow The following nodes are for all interfaces found under the Fluid Flow>Single-Phase Flow branch ( ) in the Model Wizard.M.Domain. the pressure drop over the modeled domain is probably many orders of magnitude less than atmospheric pressure. In other cases. which. clear the Reference pressure check box. the absolute pressure may be required to be solved for. but it also relates to the value of the pressure field. in a straight incompressible flow problem. For fluid flow. if included. reduces the chances for stability and convergence during the solving process for this variable. The node also provides an interface for defining the material properties of the fluid. the Fluid Properties node also adds the equations for the turbulence transport equations.Fluid Properties The Fluid Properties node adds the momentum equations solved by the interface. For the Turbulent Flow interfaces. these are typically introduced when a material requiring inputs has been applied. MODEL INPUTS Edit input variables to the fluid-flow equations if required. 224 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH .325 Pa). If the pressure field instead is an absolute pressure field. The default Absolute pressure pA (SI unit: Pa) is ppref where p defaults to the pressure variable from the Navier-Stokes equations and pref to 1[atm] (1 atmosphere 101. Using one or the other option usually depends on the system and the equations being solved for. For example. except for volume forces which are added by the Volume Force feature. The default setting is hence consistent with solving for a gauge pressure. such as where pressure is a part of an expression for gas volume or diffusion coefficients. Absolute Pressure This input appears when a material requires the absolute pressure as model input. There are usually two ways to calculate the pressure when describing fluid flow. Solve for the absolute pressure or a pressure (often denoted gauge pressure) that relates back to the absolute pressure through a reference pressure. The absolute pressure input controls the pressure used to evaluate material properties. because an upper limit on the mixing length is required. check the coupling between any interfaces using the same variable. water and air have a low viscosity. The default Dynamic viscosity (SI unit: Pa·s) uses the value From material and describes the relationship between the shear rate and the shear stresses in a fluid. Intuitively. 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.To model an incompressible fluid. set Absolute pressure pA is set to User defined and enter the desired pressure level in the edit field. it may not with physics interfaces that it is being coupled to. In such models. The default value is 1[atm]. Select User defined to define a different value or expression. While this check box maintains control over the pressure variable and instances where absolute pressure is required within this respective physics interface. FLUID PROPERTIES The default Density (SI unit: kg/m3) uses the value From material. see the settings for the Heat Transfer in Fluids node. To define the Absolute Pressure. makes it possible to define arbitrary expressions of the dynamics viscosity as a function of the shear rate. spf. Select User defined to enter a different value or expression. T H E L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R I N T E R F A C E S | 225 . k-interface. such as for gas flow governed by the gas law. Using a built-in variable for the shear rate magnitude. MIXING LENGTH LIMIT This section is available for the Turbulent Flow. and substances often described as thick (such as oil) have a higher viscosity.sr. 226 | u T + u u = – pI + u + u + F t CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH . In this case. Select how the Reference length scale lref (SI unit: m) is defined—Automatic (default) or Manual: • If Automatic is selected. DISTANCE EQUATION This section is available for a Turbulent Flow. it is recommended that it is defined manually.lim (SI unit: m) is defined—Automatic (default) or Manual: • If Automatic is selected. The default is 1 m. In this case. to incorporate the effects of gravity in a model. for example.Select how the Mixing length limit lmix. • Select Manual to define a different value or expression. Low Reynolds number k- interface since a Wall Distance interface is included. one unit length of the model unit system). a complicated system of slim entities. The default is 1 (that is. This is usually quite accurate but it can sometimes give too great a value if the geometry consists of several slim entities. If the geometry is. Use it. • Select Manual to define a different value or expression for the wall distance. Objects that are much smaller than lref will effectively be diminished while the distance to objects much larger than lref will be accurately represented. it is recommended that it is defined manually. for example. lref controls the result of the distance equation. the wall distance is automatically evaluated to one tenth of the shortest side of the geometry bounding box. Volume Force The Volume Force node specifies the volume force F on the right-hand side of the incompressible flow equation. this measure can give too big a result. the mixing length limit is automatically evaluated as the shortest side of the geometry bounding box. VO L U M E F O R C E Enter the components of the Volume force F (SI unit: N/m3).If several volume force nodes are added to the same domain. The defaults for all components are 0 N/m3. respectively. In the Turbulent Flow interfaces. By default these are specified using the predefined variables defined by the expressions in Initial Values. T H E L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R I N T E R F A C E S | 227 . also define the Turbulent kinetic energy k (SI unit: m2/s2) and the Turbulent dissipation rate ep (SI unit: m2/ s3). The Boussinesq Approximationin the COMSOL Multiphysics Reference Manual Initial Values The Initial Values node adds initial values for the velocity field and the pressure that can serve as an initial condition for a transient simulation or as an initial guess for a nonlinear solver. define the Reciprocal wall distance G (SI unit: 1/m). The Coordinate system list contains any additional coordinate systems that the model includes. The default is spf.epinit. For the Low Reynolds number k- turbulence model. INIT IA L VA LUES Enter values or expressions for the initial value of the Velocity field u (SI unit: m/s) and the Pressure p (SI unit: Pa). initial values for the turbulence variables are also specified.G0.kinit and spf. For the k- and Low Reynolds number k- turbulence models. COORDINATE SYSTEM SELECTION The Global coordinate system is selected by default. then the sum of all contributions are added to the momentum equations. The default values are spf. The default values are 0 m/s and 0 Pa. • No Slip (the default for laminar flow. the setting inherits the selection from the parent node. 228 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH . the following are also available for a k- turbulence model: • Wall Functions (the default for turbulent flow with a k- turbulence model) • Sliding Wall (Wall Functions) • Moving Wall (Wall Functions) In the COMSOL Multiphysics Reference Manual: • Slip • Sliding Wall • Moving Mesh User Interface BOUNDARY SELECTION For a default node. When nodes are added from the context menu. you can select Manual from the Selection list to choose specific boundaries or select All boundaries as required. that is. and cannot be edited. the selection is automatically selected and is the same as for the interface.Wall The Wall node includes a set of boundary conditions describing the fluid flow condition at a wall. BOUNDARY CONDITION Select a Boundary condition for the wall. that the fluid at the wall is not moving. The condition prescribes u = 0. that is. No Slip No slip is the default boundary condition for a stationary solid wall (and for the Low Reynolds number k-turbulence model). and the Low Reynolds number k- turbulence model) • Slip • Sliding Wall • Moving Wall • Leaking Wall In addition to the Slip condition. this boundary condition prescribes u = uw. Its magnitude is adjusted to be the same as the magnitude of the vector entered. no boundary layer develops. From a modeling point of view. An additional Moving Mesh interface needs to be added to physically track the wall movement in the spatial reference frame. the surface is sliding in its tangential direction. It hence implicitly assumes that there are no viscous effects at the slip wall and hence. this may be a reasonable approximation if the important effect of the wall is to prevent fluid from leaving the domain.Slip The Slip condition prescribes a no-penetration condition. Enter the components of the Fluid velocity ul (SI unit: m/s). Sliding Wall The Sliding wall boundary condition is appropriate if the wall behaves like a conveyor belt. For this reason. but the situation becomes more complicated in 3D. that is. Hence. If the velocity vector entered is not in the plane of the wall. u·n. Enter the components of the Velocity of moving wall uw (SI unit: m/s). the tangential direction is unambiguously defined by the direction of the boundary. so must the fluid. Moving Wall If the wall moves. Enter the components of the Velocity of the tangentially moving wall Uw (SI unit: m/s). For 2D models. enter the components of the Velocity of the sliding wall uw (SI unit: m/s). T H E L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R I N T E R F A C E S | 229 . For 3D models. Specifying this boundary condition does not automatically cause the associated wall to move. this boundary condition has slightly different definitions in the different space dimensions. COMSOL Multiphysics projects it onto the tangential direction. Leaking Wall Use this boundary condition to simulate a wall where fluid is leaking into or leaving through a perforated wall u = ul. The wall does not have to actually move in the coordinate system. CONSTRAINT SETTINGS To display this section. The tangential direction is determined in the same manner as in the Sliding Wall feature. The defaults are 0 m/s. For the No Slip. select an option from the Apply reaction terms on: list—All physics (symmetric) or Individual dependent variables. Enter the component values or expressions for the Velocity of sliding wall uw (SI unit: m/s). Moving Wall. The Moving wall (wall functions) boundary condition applies wall functions to a wall in a turbulent flow with prescribed velocity uw. 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. Moving Wall (Wall Functions) Specifying this boundary condition does not automatically cause the associated wall to move. Wall functions are used to model the thin region near the wall with high gradients in the flow variables. The other types of wall boundary conditions with constraints use Individual dependent variables constraints only. Enter the component values or expressions for the Velocity of moving wall uw (SI unit: m/s). and Leaking Wall boundary conditions. Select the Use weak constraints check box (not available for the Sliding Wall condition) to use weak constraints and create dependent variables for the corresponding Lagrange multipliers. The defaults are 0 m/s. click the Show button ( ) and select Advanced Physics Options. 230 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH .Wall Functions The Wall functions boundary condition applies wall functions to solid walls in a turbulent flow. Laminar Inflow. After selecting a Boundary Condition from the list. also enter the turbulent flow settings as described in More Boundary Condition Settings for the Turbulent Flow User Interfaces. or Normal Stress. if Velocity is selected. No Viscous Stress. and Normal stress sections. For the Velocity. in the Outlet type as well. some of them slightly modified. In most cases the inlet boundary conditions are available. Pressure. Theory for the Inlet Boundary ConditionIn the COMSOL Multiphysics Reference Manual: • Theory for the Pressure. T H E L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R I N T E R F A C E S | 231 . Pressure. no viscous stress. No Viscous Stress Boundary Condition • Theory for the Normal Stress Boundary Condition BOUNDARY CONDITION Select a Boundary condition for the inlet—Velocity (the default). For example.Inlet The Inlet node includes a set of boundary conditions describing the fluid flow condition at an inlet. a section with the same name displays underneath. a Velocity section displays where further settings are defined for the velocity. 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. The Velocity boundary condition is the default. • 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. Also enter the turbulent flow settings as described in More Boundary Condition Settings for the Turbulent Flow User Interfaces. The default is 0 m/s. LAMINAR INFLOW The Laminar inflow boundary condition is available for the Inlet node. The default is 0 Pa. • Also enter the turbulent flow settings as described in More Boundary Condition Settings for the Turbulent Flow User Interfaces. it sets the velocity equal to a given velocity vector u0 when u = u0. Also enter the turbulent flow settings as described in More Boundary Condition Settings for the Turbulent Flow User Interfaces. and Boundary Stress nodes. The default is 0 N/m2. PRESSURE. The default is 0 m/s. Enter the Pressure p0 (SI unit: Pa) at the boundary. Select a flow quantity for the inlet—Average velocity (the default). 232 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH . Depending on the pressure field in the rest of the domain. Open Boundary. Enter the magnitude of Normal stress f0 (SI unit: N/m2). Enter the velocity components u0 (SI unit: m/s) to set the velocity equal to a given velocity vector. It specifies vanishing viscous stress along with a Dirichlet condition on the pressure. NO VISCOUS STRESS The Pressure. This implicitly specifies that p f 0 . Enter the velocity magnitude U0 (SI unit: m/s). or Entrance pressure. an inlet boundary with this condition can become an outlet boundary. • When Average velocity is selected. NORMAL STRESS The Normal stress boundary condition is available for the Inlet.VE L O C I T Y The Velocity boundary condition is available for the Inlet and Outlet boundary nodes. • If Velocity field is selected. no viscous stress boundary condition is available for the Inlet and Outlet boundary nodes. Outlet. enter an Average velocity Uav (SI unit: m/s). Flow rate. Then for any selection. For a laminar flow. • If Entrance pressure is selected. For turbulent flow the equivalent expression is 4. The default is 0 m3/s. Lentr should be significantly greater than 0. enter the Flow rate V0 (SI unit: m3/s). no viscous stress boundary condition is the default. select All physics (symmetric). specify the entrance length and constraints: • Enter the Entrance length Lentr (SI unit: m) to define the length of the inlet channel outside the model domain. then the laminar profile has a maximum at that end.• If Flow rate is selected. 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. Otherwise the velocity is defined by the boundary condition of the adjacent boundary in the model. Or select Individual dependent variables to restrict the reaction terms as required. The default is 1 m. The Entrance length value must be large enough so that the flow can reach a laminar profile. No Viscous Stress are selected as the Boundary condition. CONSTRAINT SETTINGS To display this section.06ReD. Outlet The Outlet node includes a set of boundary conditions describing fluid flow conditions at an outlet. The default is 0 Pa. 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). Other options are based on individual licenses. For example. and to Apply reaction terms on all dependent variables. When Velocity or Pressure. Generally. if there is something interesting T H E L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R I N T E R F A C E S | 233 . if one end of a boundary with a laminar inflow condition connects to a slip boundary condition. • Select the Constrain outer edges to zero (for 3D models) or Constrain endpoints to zero (for 2D and 2D axisymmetric models) check box to force the laminar profile to go to zero at the bounding points or edges of the inlet channel. enter the Entrance pressure pentr (SI unit: Pa) at the entrance of the fictitious channel outside of the model. Selecting appropriate outlet conditions for the Navier-Stokes equations is not a trivial task. The Pressure.4Re1/6D. The Pressure. No Viscous Stress (the default). no viscous stress. Additional Theory for the Outlet Boundary Condition BOUNDARY CONDITION Select a Boundary condition for the outlet—Pressure. Theoretically. possibly slightly modified. Laminar Outflow. Velocity. 234 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH . It does however work well in most other situations as well. Pressure The Pressure boundary condition prescribes only a Dirichlet condition for the pressure p = p0. This condition does not provide sufficient information to fully specify the flow at the outlet and must at least be combined with pressure constraints on adjacent points. No Viscous Stress The No Viscous Stress condition specifies vanishing viscous stress on the outlet. No Viscous Stress. extend the computational domain to include this phenomenon. or Normal Stress. 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. the stability is guaranteed by using streamline diffusion for a flow with a cell Reynolds number Recuh 21 (h is the local mesh element size).happening at an outflow boundary. All of the formulations for the Outlet type are also available. Pressure. in other boundary types as well. Velocity. no viscous stress). and Normal stress boundary conditions are described for the Inlet node. it can be numerically unstable. While this boundary condition is flexible and seldom produces artifacts on the boundary (compared to Pressure. Enter the Pressure p0 (SI unit: Pa) at the boundary. The Exit length value must be large enough so that the flow can reach a laminar profile. • If Exit pressure is selected. if one end of a boundary with a T H E L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R I N T E R F A C E S | 235 . • If Flow rate is selected. enter the Flow rate V0 (SI unit: m3/s). 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.4Re1/6D. For turbulent flow the equivalent expression is 4. Lexit should be significantly greater than 0. The default is 1 m. A typical example is a model with volume forces that give rise to pressure gradients that are hard to prescribe in advance.If No viscous stress is selected. Laminar Outflow This section displays when Laminar outflow is selected as the Boundary condition. combine this boundary condition with a point constraint on the pressure. To make the model numerically stable. 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 example. Select the Constrain outer edges to zero (3D models) or Constrain endpoints to zero (2D models) check box to force the laminar profile to go to zero at the bounding points or edges of the inlet channel. Select a flow quantity to specify for the inlet: • If Average velocity is selected. For a laminar flow. enter an Average velocity Uav (SI unit: m/s). This condition can be useful in some situations because it does not impose any constraint on the pressure. u I n = 0 3 u + u T n = 0 using the compressible and the incompressible formulation respectively. it prescribes vanishing viscous stress: u + u T – 2 --.06ReD. Otherwise the velocity is defined by the boundary condition of the adjacent boundary in the model. enter the Exit pressure pexit (SI unit: Pa) at the end of the fictitious channel following the outlet. select All physics (symmetric). – pI + u + u T – 2 --. and the above equations are equivalent to the following equation for both the compressible and incompressible formulation: u n = 0. then the laminar profile has a maximum at that end. CONSTRAINT SETTINGS To display this section. and to Apply reaction terms on all dependent variables. Or select Individual dependent variables to restrict the reaction terms as required.Laminar inflow condition connects to a Slip boundary condition. 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 is a combination of a Dirichlet condition and a Neumann condition: u n = 0. No Viscous Stress. click the Show button ( ) and select Advanced Physics Options. The boundary condition for symmetry boundaries prescribes no penetration and vanishing shear stresses. Pressure. u I n = 0 3 u n = 0. The Dirichlet condition takes precedence over the Neumann condition. or Pressure are selected as the Boundary condition. – pI + u + u T n = 0 for the compressible and the incompressible formulation respectively. K – K n n = 0 K = u + u T n 236 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH . When Velocity. CONSTRAINT SETTINGS To display this section. For 2D axial symmetry. For the symmetry axis at r0. Boundary Stress The Boundary Stress node adds a boundary condition that represents a very general class of conditions also known as traction boundary conditions. T H E L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R I N T E R F A C E S | 237 . choose the boundaries to define. Fluid can both enter and leave the domain on boundaries with this type of condition. click the Show button ( ) and select Advanced Physics Options. a boundary condition does not need to be defined. BOUNDARY CONDITIONS Select a Boundary condition for the open boundaries—Normal Stress (the default) or No Viscous Stress.BOUNDARY SELECTION From the Selection list. Select the Use weak constraints check box to use weak constraints and create dependent variables for the corresponding Lagrange multipliers. the software automatically provides a condition that prescribes ur0 and vanishing stresses in the z direction and adds an Axial Symmetry node that implements this condition on the axial symmetry boundaries only. respectively Also enter the additional settings described in More Boundary Condition Settings for the Turbulent Flow User Interfaces. These options are described for the Inlet and Outlet nodes. Open Boundary The Open Boundary node adds boundary conditions that describe boundaries that are open to large volumes of fluid. u I n = – f 0 n. In addition to the stress condition set in the Normal Stress condition. normal flow condition also prescribes that there must be no tangential velocities on the boundary: – pI + u + u T – 2 --.The total stress on the boundary is set equal to a given stress F: – pI + u + u T – 2 --. Also enter the settings described in More Boundary Condition Settings for the Turbulent Flow User Interfaces. Normal Flow If Normal stress. Normal Stress. 3 – pI + u + u T n = – f 0 n. Normal Stress (described for the Inlet node). or Normal stress. General Stress When General stress is selected.BOUNDARY CONDITION Select a Boundary condition for the boundary stress—General stress (the default). Also enter the turbulent flow settings as described in More Boundary Condition Settings for the Turbulent Flow User Interfaces. normal flow is selected. enter the components for the Stress F (SI unit: N/ m2). u I n = F 3 – pI + u + u T n = F using the compressible and the incompressible formulation respectively. normal flow. enter the magnitude of the Normal stress f0 (SI unit: N/m2). 238 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH tu = 0 tu = 0 . Equation 6-1 states that pn·F. Also enter the turbulent flow settings as described in More Boundary Condition Settings for the Turbulent Flow User Interfaces. This boundary condition implicitly sets a constraint on the pressure that for 2D flows is u n p = 2 ---------.– n F n (6-1) If unn is small. the Normal stress. Periodic Flow Condition The Periodic Flow Condition splits its selection in two groups: one source group and one destination group. No input is required when Compressible flow (Ma<0. then to Apply reaction terms on all dependent variables. Or select Individual dependent variables to restrict the reaction terms as required. This boundary condition also implicitly sets a constraint on the pressure that for 2D flows is u n p = 2 ---------. This corresponds to a situation where the geometry is a periodic part of a larger geometry.3) is selected as the Compressibility under the Physical Model section for the interface. Fluid that leaves the domain through one of the destination boundaries enters the domain over the corresponding source boundary. CONSTRAINT SETTINGS To display this section. the velocity vector is automatically transformed. Normal Flow is selected as the Boundary condition.+ f 0 n (6-2) If unn is small. If the boundaries are not parallel to each other. T H E L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R I N T E R F A C E S | 239 . If the boundaries are curved. it is recommended to only include two boundaries.using the compressible and the incompressible formulation respectively. Equation 6-2 states that pf0. If Normal Stress. select All physics (symmetric). Select the Use weak constraints check box to use weak constraints and create dependent variables for the corresponding Lagrange multipliers. click the Show button ( ) and select Advanced Physics Options. 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. Use the Interior Fan node for interior boundaries. Enter a value or expression for the pressure difference. To set up a periodic boundary condition select both boundaries in the Periodic Flow Condition node. for example. this section is available. This pressure difference can. COMSOL automatically assigns one boundary as the source and the other as the destination.PRESSURE DIFFERENCE When Incompressible flow is selected as the Compressibility under the Physical Model section for the interface. drive the flow in a fully developed channel flow. click the Show button ( ) and select Advanced Physics Options. add a Destination Selection node to the Periodic Flow Condition node. Periodic Boundary Conditions in the COMSOL Multiphysics Reference Manual Fan Use the Fan node to define the flow direction (inlet or outlet) and the fan parameters on exterior boundaries. Select the Use weak constraints check box to use weak constraints and create dependent variables for the corresponding Lagrange multipliers. psrcpdst (SI unit: Pa). All destination sides must be connected. The default is 0 Pa. 240 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH . CONSTRAINT SETTINGS To display this section. To manually set the destination selection. This node is not available for the Turbulent Flow interfaces. When Outlet is selected as the Flow direction. After a boundary is selected. 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. or User defined.01 m3/s). the function used for the fan curve is the maximum between the user defined function and 0. To update the arrow if the selection changes. User Defined Select User defined to enter a different value or expression for the Static pressure curve. PARAMETERS When Inlet is selected as the Flow direction. phys_id is spf by default for laminar single-phase flow).V0 where phys_id is the physics interface identifier (for example. The interpolation between points given in the table is T H E L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R I N T E R F A C E S | 241 . The flow rate across the selection where this boundary condition is applied is defined by phys_id. enter values or expressions for the Static pressure at no flow pnf (SI unit: Pa) (the default is 100 Pa) and the Free delivery flow rate V0. Static Pressure Curve Data Select Static pressure curve data to enter or load data under the Static Pressure Curve Data section that displays. The default is 0 Pa. Linear For both Inlet and Outlet flow directions.FLOW DIRECTION Select a Flow direction—Inlet or Outlet. In order to avoid unexpected behavior. enter the Input pressure pinput (SI unit: Pa) to define the pressure at the fan input. an arrow displays in the Graphics window to indicate the selected flow direction. Static pressure curve data. enter the Exit pressure pexit (SI unit: Pa) to define the pressure at the fan outlet. For either flow direction.fd (SI unit: m3/s) (the default is 0. click any node in the Model Builder and then click the Fan node again to update the Graphics window. select a Static pressure curve to specify a fan curve—Linear (the default). The default is 0 Pa. if Linear is selected. Select the Interpolation function type—Linear (the default). Select Units for the Flow rate (the default SI unit is m3/s) and Static pressure curve (the default SI unit is Pa). The extrapolation method is always a constant value. STATIC PRESSURE CURVE INTERPOLATION This section is available when Static pressure curve data is selected as the Static pressure curve. In order to avoid problems with an undefined function. Then specify the Units for the Flow rate and the Static pressure curve. The Interior Fan defines a boundary condition on the slit.defined using the Interpolation function type list in the Static Pressure Curve Interpolation section. Theory for the Fan and Grille Boundary Conditions Interior Fan The Interior Fan node represents interior boundaries where a fan condition is set using the fan pressure curve to avoid an explicit representation of the fan. the other side represents the fan outlet. the function used for the boundary condition is the maximum between the interpolated function and 0. UNITS 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). One side represents a flow inlet. or Cubic spline. STATIC PRESSURE CURVE DATA This section is available when Static pressure curve data is selected as the Static pressure curve. The fan boundary condition ensures that the mass flow rate is conserved between its inlet and outlet: 242 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH . That means that the pressure and the velocity can be discontinuous across this boundary. Piecewise cubic. BOUNDARY SELECTION From the Selection list. use the Fan node instead. The pressure at the fan outlet is fixed so that the mass flow rate is conserved. an arrow displays in the Graphics window to indicate the selected flow direction. To update the arrow if the selection changes. INTERIOR FAN Define the Flow direction by selecting Along normal vector (the default) or Opposite to normal vector. inlet u n + u n = 0 outlet This boundary condition acts like a Pressure. After a boundary is selected. See Linear. On the fan inlet the pressure is set to the pressure at the fan outlet minus the pressure drop due to the fan. No Viscous Stress boundary condition on each side of the fan. click any node in the Model Builder and then click the Interior fan node again to update the Graphics window. This node is not available for the Turbulent Flow interfaces. This defines which side of the boundary is considered the fan’s inlet and outlet. Theory for the Fan Defined on an Interior Boundary T H E L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R I N T E R F A C E S | 243 . and User Defined for details. choose the boundaries to define. Static Pressure Curve Data. The pressure drop due to the fan is defined by the static pressure curve. which is usually a function of the flow rate. The rest of the settings for this section are the same as for the Fan node. To define a fan boundary condition on an exterior boundary. From a modeling point of view. The Interior Wall boundary condition is only available for single-phase flow. You can also prescribe slip conditions and conditions for a moving wall. or turbulence) across the boundary. no boundary layer develops. It allows discontinuities (velocity. It hence implicitly assumes that there are no viscous effects at both sides of the slip wall and hence. No Slip No slip is the default boundary condition for a stationary solid wall. The condition prescribes u = 0 on both sides of the boundary. Use the Interior Wall boundary condition to avoid meshing thin structures by using no-slip conditions on interior curves and surfaces instead.Interior Wall The Interior Wall boundary condition includes a set of boundary conditions describing the fluid flow condition at an interior wall. 244 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH . u·n0. or Moving wall. It is similar to the Wall boundary condition available on exterior boundaries except that it applies on both sides of an internal boundary. Slip. It is compatible with laminar and turbulent flows. this can be a reasonable approximation if the important effect of the wall is to prevent fluid from leaving the domain. pressure. choose the boundaries to define. The interior wall condition can only be applied to interior boundaries. BOUNDARY SELECTION From the Selection list. that is. the fluid at the wall is not moving. BOUNDARY CONDITION Select a Boundary condition—No slip (the default). Slip The Slip condition prescribes a no-penetration condition. Moving Wall If the wall moves, so must the fluid on both sides of the wall. Hence, this boundary condition prescribes u = uw. Enter the components of the Velocity of moving wall uw (SI unit: m/s). Specifying this boundary condition does not automatically cause the associated wall to move. An additional Moving Mesh interface needs to be added to physically track the wall movement in the spatial reference frame. In the COMSOL Multiphysics Reference Manual: • Slip • Moving Mesh User Interface CONSTRAINT SETTINGS To display this section, click the Show button ( ) and select Advanced Physics Options. For the No slip and Moving wall boundary conditions, and to Apply reaction terms on all dependent variables, select All physics (symmetric). Otherwise, select Current physics (internally symmetric) or Individual dependent variables to restrict the reaction terms as required. Grille The Grille node models the pressure drop caused by having a grille that covers the inlet or outlet. This node is not available for the Turbulent Flow interfaces. See Fan for all of the settings for the Laminar Flow interface, except for Quadratic loss, which is described here. BOUNDARY SELECTION From the Selection list, choose the boundaries to define. PARAMETERS If Quadratic loss is selected as the Static pressure curve, enter the Quadratic loss coefficient to define qlc (SI unit: kg/m7). The default value is 0 kg/m7. qlc defines the T H E L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R I N T E R F A C E S | 245 static pressure curve that is a piecewise quadratic function equal to 0 when flow rate is < 0, equal to V02qlc when flow rate is > 0. Theory for the Fan and Grille Boundary Conditions 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 and it applies to the parts of the pair boundaries where a source boundary lacks a corresponding destination boundary and vice versa. The Wall feature can be overridden by any other boundary condition that applies to exterior boundaries. Right-click the Flow Continuity node to add additional subnodes. In the COMSOL Multiphysics Reference Manual: • Continuity on Interior Boundaries • Identity and Contact Pairs Pressure Point Constraint The Pressure Point Constraint node 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. PRESSURE CONSTRAINT Enter a point constraint for the Pressure p0 (SI unit: Pa). The default is 0 Pa. CONSTRAINT SETTINGS To display this section, click the Show button ( ) and select Advanced Physics Options. To Apply reaction terms on all dependent variables, select All physics (symmetric). Or select Individual dependent variables to restrict the reaction terms as required. Select the Use weak constraints check box to replace the standard constraints with a weak implementation. 246 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH More Boundary Condition Settings for the Turbulent Flow User Interfaces For the Inlet, Open Boundary, and Boundary Stress features, the following settings are also required for the Turbulent Flow interfaces. The first sections (Turbulent Intensity and Turbulence Length Scale Parameters and Boundary Stress Turbulent Boundary Type) provide further information about the boundary conditions, and the additional settings information is described under Boundary Condition. Turbulent Intensity and Turbulence Length Scale Parameters The Turbulent intensity IT and Turbulence length scale LT values are related to the turbulence variables via the following equations, Equation 6-3 for the Inlet and Equation 6-4 for the Open Boundary: 3 2 Inlet k = --- U I T , 2 2 3 Open Boundary k = --- I T U ref , 2 3 4 k3 / 2 = C ----------LT 34 (6-3) 3 2 --- C 3 I T U ref 2 = ------------ ---------------------------- LT 2 (6-4) For the Open Boundary and Boundary Stress options, and with any turbulent flow interface, inlet conditions for the turbulence variables also need to be specified. These conditions are used on the parts of the boundary where u·n0, that is, where flow enters the computational domain. Boundary Stress Turbulent Boundary Type For Boundary Stress, first select a Turbulent boundary type to apply to the turbulence variables—Open boundary (the default), Inlet, or Outlet. • If Open boundary is selected, then expect parts of the boundary to be an outlet and parts of the boundary to be an inlet. • Select Inlet when it is expected that the whole boundary is an inlet. Under Exterior turbulence, the same options to specify turbulence variables are available for the Open boundary option is available. The difference is that, for the Inlet option, is applied to the whole boundary. • Select Outlet when it is expected that the whole boundary is an outflow. Homogeneous Neumann conditions are applied to the turbulence variables (that is, for k and ) k n = 0 n = 0 T H E L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R I N T E R F A C E S | 247 BOUNDARY CONDITION For Open Boundary and Boundary Stress>Open boundary, the following is under the Exterior turbulence subsection. For Boundary Stress, first select a Turbulent boundary type to apply to the turbulence variables—Open boundary (the default), Inlet, or Outlet. Then for Open boundary and Inlet continue entering the following parameters. Select the Specify turbulence length scale and intensity button (the default) to enter values or expressions for the: • Turbulent intensity IT (dimensionless) • Turbulence length scale LT (SI unit: m) • Reference velocity scale Uref (SI unit: m/s). This is available for most options (excluding Velocity for the Inlet node). If the Specify turbulence variables button is selected, enter values or expressions for the: • Turbulent kinetic energy k0 (SI unit: m2/s2) • Turbulent dissipation rate, 0 (SI unit: m2/s3) The default values are different for Inlet, Open Boundary, and Boundary Stress. See Table 6-1. Also, for recommendations of physically sound values see Inlet Values for the Turbulence Length Scale and Turbulent Intensity. TABLE 6-1: DEFAULT VALUES FOR THE TURBULENT INTERFACES 248 | NAME AND UNIT VARIABLE INLET OPEN BOUNDARY BOUNDARY STRESS Turbulent intensity (dimensionless) IT 0.05 0.005 0.01 Turbulence length scale (m) LT 0.01 0.1 0.1 Reference velocity scale (m/s) Uref 1 1 1 CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH TABLE 6-1: DEFAULT VALUES FOR THE TURBULENT INTERFACES NAME AND UNIT VARIABLE INLET OPEN BOUNDARY BOUNDARY STRESS Turbulent kinetic energy (m2/s2) k0 0.005 2.5 x 10-3 1 x 10-2 Turbulent dissipation rate (m2/s3) 0 0.005 1.1 x 10-4 1 x 10-3 T H E L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R I N T E R F A C E S | 249 Theory for the Laminar Flow User Interface For the basic laminar flow theory, see Theory of Laminar Flow in the COMSOL Multiphysics Reference Manual. This section discusses the theory related to the advanced features available with this module and for laminar flow. Also see Theory for the Turbulent Flow User Interfaces. In this section: • Theory for the Inlet Boundary Condition • Additional Theory for the Outlet Boundary Condition • Additional Theory for the Outlet Boundary Condition • Theory for the Fan and Grille Boundary Conditions • Non-Newtonian Flow: The Power Law and the Carreau Model Theory for the Inlet Boundary Condition LAMINAR INFLOW In order to prescribe an inlet velocity profile, this boundary condition adds a weak form contribution corresponding to one-dimensional Navier-Stokes 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 fully developed laminar flow. The “wall” boundary conditions for the fictitious domain is inherited from the real domain, , unless the 250 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH 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. pentr. this boundary condition adds a weak form contribution corresponding to one-dimensional Navier-Stokes equations projected on the boundary. unless the THEORY FOR THE LAMINAR FLOW USER INTERFACE | 251 . No Viscous Stress Boundary Condition • Theory for the Normal Stress Boundary Condition Additional Theory for the Outlet Boundary Condition LAMINAR OUTFLOW In order to prescribe an outlet velocity profile. In the COMSOL Multiphysics Reference Manual: • Prescribing Inlet and Outlet Conditions • Theory for the Pressure. . The “wall” boundary conditions for the fictitious domain is inherited from the real domain. If an average inlet velocity or inlet volume flow is specified instead of the pressure. is the actual computational domain while the dashed domain is a fictitious domain. COMSOL Multiphysics adds an ODE that calculates a pressure. • Also see Inlet for the node settings. This boundary condition uses the assumption that flow in this fictitious domain is fully developed laminar flow. such that the desired inlet velocity or volume flow is obtained. pentr Lentr Figure 6-1: An example of the physical situation simulated when using the Laminar inflow boundary condition.option to constrains outer edges or endpoints to zero is selected in which case the fictitious “walls” will be no-slip walls. the software adds an ODE that calculates pexit such that the desired outlet velocity or volume flow is obtained. It must hence be combined with at pressure point constraints on one or several points or lines surrounding the outlet. the Viscous stress condition sets the viscous stress to zero: u + u T – 2 --. u I n = 0 3 u + u T n = 0 using the compressible and the incompressible formulation. respectively. No Viscous Stress Boundary Condition (described in the COMSOL Multiphysics Reference Manual). 252 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH . and in addition to the Theory for the Pressure.option to constrains outer edges or endpoints to zero is selected in which case the fictitious “walls” will be no-slip walls. NO VISCOUS STRESS For this module. If the average outlet velocity or outlet volume flow is specified instead of the pressure. is the actual computational domain while the dashed domain is a fictitious domain. The condition is not a sufficient outlet condition since it lacks information about the outlet pressure. pexit Lexit Figure 6-2: An example of the physical situation simulated when using Laminar outflow boundary condition. for example. The pressure drop is calculated from a lumped curve using the flow rate evaluated on dev_in. In both cases. • Also see Outlet for the node settings.This boundary condition is numerically the least stable outlet condition. • For the inlet boundary condition. See Interior Fan for node settings. the boundary condition implementation specifies vanishing viscous stress along with a Dirichlet condition on the pressure. In the COMSOL Multiphysics Reference Manual: • Prescribing Inlet and Outlet Conditions • Theory for the Pressure. the inlet and outlet of the device are both interior boundaries (see Figure 6-3). a nonlinear volume force. THEORY FOR THE LAMINAR FLOW USER INTERFACE | 253 . the pressure value is set so that the flow rate is equal on dev_in and dev_out. but can still be beneficial if the outlet pressure is nonconstant due to. An ODE is added to compute the pressure value. The boundary conditions are described as follows: • The inlet of the device is an outlet boundary condition for the modeled domain. on dev_out. No Viscous Stress Boundary Condition • Theory for the Normal Stress Boundary Condition Theory for the Fan Defined on an Interior Boundary In this case. The boundaries are called dev_in and dev_out. on dev_in. For this outlet boundary condition. 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. Theory for the Fan and Grille Boundary Conditions Fans. pumps. These simplifications also imply some assumptions. See Fan and Grille for node settings. the device’s inlet is an external boundary.Figure 6-3: A device between two boundaries. Such a boundary should not be a mix of inlets/outlets. and the two cubes are the domain that are modeled with a particular inlet boundary condition to account for the device. The device’s outlet is an interior face situated between the green and blue domains in Figure 6-4. The red arrows represent the flow direction. The lumped curve gives the flow rate as a function of the pressure difference between the external 254 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH . it is assumed that a given boundary can only be either an inlet or an outlet. or grilles (devices) can be represented using lumped curves implemented as boundary conditions. Manufacturers usually provide curves that describe the static pressure as a function of flow rate for a fan. In particular. the cylindrical part represents the device (that should be not be part of the model). represented by the external circular boundary of the green domain on Figure 6-4. nor should it change during a simulation. DEFINING A DEVICE AT AN INLET In this case. This boundary condition implementation specifies vanishing viscous stress along with a Dirichlet condition on the pressure. Dz. is used to define the flow rate. pinput is the pressure at the device’s inlet. In 2D the thickness in the third direction. The Fan boundary condition sets the following conditions: u + u T – 2 --. Fans are modeled as rectangles in this case. and pfanV0) and pgrille(V0) are the static pressure functions of flow rate for the fan and the grille. 3 T u + u n = 0.boundary and the interior face. u I n = 0. u I n = 0. THEORY FOR THE LAMINAR FLOW USER INTERFACE | 255 . 3 T u + u n = 0. p = p input – p grill V 0 p = p input – p grill V 0 (6-7) (6-8) where V0 is the flow rate across the boundary. Equation 6-6 and Equation 6-8 correspond to the incompressible formulation. Equation 6-5 and Equation 6-7 correspond to the compressible formulation. p = p input + p fan V 0 p = p input + p fan V 0 (6-5) (6-6) The Grille boundary condition sets the following conditions: u + u T – 2 --. the fan’s inlet is the interior face situated between the blue (cube) and green (circle) domain while its outlet is an external boundary. This boundary condition implementation specifies vanishing viscous stress along with a Dirichlet condition on the pressure. 3 T u + u n = 0.Figure 6-4: A device at the inlet. u I n = 0. DEFINING A DEVICE AT AN OUTLET In this case (see Figure 6-5). The Fan boundary condition sets the following conditions: u + u T – 2 --. the green circle represents the device (that should not be part of the model). p = p ext – p fan V 0 p = p ext – p fan V 0 (6-9) (6-10) The Grille boundary condition sets the following conditions: u + u T – 2 --. 256 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH p = p input + p grill V 0 p = p input + p grill V 0 (6-11) (6-12) . u I n = 0. here the circular boundary of the green domain. and the blue cube represents the modeled domain with an inlet boundary condition described by a lumped curve for the attached device. The lumped curve gives the flow rate as a function of the pressure difference between the interior face and the external boundary. The arrow represents the flow direction. 3 T u + u n = 0. Equation 6-11. The arrow represents the flow direction. is used to define the flow rate. Equation 6-10. and Equation 6-12 correspond to the incompressible formulation. Equation 6-10. Non-Newtonian Flow: The Power Law and the Carreau Model The viscous stress tensor is directly dependent on the shear rate tensor and can be written as: 2 = · – --. and the blue cube represents the modeled domain with an outlet boundary condition described by a lumped curve for the attached fan. Fans are modeled as rectangles in this case. the green circle represents the fan (that should not be part of the model). Equation 6-9. and pfan(V0). and Equation 6-11 correspond to the compressible formulation. Figure 6-5: A fan at the outlet. u I 3 = · THEORY FOR THE LAMINAR FLOW USER INTERFACE | 257 . the vacuum pump. Dz. and pgrille(V0) are the static pressure function of flow rate for the fan. and the grille.where V0 is the flow rate across the boundary. pvacuum pump(V0). pext is the pressure at the device outlet. In 2D the thickness in the third direction. A value of n equal to one gives the expression for a Newtonian fluid. respectively. This is however never the case physically. Here · denotes the engineering strain-rate tensor defined by: · = u + u T Its magnitude. Instead. but it can be changed in the equation view. it describes a shear thinning (pseudoplastic) fluid. is: · = · = 1 --. For n1.using the compressible and incompressible formulation. the power law describes a shear thickening (dilatant) fluid. The default value for · min is 10-2 s-1. It prescribes · = m n – 1 (6-13) where m and n are scalars that can be set to arbitrary values. COMSOL implements the power law as · · = mmax min n – 1 (6-14) where · min is a lower limit for the evaluation of the shear rate magnitude. For n1. the dynamic viscosity is assumed to be a function of the shear rate: · = The Laminar Flow interfaces have the following predefined models to prescribe a non-Newtonian viscosity—the power law and the Carreau model. Equation 6-13 predicts infinite viscosity at zero shear rate. 258 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH . the shear rate. most fluids have constant viscosity for shear rates smaller than 10-2 s-1 (Ref.· :· 2 where the contraction operator “:” is defined by a:b = anm bnm n m For a non-Newtonian fluid. POWER LAW The power law is an example of a generalized Newtonian model. 1). Since infinite viscosity also makes models using Equation 6-13 difficult to solve. 0 is the zero shear rate viscosity. This expression is able to describe viscosity for mostly stationary polymer flow. THEORY FOR THE LAMINAR FLOW USER INTERFACE | 259 .CARREAU MODEL The Carreau expression gives the viscosity by the following four-parameter equation n – 1 · ---------------- = + 0 – inf 1 + 2 2 (6-15) where is a parameter with units of time. inf is the infinite shear rate viscosity. and n is a dimensionless parameter. Turbulent Flow interfaces theory is described in this section: • Turbulence Modeling • The k-Turbulence Model • The Low Reynolds Number k- Turbulence Model • Inlet Values for the Turbulence Length Scale and Turbulent Intensity • Theory for the Pressure. The small scales are then modeled using a turbulence model with 260 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH . which depend on the geometry. the density. The tendency for an isothermal flow to become turbulent is measured by the Reynolds number UL Re = ----------- (6-16) where is the dynamic viscosity. although this would require a large number of elements to capture the wide range of scales in the flow. 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. and all the scales in between. respectively. the smallest quickly fluctuating scales. No Viscous Stress Boundary Condition • Solvers for Turbulent Flow • Pseudo Time Stepping for Turbulent Flow Models • References for the Single-Phase Flow. and U and L are velocity and length scales of the flow. User Interfaces Theory for the Laminar Flow User Interface Turbulence Modeling Turbulence is a property of the flow field and it is mainly characterized by a wide range of flow scales: the largest occurring scales. An alternative approach is to divide the flow in large resolved scales and small unresolved scales.Theory for the Turbulent Flow User Interfaces The Single-Phase Flow. and an average part. Because the flow field also varies T H E O R Y F O R T H E TU R B U L E N T F L O W U S E R I N T E R F A C E S | 261 . the mean value can vary in space and time. there is a fluctuating part. This module includes Reynolds-averaged Navier-Stokes (RANS) models which is the model type most commonly used for industrial flow applications. ui. In general. The unfiltered flow has a time scale t1. which shows time averaging of one component of the velocity vector for nonstationary turbulence. REYNOLDS-AVERAGED NAVIER-STOKES (RANS) EQUATIONS The information below assumes that the flow fluid is incompressible and Newtonian in which case the Navier-Stokes equations take the form: T u + u u = – pI + u + u + F t u = 0 (6-17) Once the flow has become turbulent. This is exemplified in Figure 6-6. After a time filter with width t2 t1 has been applied. Ui.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. all quantities fluctuate in time and space. An averaged representation often provides sufficient information about the flow. It is seldom worth the extreme computational cost to obtain detailed information about the fluctuations. = + where can represent any scalar quantity of the flow. The Reynolds-averaged representation of turbulent flows divides the flow quantities into an averaged value and a fluctuating part. Ui is still time dependent but is much smoother than the unfiltered velocity ui. information about the small-scale structure of the flow is needed.trace u' u' I = – T U + U 3 262 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH . This term represents interaction between the fluctuating velocities and is called the Reynolds stress tensor. Figure 6-6: The unfiltered velocity component ui. Ui. and the averaged velocity component. EDDY VISCOSITY The most common way to model turbulence is to assume that the turbulence is of a purely diffusive nature.on a time scale longer than t2. Decomposition of flow fields into an averaged part and a fluctuating part. with a time scale t1. In this case. followed by insertion into the Navier-Stokes equation. with time scale t2. The deviating part of the Reynolds stress is then expressed by T u' u' – --. gives the Reynolds-averaged Navier-Stokes (RANS) equations: U T + U U + u' u' = – P + U + U + F t U = 0 (6-18) where U is the averaged velocity field and is the outer vector product. then averaging. A comparison with Equation 6-17 indicates that the only difference is the appearance of the last term on the left-hand side of Equation 6-18. that information is the correlation between fluctuations in different directions. This means that to obtain the mean flow characteristics. To avoid this. also known as the turbulent viscosity.+ -------- – --. ui. 8). a density-based average. The spherical part can be written --. terms of the form u appear and need to be modeled. this term is included in the pressure.)u i (x.lim --- T T (x. In simulations of incompressible flows. and a fluctuating component. is decomposed in a mass-averaged component. known as the Favre average.trace u' u' I = 2 --. Equation 6-19 can be written in the form -----.+ ------.where T is the eddy viscosity. but when the absolute pressure level is of importance (in compressible flows. u˜ i . according to u i = u˜ i + u i (6-21) Using Equation 6-20 and Equation 6-21 along with some modeling assumption for compressible flows (Ref.k 3 3 where k is the turbulent kinetic energy. for example) this term must be explicitly included. ij – u j u i + F i t x j x i x j x j x i 3 x k (6-22) The Favre-averaged Reynolds stress tensor is modeled using the same argument as for incompressible flows: T H E O R Y F O R T H E TU R B U L E N T F L O W U S E R I N T E R F A C E S | 263 . -------. --------.= – ------. u˜ i = 0 t x i ˜ ˜ ˜ u˜ i u˜ i u i u j 2 u k p -------. is introduced: t+T 1 1 u˜i = --. TU R B U L E N T C O M P R E S S I B L E F L OW If the Reynolds average is applied to the compressible form of the Navier-Stokes.) d (6-19) t It follows from Equation 6-19 that u˜ i = u i (6-20) and a variable.+ u˜ j -------. ui.+ ------. 2). k. The turbulent transport equations are used in their fully compressible formulations (Ref. u˜ i u˜ j 2 u˜ k – u j u i = T -------.k u 3 3 The transport equation for reads: 264 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH (6-25) . The k-Turbulence Model The k- model is one of the most used turbulence models for industrial applications. and the dissipation rate of turbulence energy. Comparing Equation 6-22 to its incompressible counterpart (Equation 6-18). it can be seen that except for the term – 2 3 k ij the compressible and incompressible formulations are exactly the same. T --------.+ u k = + ------ k + P k – t k (6-24) where the production term is 2 2 P k = T u: u + u T – --. The transport equation for k reads: T k -----. . u 2 – --.+ -------- – --. Turbulent viscosity is modeled by k2 T = C ----- (6-23) where C is a model constant. 8.+ k ij x j x i 3 x k where k is the turbulent kinetic energy. This module includes the standard k- model (Ref. 2 and Ref. This introduces two additional transport equations and two dependent variables: the turbulent kinetic energy. 9). except that the free variables are u˜ i instead of Ui = ui More information about modeling turbulent compressible flows is in Ref. 0 1. To assert that u i u i 0 i the turbulent viscosity is subjected to a realizability constraint.3 MIXING LENGTH LIMIT Equation 6-24 and Equation 6-26 cannot be implemented directly as written.92 k 1. for example. The implementation includes an upper limit on lim : the mixing length. TABLE 6-2: MODEL CONSTANTS CONSTANT VALUE C 0. Equation 6-24.T 2 ----. There is. l mix active in a converged solution but is merely a tool to obtain convergence. l mix 3/2 k lim l mix = max C ----------- l mix (6-27) lim should not be The mixing length is used to calculated the turbulent viscosity. 2) and the values are listed in Table 6-2. The constraint for 2D and 2D axisymmetry is: T H E O R Y F O R T H E TU R B U L E N T F L O W U S E R I N T E R F A C E S | 265 . 10. nothing that prevents division by zero. The equations are instead implemented as suggested in Ref.P – C 2 ---- k k k t (6-26) The model constants in Equation 6-23. and Equation 6-26 are determined from experimental data (Ref. but calculating T from Equation 6-23 does not guarantee this.k ij 3 where ij is the Kronecker delta and Sij is the strain-rate tensor.44 C2 1.+ u = + ------ + C 1 --.09 C1 1. The diagonal elements of the Reynolds stress tensor must be nonnegative. REALIZABILITY CONSTRAINTS The eddy-viscosity model of the Reynolds stress tensor can be written 2 u i u j = – 2 T S ij + --. in the description of rotating flows. Furthermore.k 2 T ----------------------3 S ij S ij (6-28) and for 3D and 2D axisymmetry with swirl flow it reads: k T --------------------------6 S ij S ij (6-29) Swirl flow is not available with the Heat Transfer Module. 2). and Ref.------------------3 S ij S ij (6-30) Equivalently. the most important of which is that the Reynolds number is high enough. MODEL LIMITATIONS The k- turbulence model relies on several assumptions. the limited accuracy is a fair 266 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH . Ref. 3). 5. To avoid such artifacts. It does not. Combining equation Equation 6-28 with Equation 6-23 and the definition of the mixing length gives a limit on the mixing length scale: k l mix 2 --. for example. 6. which means that production equal dissipation. 7. The effect is most clearly visible in stagnation points. In most cases. The effect of not applying a realizability constraint is typically excessive turbulence production. the realizability constraint is always applied for the RANS models. combining Equation 6-29 with Equation 6-23 and Equation 6-27 gives: 1 k l mix ------. More details can be found in Ref. It is also important that the turbulence is in equilibrium in boundary layers. respond correctly to flows with adverse pressure gradients that can result in under-predicting the spatial extension of recirculation zones (Ref.------------------6 S ij S ij (6-31) This means there are two limitations on lmix: the realizability constraint and the imposed limit via Equation 6-27. the model often shows poor agreement with experimental data (Ref. These assumptions limit the accuracy of the model because they are not always true. which becomes 11. this is not always desirable because of the very high resolution requirements that follow. This means that the assumptions used to derive the k- model are not valid close to walls. Mesh cells w Solid wall Figure 6-7: The computational domain starts a distance w from the wall for wall functions. w is limited from below so that it never becomes smaller than half of the height of the boundary T H E O R Y F O R T H E TU R B U L E N T F L O W U S E R I N T E R F A C E S | 267 . 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). analytical expressions are used to describe the flow at the walls. Instead.trade-off for the amount of computational resources saved compared to more complicated turbulence models. The wall functions in COMSOL Multiphysics are such that the computational domain is assumed to start a distance w from the wall (see Figure 6-7). 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). WAL L F UN C T IO NS The flow close to a solid wall is for a turbulent flow and is very different compared to the free stream.06. These expressions are known as wall functions. The distance w is automatically computed so that + = u w w where uC1/4k is the friction velocity. 2. w. + is much higher over a significant part of the walls.ln w + B v where in turn. are available as results and analysis variables. 10). INITIAL VALUES The default initial values for a stationary simulation are (Ref. 11 for further details. the accuracy If w might become compromised. This means that w coarse. 268 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH . Also check that w walls. v. w The boundary conditions for the velocities are a no-penetration condition u n = 0 and a shear stress condition u n – n n n = – u ------. Always investigate the solution to check that w is small compared to the + is 11.06 on most of the dimension of the geometry. and the wall + .+ can become higher than 11.41) and B is a constant that by default is set to 5.06 if the mesh is relatively mesh cell. The turbulent kinetic energy is subject to a homogeneous Neumann condition n k = 0 and the boundary condition for reads: C 3 / 4 k 3 / 2 = ----------------------v w See Ref. lift-off in viscous units. Both the wall lift-off. is the von Kárman constant (default value 0. 10 and Ref.max C 1 / 4 k u u where = u + u T is the viscous stress tensor and u u = ----------------------------1 ----. the initial value where l mix for k is instead 2 k = ------------------------------ 0.1 l mix lim is the mixing length limit.1 l lim mix SCALING FOR TIME-DEPENDENT SIMULATIONS The k- equations are derived under the assumption that the flow has a high enough Reynolds number. the default time-dependent solver for the k- model employs unscaled absolute tolerances for k and . Their default values are 1 ms and 0. To sort out numerical fluctuations in k and during such periods. The tolerances are set to k scale = 0.01U scale 2 3/2 L scale = 0. both k and will have very small magnitudes and chaotically in the manner that the relative values of k and can change relatively much just because of small changes in the flow field.u=0 p=0 10 2 k = ------------------------------lim 0.09k sclae fact l bb min (6-32) where Uscale and Lfact are input parameters available in the Advanced section of the physics interface node. lbb. A time-dependent simulation of a turbulent flow can include a period when the flow is not fully turbulent. For time dependent simulations.min is the shortest side of the geometry bounding box.035 respectively. The practical implication of Equation 6-32 is that variations in k and smaller than kscale and scale respectively. will be regarded will be regarded as numerical noise. If this assumption is not fulfilled. Equation 6-32 is closely related to the expressions for k and on inlet boundaries (see Equation 6-37).1 l mix 3/2 C k init = ---------------------lim 0. A typical example is the startup phase when for example an inlet velocity or a pressure difference is gradually increased. T H E O R Y F O R T H E TU R B U L E N T F L O W U S E R I N T E R F A C E S | 269 . Kondoh.+ u k = + ------ k + P k – k t T 2 ----. is provided by a mathematical Wall Distance interface that is included when using the low Reynolds number k- model. The distance to objects 270 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH .4 = 1. a so called low Reynolds number model can be used. Ref. and Nagano. Most low Reynolds number k- models adapt the turbulence transport equations by introducing damping functions. lw.k u 3 3 k2 T = f C ----- 5 – R t 200 2 * 2 -e f = 1 – e – l 14 1 + ---------- R t3 / 4 f = 1 – e –l * 3. “Low Reynolds number” refers to the region close to the wall where the viscous effects dominate. lw is the distance to the closest wall.4 (6-35) Also.3e – Rt 6. This module includes the AKN model (after the inventors Abe. The solution to the wall distance equation is controlled using the parameter lref.5 2 l * = u l w R t = k 2 u = 1 / 4 and C 1 = 1. WA LL DIS TANCE The wall distance variable. 12): T k -----.9 C = 0. 12).1 2 (6-34) 1 – 0.09 k = 1. 9 and Ref.+ u = + ------ + C 1 --. Realizability Constraints are applied to the low Reynolds number k- model.P – f C 2 ---- t k k k (6-33) where 2 2 P k = T u: u + u T – --. The AKN k- model for compressible flows reads (Ref.The Low Reynolds Number k- Turbulence Model When the accuracy provided by wall functions in the k- model is not enough.5 C 2 = 1. u 2 – --. Since all velocities must disappear on the wall.larger than lref is represented accurately. In the COMSOL Multiphysics Reference Manual: • The Wall Distance User Interface • Stationary with Initialization • Transient with Initialization • Wall Distance Initialization WAL L B O UN D A R Y C ON D I TIO N S The damping terms in the equations for k and allows a no slip condition to be applied to the velocity.5 T H E O R Y F O R T H E TU R B U L E N T F L O W U S E R I N T E R F A C E S | 271 . This is a desirable feature in turbulence modeling since small objects would get too large an impact on the solution if the wall distance were measured exactly. is not solved for in the cells adjacent to a solid wall and the analytical relation k = 2 --. That condition is however numerically very unstable. A Wall Distance Initialization study type is provided for this purpose and should be added before the actual Stationary or Transient study step. k0 on the wall. Hence. it is required that l c* 0. Equation 6-36 can be derived as the first term in a series expansion of 2 k n 2 For the expansion to be a valid.----2 lw (6-36) is prescribed in those cells. The most convenient way to handle the wall distance variable is to solve for it in a separate study step. that is u0. Instead. so must k. while objects smaller than lref will effectively be diminished by appearing to be farther away than they actually are. The correct wall boundary condition for is 2 k n 2 where n is the wall normal direction. S= 272 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH 2S ij S ij = 2 ij ij . 3 Add a new Stationary with Initialization study. INITIAL VALUES The low-Reynolds number k- model has the same default initial guess as the standard lim replaced by l . The boundary variable Dimensionless distance to cell center is available to ensure that the mesh is fine enough.5 I N L E T VA 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 Turbulent Intensity for selecting turbulence length scale. IT. In some cases. 2 Switch to the low-Reynolds number k- model. 5 Solve the new study. k- model (see Initial Values) but with l mix ref The default initial value for the wall distance equations (which solves for the reciprocal wall distance) is 2lref. apply also to the low-Reynolds number k- model. a fast way to convergence is to first solve the model using an ordinary k- model and then use that solution as initial guess for the low-Reynolds number k- model. SCALING FOR TIME-DEPENDENT SIMULATIONS The low-Reynolds number k- model uses absolute scales of the same type as the k- model (see Scaling for Time-Dependent Simulations). specially for stationary solutions. Observe though that it is unlikely that a solution is obtained at all if l c* » 0.l c* is the distance. This is to propagate the solution from the first study down to the second step in the new study. The procedure is then as follows: 1 Solve the model using the k- model. and the turbulence intensity. measured in viscous units. from the wall to the center of the wall adjacent cell. set Values of variables not solved for to Solution from the first study. LT. 4 In the Wall Distance Initialization study step. 1% is a low turbulence intensity IT. crude approximations for k and can be obtained from the following formulas: 2 3 k = --. Fully turbulent flows usually have intensities between five and ten percent. The length scale cannot be zero. A value of 0.07L Layer width Plane jet 0.07L (fully developed flows) Boundary layer thickness Pipe radius or channel half width T H E O R Y F O R T H E TU R B U L E N T F L O W U S E R I N T E R F A C E S | 273 .075L Jet half width Boundary layer (px0) – Viscous sublayer and log-layer + l w 1 – exp – l w 26 – Outer layer 0. For free-stream flows these are typically very small (in the order of centimeters).05%. Use Table 6-3 as a guideline when specifying LT (Ref. however. U I T 2 = (6-37) 3 4 k3 / 2 C ----------LT where IT is the turbulence intensity and LT is the turbulent length scale. because that would imply infinite dissipation.09L Jet half width Wake 0.08L Wake width Axisymmetric jet 0. 4) where lw is the wall distance. Good wind tunnels can produce values of as low as 0. and + lw = lw l* TABLE 6-3: TURBULENT LENGTH SCALES FOR TWO-DIMENSIONAL FLOWS FLOW CASE LT L Mixing layer 0.09L Pipes and channels 0. The turbulent length scale LT is a measure of the size of the eddies that are not resolved.Inlet Values for the Turbulence Length Scale and Turbulent Intensity If inlet data for the turbulence variables are not available. two or three iterations are made for the turbulence transport equations. and reference velocity scale Uref values are related to the turbulence variables via 2 3 k = --. Turbulent flows are therefore solved using a segregated approach (Ref. turbulence length scale LT. I T U ref . 13): Navier-Stokes in one group and the turbulence transport equations in another.Theory for the Pressure. LT 2 = I T U ref 3 --. No Viscous Stress Boundary Condition (described in the COMSOL Multiphysics Reference Manual). and in addition to the Theory for the Pressure. Also see Inlet and Outlet for the node settings. Solvers for Turbulent Flow The non-linear system that Navier-Stokes and the turbulence transport equations constitute can become ill-conditioned if solved using a fully coupled solver. ---------------------------- .-------------------------2 0* 1 / 4 L T For recommendations of physically sound values see Inlet Values for the Turbulence Length Scale and Turbulent Intensity. This is necessary to make sure that the very non-linear source terms in the turbulence transport equations are in balance before making another iteration for the Navier-Stokes group. For each iteration in the Navier-Stokes group. No Viscous Stress Boundary Condition For this module. This both improves the robustness of the non-linear iterations as well as the condition number for the linear equation system. 274 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH . The default stationary solver is solved using pseudo time stepping. 2 34 3 2 --- C 3 I T U ref 2 = -----------. the turbulent intensity IT. The latter is specially important for large 3D models where iterative solvers must be applied. for 2D models: 1.3 min niterCMP – 30 9 0 + if niterCMP 60 90 1.3 min niterCMP – 25 9 0 + if niterCMP 50 90 1.3 min niterCMP-1 9 + if niterCMP 25 9 1. The turbulence equations use the same t˜ as the momentum equations. The default manual expression for CFLloc is.The default iterative solver for the turbulence transport equations is GMRES accelerated by Geometric Multigrid. for both 2D and 3D models.3 min niterCMP – 50 9 0 and for 3D models: 1.3 min niterCMP – 60 9 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 U S E R I N T E R F A C E S | 275 .3 min niterCMP-1 9 + if niterCMP 30 9 1. Pseudo Time Stepping for Turbulent Flow Models In the COMSOL Multiphysics Reference Manual: • Multigrid • Stationary Solver • Iterative • SOR Line Pseudo Time Stepping for Turbulent Flow Models Pseudo time stepping is by default applied to the turbulence equations for stationary problems. The default smoother is SOR Line. “Non-Newtonian Flow and Applied Rheology”.” AIAA Journal. Chalmers University of Technology. An Introduction to Computational Fluid Dynamics. Ilinca. John Wiley & Sons. “Wall Functions for General Application CFD Codes. Richardson. Turbulence Transport Modeling in Gas Turbine Related Applications. Chhabra and J. 193–206. pp. Prentice Hall. 9. Park. Malalasekera. doctoral dissertation. 2006.R. 1998.” International Journal of Computational Fluid Dynamics. Svenningsson. 1986. Kuzmin. 6. Mierka. 1998.” Computer Methods in Applied Mechanics and Engineering. 2007. Proceedings of the Fourth European Computational Fluid Dynamics Conference. pp. vol. H. 23. DCW Industries. Ignat. 79– 86.K. 2:nd ed. 19.M.” ECCOMAS 98. “On the Implementation of the k- Turbulence Model in Incompressible Flow Solvers Based on a Finite Element Discretization. 276 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH . R. O. vol. 2–4. D.” doctoral dissertation. Wilcox. Durbin. Driver and H. and S. “On the Limiters of Two-equation Turbulence Models. pp. 8. Pelletier. Turbulence Modeling for CFD.” International Journal of Computing Science and Mathematics. Elsivier. 5. Department of Applied Mechanics.. 17. 10. 2nd ed. “On the k- Stagnation Point Anomality. Turek.P. 1998. 1995. A.References for the Single-Phase Flow. no. 4. “A Universal Formulation of Two-equation Models for Adaptive Computation of Turbulent Flows. “Features of a Reattaching Turbulent Shear Layer in Diverging Channel Flow. J. A. Chalmers University of Technology. 1.O. 1112–1117. 7. 3. Menter.F. H. 1119–1139. 2005. User Interfaces 1.” International Journal of Heat and Fluid Flow. 189. and F. Park and S. Seegmiller. 2. 163–171. Sweden. 1. D. 11. Grotjans and F. pp. H. D. 1985. vol. Numerical Simulation of Turbulent Flows for Turbine Blade Heat Transfer. pp. 2000. vol. vol. Larsson.L. C.C. Versteeg and W. 2008. L. pp. D. 89–90. no. F.” International Journal of Heat and Mass Transfer. 555–579. Abe. M. 139–151. U S E R I N T E R F A C E S | 277 . Mallet. T. vol. and Y. Nagano. Chalot. “The Robustness Issue on Multigrid Schemes Applied to the Navier-Stokes Equations for Laminar and Turbulent. “A New Turbulence Model for Predicting Fluid Flow and Heat Transfer in Separating and Reattaching Flows—I. 45. pp.P H A SE F L OW. Vázquez. M. Flow Field Calculations. 13. R E FE RE N C E S FO R T H E S I N G L E . and M. no. vol. K.12. 1. 2004. 37. Kondoh. pp. Ravachol. 1994.” International Journal for Numerical Methods in Fluids. Incompressible and Compressible Flows. 278 | CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH . 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. Laminar Flow and Turbulent Flow User Interfaces • Theory for the Non-Isothermal Flow and Conjugate Heat Transfer User Interfaces • References for the Non-Isothermal Flow and Conjugate Heat Transfer User Interfaces 279 . which are also under the Fluid Flow branch as Non-Isothermal Flow interfaces. In this chapter: • About the Conjugate Heat Transfer User Interfaces • The Non-Isothermal Flow and Conjugate Heat Transfer. This chapter discusses applications involving the Conjugate Heat Transfer branch ( ).7 The Conjugate Heat Transfer Branch The Heat Transfer Module has interfaces for conjugate heat transfer. turbulent. which can be laminar. 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. k- RANS k- Kays-Crawford C H A P T E R 7 : T H E C O N J U G A T E H E A T TR A N S F E R B R A N C H . Solving this coupled system of equation usually requires numerical stabilization.About the Conjugate Heat Transfer User Interfaces In this section: • Selecting the Right User Interface • The Non-Isothermal Flow Options • Conjugate Heat Transfer Options Selecting the Right User Interface There are several variations of the same predefined multiphysics interface (all with the interface identifier nitf). Laminar Flow None N/A N/A Non-Isothermal Flow. interfaces use the same definition of the density. or Stokes flow. which are selected during Model Wizard selection. In particular. The settings vary only by one or two default settings (see Table 7-1). in combination with heat transfer. Figure 7-1 is an example that compares the two settings windows. which can therefore be a function of both pressure and temperature. ) and Heat The user interfaces found under the Fluid Flow>Non-Isothermal Flow ( Transfer>Conjugate Heat Transfer ( ) branches are multiphysics interfaces and contain the features for modeling fluid flow. TABLE 7-1: THE NON-ISOTHERMAL FLOW AND CONJUGATE HEAT TRANSFER PHYSICAL MODEL DEFAULT SETTINGS* 280 | USER INTERFACE (NITF) TURBULENCE MODEL TYPE TURBULENCE MODEL HEAT TRANSPORT TURBULENCE MODEL Non-Isothermal Flow. which the predefined multiphysics interface also sets up. Turbulent Flow. or from a check box or list under the Physical Model section for the interface. Turbulent Flow. Turbulent Flow. Figure 7-1: On the left is the settings window for the Non-Isothermal Flow. or Stokes flow. the Neglect initial term (Stokes flow) check box is not selected by default. Low Re k- RANS Low Reynolds number k- Kays-Crawford *For all the interfaces. Low Re k- RANS Low Reynolds number k- Kays-Crawford Conjugate Heat Transfer.TABLE 7-1: THE NON-ISOTHERMAL FLOW AND CONJUGATE HEAT TRANSFER PHYSICAL MODEL DEFAULT SETTINGS* USER INTERFACE (NITF) TURBULENCE MODEL TYPE TURBULENCE MODEL HEAT TRANSPORT TURBULENCE MODEL Non-Isothermal Flow. Turbulent Flow. k- RANS k- Kays-Crawford Conjugate Heat Transfer. in combination with A B O U T T 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 | 281 . Laminar Flow None N/A N/A Conjugate Heat Transfer. You can model laminar and turbulent flow. Turbulent Flow interface. you may know exactly how a fluid behaves and which equations. Usually you start with the simplest-to-set-up physics interface. If the type of flow to model is known. Stokes’ law (creeping flow) can be activated from the Non-Isothermal Flow. 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. then select it directly from the Model Wizard. The interface solves the Navier-Stokes equations together with an energy balance assuming heat flux through convection and conduction. Processes where natural convection are an important component are classic areas for such modeling. The density term is assumed to be affected by temperature and flow is always assumed to be compressible. The various forms of the Non-Isothermal Flow interfaces are. Low Reynold’s k- interface. 282 | C H A P T E 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 . In other cases. or physics interfaces best describe it. Laminar Flow User Interface ( ) is used primarily to model slow-moving flow in environments where energy transport is also an important part of the system and application. This can be the case when you know that the flow is essentially turbulent in nature. and other interfaces may be better to fine-tune the solution process for the more complex problem. you can start instead with a simplified model and add complexity as you build the model. Laminar Flow interface. On the right is the settings window for the Conjugate Heat Transfer. The next sections are a brief overview of each of the interfaces.heat transfer. Turbulent Flow. Laminar Flow interface if wanted. Choose to model laminar and turbulent flow in combination with heat transfer.I S O T H E R M A L F L O W. when you are not certain of the flow type. 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. models. L A M I N A R F L O W The Non-Isothermal Flow. by default. or because it is difficult to reach a solution easily. which in most cases in non-isothermal flow is the Non-Isothermal Flow. found under the Fluid Flow branch. N O N . The Non-Isothermal Flow Options Different types of flow require different equations to describe them. However. and must coupled or connected to the fluid flow in some way. I S O T H E R M A L F L O W. The density term is assumed to be affected by temperature and flow is always assumed to be compressible. and eddies. These are used to set up and model heat transfer throughout a fluid in collaboration with a solid where heat is transferred by conduction. The interface solves the Navier-Stokes equations together with an energy balance assuming heat flux through convection and conduction. by default. Once again. See Table 7-1 for details. The reason is that the local equilibrium assumption on which the wall functions rely is seldom fulfilled when there are temperature gradients present. 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. A B O U T T 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 | 283 . Laminar Flow User Interface ( ) is used primarily to model slow-moving flow in environments where temperature and energy transport are also an important part of the system and application. In addition to the properties for the different turbulence models mentioned in Theory for the Non-Isothermal Flow and Conjugate Heat Transfer User Interfaces. the interface includes added functionality for calculating the added dispersion of heat transfer due to turbulence. found under the Heat Transfer branch. Conjugate Heat Transfer Options The various forms of the Conjugate Heat Transfer interfaces are. TU R B U L E N T F L O W The Turbulent Flow. and must coupled or connected to the fluid-flow in some way. or by including your own turbulent Prandtl number.N O N . 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. C O N J U G A T E H E A T TR A N S F E R. Stokes’ law (creeping flow) can be activated from the Conjugate Heat Transfer. Laminar Flow interface if required. This is particularly relevant for applications in non-isothermal flow where the heat flux at solid-liquid interfaces is important to the final solution. L A M I N A R F L OW The Conjugate Heat Transfer. Process or component cooling are classic examples. k- and Turbulent Flow Low Re k-User Interfaces( ) model flows that are relatively fast-moving and geometries that change significantly to induce disorder. For this reason. This is represented by one of the Kays-Crawford or Extended Kays-Crawford Turbulence heat transport models. Processes where natural convection are an important component are classic areas for such modeling. vortices. and each use the Reynolds-Averaged Navier-Stokes (RANS) equations. vortices. and must be coupled or connected to the fluid-flow in some way. 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. k- and Turbulent Flow Low Re k-User Interfaces ( ) are used to model flows that are relatively fast-moving and/or geometries that change significantly to induce disorder. Each interface includes added functionality for calculating the added dispersion of heat transfer due to turbulence. The Heat Transfer Branch 284 | C H A P T E 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 . Turbulent Flow interfaces. and eddies. The interfaces are set up assuming that temperature and energy transport are also an important part of the system and application.There are different versions of the Conjugate Heat Transfer. The Turbulent Flow. Process or component cooling are classic examples. along with the k- model. solving for the mean velocity field and pressure. these default nodes are also added to the Model Builder— Non-Isothermal Flow. Thermal Insulation. Laminar Flow User Interface The Non-Isothermal Flow version of the Laminar Flow (nitf) user interface ( ). L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R . and Pair Nodes Settings for the NITF User Interfaces • Theory for the Non-Isothermal Flow and Conjugate Heat Transfer User Interfaces The Non-Isothermal Flow. Laminar Flow User Interface • The Turbulent Flow. found ) of the Model Wizard.The Non-Isothermal Flow and Conjugate Heat Transfer. and Initial Values. for example. When this interface is added. INTERFACE IDENTIFIER The interface identifier is used primarily as a scope prefix for variables defined by the physics user interface. volume forces. using a compressible formulation. boundary conditions. k- and Turbulent Flow Low Re k-User Interfaces • About the Conjugate Heat Transfer User Interfaces • Selecting the Right User Interface • Domain. Refer to such interface variables in expressions using the pattern T H E N O N . in combination with a Heat Transfer interface. Boundary. or heat sources.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 . Laminar Flo w a nd T ur bu l en t Fl ow U ser Interfaces In this section: • The Non-Isothermal Flow. Edge. Fluid. Laminar Flow User Interface • The Conjugate Heat Transfer. Right-click any node to add other features that implement. Wall. Point. is a under the Fluid Flow>Non-Isothermal Flow branch ( predefined multiphysics coupling consisting of a single-phase flow interface. This adds a Radiation Settings section. This adds a Participating media Settings section. no turbulence model is needed when studying laminar flows. DOMAIN SELECTION The default setting is to include All domains in the model to define the dependent variables and the equations. The default Turbulence model type is None. the additional turbulence model settings are made available. PHYSICAL MODEL Define interface properties to control the overall type of model. select the Surface-to-surface radiation check box. To choose specific domains. The default identifier (for the first interface in the model) is nitf. numbers and underscores (_) are permitted in the Identifier field. select Manual from the Selection list. Only letters. Neglect Inertial Term (Stokes Flow)—All Interfaces Select the Neglect inertial term (Stokes flow) check box to model flow at very low Reynolds numbers where the inertial term in the Navier-Stokes equations can be neglected. However.<variable_name>. where the flow length scales are very small. if the default Turbulence model type selected is RANS. Instead use the linear Stokes equations. The node is still called Non-Isothermal Flow or Conjugate Heat Transfer with a number added at the end of the name to indicate the change. To enable surface-to-surface radiation.<identifier>. In order to distinguish between variables belonging to different physics user interfaces. the identifier string must be unique. This flow type is referred to as creeping flow or Stokes flow and can occur in microfluidics (and MEMS devices). 286 | C H A P T E 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 . select the Heat Transfer in biological tissue check box. The first character must be a letter. To enable the Biological Tissue feature. To enable radiation in participating media. select the Radiation in participating media check box. Turbulence Model Type By definition. and the second term is the order for the pressure. For turbulence modeling and heat radiation. and the Temperature T (SI unit: K). The names can be changed but the names of fields and dependent variables must be unique within a model. See Radiation Settings as described for The Heat Transfer Interface. click the Show button ( ) and select Discretization. See Participating Media Settings as described for The Heat Transfer Interface. The first term describes the element order for the velocity components. The element order for the temperature is set to follow the velocity order. DEPENDENT VA RIA BLES The dependent variables (field variables) are for the Velocity field u (SI unit: m/s). PARTICIPATING MEDIA SETTINGS This section is available when the Radiation in participating media check box is selected. and 293. Select a Surface radiosity—Linear (the default). P2+P1. so the temperature order is 1 for P1+P1. or Quintic (2D axisymmetric and 2D models only). Select a Discretization of fluids—P1+P1 (the default). The default values are 0 m/s for the velocity.15 K for the temperature. L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R .I 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 . T H E N O N . and 3 for P3+P2. Quartic. or P3+P2. turbulent kinetic energy. Specify the Value type when using splitting of complex variables—Real (the default) or Complex. reciprocal wall distance. DISCRETIZATION To display this section. 2 for P2+P1. and surface radiosity. 0 Pa for the pressure. there are additional dependent variables for the turbulent dissipation rate.RADIATION SETTINGS This section is available when the Surface-to-surface radiation check box is selected. Quadratic. the Pressure p (SI unit: Pa). Cubic. Laminar Flow User Interface Viscous Heating in a Fluid Damper: Model Library path Heat_Transfer_Module/Buildings_and_Constructions/fluid_damper 288 | C H A P T E 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 . ) and select Stabilization. see The Heat Transfer Interface • About Pseudo Time Stepping in the COMSOL Multiphysics Reference Manual • The Conjugate Heat Transfer. Edge. When selected. ADVANCED SETTINGS To display this section. If Manual is selected. click the Show button ( unique to this interface are listed below. • About Handling Frames in Heat Transfer • Show More Physics Options • Domain. Any settings • The consistent stabilization methods are applicable to the Heat and flow equations. It adds pseudo time derivatives to the equation momentum and heat equations when the Stationary equation form is used. The Use pseudo time stepping for stationary equation form check box is active per default for the Non-Isothermal Flow interface.CONSISTENT AND INCONSISTENT STABILIZATION To display this section. By default the Enable conversions between material and spatial frames check box is selected. Automatic sets the local CFL number (from the Courant–Friedrichs–Lewy condition) to the built-in variable CFLCMP which in turns trigger a PID regulator for the CFL number. click the Show button ( ) and select Advanced Physics Options. also choose a CFL number expression—Automatic (the default) or Manual. Point. and Pair Nodes Settings for the NITF User Interfaces • For settings window details for the Heat Transfer in Solids feature. • The Isotropic diffusion inconsistent stabilization method can be activated for both the Heat equation and the Navier-Stokes equations. Normally these settings do not need to be changed. Boundary. enter a Local CFL number CFLloc. Viscous Heating in a Fluid Damper: Model Library path Heat_Transfer_Module/Buildings_and_Constructions/fluid_damper The Turbulent Flow. or heat sources. Most of the setting options are the same as for The Non-Isothermal Flow. Fluid (with empty initial domain selection). Laminar Flow User Interface The Conjugate Heat Transfer version of the Laminar Flow (nitf) user interface ( ). Laminar Flow User Interface • Domain. volume forces. PHYSICAL MODEL The default Turbulence model for the Turbulent flow. boundary conditions. Low Re k- interface it is Low Reynolds number k-. for example. When this interface is added. For the Turbulent flow. Right-click any node to add other features that implement. Laminar Flow User Interface. in combination with a Heat Transfer interface. using a compressible formulation. k- interface is k-. Thermal Insulation. • Show More Physics Options • The Non-Isothermal Flow. L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R . in combination with a Heat Transfer interface. found under the Heat Transfer>Conjugate Heat Transfer branch ( ). except where noted below. or heat sources. Point. using a compressible formulation. k- and Turbulent Flow Low Re k-User Interfaces These predefined multiphysics couplings consist of a turbulent flow interface.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 . Wall. is a predefined multiphysics coupling consisting of a single-phase flow interface. boundary conditions. these default nodes are also added to the Model Builder—Conjugate Heat Transfer. Right-click any node to add other features that implement. Heat Transfer in Solids (with domain selection set to All domains). volume forces. T H E N O N .The Conjugate Heat Transfer. Boundary. for example. and Initial Values. and Pair Nodes Settings for the NITF User Interfaces • The Heat Transfer Interface for settings window details for the Heat Transfer in Solids feature. Edge. The default values are 0 m/s for the velocity. and 293. 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. For a description of the turbulence model and the included model parameters see Theory for the Turbulent Flow User Interfaces. 290 | C H A P T E 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 .For all the turbulent interfaces. For turbulence modeling and heat radiation. the Pressure p (SI unit: Pa). The Extended Kays-Crawford model requires a Reynolds number at infinity. and the Temperature T (SI unit: K). 0 Pa for the pressure. 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. The names can be changed but the names of fields and dependent variables must be unique within a model. That input is given in the Model Inputs section of the Fluid feature node. Other Heat transport turbulence model options are Extended Kays-Crawford or User-defined turbulent Prandtl number. better performance can be obtained by tuning the model parameters. Enter the user-defined value or expression for the turbulence Prandtl number in the Model Inputs section of the Fluid feature node. DEPENDENT VARIABLES The dependent variables (field variables) are for the Velocity field u (SI unit: m/s). but for some special cases. It is always possible to specify a user-defined model for the turbulence Prandtl number. the default Turbulence model type is RANS and the default Heat transport turbulence model is Kays-Crawford.15 K for the temperature. enter a Tuning parameter Ck for one or both of the Heat and flow equations and Turbulence Equations. units or parameters defined under Global Definitions. the new values will take effect the next time you generate a new default solver. and the Turbulence equations. The Use pseudo time stepping for stationary equation form check box adds pseudo time derivatives not only to the equation momentum and heat equations when the Stationary equation form is used. Navier-Stokes equations. select the Isotropic diffusion check box and enter a Tuning parameter id for one or all of Heat equation. or Turbulence equations.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 . Navier-Stokes equations. ADVANCED SETTINGS To display this section.25. Any settings • The consistent stabilization methods are applicable to the Heat and flow equations and the Turbulence Equations. click the Show button ( ) and select Advanced Physics Options. If you change the parameters. and 1 for the Turbulence equations.CONSISTENT AND INCONSISTENT STABILIZATION To display this section. Normally these settings do not need to be changed. The defaults are 0.5. • By default there is no isotropic diffusion selected. The section is only visible when Turbulence model type in the Physical Model section is set to RANS. The scaling parameters cannot contain variables. The scaling parameters must only contain numerical values. ) and select Stabilization. T H E N O N . The parameters are used when a new default solver for a transient study step is generated. • When the Crosswind diffusion check box is selected. If required. • The Isotropic diffusion inconsistent stabilization method can be activated for the Heat equation. The default for the Heat and flow equations is 0. The Turbulence variables scale parameters subsection contains the parameters Uscale and Lfact that are used to calculate absolute tolerances for the turbulence variables. L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R . click the Show button ( unique to this interface are listed below. but to the turbulence equations as well. By default the Enable conversions between material and spatial frames check box is selected. Below are links to the domain. Point. Flow • Initial Values • Symmetry. almost every physics node is shared with. boundary. edge. To locate and search all the documentation. Also because these are all multiphysics interfaces. and Pair Nodes Settings for the NITF User Interfaces • Turbulent Non-Isothermal Flow Theory Turbulent Flow Through a Shell-and-Tube Heat Exchanger: Model Library path Heat_Transfer_Module/Heat_Exchangers/ turbulent_heat_exchanger Domain. point. and pair physics as indicated. Boundary. in COMSOL Multiphysics. and Pair Nodes Settings for the NITF User Interfaces All the versions of the Non-Isothermal Flow and Conjugate Heat Transfer interfaces have shared domain. and pair physics features based on the selections made for the model. Heat • Interior Wall • Viscous Heating • Open Boundary • Wall • Pressure Work 292 | C H A P T E 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 . Edge. Point. Edge. other interfaces. point. select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree. • About Handling Frames in Heat Transfer • Show More Physics Options • Domain. and described for. boundary. Boundary. edge. These features are described in this section: • Fluid • Symmetry. or viscous heating. Laminar Flow and Turbulent Flow User Interfaces • Theory for the Non-Isothermal Flow and Conjugate Heat Transfer User Interfaces Fluid The Fluid node adds both the momentum equations and the temperature equation but without volume forces.These features are described for the Single Phase Flow interface (listed in alphabetical order): • Boundary Stress • Outlet • Interior Fan • Periodic Flow Condition • Flow Continuity • Pressure Point Constraint • Inlet • Volume Force • Interior Wall These physics are described for the Heat Transfer interface (listed in alphabetical order): • Boundary Heat Source • Outflow • Convective Heat Flux • Periodic Heat Condition • Continuity • Point Heat Source • Heat Flux • Surface-to-Ambient Radiation • Heat Source • Temperature • Heat Transfer in Solids • Thermal Contact • Highly Conductive Layer • Thermal Insulation • Inflow Heat Flux • Thin Thermally Resistive Layer • Line Heat Source • The Heat Transfer Interface • The Non-Isothermal Flow and Conjugate Heat Transfer. pressure work. L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R . heat sources.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 . You can add volume forces and heat sources as separate features. and Viscous Heating and Pressure T H E N O N . In other cases. Using one or the other option usually depends on the system and the equations being solved for. reduces the chances for stability and convergence during the solving process for this variable. There are usually two ways to calculate the pressure when describing fluid flow and mass and heat transfer. that is. the Fluid node also adds the equations for k and . MODEL INPUTS This section controls both the variable as well as any property value (reference pressures) used when solving for pressure. in a straight incompressible flow problem. and cannot be edited. the setting inherits the selection from the parent node. the absolute pressure may be required to be solved for. When the turbulence model type is set to RANS. DOMAIN SELECTION For a default node. the selection is automatically selected and is the same as for the interface. which. Solve for the absolute pressure or a pressure (often denoted gauge pressure) that relates back to the absolute pressure through a reference pressure. 294 | C H A P T E 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 . if included. such as where pressure is a part of an expression for gas volume or diffusion coefficients. you can select Manual from the Selection list to choose specific domains or select All domains as required. For example. When nodes are added from the context menu. the pressure drop over the modeled domain is probably many orders of magnitude less than atmospheric pressure.Workcan be added as subnodes. The Ratio of specific heats is the ratio of heat capacity at constant pressure. 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.325 Pa). I the identity matrix and T the thermal turbulent conductivity defined by Cp T T = -------------Pr T THERMODYNAMICS Select a Fluid type—Gas/Liquid (the default). to heat capacity at constant volume. When the turbulence model type is set to RANS. Ideal Gas.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 . choose Isotropic. While this check box maintains control over the pressure variable and instances where absolute pressure is required within this respective physics interface. HEAT CONDUCTION The default Thermal conductivity k (SI unit: W/(m·K)) uses values From material. L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R . Cv. Diagonal. Cp. the conductive heat flux includes the turbulent contribution: q = k+TI)T where k is the thermal conductivity tensor. such as for gas flow governed by the gas law. or Anisotropic based on the characteristics of the thermal conductivity and enter other values or expressions in the field or matrix. The Reference pressure check box is selected by default and the default value of pref is 1[atm] ((101. or Moist Air. The Heat capacity at constant pressure Cp describes the amount of heat energy required to produce a unit temperature change in a unit mass. it may not with physics interfaces that it is being coupled to. check the coupling between any interfaces using the same variable.The default Absolute pressure p (SI unit: Pa) is Pressure (nitf/fluid). If User defined is selected. In such models. 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. T H E N O N . Enter this quantity as power per length and temperature. Symmetric. and Ratio of specific heats (dimensionless) use values From material.314 J/(mol·K). the ideal gas law is used to describe the fluid. dependent. For an ideal gas the density is defined in the following equation where pA is the absolute pressure. and T the temperature: pA Mn pA = ---------------. is used in the streamline stabilization and in the results and analysis variables for heat fluxes and total energy fluxes. the universal gas constant R 8. the default uses data From material.Gas/Liquid If Gas/Liquid is selected as the Fluid type.4 is the standard value. Heat capacity at constant pressure Cp (SI unit: J/(kg·K)). For common diatomic gases such as air. Select User defined to enter other values or expressions. In both cases. properties of a non-ideal gas or liquid can be used. It is also used in the ideal gas law. By default the Density (SI unit: kg/m3). Most liquids have 1. Select an option from the Specify Cp or list—Heat capacity at constant pressure Cp (SI unit: J/(kg·K)) or Ratio of specific heats (dimensionless). For an ideal gas. specifying is enough to evaluate Cp.= ----------RT Rs T Select a Gas constant type—Specific gas constant Rs (SI unit: J/(kg·K)) or Mean molar mass Mn (SI unit: kg/mol). 296 | C H A P T E 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 . is also used. but not both since they are. Select User defined to enter other values or expressions. you choose to specify either Cp or . 1. Select User defined to enter another value or expression. the default uses the value From material. When using the ideal gas law to describe a fluid. If Mean molar mass is selected. which is a built-in physical constant. Ideal Gas If Ideal gas is selected as the Fluid type.1 while water has 1. in that case.0. For both options. . • Moisture content xvap (SI unit: dimensionless.Reference relative humidity ref (dimensionless). a Concentration model input is automatically added in the Models Inputs section. water and air have a low viscosity. .Reference pressure pref (SI unit: Pa). kg of water vapor/ kg of dry air). • Select Relative humidity to define the quantity of vapor from the following reference values. The default is 1 atm. Select an Input quantity: • Vapor mass fraction (the default) (SI unit: dimensionless. and polymer suspensions. which are used to estimate the mass fraction of vapor that is used to define the thermodynamic properties of the moist air: .15 K. have a higher viscosity.Moist Air If Moist air is selected as the Fluid type. The default is 0. such as oil. Non-Newtonian fluids have a shear-rate dependent viscosity.Reference temperature Tref (SI unit: K). Examples of non-Newtonian fluids include yoghurt. L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R . T H E N O N . Intuitively. (kg of water vapor) / (total mass in kg = kg of dry air + kg of water vapor)) • Concentration (SI unit: mol/m3) When Concentration is selected. paper pulp.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 . DYNAMIC VISCOSITY This section is not available if Moist air is selected as the Fluid type. The dynamic viscosity describes the relationship between the shear rate and the shear stresses in a fluid. and substances often described as thick. the thermodynamics properties are defined as a function of the quantity of vapor in the moist air. The default is 293. or • Microfluidics Module plus either the CFD Module or Heat Transfer Module Non-Newtonian Flow: The Power Law and the Carreau Model Non-Newtonian Power Law This option is available with the CFD Module or the Heat Transfer Module plus the Microfluidics Module.The default Dynamic viscosity (SI unit: Pa·s) uses values From material. enter the Power law model parameter m and Model parameter n (both dimensionless). spf. which makes it possible to define arbitrary expressions of the dynamic viscosity. · min is per default 1·10-2 1s. If Non-Newtonian power law is selected. This selection uses the power law as the viscosity model for a non-Newtonian fluid where the following equation defines dynamic viscosity: · · = m max min n – 1 where · is the shear rate and · min is a lower limit for the shear rate evaluation. additional options are available— Non-Newtonian power law and Non-Newtonian Carreau model. • CFD Module • Heat Transfer Module plus the Microfluidics Module. 298 | C H A P T E 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 .sr. Or select User defined to use a built-in variable for the shear rate magnitude. Non-Newtonian Carreau Model This option is available with the CFD Module or the Heat Transfer Module plus the Microfluidics Module. If you also have the following modules. enter a value or expression for the Mixing length limit l mix (SI unit: m).5l l mix bb (7-1) where lbb is the shortest side of the geometry bounding box. Select a Mixing length limit—Automatic (the default) or Manual. which needs an upper limit on the mixing length. Equation 7-1 tends to give a lim manually. the mixing length limit is automatically evaluated as: lim = 0. which needs the distance to the closest wall. If the geometry is for example a complicated system of very slender entities. L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R .If Non-Newtonian Carreau model is selected. result that is too large. Then define l mix lim • If Manual is selected. • If Automatic is selected. the reference length scale is automatically evaluated as: T H E N O N . 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 available for the low-Reynolds number k- model. • If Automatic is selected. Select a Reference length scale—Automatic (the default) or Manual.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 . enter these Carreau model parameters: • The Zero shear rate viscosity 0 (SI unit: Pa·s) • The Infinite shear rate viscosity inf (SI unit: Pa·s) • The Model parameters (SI unit: s) and n (dimensionless) This selection uses the Carreau model as the viscosity model for a non-Newtonian fluid where the following equation defines the dynamic viscosity: n – 1 · ---------------- = + 0 – inf 1 + 2 2 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 available for the k- model. there is a theoretical gap between the solid wall and the computational domain of the fluid. 300 | C H A P T E 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 . Then define l ref manually. Ts. If a temperature is prescribed to an internal wall. The approach is slightly different depending on what type of wall the condition applies to. • If Manual is selected. and one for the fluid. On external walls. that is. On internal walls. but it must nevertheless be considered in the equations for the temperature field. If the geometry is for example a complicated system of very slender entities. About the Thermal Wall Function Whenever wall functions are used. Figure 0-1 shows the difference between internal and external walls. Equation 7-1 tends to give a result that is too large. In these cases.l ref = 0. the Wall node is identical to the single-phase flow settings (the Boundary condition defaults to No slip). Tw is a variable that is solved for and the equation for Tw is q wf = q tot where qtot is the total heat flux prescribed to the boundary. the constraint is applied to the temperature for the solid. This gap is often ignored in so much that it is ignored when the computational geometry is drawn. Wall For laminar flow. to Ts.1l bb (7-2) where lbb is the shortest side of the geometry bounding box. Tf. The settings below are for the k- turbulence model. enter a value or expression for the Reference length scale l ref (SI unit: m). continuity of the temperature is enforced on internal walls separating a fluid and solid domain. there are two temperatures. the low Reynolds number k- turbulence model. one for the solid. the temperature T is the temperature of the fluid while the wall temperature is represented by the dependent variable Tw. Any wall feature that utilizes wall functions automatically detects internal and external walls. but for some special cases. enter the coordinates for the Velocity of sliding wall uw (SI unit: m/s). If a temperature is prescribed to an external wall. instead of the wall temperature. the constraint is applied to the wall temperature Tw. Turbulence model parameters are optimized to fit as many flow types as possible. the Boundary condition defaults to Wall functions. For a description of the turbulence model and the included model parameters see Theory for the Turbulent Flow User Interfaces. L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R . enter the coordinates for the Velocity of moving wall uw (SI unit: m/s). T H E N O N . Tw. Ts. Sliding wall (wall functions). or Moving wall (wall functions)—the wall functions for the temperature field is also prescribed. BOUNDARY CONDITION When using the k- turbulence model. • If Moving wall (wall functions) is selected.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 . The other options available are Slip. Any other heat boundary condition applied to an external wall is wrong in the sense that it acts on the fluid temperature. If any one of these options are selected—Wall functions. which is called a thermal wall functions. and Moving wall (wall functions). • If Sliding wall (wall functions) is selected. Sliding wall (wall functions). better performance can be obtained by tuning the model parameters.External wall Outflow Inflow Internal wall Fluid Solid Figure 7-2: A simple example that includes both an external wall and an internal wall. the low Reynolds number k- turbulence model. but it must nevertheless be considered in the equations for the temperature field. Tf. the constraint is applied to the temperature for the wall. and one for the fluid on up and down sides of the wall. The settings below are for the k- turbulence model. which is called a thermal wall functions. If any one of these options are selected—Wall functions or Moving wall (wall functions)— the wall functions for the temperature field is also prescribed. If a temperature is prescribed to an internal wall. Tw. there is a theoretical gap between the internal wall and the computational domain of the fluid. This gap is often ignored in so much that it is ignored when the computational geometry is drawn. The other options available are Slip. d. u and Tf. the Interior Wall node is identical to the single-phase flow settings (the Boundary condition defaults to No slip).About the Thermal Wall Function Interior Wall For laminar flow. On internal walls. the Boundary condition defaults to Wall functions. to Tw. In these cases. BOUNDARY CONDITION When using the k- turbulence model. there are three temperatures. that is. 302 | C H A P T E 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 . About the Thermal Wall Function Whenever wall functions are used. and Moving wall (wall functions). continuity of the temperature is enforced across internal walls separating two fluid domains. one for the wall. the Pressure p (SI unit: Pa). When nodes are added from the context menu. but for some special cases. default initial values for the turbulence variables. the setting inherits the selection from the parent node. L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R . Open Boundary Use the Open Boundary node to set up heat and momentum transport across boundaries where both convective inflow and outflow can occur. defined for each turbulence model in Theory for the Turbulent Flow User Interfaces. you can select Manual from the Selection list to choose specific domains or select All domains as required. For turbulence models. and the Temperature T (SI unit: K).15 K for the temperature. that is. About the Thermal Wall Function Initial Values The Initial Values node adds initial values for the velocity field. and cannot be edited.If Moving wall (wall functions) is selected. and 293.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 . Turbulence model parameters are optimized to fit as many flow types as possible. 0 Pa for the pressure. better performance can be obtained by tuning the model parameters. enter the coordinates for the Velocity of moving wall uw (SI unit: m/s). For turbulent flow there are also initial values for the turbulence model variables. INIT IA L VA LUES Enter values or expressions for the initial value of the Velocity field u (SI unit: m/s). For a description of the turbulence model and the included model parameters see Theory for the Turbulent Flow User Interfaces. DOMAIN SELECTION For a default node. the selection is automatically selected and is the same as for the interface. are applied. The default values are 0 m/s for the velocity. The node specifies a T H E N O N . and temperature that can serve as an initial condition for a transient simulation or as an initial guess for a nonlinear solver. the pressure. 304 | C H A P T E 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 (7-4) . the Pressure Work node adds the following term to the right-hand side of the Heat Transfer in Solids equation: – T ----.------. choose a fluid flow condition for the open boundary the boundaries to apply the open boundary condition. BOUNDARY CONDITION From the Boundary Condition list. When added under Fluid node. Pressure Work Right-click the Heat Transfer in Solidsor Fluid node to add this subnode. the Pressure Work feature adds the following contribution to the right-hand side of the Heat Transfer in Fluids equation: T p – ---. -----. When added under a Heat Transfer in Solids node. Select • Select Normal stress (the default) and enter the normal stress f0 (SI unit: N/m2). choose the boundaries to define.+ u p T p t The software computes the pressure work using the absolute pressure. together with an exterior temperature to be applied on the parts of the boundary where fluid flows into the domain. BOUNDARY SELECTION From the Selection list. E X T E R I O R TE M P E R A T U RE Enter a value or expression for the external temperature (SI unit: K). This implicitly specifies that p f0 • Select No viscous stress to prescribe a vanishing normal viscous stress on the boundary.fluid flow condition.S el t (7-3) where Sel is the elastic contribution to entropy. The direction of the flow across the boundary is typically calculated by a Fluid Flow branch interface and is entered as Model Inputs. By default. If you apply a Symmetry. Viscous Heating The Viscous Heating subnode adds the following term to the right-hand side of the heat transfer in fluids equation: :S (7-5) Here is the viscous stress tensor and S is the strain rate tensor. Flow node to that boundary since physically.DOMAIN SELECTION From the Selection list. By default. you should also consider adding a Symmetry. or Low Mach number formulation. select a Pressure work formulation—Full formulation (the default). which is small for most flows with a low Mach number. T H E N O N . P RE S S U RE WO R K For the Heat Transfer in Solids model. choose the domains to define. For the Fluid model. Symmetry. the selection is the same as for the Fluid node that it is attached to. Heat feature to a boundary adjacent to a fluid domain. Equation 7-5 represents the heating caused by viscous friction within the fluid. The latter excludes the term u · p from Equation 7-4. choose the domains to define. L A M I N A R F L O W A N D TU R B U L E N T F L O W U S E R . the selection is the same as for the parent node to which it is attached (Heat Transfer in Solidsor Fluid node). The default is 0 Jm3·K). Heat node provides a boundary condition for symmetry boundaries. This boundary condition is similar to a Thermal Insulation condition. DOMAIN SELECTION From the Selection list. enter a value or expression for the Elastic contribution to entropy Ent (SI unit: Jm3·K)). Heat The Symmetry.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 . and it means that there is no heat flux across the boundary. u I n = 0 3 u n = 0. Heat node because it is not possible to have symmetry in the velocity and pressure without having symmetry in the temperature as well. the boundary should also be supplemented with a Symmetry. and the above equations are equivalent to the following equation for both the compressible and incompressible formulation: u n = 0. – pI + u + u T n = 0 for the compressible and the incompressible formulation respectively. The symmetry condition only applies to the temperature field. The Dirichlet condition takes precedence over the Neumann condition. the node does not require any user input. Symmetry.you cannot have symmetry in the temperature without having symmetry in the velocity and pressure as well. K – K n n = 0 K = u + u T n If you apply a Symmetry. – pI + u + u T – 2 --. BOUNDARY SELECTION In most cases. Flow feature to a boundary. If required. Flow The Symmetry. The boundary condition is a combination of a Dirichlet condition and a Neumann condition: u n = 0. Flow node adds a boundary condition that describes symmetry boundaries in a fluid flow simulation. 306 | C H A P T E 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 . It has no effect on the radiosity (surface-to-surface radiation) and on the radiative intensity (radiation in participating media). The boundary condition for symmetry boundaries prescribes no penetration and vanishing shear stresses. define the symmetry boundaries. For 2D axial symmetry. For the symmetry axis at r0. CONSTRAINT SETTINGS To display this section. choose the boundaries to define. the software automatically provides a condition that prescribes ur0 and vanishing stresses in the z direction and adds an Axial Symmetry node that implements this condition on the axial symmetry boundaries only.BOUNDARY SELECTION From the Selection list.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 A N D TU R B U L E N T F L O W U S E R . click the Show button ( ) and select Advanced Physics Options. T H E N O N . a boundary condition does not need to be defined. Select the Use weak constraints check box to use weak constraints and create dependent variables for the corresponding Lagrange multipliers. + u T = – q + :S – ---. for instance where reactants associate or dissociate. The Non-Isothermal Flow and Conjugate Heat Transfer interfaces contain the fully compressible formulation of the continuity equation and momentum equations: -----.+ u u = – p + u + u T – --.+ u p + Q t T p t where in addition to the quantities above • Cp is the specific heat capacity at constant pressure (SI unit: J/(kgK)) 308 | C H A P T E 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 (7-6) . These variations can have a number of different sources but the most common one is the presence of an inhomogeneous temperature field.------.+ u = 0 t 2 u ------. This module includes the Non-Isothermal Flow predefined multiphysics coupling to simulate systems where density varies with temperature. which for a fluid is T T p C p ------. -----. u I + F 3 t where • is the density (SI unit: kg/m3) • u is the velocity vector (SI unit: m/s) • p is pressure (SI unit: Pa) • is the dynamic viscosity (SI unit: Pa·s) • F is the body force vector (SI unit: N/m3) It also solves the heat equation. Other situations where the density might vary includes chemical reactions.Theory for the Non-Isothermal Flow a nd C o njug a te H eat Tran sfer User Interfaces In industrial applications it is common that the density of a process fluid varies. I S O T H E R M A L F L O W A N D C O N J U G A T E H E A T TR A N S F E R U S E R I N T E R F A C E S | 309 . These can. 3.= – q – T ------. see Ref. • The Heat Equation • Turbulent Non-Isothermal Flow Theory • References for the Non-Isothermal Flow and Conjugate Heat Transfer User Interfaces T H E O R Y F O R T H E N O N . the pressure work term E T ------t is not included by default but must be added as a subfeature. however.• 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) 1 S = --. u + u T 2 • Q contains heat sources other than viscous heating (SI unit: W/m3) The pressure work term T .+ Q t t where E is the elastic contribution to entropy (SI unit: J/(m3·K)). be added as subnodes to the Fluid node. The interface also supports heat transfer in solids: T E C p ------. As in the case of fluids. For a detailed discussion of the fundamentals of heat transfer in fluids. ----p---.-----+ u p T p t and the viscous heating term :S are not included by default because they are commonly negligible. ˜ ---------.u j h + ----------.+ ------. e. The stress tensor – u i''u'' j is model with the Boussinesq approximation: 310 | C H A P T E 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 . Equations for compressible turbulence are derived using the Favre average.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.= e+ t 2 2 x j 2 2 u j''u i''u i'' - -----– q – u j''h'' + ij u i'' – --------------------------- + ------. The vector T q j = – ------x j (7-8) is the laminar conductive heat flux and 2 u k ij = 2S ij – --. The full field is then decomposed as ˜ T = T + T'' With these notations the equation for total internal energy. Notice that the thermal conductivity is denoted . The modeling assumptions are in large part analogous to incompressible turbulence modeling. viscous stress tensor. ij 3 x k is the laminar. becomes u˜ i u˜ i u i''u i'' ˜ ˜ u˜ i u˜ i ˜ u i''u i'' ----. --------. u˜ i ij – u i''u j'' x j x j j 2 (7-7) where h is the enthalpy.+ ------------------. The Favre ˜ average of a variable T is denoted T and is defined by ˜ T T = ------ where the bar denotes the usual Reynolds average.+ u j ------------------. --------. u˜ i ij + T ij x j j k x j x j (7-13) The Favre average can also be applied to the momentum equation. u j''u i''u i'' 2 are modeled by a generalization of the molecular diffusion and turbulent transport term found in the incompressible k equation T k u j''u i''u i'' ij u i'' – --------------------------. ij – --.I S O T H E R M A L F L O W A N D C O N J U G A T E H E A T TR A N S F E R U S E R I N T E R F A C E S | 311 .˜ ˜ 1 u k 2 – u i''u'' j = T ij = 2 T S ij – --. can be written ----.k ij 3 x k 3 (7-9) where k is the turbulent kinetic energy.+ ------.u i''u i'' 2 (7-10) The correlation between u j'' and h'' in Equation 7-7 is the turbulent transport of heat. Equation 7-10. ˜ u i u i e + ----------. ij u i'' and turbulent transport term. u˜ j u˜ i = – ------. which. which in turned is defined by 1 k = --. Equation 7-9.= – -------------. using Equation 7-9.------x j Pr T x j (7-11) The molecular diffusion. It is model analogous to the laminar conductive heat flux ˜ ˜ T C p T T u j''h'' = q T j = – T ------. Equation 7-11 and Equation 7-12 into Equation 7-7 gives ˜ ˜ u˜ i u˜ i ----.= + ------ ------ 2 k x j (7-12) Inserting Equation 7-8. ij + T ij x j t x j x j (7-14) T H E O R Y F O R T H E N O N . u˜ j h˜ + ----------.+ k + ------.+ k = x j 2 2 t T k - -----– q – q T j + + ------ ------- + ------.˜ p u i + ------. it will follow that ij ˜ ij and consequently ˜ u˜ j T ----.˜ Tot e + ------. 1).+ ------. u˜ j e˜ = – p -------. u˜ i ij + T ij x j k x j x j (7-15) where the relation e˜ = h˜ + p has been used.+ x j t x j T k - -----– q j – q T j + + ------ ------- + ------. – q j – q T j + ------.+ ------. This gives the following approximation of Equation 7-15 is u˜ j ----. u˜ i ij + T ij x j x j t x j x j (7-16) Larsson (Ref.Taking the inner product between u˜ i and Equation 7-14 results in an equation for the resolved kinetic energy. it is usually a good approximation to neglect the contributions of k for flows with Mach numbers up to the supersonic range. u˜ j e˜ = – p -------. + T ------- + ------. u˜ i ˜ ij x j x j t x j x j x j where 312 | C H A P T E 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 (7-17) . 2) suggests to make the split ij = ˜ ij + ij'' Since ˜ ij » ij'' for all applications of engineering interest. u˜ j e˜ + k = – p -------. According to Wilcox (Ref.˜ e + ------. which can be subtracted from Equation 7-13 with the following result: u˜ j ---- e˜ + k + ------. TU R B U L E N T C O N D U C T I V I T Y Kays-Crawford This is a relatively exact model for PrT. 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). ij 3 x k Equation 7-17 is completely analogous to the laminar energy equation and can be expanded using the same theory (see for example Ref. This gap is often ignored in so much that it is ignored when the computational geometry is drawn. T H E O R Y F O R T H E N O N . 4. + T ------- + ˜ ij S ij – ---.-------------. there is a theoretical gap between the solid wall and the computational domain of the fluid also for the temperature field.888 C p Re where Re.– 0.------. the Reynolds number at infinity must be provided either as a constant or as a function of the flow field.3 C p T Pr T = ----------------.3C p T 2Pr Pr T T –1 (7-18) Pr T where the Prandtl number at infinity is PrT0.˜ ˜ 2 u k Tot ˜ ij = + T 2S ij – --. 5) suggested an extension of Equation 7-18 to liquid metals by introducing 100 Pr T = 0. -----. This is entered in the Model Inputs section of the Fluid feature. Extended Kays-Crawford Weigand and others (Ref.+ u˜ j ------- ˜ t x j x j x j x j T t p which is the temperature equation solved in the turbulent Non-Isothermal Flow and Conjugate Heat Transfer interfaces.--------. 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.85 and is the conductivity.I S O T H E R M A L F L O W A N D C O N J U G A T E H E A T TR A N S F E R U S E R I N T E R F A C E S | 313 . still simple. 3): ˜ ˜ ˜ ˜ T ˜ T p T T p C p ------.+ u˜ j ------- = ------.85 + ------------------------------0.3 -------------- 1 – e – 0.+ -----------------. The model is given by Cp T 2 1 0. In Ref. is: C p C 1 / 4 k 1 / 2 T w – T f q wf = ---------------------------------------------------------T+ where is the fluid density.41. The distance w is available as a variable on boundaries. T is the dimensionless temperature and is given by (Ref.ln w + + for w2 w where in turn w C 1 / 2 k + = -----------------------------w + = 10 10 ------- w2 Pr T 10 + = ----------- w1 Pr 1 / 3 Cp Pr = --------- Pr T = 15Pr 2 / 3 – --------. C is a turbulence modeling constant. Cp is the fluid heat capacity. 6): + + for w w1 + Pr w 2 / 3 – 500 ---------- T + = 15Pr + + + + for w1 w w2 w2 Pr + + ----. and is the von Karman constant equal to 0. The computational result should be checked so that the distance between the computational fluid domain and the wall.The heat flux between the fluid with temperature Tf and a wall with temperature Tw. w. is almost everywhere small compared to any geometrical quantity of interest. and k is the turbulent kinetic energy. 314 | C H A P T E 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 . 1 + ln 1000 --------- 2 Pr T where in turn is the thermal conductivity. I S O T H E R M A L F L O W A N D C O N J U G A T E H E A T TR A N S F E R U S E R I N T E R F A C E S | 315 .M. Larsson. Turgeon. 4191–4196.R. 284–295. W. R. Inc.. 40.L. J. pp. and D. Wilcox. 3. Sweden. 1994. DCW Industries. 1998. 1996. “Turbulent Prandtl Number — Where Are We?”. pp. John Wiley & Sons. 4.” International Journal of Thermal Sciences. Wall Functions and Adaptivity. 43.. D. Lacasse. Pelletier. 2nd ed. Incompressible Flow. D. J.References for the Non-Isothermal Flo w a nd C o nju gat e H eat Tran sfer User Interfaces 1. 1998. ASME Journal of Heat Transfer. 1997. Doctoral Thesis for the Degree of Doctor of Philosophy. 6. Weigand. “On the Judicious Use of the k— Model. Ferguson. 2. 2004.. 925–938. 5.E. B. and M. pp. Kays. R E F E R E N C E S F O R T H E N O N . no. Turbulence Modeling for CFD.C. vol. 17. È. Numerical Simulation of Turbulent Flows for Turbine Blade Heat Transfer. Panton. Crawford. 2nd ed. 116. “An Extended Kays and Crawford Turbulent Prandtl Number Model. vol.” International Journal of Heat and Mass Transfer. Chalmers University of Technology. 316 | C H A P T E 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 . and CAD. see the glossary in the COMSOL Multiphysics Reference Manual. For references to more information about a term. geometry. For information about terms relating to finite element modeling. 317 . mathematics.8 Glossary This Glossary of Terms contains application-specific terms used in the Heat Transfer Module software and documentation. see the Index in this or other manuals. emissivity A dimensionless factor between 0 and 1 that specifies the ability of a surface to emit radiative energy. convection The term convection is used for the heat dissipation from a solid surface to a fluid. conduction Heat conduction takes place through different mechanisms in different media. conduction takes place through collisions of molecules in a gas.Glossary of Terms anisotropy The condition of exhibiting properties with different values when measured in different directions. that is. where the heat transfer coefficient and the temperature difference across a fictitious film describes the flux. Typical for heat conduction is that the heat flux is proportional to the temperature gradient. Theoretically. discrete ordinates method This order defines the discretization of the radiative intensity direction. it does not reflect radiation. The blackbody also emits the maximum possible radiation. blackbody A blackbody is a surface that absorbs all incoming radiation. bioheat equation An alternative form of the heat equation that incorporates the effects of blood perfusion. which emits the maximum possible radiative energy. highly conductive layer A highly conductive layer is a thin layer on a boundary. heat capacity See specific heat. The equation describes heat transfer in tissue. and by the electrons carrying heat in metals or by molecular motion in other solids. It has much higher thermal conductivity than the material in the adjacent domain. advection Heat advection takes place through the net displacement of a fluid. This allows for the assumption that the temperature is constant across the layer’s thickness. metabolism. which translates the heat content in a fluid through the fluid's own velocity. through oscillations of each molecule in a “cage” formed by its nearest neighbors in a fluid. and external heating. 318 | CHAPTER 8: GLOSSARY . The value 1 corresponds to an ideal surface. In this equation. 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.The general Heat Transfer physics interface supports heat transfer in highly conductive layers. to allow for the assumption that the temperature is constant across the shell’s thickness. the thermal conductivity is the proportional constant. It has units of energy per mass per degree. that is. The Heat Transfer Module includes surface-to-surface radiation. thermal conductivity The definition of thermal conductivity is given by Fourier’s law. emit. that is. radiation Heat transfer by radiation takes place through the transport of photons. both the emitted and the reflected radiation. thin conductive shell “Thin” means that the shell is thin enough. which relates the heat flux to the temperature gradient. that is. irradiation The total radiation that arrives at a surface. opaque material An opaque body does not transmit any radiative heat flux. See also highly conductive layer. specific heat capacity See specific heat. the surface of an opaque body has a transmissivity equal to 0. The most general versions of Navier-Stokes equations do however describe fully compressible flows. transparent material A transparent body transmits radiative heat flux. which accounts for effects of shading and reflections between radiating surfaces. This quantity is also called specific heat or specific heat capacity. the surface of a transparent body has a transmissivity greater than 0. which can be absorbed or reflected on solid surfaces. G L O S S A R Y O F TE R M S | 319 . radiosity The total radiation that leaves a surface. participating media A media that can absorb. or has high enough thermal conductivity. It also includes surface-to-ambient radiation where the ambient radiation can be fixed or given by an arbitrary function. and scatter thermal radiation. 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. 320 | CHAPTER 8: GLOSSARY . and 34 heat transfer coefficients. definition Clausius-Clapeyron formulation 42 207 blood. and 218 CFL number. natural and forced 54 INDEX| i . 224–225. 299 arterial blood temperature 134 cell Reynolds number 234 axisymmetric geometries 166. 295 acceleration of gravity 55 boundary stress (node) 237 accurate flux variables 32 Brinell hardness 50 advanced settings 9 bulk velocity 55 AKN model 270 buoyancy force 55 apparent heat capacity 46 C Carreau model 259. pseudo time stepping.I n d e x 1D and 2D models nitf interfaces 292 out-of-plane heat transfer 156 radiation in participating media 201 3D models spf 223 thin conductive shells 146 A absolute pressure 91. and 53 radiation groups 197 coefficient of volumetric thermal expansion 56 conductive heat flux variable 26 conjugate heat transfer interface theory 308 conjugate heat transfer interfaces 285 boundary heat source (node) 104 conservation of energy 19 boundary heat source variable 33 consistent stabilization settings 9 boundary nodes constraint settings 9 bioheat interface 132 heat transfer 84 heat transfer in porous media 137 heat transfer in thin shells 148 continuity (node) heat transfer 106 continuity on interior boundary (node) 206 convection. 197 CFL number azimuthal sectors 193 B surface-to-surface radiation 167 boundary selection 10 bioheat (node) 134 bioheat transfer interface 132 theory 65 biological tissue 133 black walls 205 blackbody intensity 208–209 pseudo time stepping. and 288 change effective thickness (node) 154 change thickness (node) heat transfer in thin shells 152 out-of-plane heat transfer 130 blackbody radiation 175 characteristic length 55 blackbody radiation intensity 209 Charron’s relation 52 blackbody radiation intensity. 139. 193. bioheat properties 133 boundary conditions heat equation. convective heat flux (node) 116 nitf interfaces 292 convective heat flux variable 27, 30 radiation in participating media 201 convective out-of-plane heat flux varia- spf interfaces 223 ble 28 surface-to-surface radiation 167 Cooper-Mikic Yovanovich (CMY) correlation 49 domain selection 10 E coordinate system selection 10 edge heat flux (node) 121 crosswind diffusion edge heat source (node) 154 definition 39 edge nodes crosswind diffusion, consistent stabiliza- heat transfer 84 tion method 39 heat transfer in porous media 137 curves, fan 254 D heat transfer in thin shells 148 Dalton’s law 41 nitf interfaces 292 del operators 62 radiation in participating media 201 density, blood 135 surface-to-surface radiation 167 dev_in and dev_out variables 253 edge selection 10 diffuse gray radiation model 165 edge surface-to-ambient radiation diffuse mirror (node) 173 (node) 125 diffuse spectral radiation model 165 edges diffuse-gray surface 188–189 heat flux 121 diffuse-spectral 190 temperature 123 diffuse-spectral surface 190 effective volumetric heat capacity 88 dimensionless distance to cell center var- elastic contribution to entropy 113, 304 iable 272 elevation 193 Dirac pulse 47 emailing COMSOL 12 direct area integration, axisymmetric ge- emission, radiation and 207 ometry and 193 equation view 9 direct area integration, radiation settings equivalent thermal conductivity 66 equivalent volumetric heat capacity 66 165 Dirichlet condition 255 evaluating view factors 192 discrete ordinates method (DOM) 211 exit length 235 discretization 9 expanding sections 9 dispersivities, porous media 143 external radiation source (node) 178 documentation 11 domain heat source variable 33 domain nodes heat transfer 84 heat transfer in porous media 137 ii | I N D E X eddy viscosity 262 F fan (node) 240 fan curves inlet boundary condition 241 theory 254 Favre average 263, 310 first law of thermodynamics 19 heat transfer coefficients flow continuity (node) 246 out-of-plane heat transfer 128 fluid (node) 293 theory 54 fluid flow heat transfer in fluids (node) 89 heat transfer in participating media inter- selecting interfaces 280 turbulent flow theory 260 face 161 fluid properties (node) 224 heat transfer in porous media (node) 137 Fourier’s law 20 heat transfer in porous media interface frames, moving 68 G 136 theory 66 Galerkin constraints 100 heat transfer in solids (node) 86 gap conductance 51 heat transfer in thin shells interface 146 general stress (boundary stress condi- theory 156 tion) 238 heat transfer interfaces 78, 81 geometric entity selection 10 selecting 280 Grashof number 55 theory 18 gravity 55 heat transfer with phase change (node) gray walls 204 96 graybody radiation 175, 190–191 heat transfer with surface-to-surface ra- gray-diffuse parallel plate model 51 diation interface 160 grille (node) 245 Heaviside function 47 grouping boundaries 197 hemicubes, axisymmetric geometry and guidelines, solving surface-to-surface ra- 193 diation problems 196 hemicubes, radiation settings 165 H heat equation, highly conductive layers hide button 9 and 62 highly conductive layer (node) 118 heat flux (node) 101 highly conductive layers, defined 61 heat transfer in thin shells 149 heat flux, theory 21 heat source (node) heat transfer 94 heat transfer in thin shells 151 heat sources defining as total power 94, 105, 111, 155 edges, thin shells 154 highly conductive layers 120 line and point 111 point, thin shells 155 I incident intensity (node) 205 inconsistent stabilization settings 9 inflow heat flux (node) 114 initial values (node) heat transfer 93 heat transfer in thin shells 152 nitf interfaces 303 spf interfaces 227 inlet (boundary stress condition) 247 inlet (node) 231 insulation/continuity (node) 153 INDEX| iii interior fan (node) 242 mechanisms of heat transfer 18 interior wall (node) 302 metabolic heat source 135 spf interfaces 244 Mikic elastic correlation 50 internal boundary heat flux variables 30 Model Library 12 Internet resources 11 Model Library examples isotropic diffusion, inconsistent stabiliza- bioheat transfer interface 132 consistent stabilization 39 tion methods 40 K convective heat flux 117 Karman constant 314 heat transfer in fluids 90 Kays-Crawford models 283, 313 heat transfer in porous media 136 k-epsilon turbulence model 219, 264 heat transfer in solids 88 knowledge base, COMSOL 13 heat transfer in thin shells 148 L laminar flow (nitf) interface, conjugate heat transfer with surface-to-surface heat transfer 289 radiation 167 laminar flow (nitf) interface, non-isother- highly conductive layers 118 mal flow 285 laminar flow 219 laminar flow interface 216 nitf interfaces 288–289 laminar inflow (inlet boundary condition) out-of-plane convective heat flux 127 232 radiation in participating media 201 laminar outflow (outlet boundary condi- surface-to-ambient radiation 104 tion) 235 thermal contact 52 Latitude 180 thermodynamics 88 latitude 193 translational motion 89 layer heat source (node) 120 turbulent flow, k-epsilon (nitf) inter- leaking wall, wall boundary condition 229 face 292 line heat source (node) 111 turbulent flow, k-epsilon (spf) 221 line heat source variable 33 moist air 93, 141 local CFL number 218, 275, 288 moisture content 93, 141 Longitude 180 moving frames 68 longitude 193 moving mesh, heat transfer and 48 low re k-epsilon turbulence model 221 moving wall (wall functions), boundary low Reynolds number condition 230 k-epsilon turbulence theory 270 moving wall, wall boundary condition neglect inertial term 286 M 229, 245 material frame MPH-files 12 mutual irradiation 194 heat transfer 73 mean effective thermal conductivity 87 mean effective thermal diffusivity 87 iv | I N D E X N nabla operators 62 natural and forced convection 54 Neumann condition 268 Newtonian model 258 no slip, interior wall boundary condition 244 pair boundary heat source (node) 104 pair nodes heat transfer 84 heat transfer in porous media 137 no slip, wall boundary condition 228 heat transfer in thin shells 148 no viscous stress (outlet boundary con- nitf interfaces 292 dition) 234 non-isothermal flow interface theory 308 radiation in participating media 201 spf interfaces 223 surface-to-surface radiation 167 non-isothermal flow interfaces 285 pair selection 10 non-Newtonian power law and Carreau pair thermal contact (node) 108 model 298 normal conductive heat flux variable 29 normal convective heat flux variable 29 normal stress (boundary condition) 232 normal stress, normal flow (boundary stress condition) 238 O P pair thin thermally resistive layer (node) 106 participating media, radiative heat transfer 207 Pennes’ approximation 65 perfusion rate, blood 134 normal total energy flux variable 30 periodic flow condition (node) 239 normal translational heat flux variable 29 periodic heat condition (node) 104 Nusselt number 55 phase transitions 97 open boundary (boundary stress condition) 247 open boundary (node) heat transfer 115 single-phase flow 237 outflow (node) 100 outlet (boundary stress condition) 247 outlet (node) 233 out-of-plane convective heat flux (node) 127 out-of-plane heat flux (node) 130 out-of-plane heat transfer change thickness 130 general theory 63 thin shells theory 156 out-of-plane inward heat flux variable 29 out-of-plane radiation (node) 129 override and contribution 9 point heat flux (node) 122 point heat source (node) 155 heat transfer 112 point heat source variable 33 point nodes heat transfer 84 heat transfer in porous media 137 heat transfer in thin shells 148 nitf interfaces 292 radiation in participating media 201 spf interfaces 223 surface-to-surface radiation 167 point selection 10 point surface-to-ambient radiation (node) 125 point temperature (node) 124 points heat flux 122 INDEX| v temperature 124 radiosity method 188 power law, non-Newtonian 298 RANS power law, single-phase flow theory 258 theory, single-phase flow 261 Prandtl number 55, 283, 313 ratio of specific heats 89, 138 prescribed radiosity (node) 174 Rayleigh number 55 pressure (outlet boundary condition) Refractive index 200 refractive index 189, 208–209 234 pressure point constraint (node) 246 relative humidity 93, 141 pressure work (node) Reynolds number 55 heat transfer 112 extended Kays-Crawford 313 nitf interfaces 304 low, turbulence theory 270 pressure, no viscous stress (inlet and turbulent flow theory 260 Reynolds stress tensor 262, 265 outlet boundary conditions) 232 pseudo time stepping Reynolds-averaged Navier-Stokes. See advanced settings 218, 288, 291 RANS. turbulent flow theory 275 pseudoplastic fluids 258 pumps, lumped curves and 254 R radiation axisymmetric geometries, and 166, 193, 197 participating media 207 radiation group (node) 177 S scalar density variables, frames and 72 scattering, radiation and 207 sectors, azimuthal 193 selecting conjugate heat transfer interfaces 280 heat transfer interfaces 78, 81 non-isothermal flow interfaces 280 radiation groups 197 settings windows 9 radiation in participating media (node) shear rate magnitude variable 225 heat transfer 202 radiation in participating media interface 199 shear thickening fluids 258 shell thickness 146 shells, conductive 156 radiation intensity, for blackbody 207 show button 9 radiation, out-of-plane 129 single-phase flow radiative conductance 51 radiative heat flux variable 30 radiative heat, theory 36 radiative out-of-plane heat flux variable 28 vi | I N D E X Rodriguez formula 209 turbulent flow theory 260 single-phase flow interface laminar flow 216 sliding wall (wall functions), boundary condition 230 radiative transfer equation 208 sliding wall, wall boundary condition 229 radiosity 189 slip, wall boundary condition 229, 244 radiosity expressions 175 Solar position 193 solving surface-to-surface radiation problems 196 heat transfer 18 source terms, bioheat 134 heat transfer coefficients 54 spatial frame heat transfer in porous media 66 heat transfer and 73 heat transfer in thin shells 156 specific heat capacity, definition 20 non-isothermal flow 308 specific heat, blood 134 out-of-plane heat transfer 63 spf.sr variable 225 surface-to-surface radiation 194 stabilization settings 9 turbulent flow k-epsilon model 260 stabilization techniques turbulent flow low re k-epsilon model crosswind diffusion 39 static pressure curves 241 strain-rate tensors 113 streamline diffusion, consistent stabilization methods 39 sun position 181 surface emissivity 189 surface-to-ambient radiation (node) 103, 153 edges and points 125 260 thermal conductivity components, thin shells 157 thermal conductivity, frames and 73 thermal conductivity, mean effective 87 thermal contact (node) 108 theory 48 thermal diffusivity 87 thermal dispersion (node) 143 thermal expansivity 55 surface-to-surface radiation (node) 168 thermal friction 51 surface-to-surface radiation interface thermal insulation (node) 99 164 theory 194 T heat equation definition 19 thin conductive layer (node) 150 thin conductive layers, definition 61 swirl flow 266 thin thermally resistive layer (node) 106 symmetry (node) 236 Time zone 180 heat transfer 101 time zone 193 symmetry, flow (node) 306 total energy flux variable 28 symmetry, heat (node) 305 total heat flux 102 technical support, COMSOL 12 temperature (node) 99, 123 tensors Reynolds stress 265 strain-rate 113 viscous stress theory 257 theory bioheat transfer 65 conjugate heat transfer interface 308 total heat flux variable 26 total normal heat flux variable 29 total power 94, 105, 111, 155 traction boundary conditions 237 translational heat flux variable 27 translational motion (node) 88 turbulence models k-epsilon 219, 264 low re k-epsilon 221 INDEX| vii theory 257 viii | I N D E X Z zenith 193 zero shear rate viscosity 259 .nitf interfaces 289 viscous stress. turbulent flow 267 turbulent flow k-epsilon interface wall functions. COMSOL 13 V vapor mass fraction 93. COMSOL 13 turbulent flow low re k-epsilon (spf) interface 221 turbulent flow low re k-epsilon interface theory 260 turbulent heat flux variable 27 turbulent kinetic energy theory k-epsilon model 264 RANS 264 turbulent length scale 273 turbulent non-isothermal flow interfaces theory 310 turbulent Prandtl number 313 U user community. theory 252 single-phase flow 217 volume force (node) 226 turbulent compressible flow 263 turbulent conjugate heat transfer volumetric heat capacity 88 W wall (node) theory 310 heat transfer 203 turbulent flow k-epsilon (nitf) interface nitf interfaces 300 289 single-phase flow 228 turbulent flow k-epsilon (spf) interface wall distance initialization study step 271 219 wall functions. wall boundary condition theory 260 230 turbulent flow low re k-epsilon (nitf) in- weak constraint settings 9 terface 289 web sites. 141 variables dimensionless distance to cell center 272 velocity (inlet and outlet boundary conditions) 232 view factors 192 viscous force 55 viscous heating (node) heat transfer 113 nitf interfaces 305 viscous stress tensors.
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