Heat Transfer Module. User s Guide VERSION 4.3

Transcription

1 Heat Transfer Module User s Guide VERSION 4.3

2 Heat Transfer Module User s Guide 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 ( and may be used or copied only under the terms of the license agreement. COMSOL, COMSOL Desktop, COMSOL Multiphysics, and LiveLink are registered trademarks or trademarks of COMSOL AB. Other product or brand names are trademarks or registered trademarks of their respective holders. Version: May 2012 COMSOL 4.3 Contact Information Visit for a searchable list of all COMSOL offices and local representatives. From this web page, search the contacts and find a local sales representative, go to other COMSOL websites, request information and pricing, submit technical support queries, subscribe to the monthly enews newsletter, and much more. If you need to contact Technical Support, an online request form is located at Other useful links include: Technical Support Software updates: Online community: Events, conferences, and training: Tutorials: Knowledge Base: Part No. CM020801

3 Contents Chapter 1: Introduction About the Heat Transfer Module 12 Why Heat Transfer is Important to Modeling How the Heat Transfer Module Improves Your Modeling Heat Transfer Module Physics Interface Guide The Heat Transfer Module Study Capabilities by Interface Model Builder Options for Physics Feature Node Settings Windows Where Do I Access the Documentation and Model Library? Typographical Conventions Overview of the User s Guide 26 Chapter 2: Heat Transfer Theory Theory for the Heat Transfer Interfaces 30 What is Heat Transfer? The Heat Equation A Note on Heat Flux Heat Flux Variables and Heat Sources About the Boundary Conditions for the Heat Transfer Interfaces Radiative Heat Transfer in Transparent Media Consistent and Inconsistent Stabilization Methods for the Heat Transfer Interfaces References for the Heat Transfer Interfaces About Infinite Elements 49 Modeling Unbounded Domains Known Issues When Modeling Using Infinite Elements About the Heat Transfer Coefficients 53 Heat Transfer Coefficient Theory CONTENTS 3

4 Nature of the Flow the Grashof Number Available Heat Transfer Coefficients References for the Heat Transfer Coefficients About Highly Conductive Layers 61 Theory of Out-of-Plane Heat Transfer 64 Equation Formulation Activating Out-of-Plane Heat Transfer and Thickness Theory for the Bioheat Transfer Interface 66 Reference for the Bioheat Interface Theory for the Heat Transfer in Porous Media Interface 67 Chapter 3: Heat Transfer Branch The Heat Transfer Interfaces 70 Accessing the Heat Transfer Interfaces via the Model Wizard The Heat Transfer Interface 73 Heat Transfer in Solids Translational Motion Pressure Work Opaque Heat Transfer in Fluids Viscous Heating Heat Source Radiation in Participating Media Infinite Elements Manual Scaling Initial Values Boundary Conditions for the Heat Transfer Interfaces Temperature Thermal Insulation Outflow C ONTENTS

5 Symmetry Heat Flux Inflow Heat Flux Open Boundary Surface-to-Ambient Radiation Periodic Heat Condition Boundary Heat Source Heat Continuity Pair Thin Thermally Resistive Layer Thin Thermally Resistive Layer Opaque Surface Incident Intensity Continuity on Interior Boundary Line Heat Source Point Heat Source Convective Cooling Highly Conductive Layer Features 103 Highly Conductive Layer Layer Heat Source Edge Heat Flux or Point Heat Flux Edge Temperature or Point Temperature Edge Surface-to-Ambient or Point Surface-to-Ambient Radiation Out-of-Plane Heat Transfer Features 109 Out-of-Plane Convective Cooling Out-of-Plane Radiation Out-of-Plane Heat Flux Change Thickness The Bioheat Transfer Interface 114 Biological Tissue Bioheat Boundary Conditions for the Bioheat Transfer Interface The Heat Transfer in Porous Media Interface 118 Porous Matrix Heat Transfer in Fluids CONTENTS 5

6 Thermal Dispersion Heat Source Chapter 4: Heat Transfer in Thin Shells The Heat Transfer in Thin Shells Interface 124 Thin Conductive Layer Heat Source Initial Values Change Thickness Other Boundary Conditions Edge and Point Conditions Insulation/Continuity Radiation Change Effective Thickness Edge Heat Source Point Heat Source Theory for the Heat Transfer in Thin Shells Interface 131 About Thin Conductive Shells Heat Transfer Equation in Thin Conductive Shell Thermal Conductivity Tensor Components Chapter 5: Radiation Heat Transfer Branch The Surface-To-Surface Radiation Interface 136 Surface-to-Surface Radiation (Boundary Condition) Opaque Initial Values Reradiating Surface Prescribed Radiosity Radiation Group External Radiation Source C ONTENTS

7 The Radiation in Participating Media Interface 145 Radiation in Participating Media Opaque Surface Incident Intensity Initial Values The Heat Transfer with Radiation in Participating Media Interface 151 Domain and Boundary Conditions Edge, Point, and Pair Conditions Theory for the Radiative Heat Transfer Interfaces 154 The Radiosity Method View Factor Evaluation Radiation and Participating Media Interactions Radiative Transfer Equation Boundary Condition for the Transfer Equation Heat Transfer Equation in Participating Media Discrete Ordinates Method Theory for the Surface-to-Surface Radiation Interface 162 About Surface-to-Surface Radiation Solving for the Radiosity About the Surface-to-Surface Radiation Boundary Conditions Guidelines for Solving Surface-to-Surface Radiation Problems Radiation Group Boundaries References for the Surface-to-Surface Radiation Interface Chapter 6: Single-Phase Flow Branch The Single-Phase Flow, Laminar Flow Interface 170 The Laminar Flow Interface Fluid Properties Volume Force Initial Values CONTENTS 7

8 The Single-Phase Flow, Turbulent Flow Interfaces 177 The Turbulent Flow, k- Interface The Turbulent Flow, Low Re k- Interface Boundary Conditions for the Single-Phase Flow Interfaces 180 Wall Interior Wall Inlet Outlet Symmetry Open Boundary Boundary Stress Periodic Flow Condition Flow Continuity Pressure Point Constraint Fan Theory for the Laminar Flow Interface 204 Theory for the Pressure, No Viscous Stress Condition Theory for the Laminar Inflow Condition Theory for the Laminar Outflow Condition Theory for the Fan Defined on an Interior Boundary Theory for the Fan and Grill Inlet and Outlet Condition Theory for the No Viscous Stress Condition Theory for the Turbulent Flow Interfaces 211 Turbulence Modeling The k-turbulence Model The Low Reynolds Number k- Turbulence Model Inlet Values for the Turbulence Length Scale and Intensity Pseudo Time Stepping for Turbulent Flow Models References for the Single-Phase Flow, Turbulent Flow Interfaces C ONTENTS

9 Chapter 7: Conjugate Heat Transfer Branch The Conjugate Heat Transfer Interfaces 226 The Non-Isothermal Flow and Conjugate Heat Transfer, Laminar Flow Interfaces 228 The Non-Isothermal Flow, Laminar Flow Interface The Conjugate Heat Transfer, Laminar Flow Interface The Non-Isothermal Flow and Conjugate Heat Transfer, Turbulent Flow Interfaces 233 The Turbulent Flow, k- and Turbulent Flow Low Re k- Interfaces Shared Interface Features 236 Fluid Wall Initial Values Pressure Work Viscous Heating Theory for the Non-Isothermal Flow and Conjugate Heat Transfer Interfaces 244 Turbulent Non-Isothermal Flow Theory References for the Non-Isothermal Flow and Conjugate Heat Transfer Interfaces 250 Chapter 8: Materials Material Library and Databases 252 About the Material Databases About Using Materials in COMSOL Opening the Material Browser Using Material Properties CONTENTS 9

10 Liquids and Gases Material Database 259 Liquids and Gases Materials References for the Liquids and Gases Material Database Chapter 9: Glossary Glossary of Terms C ONTENTS

11 1 Introduction This guide describes the Heat Transfer Module, an optional package that extends the COMSOL Multiphysics modeling environment with customized physics interfaces for the analysis of heat transfer. This chapter introduces you to the capabilities of this module. A summary of the physics interfaces and where you can find documentation and model examples is also included. The last section is a brief overview with links to each chapter in this guide. About the Heat Transfer Module Overview of the User s Guide 11

12 About the Heat Transfer Module In this section: Why Heat Transfer is Important to Modeling How the Heat Transfer Module Improves Your Modeling Heat Transfer Module Physics Guide The Heat Transfer Module Study Capabilities Show More Physics Options Where Do I Access the Documentation and Model Library? Typographical Conventions See Also Overview of the Physics Interfaces and Building a COMSOL Model in the COMSOL Multiphysics User s Guide Why Heat Transfer is Important to Modeling The Heat Transfer Module is an optional package that extends the COMSOL Multiphysics modeling environment with customized user interfaces and functionality optimized for the analysis of heat transfer. It is developed for a wide audience including researchers, developers, teachers, and students. To assist users at all levels of expertise, this module comes with a library of ready-to-run example models that appear in the companion Heat Transfer Module Model Library. Heat transfer is involved in almost every kind of physical process, and can in fact be the limiting factor for many processes. Therefore, its study is of vital importance, and the need for powerful heat transfer analysis tools is virtually universal. Furthermore, heat transfer often appears together with, or as a result of, other physical phenomena. The modeling of heat transfer effects has become increasingly important in product design including areas such as electronics, automotive, and medical industries. Computer simulation has allowed engineers and researchers to optimize process efficiency and explore new designs, while at the same time reducing costly experimental trials. 12 CHAPTER 1: INTRODUCTION

13 How the Heat Transfer Module Improves Your Modeling The Heat Transfer Module has been developed to greatly expand upon the base capabilities available in COMSOL Multiphysics. The module supports all fundamental mechanisms including conductive, convective, and radiative heat transfer. Using the physics interfaces in this module along with the inherent multiphysics capabilities of COMSOL Multiphysics, you can model a temperature field in parallel with other physics a versatile combination increasing the accuracy and predicting power of your models. This User s Guide introduces the basic modeling process. The different physics interfaces are described and the modeling strategy for various cases is discussed. These sections cover different combinations of conductive, convective, and radiative heat transfer. This guide also reviews special modeling techniques for highly conductive layers, thin conductive shells, participating media, and out-of-plane heat transfer. Throughout the guide the topics and examples increase in complexity by combining several heat transfer mechanisms and also by coupling these to physics interfaces describing fluid flow conjugate heat transfer. Another source of information is the Heat Transfer Module Model Library, a set of fully-documented models that is divided into broadly defined application areas where heat transfer plays an important role electronics and power systems, processing and manufacturing, and medical technology and includes tutorial and verification models. Most of the models involve multiple heat transfer mechanisms and are often coupled to other physical phenomena, for example, fluid dynamics or electromagnetics. The authors developed several state-of-the art examples by reproducing models that have appeared in international scientific journals. See Where Do I Access the Documentation and Model Library?. Heat Transfer Module Physics Guide The table below lists all the interfaces available specifically with this module. Having this module also enhances these COMSOL basic interfaces: Heat Transfer in Fluids, Heat Transfer in Solids, Joule Heating, and the Single-Phase Flow, Laminar interface. ABOUT THE HEAT TRANSFER MODULE 13

14 If you have an Subsurface Flow Module combined with the Heat Transfer Module, this also enhances the Heat Transfer in Porous Media interface. Note The Non-Isothermal Flow, Laminar Flow (nitf) and Non-Isothermal Flow, Turbulent Flow (nitf) interfaces found under the Fluid Flow>Non-Isothermal Flow branch are identical to the Conjugate Heat Transfer interfaces (Laminar Flow and Turbulent Flow) found under the Heat Transfer>Conjugate Heat Transfer branch. The only difference is that Fluid is selected as the Default model in the former case. If Heat transfer in solids is selected as the default model, the interface changes to a Conjugate Heat Transfer interface. Study Types in the COMSOL Multiphysics Reference Guide See Also Available Study Types in the COMSOL Multiphysics User s Guide PHYSICS ICON TAG SPACE DIMENSION PRESET STUDIES Fluid Flow Single-Phase Flow Single-Phase Flow, Laminar Flow* spf 3D, 2D, 2D axisymmetric stationary; time dependent Turbulent Flow, k- spf 3D, 2D, 2D axisymmetric Turbulent Flow, Low Re k- spf 3D, 2D, 2D axisymmetric Non-Isothermal Flow stationary; time dependent stationary with initialization; transient with initialization Laminar Flow nitf 3D, 2D, 2D axisymmetric Turbulent Flow, k- nitf 3D, 2D, 2D axisymmetric stationary; time dependent stationary; time dependent 14 CHAPTER 1: INTRODUCTION

15 PHYSICS ICON TAG SPACE DIMENSION Turbulent Flow, Low Re k- nitf 3D, 2D, 2D axisymmetric Heat Transfer PRESET STUDIES stationary with initialization; transient with initialization Heat Transfer in Solids* ht all dimensions stationary; time dependent Heat Transfer in Fluids* ht all dimensions stationary; time dependent Heat Transfer in Porous Media ht all dimensions stationary; time dependent Bioheat Transfer ht all dimensions stationary; time dependent Heat Transfer in Thin Shells (also called Thin Conductive Shell) Conjugate Heat Transfer htsh 3D stationary; time dependent Laminar Flow nitf 3D, 2D, 2D axisymmetric Turbulent Flow, k- nitf 3D, 2D, 2D axisymmetric Turbulent Flow, Low Re k- nitf 3D, 2D, 2D axisymmetric Radiation stationary; time dependent stationary; time dependent stationary with initialization; transient with initialization Heat Transfer with Surface-to-Surface Radiation Heat Transfer with Radiation in Participating Media ht all dimensions stationary; time dependent ht 3D, 2D stationary; time dependent Surface-to-Surface Radiation rad all dimensions stationary; time dependent Radiation in Participating Media rpm 3D, 2D stationary; time dependent ABOUT THE HEAT TRANSFER MODULE 15

16 PHYSICS ICON TAG SPACE DIMENSION Electromagnetic Heating PRESET STUDIES Joule Heating* jh all dimensions stationary; time dependent * This is an enhanced interface, which is included with the base COMSOL package but has added functionality for this module. The Heat Transfer Module Study Capabilities Table 1-1 lists the Preset Studies available for the interfaces most relevant to this module. See Also Study Types in the COMSOL Multiphysics Reference Guide Available Study Types in the COMSOL Multiphysics User s Guide TABLE 1-1: HEAT TRANSFER MODULE DEPENDENT VARIABLES AND PRESET STUDY AVAILABILITY PHYSICS TAG DEPENDENT VARIABLES PRESET STUDIES* STATIONARY TIME DEPENDENT STATIONARY WITH INITIALIZATION TRANSIENT WITH INITIALIZATION FLUID FLOW>SINGLE-PHASE FLOW Laminar Flow spf u, p Turbulent Flow, k- spf u, p, k, ep Turbulent Flow, Low Re k- spf u, p, k, ep, G FLUID FLOW>NON-ISOTHERMAL FLOW Laminar Flow nitf u, p, T Turbulent Flow, k- nitf u, p, k, ep, T 16 CHAPTER 1: INTRODUCTION

17 TABLE 1-1: HEAT TRANSFER MODULE DEPENDENT VARIABLES AND PRESET STUDY AVAILABILITY PHYSICS TAG DEPENDENT VARIABLES PRESET STUDIES* Turbulent Flow, Low Re k- nitf u, p, k, ep, G, T HEAT TRANSFER Heat Transfer in Solids** ht T Heat Transfer in Fluids** ht T Heat Transfer in Porous ht T Media** Bioheat Transfer** ht T Heat Transfer in Thin Shells htsh T HEAT TRANSFER>CONJUGATE HEAT TRANSFER Laminar Flow** nitf u, p, T Turbulent Flow, k-** nitf u, p, k, ep, T Turbulent Flow, Low Re k-** nitf u, p, k, ep, G, T HEAT TRANSFER>RADIATION Heat Transfer with Surface-to-Surface Radiation** Heat Transfer with Radiation in Participating Media** ht T, J ht T, I (radiative intensity) Surface-to-Surface Radiation rad J Radiation in Participating rpm I (radiative intensity) Media HEAT TRANSFER>ELECTROMAGNETIC HEATING Joule Heating** jh T, V * Custom studies are also available based on the interface. ** For these interfaces, it is possible to enable surface to surface radiation and/or radiation in participating media. In these cases, J and I are dependent variables. STATIONARY TIME DEPENDENT STATIONARY WITH INITIALIZATION TRANSIENT WITH INITIALIZATION ABOUT THE HEAT TRANSFER MODULE 17

18 Show More Physics Options There are several features available on many physics interfaces or individual nodes. This section is a short overview of the options and includes links to the COMSOL Multiphysics User s Guide or COMSOL Multiphysics Reference Guide where additional information is available. Important The links to the features described in the COMSOL Multiphysics User s Guide and COMSOL Multiphysics Reference Guide do not work in the PDF, only from within the online help. Tip To locate and search all the documentation for this information, in COMSOL, select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree. To display additional features for the physics interfaces and feature nodes, click the Show button ( ) on the Model Builder and then select the applicable option. After clicking the Show button ( ), some sections display on the settings window when a node is clicked and other features are available from the context menu when a node is right-clicked. For each, the additional sections that can be displayed include Equation, Advanced Settings, Discretization, Consistent Stabilization, and Inconsistent Stabilization. You can also click the Expand Sections button ( ) in the Model Builder to always show some sections or click the Show button ( ) and select Reset to Default to reset to display only the Equation and Override and Contribution sections. For most physics nodes, both the Equation and Override and Contribution sections are always available. Click the Show button ( ) and then select Equation View to display the Equation View node under all physics nodes in the Model Builder. Availability of each feature, and whether it is described for a particular physics node, is based on the individual physics selected. For example, the Discretization, Advanced 18 CHAPTER 1: INTRODUCTION

19 Settings, Consistent Stabilization, and Inconsistent Stabilization sections are often described individually throughout the documentation as there are unique settings. SECTION CROSS REFERENCE LOCATION IN COMSOL MULTIPHYSICS USER GUIDE OR REFERENCE GUIDE Show More Options and Expand Sections Discretization Discretization - Splitting of complex variables Pair Selection Consistent and Inconsistent Stabilization Showing and Expanding Advanced Physics Sections The Model Builder Window Show Discretization Element Types and Discretization Finite Elements Discretization of the Equations Compile Equations Identity and Contact Pairs Specifying Boundary Conditions for Identity Pairs Show Stabilization Stabilization Techniques Numerical Stabilization User s Guide User s Guide Reference Guide Reference Guide User s Guide User s Guide Reference Guide Geometry Working with Geometry User s Guide Constraint Settings Using Weak Constraints User s Guide Where Do I Access the Documentation and Model Library? A number of Internet resources provide more information about COMSOL Multiphysics, including licensing and technical information. The electronic ABOUT THE HEAT TRANSFER MODULE 19

20 documentation, Dynamic Help, and the Model Library are all accessed through the COMSOL Desktop. Important If you are reading the documentation as a PDF file on your computer, the blue links do not work to open a model or content referenced in a different user s guide. However, if you are using the online help in COMSOL Multiphysics, these links work to other modules, model examples, and documentation sets. THE DOCUMENTATION The COMSOL Multiphysics User s Guide and COMSOL Multiphysics Reference Guide describe all interfaces and functionality included with the basic COMSOL Multiphysics license. These guides also have instructions about how to use COMSOL Multiphysics and how to access the documentation electronically through the COMSOL Multiphysics help desk. To locate and search all the documentation, in COMSOL Multiphysics: Press F1 for Dynamic Help, Click the buttons on the toolbar, or Select Help>Documentation ( ) or Help>Dynamic Help ( ) from the main menu and then either enter a search term or look under a specific module in the documentation tree. THE MODEL LIBRARY Each model comes with documentation that includes a theoretical background and step-by-step instructions to create the model. The models are available in COMSOL as MPH-files that you can open for further investigation. You can use the step-by-step instructions and the actual models as a template for your own modeling and applications. SI units are used to describe the relevant properties, parameters, and dimensions in most examples, but other unit systems are available. To open the Model Library, select View>Model Library ( ) from the main menu, and then search by model name or browse under a module folder name. Click to highlight any model of interest, and select Open Model and PDF to open both the model and the documentation explaining how to build the model. Alternatively, click the Dynamic 20 CHAPTER 1: INTRODUCTION

21 Help button ( ) or select Help>Documentation in COMSOL to search by name or browse by module. The model libraries are updated on a regular basis by COMSOL in order to add new models and to improve existing models. Choose View>Model Library Update ( ) to update your model library to include the latest versions of the model examples. If you have any feedback or suggestions for additional models for the library (including those developed by you), feel free to contact us at info@comsol.com. CONTACTING COMSOL BY For general product information, contact COMSOL at info@comsol.com. To receive technical support from COMSOL for the COMSOL products, please contact your local COMSOL representative or send your questions to support@comsol.com. An automatic notification and case number is sent to you by . COMSOL WEB SITES Main Corporate web site Worldwide contact information Technical Support main page Support Knowledge Base Product updates COMSOL User Community Typographical Conventions All COMSOL user guides use a set of consistent typographical conventions that make it easier to follow the discussion, understand what you can expect to see on the Graphical User Interface (GUI), and know which data must be entered into various data-entry fields. In particular, these conventions are used throughout the documentation: Click text highlighted in blue to go to other information in the PDF. When you are using the online help desk in COMSOL Multiphysics, these links also work to other modules, model examples, and documentation sets. ABOUT THE HEAT TRANSFER MODULE 21

22 A boldface font indicates that the given word(s) appear exactly that way on the COMSOL Desktop (or, for toolbar buttons, in the corresponding tooltip). For example, the Model Builder window ( ) is often referred to and this is the window that contains the model tree. As another example, the instructions might say to click the Zoom Extents button ( ), and this means that when you hover over the button with your mouse, the same label displays on the COMSOL Desktop. The names of other items on the COMSOL Desktop that do not have direct labels contain a leading uppercase letter. For instance, the Main toolbar is often referred to the horizontal bar containing several icons that are displayed on top of the user interface. However, nowhere on the COMSOL Desktop, nor the toolbar itself, includes the word main. The forward arrow symbol > is instructing you to select a series of menu items in a specific order. For example, Options>Preferences is equivalent to: From the Options menu, choose Preferences. A Code (monospace) font indicates you are to make a keyboard entry in the user interface. You might see an instruction such as Enter (or type) 1.25 in the Current density field. The monospace font also is an indication of programming code. or a variable name. An italic Code (monospace) font indicates user inputs and parts of names that can vary or be defined by the user. An italic font indicates the introduction of important terminology. Expect to find an explanation in the same paragraph or in the Glossary. The names of other user guides in the COMSOL documentation set also have an italic font. THE DIFFERENCE BETWEEN NODES, BUTTONS, AND ICONS Node: A node is located in the Model Builder and has an icon image to the left of it. Right-click a node to open a context menu and to perform actions. Button: Click a button to perform an action. Usually located on a toolbar (the main toolbar or the Graphics toolbar, for example), or in the upper-right corner of a settings window. Icon: An icon is an image that displays on a window (for example, the Model Wizard or Model Library) or displays in a context menu when a node is right-clicked. Sometimes selecting an item with an icon from a node s context menu adds a node with the same image and name, sometimes it simply performs the action indicated (for example, Delete, Enable, or Disable). 22 CHAPTER 1: INTRODUCTION

23 KEY TO THE GRAPHICS Throughout the documentation, additional icons are used to help navigate the information. These categories are used to draw your attention to the information based on the level of importance, although it is always recommended that you read these text boxes. Caution A Caution icon is used to indicate that the user should proceed carefully and consider the next steps. It might mean that an action is required, or if the instructions are not followed, that there will be problems with the model solution, for example: Caution This may limit the type of boundary conditions that you can set on the eliminated species. The species selection must be carefully done. Important An Important icon is used to indicate that the information provided is key to the model building, design, or solution. The information is of higher importance than a note or tip, and the user should endeavor to follow the instructions, for example: Important Do not select any domains that do not conduct current, for example, the gas channels in a fuel cell. Note A Note icon is used to indicate that the information may be of use to the user. It is recommended that the user read the text, for example: Note Undo is not possible for nodes that are built directly, such as geometry objects, solutions, meshes, and plots. ABOUT THE HEAT TRANSFER MODULE 23

24 Tip A Tip icon is used to provide information, reminders, short cuts, suggestions of how to improve model design, and other information that may or may not be useful to the user, for example: Tip It can be more accurate and efficient to use several simple models instead of a single, complex one. See Also The See Also icon indicates that other useful information is located in the named section. If you are working on line, click the hyperlink to go to the information directly. When the link is outside of the current document, the text indicates this, for example: Theory for the Single-Phase Flow Interfaces See Also The Laminar Flow Interface in the COMSOL Multiphysics User s Guide Model The Model icon is used in the documentation as well as in COMSOL Multiphysics from the View>Model Library menu. If you are working online, click the link to go to the PDF version of the step-by-step instructions. In some cases, a model is only available if you have a license for a specific module. These examples occur in the COMSOL Multiphysics User s Guide. The Model Library path describes how to find the actual model in COMSOL Multiphysics. Acoustics of a Muffler: Model Library path COMSOL_Multiphysics/ Acoustics/automotive_muffler Model If you have the RF Module, see Radar Cross Section: Model Library path RF_Module/Tutorial_Models/radar_cross_section Space Dimension Icons Another set of icons are also used in the Model Builder the model space dimension is indicated by 0D, 1D, 1D axial symmetry, 2D, 2D axial symmetry, and 3D icons. These icons are also used in the documentation to clearly list 24 CHAPTER 1: INTRODUCTION

25 the differences to an interface, feature node, or theory section, which are based on space dimension. The following tables are examples of these space dimension icons. 3D 3D models often require more computer power, memory, and time to solve. The extra time spent on simplifying a model is time well spent when solving it. 2D Remember that modeling in 2D usually represents some 3D geometry under the assumption that nothing changes in the third dimension. ABOUT THE HEAT TRANSFER MODULE 25

26 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. Instructions how to use COMSOL in general are included with the COMSOL Multiphysics User s Guide. Tip As detailed in the section Where Do I Access the Documentation and Model Library? this information is also searchable from the COMSOL Multiphysics software Help menu. TABLE OF CONTENTS, GLOSSARY, AND INDEX To help you navigate through this guide, see the Contents, Glossary, and Index. HEAT TRANSFER THEORY The Heat Transfer Theory chapter starts with the general theory underlying the heat transfer interfaces used in this module. It then discusses theory about infinite elements, heat transfer coefficients, highly conductive layers, and out-of-plane heat transfer. The last three sections briefly describe the underlying theory for the Bioheat Transfer, Heat Transfer in Thin Shells, and Heat Transfer in Porous Media interfaces. THE HEAT TRANSFER BRANCH INTERFACES The module includes interfaces for the simulation of heat transfer. As with all other physical descriptions simulated by COMSOL Multiphysics, any description of heat transfer can be directly coupled to any other physical process. This is particularly relevant for systems based on fluid-flow, as well as mass transfer. General Heat Transfer The Heat Transfer Branch chapter details the variety of Heat Transfer interfaces that form the fundamental interfaces in this module. It covers all the types of heat transfer conduction, convection, and radiation for heat transfer in solids and fluids. About the Heat Transfer Interfaces provides a quick summary of each interface, and the rest of the chapter describes these interfaces in details. This includes the highly conductive layer and out-of-plane heat transfer features and the Heat Transfer in Porous Media interface. The Heat Transfer with Participating Media (ht) interface is also described as it is a Heat Transfer interface where surface-to-surface radiation is active by default. 26 CHAPTER 1: INTRODUCTION

27 Bioheat Transfer The Bioheat Transfer Interface section discusses modeling heat transfer within biological tissue using the Bioheat Transfer interface. Heat Transfer in Thin Shells Heat Transfer in Thin Shells chapter describes the Thin Conductive Shell interface, which opens after selecting Heat Transfer in Thin Shells in the Model Wizard. It is suitable for solving thermal-conduction problems in thin structures. Radiative Heat Transfer The The Radiation Heat Transfer Branch chapter describes the Surface-to-Surface Radiation, the Heat Transfer with Surface-to-Surface Radiation, and the Radiation in Participating Media interfaces. THE CONJUGATE HEAT TRANSFER INTERFACES The The Conjugate Heat Transfer Branch chapter describes the Non-Isothermal Flow Laminar Flow (nitf) and Turbulent Flow (nitf) interfaces found under the Fluid Flow branch, which are identical to the Conjugate Heat Transfer interfaces. Each section describes the applicable interfaces in detail and concludes with the underlying theory for the interfaces. THE FLUID FLOW BRANCH INTERFACES The Single-Phase Flow Branch chapter describe the single-phase laminar and turbulent flow interfaces in detail. Each section describes the applicable interfaces in detail and concludes with the underlying theory for the interfaces. MATERIALS The Materials chapter has details about the Liquids and Gases material database included with this module. OVERVIEW OF THE USER S GUIDE 27

28 28 CHAPTER 1: INTRODUCTION

29 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 Interfaces About Infinite Elements About the Heat Transfer Coefficients About Highly Conductive Layers Theory of Out-of-Plane Heat Transfer Theory for the Bioheat Transfer Interface Theory for the Heat Transfer in Porous Media Interface 29

30 Theory for the Heat Transfer Interfaces This section reviews the theory about the heat transfer equations. For more detailed discussions of the fundamentals of heat transfer, see Ref. 1 and Ref. 3. The Heat Transfer Interface theory is described in this section: What is Heat Transfer? The Heat Equation A Note on Heat Flux Heat Flux Variables and Heat Sources About the Boundary Conditions for the Heat Transfer Interfaces Radiative Heat Transfer in Transparent Media Consistent and Inconsistent Stabilization Methods for the Heat Transfer Interfaces References for the Heat Transfer Interfaces What is Heat Transfer? Heat transfer is defined as the movement of energy due to a difference in temperature. It is characterized by the following mechanisms: Conduction Heat conduction takes place through different mechanisms in different media. Theoretically it takes place in a gas through collisions of the molecules; in a fluid through oscillations of each molecule in a cage formed by its nearest neighbors; in metals mainly by electrons carrying heat and in other solids by molecular motion which in crystals take the form of lattice vibrations known as phonons. Typical for heat conduction is that the heat flux is proportional to the temperature gradient. Convection Heat convection (sometimes called heat advection) takes place through the net displacement of a fluid, which transports the heat content in a fluid through the fluid s own velocity. The term convection (especially convective cooling 30 CHAPTER 2: HEAT TRANSFER THEORY

31 and convective heating) also refers to the heat dissipation from a solid surface to a fluid, typically described by a heat transfer coefficient. Radiation Heat transfer by radiation takes place through the transport of photons. Participating (or semitransparent) media absorb, emit and scatter photons. Opaque surfaces absorb or reflect them. The Heat Equation The fundamental law governing all heat transfer is the first law of thermodynamics, commonly referred to as the principle of conservation of energy. However, internal energy, U, is a rather inconvenient quantity to measure and use in simulations. Therefore, the basic law is usually rewritten in terms of temperature, T. For a fluid, the resulting heat equation is: T C p u T t q :S T p = u p T t + Q p (2-1) where is the density (SI unit: kg/m 3 ) C p 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/m 2 ) 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 = -- u + u 2 T Q contains heat sources other than viscous heating (SI unit: W/m 3 ) THEORY FOR THE HEAT TRANSFER INTERFACES 31

32 For a detailed discussion of the fundamentals of heat transfer, see Ref. 1. Note 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. In deriving Equation 2-1, a number of thermodynamic relations have been used. The equation also assumes that mass is always conserved, which means that density and velocity must be related through: + v = 0 t The heat transfer interfaces use Fourier s law of heat conduction, which states that the conductive heat flux, q, is proportional to the temperature gradient: q i = k T x i (2-2) where k is the thermal conductivity (SI unit: W/(m K)). In a solid, the thermal conductivity can be anisotropic (that is, it has different values in different directions). Then k becomes a tensor k = k xx k xy k xz k yx k yy k yz k zx k zy k zz and the conductive heat flux is given by q i = j T k ij x j The second term on the right of Equation 2-1 represents viscous heating of a fluid. An analogous term arises from the internal viscous damping of a solid. The operation : is a contraction and can in this case be written on the following form: a:b = n m a nm b nm 32 CHAPTER 2: HEAT TRANSFER THEORY

33 The third term represents pressure work and is responsible for the heating of a fluid under adiabatic compression and for some thermoacoustic effects. It is generally small for low Mach number flows. A similar term can be included to account for thermoelastic effects in solids. Inserting Equation 2-2 into Equation 2-1, reordering the terms and ignoring viscous heating and pressure work puts the heat equation into a more familiar form: T C p C t p u T = kt+ Q The Heat Transfer interface with the Heat Transfer in Fluids feature solves this equation for the temperature, T. If the velocity is set to zero, the equation governing pure conductive heat transfer is obtained: T C p kt = 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. The conserved property is instead the total energy. There is hence heat flux and energy flux which are similar, but not identical. This section briefly describes the theory for the variables for Total heat flux and Total energy flux. The approximations made do not affect the computational results, only variables available for results analysis and visualization. TOTAL ENERGY FLUX The total energy flux for a fluid is equal to (Ref. 4, chapter 3.5) uh 0 + kt+ u + q r (2-3) Above, H 0 is the total enthalpy H 0 = H u u where in turn H is the enthalpy. In Equation 2-3 is the viscous stress tensor and q r is the radiative heat flux. in Equation 2-3 is the force potential. It can be formulated in some special cases, for example, for gravitational effects (Chapter 1.4 in Ref. 4), but it is in general rather difficult to derive. Potential energy is therefore often excluded and the total energy flux is approximated by THEORY FOR THE HEAT TRANSFER INTERFACES 33

34 u H u u k T + u + q r (2-4) For a simple compressible fluid, the enthalpy, H, has the form (Ref. 5) T p H = H ref + C p dt + T ref p ref T T dp p (2-5) where p is the absolute pressure. The reference enthalpy, H ref, is the enthalpy at reference temperature, T ref, and reference pressure, p ref. In COMSOL, T ref is K and p ref is one atmosphere. In theory, any value can be assigned to H ref (Ref. 7), but for practical reasons, it is given a positive value according to the following approximations Solid materials and ideal gases: H ref = C p ref T ref Gasliquid: H ref = C p ref ref T ref + p ref ref where subscript ref indicates that the property is evaluated at the reference state. The two integrals in Equation 2-5 are sometimes referred to as the sensible enthalpy (Ref. 7). These are evaluated in COMSOL by numerical integration. The second integral is only included for gas/liquid since it is commonly much smaller than the first integral for solids and it is identically zero for ideal gases. Note For the evaluation of H to work, it is important that the dependence of C p, and on the temperature are prescribed either via model input or as a function of the temperature variable. If C p, or depend on the pressure, the dependency must be prescribed either via model input or by using the variable pa which is the variable for the absolute pressure. HEAT FLUX The total heat flux vector is defined as (Ref. 6): uu kt+ q r (2-6) where U is the internal energy. It is related to the enthalpy via H = U + p -- (2-7) 34 CHAPTER 2: HEAT TRANSFER THEORY

35 What is the difference between Equation 2-4 and Equation 2-7? As an example, consider a channel with fully developed incompressible flow with all properties of the fluid independent of pressure and temperature. The walls are assumed to be insulated. Assume that the viscous heating is neglected. This is a common approximation for low-speed flows. There will be a pressure drop along the channel that drives the flow. Since there is no viscous heating and the walls are isolated, Equation 2-5 will give that H in H out. Since everything else is constant, Equation 2-4 shows that the energy flux into the channel is higher than the energy flux out of the channel. On the other hand U in U out, so the heat flux into the channel is equal to the heat flux going out of the channel. If the viscous heating on the other hand is included, then H in H out (first law of thermodynamics) and U in U out (since work has been converted to heat). Heat Flux Variables and Heat Sources This section lists some predefined variables that are available to compute heat fluxes and sources. All the variable names start with the physics interface prefix. By default the Heat Transfer interface prefix is ht. As an example, the variable named tflux can be analyzed using ht.tflux (as long as the physics interface prefix is ht). TABLE 2-1: HEAT FLUX VARIABLES VARIABLE NAME GEOMETRIC ENTITY LEVEL tflux Total heat flux domains, boundaries dflux Conductive heat flux domains, boundaries turbflux Turbulent heat flux domains, boundaries aflux Convective heat flux domain, boundaries trlflux Translation heat flux domains, boundaries teflux Total energy flux domains, boundaries ccflux_u ccflux_d Convective out-of-plane heat flux out-of-plane domains (1D and 2D) ccflux_z rflux_u rflux_d Radiative out-of-plane heat flux out-of-plane domains (1D and 2D), boundaries rflux_z q0_u q0_d q0_z Out-of-plane inward heat flux out-of-plane domains (1D and 2D) THEORY FOR THE HEAT TRANSFER INTERFACES 35

36 TABLE 2-1: HEAT FLUX VARIABLES VARIABLE NAME GEOMETRIC ENTITY LEVEL ntflux Total normal heat flux boundaries ndflux Normal conductive heat flux boundaries naflux Normal convective heat flux boundaries ntrlflux Normal translational heat flux boundaries nteflux Normal total energy flux boundaries ccflux Convective heat flux boundaries Qtot Domain heat source domains Qbtot Boundary heat source boundaries Ql Line heat source edges Qp Point heat source points DOMAIN HEAT FLUXES On domains the heat fluxes are vector quantities. Their definition can vary depending on the active features and selected properties. Total Heat Flux On domains the total heat flux, tflux, corresponds to the conductive and convective heat flux. For accuracy reasons the radiative heat flux is not included. Tip See Radiative Heat Flux to evaluate the radiative heat flux. For solid domains, for example heat transfer in solids and biological tissue domains, the total heat flux is defined by: tflux = trlflux + dflux For fluid domains (for example, heat transfer in fluids), the total heat flux is defined by: tflux = aflux + dflux Conductive Heat Flux The conductive heat flux variable, dflux is evaluated using the temperature gradient and the effective thermal conductivity: dflux = k eff T 36 CHAPTER 2: HEAT TRANSFER THEORY

37 When out-of-plane property is activated (1D and 2D only) the conductive heat flux is defined by in 2D (d z is the domain thickness) in 1D (A c is the cross-section area) dflux = d z k eff T dflux = A c k eff T In the general case k eff is the thermal conductivity, k. For heat transfer in fluids with turbulent flow k eff = k + k T where k T is the turbulent thermal conductivity. For heat transfer in porous media, k eff = k eq where k eq is the equivalent conductivity defined in the Porous Matrix feature. Turbulent Heat Flux The turbulent heat flux variable turbflux enables to access the part of the conductive heat flux that is due to the turbulence. turbflux = k T T Convective Heat Flux The conductive heat flux variable aflux is defined using the internal energy: aflux = ue When out-of-plane property is activated (1D and 2D only) the convective heat flux is defined as aflux = d z ue in 2D (d z is the domain thickness) aflux = A c ue in 1D (A c is the cross-section area) E is the internal energy defined by: EC p T for solid domains, EC p T for ideal gas fluid domains, THEORY FOR THE HEAT TRANSFER INTERFACES 37

38 EHp for other fluid domains. H is the enthalpy defined by: HC p T for solid domains, HC p Tp for ideal gas fluid domains, HC p Tp for other fluid domains. Translational Heat Flux Similar to convective heat flux but defined for solid domains with translation. The variable name is trlflux. Total Energy Flux The total energy flux, teflux, is defined when viscous heating is enabled: where the total enthalpy, H 0, is defined as: teflux = uh 0 + dflux + u H 0 = H + u u 2 Radiative Heat Flux In participating media, the radiative heat flux, q r, is not available for analysis on domains because it is much more accurate to evaluate Q r = q r the radiative heat source. OUT-OF-PLANE DOMAIN FLUXES When out-of-plane property is activated (1D and 2D only), out-of-plane domain fluxes are defined. If there are no out-of-plane features, they are evaluated to zero. Convective Out-of-Plane Heat Flux The convective out-of-plane heat flux, ceflux, is generated by the Out-of-Plane Convective Cooling feature. In 2D: upside: ccflux_u = h u T ext u T downside: ccflux_d = h d T ext d T 38 CHAPTER 2: HEAT TRANSFER THEORY