Heat Transfer Module User’s Guide Heat Transfer Module User’s Guide © 1998–2019 COMSOL Protected by patents listed on www.comsol.com/patents, and U.S. Patents 7,519,518; 7,596,474; 7,623,991; 8,457,932; 8,954,302; 9,098,106; 9,146,652; 9,323,503; 9,372,673; 9,454,625; and 10,019,544. Patents pending. This Documentation and the Programs described herein are furnished under the COMSOL Software License Agreement (www.comsol.com/comsol-license-agreement) and may be used or copied only under the terms of the license agreement. COMSOL, the COMSOL logo, COMSOL Multiphysics, COMSOL Desktop, COMSOL Compiler, COMSOL Server, 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/trademarks. Version: COMSOL 5.5 Contact Information Visit the Contact COMSOL 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 • Product Download: www.comsol.com/product-download • Product Updates: www.comsol.com/support/updates • COMSOL Blog: www.comsol.com/blogs • Discussion Forum: www.comsol.com/community • Events: www.comsol.com/events • COMSOL Video Gallery: www.comsol.com/video • Support Knowledge Base: www.comsol.com/support/knowledgebase Part number: CM020801 Contents Chapter 1: Introduction About the Heat Transfer Module 20 Why Heat Transfer Is Important to Modeling . 20 How the Heat Transfer Module Improves Your Modeling . 21 The Heat Transfer Module Physics Interface Guide . 21 Common Physics Interface and Feature Settings and Nodes . 29 The Heat Transfer Module Study Capabilities . 29 Additional Materials Database . 33 Where Do I Access the Documentation and Application Libraries? . 34 Overview of the User’s Guide 38 Chapter 2: Notations Symbols 44 Chapter 3: Modeling with the Heat Transfer Module Heat Transfer Variables 60 Predefined Variables . 60 Global Variables . 63 Domain Fluxes. 65 Out-of-Plane Domain Fluxes . 67 Boundary Fluxes (Heat Transfer Interface) . 68 Internal Boundary Heat Fluxes. 69 Domain Heat Sources . 71 Boundary Heat Sources . 71 Line and Point Heat Sources . 72 Moist Air Variables . 72 CONTENTS | 3 Moisture Transport Variables 75 Predefined Variables . 75 Moist Air Properties . 76 Domain Moisture Fluxes . 78 Boundary Moisture Fluxes . 78 Domain Moisture Source. 80 Using the Boundary Conditions for the Heat Transfer Interfaces 81 Temperature and Heat Flux Boundary Conditions . 81 Overriding Mechanism for Heat Transfer Boundary Conditions . 82 Handling Frames in Heat Transfer 85 Heat Transfer Analysis with Moving Frames. 85 Material Density in Features Defined in the Material Frame . 90 Heat Transfer Consistent and Inconsistent Stabilization Methods 91 Consistent Stabilization . 91 Inconsistent Stabilization . 92 Heat Transfer and Fluid Flow Coupling 93 Coupling Based on Model Inputs . 93 Adding Nonisothermal Flow Coupling in an Existing Model . 94 Nonisothermal Flow and Conjugate Heat Transfer Multiphysics Interfaces . 94 Boundary Wall Temperature 96 Solver Settings 101 Linearity Property of the Temperature Equation . 101 Linear Solver . 102 Nonlinear Solver . 103 Time-Dependent Study Step . 107 Guidelines for Solving Surface-to-Surface Radiation Problems . 109 Guidelines for Solving Multiphysics Problems . 109 4 | CONTENTS Plotting and Evaluating Results in Layered Materials 112 Plotting Along and Through the Layered Material . 112 Built-in Operators for Evaluation in the Layered Material . 113 Ambient Properties 117 Settings for the Ambient Properties. 117 Processing of ASHRAE Data . 121 Ambient Data Interpolation . 122 Ambient Variables and Conditions . 123 Use of Ambient Data From the Features . 131 Modeling Heat Transfer by Radiation 133 Interfaces and Features for Radiation in Transparent Media . 135 Interfaces and Features for Radiation in Participating Media . 137 Specifying the Temperature at a Flow Inlet 140 Using the Temperature Condition . 140 Using the Inflow Condition . 140 Temperature Contribution to the Inflow Boundary Condition. 142 Pressure Contribution to the Inflow Boundary Condition . 143 Choosing Between the Temperature and the Inflow Conditions . 143 Heat Part Library 144 Heat Part Library Contents . 144 Heat Sink — Parameterized Fin Types Part . 147 Heat Sink — Dissimilar Border Pins Part . 150 Heat Sink — Pin Fins Part . 150 Heat Sink — Straight Fins Part. 150 References 151 Chapter 4: Theory for the Heat Transfer Module Foundations of the General Heat Transfer Equation 153 Thermodynamic Description of Heat Transfer. 153 The Physical Mechanisms Underlying Heat Transfer. 157 CONTENTS | 5 The Heat Balance Equation . 158 Consistency with Mass and Momentum Conservation Laws. 161 Theory for Heat Transfer in Solids 163 Theory for Heat Transfer in Fluids 164 Theory for the Inflow Boundary Condition . 165 Theory for Bioheat Transfer 167 The Bioheat Equation . 167 Damaged Tissue . 167 Theory for Heat Transfer in Porous Media 171 When Should Local Thermal Equilibrium and Nonequilibrium Be Considered? . 171 Local Thermal Equilibrium . 172 Local Thermal Nonequilibrium . 174 Theory for Heat Transfer with Phase Change 177 Phase Change in Solid Materials . 179 Theory for Heat Transfer in Building Materials 181 Theory for Harmonic Heat Transfer 183 Theory for Lumped Isothermal Domain 185 Theory for Heat Transfer in Thin Structures 188 Modeling Layered Materials . 188 Theoretical Background of the Different Formulations . 192 Thin Layer . 197 Thin Film . 200 Fracture . 202 Thin Rod . 203 Theory for the Lumped Thermal System Interface 205 Temperature Differences and Heat Rates in the Lumped Thermal System . 205 6 | CONTENTS Theory for the Conductive Thermal Resistor Component . 207 Theory for the Radiative Thermal Resistor Component . 211 Theory for the Thermal Capacitor and Thermal Mass Components . 213 Theory for the Heat Pipe Component . 214 Theory for the Thermoelectric Module Component . 216 Theory for Surface-to-Surface Radiation 220 Deriving the Radiative Heat Flux for Opaque Surfaces . 220 Deriving the Radiative Heat Flux for Semi-Transparent Surfaces . 221 Wavelength Dependence of Surface Emissivity and Absorptivity . 223 The Radiosity Method for Diffuse-Gray Surfaces . 229 The Radiosity Method for Diffuse-Spectral Surfaces. 232 View Factor Evaluation . 235 Theory for Radiation in Participating Media 240 Radiation and Participating Media Interactions . 240 Radiative Transfer Equation . 241 Boundary Condition for the Radiative Transfer Equation . 243 Heat Transfer Equation in Participating Media . 244 Discrete Ordinates Method (DOM) . 245 Discrete Ordinates Method Implementation in 2D . 246 P1 Approximation Theory . 248 Radiation in Absorbing-Scattering Media Theory . 251 Polychromatic Radiation . 253 Radiative Beam in Absorbing Media Theory . 254 Rosseland Approximation Theory . 254 Theory for Moisture Transport 256 Theory for Moisture Transport in Building Materials . 256 Theory for Moisture Transport in Air . 256 Theory for the Heat Transfer Multiphysics Couplings 259 Theory for the Nonisothermal Flow and Conjugate Heat Transfer Interfaces . 259 Theory for the Moisture Flow Interface . 267 Theory for the Thermoelectric Effect Interface . 271 Theory for the Local Thermal Nonequilibrium Interface . 274 Theory for the Building Materials Version of the Heat and Moisture CONTENTS | 7 Transport Interface . 275 Theory for the Moist Air Version of the Heat and Moisture Transport Interface . 276 Theory for the Heat and Moisture Flow Interfaces . 277 Theory for the Electromagnetic Heating Interfaces . 277 Theory for the Thermal Stress Interface . 277 Theory for Thermal Contact 278 Theory for the Thermal Contact Feature . 278 Moist Air Fluid Type 283 Humidity . 283 Saturation State . 284 Moist Air Properties . 285 Out-of-Plane Heat Transfer 290 Equation Formulation . 290 The Heat Transfer Coefficients 293 Defining the Heat Transfer Coefficients . 294 Nature of the Flow — The Grashof Number . 295 Heat Transfer Coefficients — External Natural Convection . 297 Heat Transfer Coefficients — Internal Natural Convection . 304 Heat Transfer Coefficients — External Forced Convection . 305 Heat Transfer Coefficients — Internal Forced Convection . 308 Using the Heat and Mass Transfer Analogy for the Evaluation of Moisture Transfer Coefficients . 308 Equivalent Thermal Conductivity Correlations 310 Horizontal Cavity With Bottom Heating . 311 Vertical Cavity With Sidewall Heating . 311 Temperature Dependence of Surface Tension 313 Heat Flux and Heat Balance 314 Total Heat Flux and Energy Flux . 314 Heat and Energy Balance . 315 8 | CONTENTS Frames for the Heat Transfer Equations 318 Material and Spatial Frames . 318 Conversion Between Material and Spatial Frames . 319 References 323 Chapter 5: The Heat Transfer Module Interfaces About the Heat Transfer Interfaces 329 Space Dimensions . 329 Study Types . 330 Versions of the Heat Transfer Physics Interface . 330 Benefits of the Different Heat Transfer Interfaces . 331 Versions of the Heat Transfer in Shells Physics