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 [email protected]. CONTACTING COMSOL BY For general product information, contact COMSOL at [email protected]. To receive technical support from COMSOL for the COMSOL products, please contact your local COMSOL representative or send your questions to [email protected]. An automatic notification and case number is sent to you by . 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

39 In 1D: ccflux_z = h z T ext z T Radiative Out-of-Plane Heat Flux The radiative out-of-plane heat flux, rflux, is generated by the Out-of-Plane Radiation feature. In 2D: upside: rflux_u = 4 u T amb u T 4 downside: rflux_d = 4 d T amb d T 4 In 1D: rflux_z = 4 z T amb z T 4 Out-of-Plane Inward Heat Flux The convective out-of-plane heat flux, q0, is generated by the Out-of-Plane Heat Flux feature. 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. In addition normal boundary heat fluxes (scalar quantity) are available on boundaries. Total Normal Heat Flux The variable ntflux is defined by: ntflux = meantflux n THEORY FOR THE HEAT TRANSFER INTERFACES 39

40 Normal Conductive Heat Flux The variable ndflux is defined by: ndflux Normal Convective Heat Flux The variable naflux is defined by: naflux Normal Translational Heat Flux The variable ntrlflux is defined by: ntrlflux Normal Total Energy Flux The variable nteflux is defined by: nteflux = = = = meandflux n meanaflux n meantrlflux n meanteflux n Radiative Heat Flux On boundaries the radiative heat flux, rflux, is a scalar quantity defined as: rflux = 4 T amb T 4 + G T 4 + q w where the terms respectively account for surface-to-ambient radiative flux, surface-to-surface radiative flux and radiation in participating net flux. Convective Heat Flux Convective heat flux, ccflux, is defined as the contribution from Convective Cooling boundary condition: When out-of-plane property is activated (1D and 2D only) the convective cooling heat flux is defined as in 2D (d z is the domain thickness): in 1D (A c is the cross section area): ccflux = ht ext T ccflux = d z ht ext T ccflux = A c ht ext T 40 CHAPTER 2: HEAT TRANSFER THEORY

41 DOMAIN HEAT SOURCES The sum of the domain heat sources added by different features are available in one variable, Q tot (SI unit: W/m 3 ). This variable Qtot is the sum of: Q which is the heat source added by Heat Source and Electromagnetic Heat Source features. Q met which is the heat source added by the Bioheat feature. Note The out-of-plane contributions (convective cooling, heat flux, and radiation), and the blood contribution in Bioheat are considered flux so that they are not added to Q tot. BOUNDARY HEAT SOURCES The sum of the boundary heat sources added by different boundary conditions is available in one variable, Q b,tot (SI unit: W/m 2 ). This variable Qbtot is the sum of: Q b which is the boundary heat source added by Boundary heat Source, Electrochemical reaction heat flux and Reaction heat flux boundary conditions. Q sh which is the boundary heat source added by Boundary Electromagnetic Heat Source boundary condition. Q s : which is the boundary heat source added by Layer heat source subfeature of Highly conductive layer. LINE AND POINT HEAT SOURCES The sum of the line heat sources is available in a variable called Ql (SI unit: W/m). The sum of the point heat sources is available in a variable called Qp (SI unit: W). 2D Axi In 2D axisymmetric models, the SI unit for the variable Qp is W/m. THEORY FOR THE HEAT TRANSFER INTERFACES 41

42 About the Boundary Conditions for the Heat Transfer Interfaces TEMPERATURE AND HEAT FLUX BOUNDARY CONDITIONS The heat equation accepts two basic types of boundary conditions: specified temperature and specified heat flux. The former is of a constraint type and prescribes the temperature at a boundary: while the latter specifies the inward heat flux where T = T 0 on n q = q 0 on q is the conductive heat flux vector (SI unit: W/m 2 ) where q = kt. n is the normal vector of the boundary. q 0 is inward heat flux (SI unit: W/m 2 ), normal to the boundary. The inward heat flux, q 0, is often a sum of contributions from different heat transfer processes (for example, radiation and convection). The special case q 0 0 is called thermal insulation. A common type of heat flux boundary conditions are those where q 0 h T inf T, where T inf is the temperature far away from the modeled domain and the heat transfer coefficient, h, represents all the physics occurring between the boundary and far away. It can include almost anything, but the most common situation is that h represents the effect of an exterior fluid cooling or heating the surface of solid, a phenomenon often referred to as convective cooling or heating. See Also The Heat Transfer Module contains a set of correlations for convective cooling and heating. See About the Heat Transfer Coefficients. OVERRIDING MECHANISM FOR HEAT TRANSFER BOUNDARY CONDITIONS Many boundary conditions are available in heat transfer. Some of them can be associated (for example, Heat Flux and Highly Conductive Layer). Others cannot be associated (for example, Heat Flux and Thermal Insulation). 42 CHAPTER 2: HEAT TRANSFER THEORY

43 Several categories of boundary condition exist in heat transfer. Table 2-2 gives the overriding rules for these groups. Temperature, Convective Outflow, Open Boundary, Inflow Heat Flux Thermal Insulation, Symmetry, Periodic Heat Condition Highly Conductive Layer Heat Flux, Convective Cooling Boundary Heat Source, Electrochemical Reaction Heat Flux, Reaction Heat Flux, Radiation Group Surface-to-Surface Radiation, Re-radiating Surface, Prescribed Radiosity, Surface-to-Ambient Radiation Opaque Surface, Incident Intensity, Continuity on interior boundaries Thin Thermally Resistive Layers TABLE 2-2: OVERRIDING RULES FOR HEAT TRANSFER BOUNDARY CONDITIONS A\B Temperature X X X X 2-Thermal Insulation X X X 3-Highly Conductive X X Layer 4-Heat Flux X X 5-Boundary heat source 6-Surface-to-surface X X radiation 7-Opaque Surface X 8-Thin Thermally Resistive Layer 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, use Table 2-2 to determine if A is overridden by B or not: Locate the line that corresponds to A group (see above the definition of the groups). In the table above only the first member of the group is displayed. Locate the column that corresponds to the group of B. THEORY FOR THE HEAT TRANSFER INTERFACES 43

44 If the corresponding cell is empty A and B contribute. If it contains an X, B overrides A. Important Group 4 and group 5 boundary conditions are always contributing. That means that they never override any other boundary condition. But they might be overridden. Example 1 Consider a boundary where Temperature is applied. Then a Surface-to-surface Radiation boundary condition is applied on the same boundary afterward. Temperature belongs to group 1 Surface-to-surface radiation belongs to group 6. The cell on the line of group 1 and the column of group 6 is empty so Temperature and Surface-to-Surface radiation contribute. Example 2 Consider a boundary where Convective Cooling is applied. Then a Symmetry boundary condition is applied on the same boundary afterward. Convective Cooling belongs to group 4. Symmetry belongs to group 2 The cell on the line of group 4 and the column of group 2 contains an X so Convective Cooling is overridden by Symmetry. Note In Example 2 above, if Symmetry followed by Convective Cooling is added, the boundary conditions contribute. Radiative Heat Transfer in Transparent Media This discussion so far has considered heat transfer by means of conduction and convection. The third mechanism for heat transfer is radiation. Consider an environment with fully transparent or fully opaque objects. Thermal radiation denotes the stream of electromagnetic waves emitted from a body at a certain temperature. 44 CHAPTER 2: HEAT TRANSFER THEORY

45 DERIVING THE RADIATIVE HEAT FLUX J =G + T 4 G x,t x,t Figure 2-1: Arriving irradiation (left), leaving radiosity (right). Consider Figure 2-1. A point x is located on a surface that has an emissivity, reflectivity, absorptivity, and temperature T. Assume the body is opaque, which means that no radiation is transmitted through the body. This is true for most solid bodies. The total arriving radiative flux at x is named the irradiation, G. The total outgoing radiative flux x is named the radiosity, J. 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, q, 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. q = 1 G T 4 (2-10) Most opaque bodies also behave as ideal gray bodies, meaning that the absorptivity and emissivity are equal, and the reflectivity is therefore given from the following relation: = = 1 (2-11) Thus, for ideal gray bodies, q is given by: q = G T 4 (2-12) THEORY FOR THE HEAT TRANSFER INTERFACES 45

46 This is the equation used as a radiation boundary condition. RADIATION TYPES It is common to differentiate between two types of radiative heat transfer: surface-to-ambient radiation and surface-to-surface radiation. Equation 2-12 holds for both radiation types, but the irradiation term, G, is different for each of them. The Heat Transfer interface supports both types of radiation. SURFACE-TO-AMBIENT RADIATION Surface-to-ambient radiation assumes the following: The ambient surroundings in view of the surface have a constant temperature, T amb. The ambient surroundings behave as a blackbody. This means that the emissivity and absorptivity are equal to 1, and zero reflectivity. These assumptions allows the irradiation to be explicitly expressed as G = 4 T amb (2-13) Inserting Equation 2-13 into Equation 2-12 results in the net inward heat flux for surface-to-ambient radiation q = 4 T amb T 4 (2-14) For boundaries where a surface-to-ambient radiation is specified, COMSOL Multiphysics adds this term to the right-hand side of Equation See Also Theory for the Surface-to-Surface Radiation Interface Theory for the Radiative Heat Transfer Interfaces Radiation and Participating Media Interactions Consistent and Inconsistent Stabilization Methods for the Heat Transfer Interfaces Several of the Heat Transfer interfaces have this advanced option to set the stabilization method parameters. Below is some information pertaining to these options. 46 CHAPTER 2: HEAT TRANSFER THEORY

47 To display this section, click the Show button ( ) and select Stabilization. Show Stabilization in the COMSOL Multiphysics User s Guide See Also Stabilization Techniques and Numerical Stabilization in the COMSOL Multiphysics Reference Guide CONSISTENT STABILIZATION This section contains two consistent stabilization methods: streamline diffusion and crosswind diffusion. These are consistent stabilization methods, which means that they do not perturb the original transport equation. The consistent stabilization methods take effect for fluids and for solids with Translational Motion. A stabilization method is active when the corresponding check box is selected. Model Continuous Casting: Model Library path Heat_Transfer_Module/ Process_and_Manufacturing/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. Crosswind Diffusion The crosswind diffusion provides extra diffusion in the region of sharp gradients. The added diffusion is orthogonal to the streamline diffusion, so streamline diffusion and crosswind diffusion can be used simultaneously. When Crosswind diffusion is selected, enter a Lower gradient limit g lim (SI unit: K/m). The default is 0.01[K]/jh.helem. The variable g lim is needed because both Equation 2-15 and Equation 2-16 contain terms of the form 1T, which become singular if T0. Hence, all occurrences of 1T are replaced by 1maxTg lim where g lim is a measure of a small gradient. The method in the Heat Transfer interfaces adds the following contribution to the weak formulation (see Codina in Ref. 2): THEORY FOR THE HEAT TRANSFER INTERFACES 47

48 N el e = 1 e 1 --max0 C 2 e 2k hr u u h Tˆ I T Td (2-15) where R is the PDE residual, Tˆ is the test function for T, h is the element size, and is defined as u 2 = C p u T T 2 T if T 0 0 if T = 0 (2-16) INCONSISTENT STABILIZATION This section contains one inconsistent stabilization method: isotropic diffusion. Adding isotropic diffusion is equivalent to adding a term to the physical diffusion coefficient. This means that the original problem is not solved, which is why isotropic diffusion is an inconsistent stabilization method. Still, the added diffusion definitely dampens the effects of oscillations, but try to minimize the use of isotropic diffusion. By default there is no isotropic diffusion. To add isotropic diffusion, select the Isotropic diffusion check box. The field for the tuning parameter id then becomes available. The default value is 0.25; increase or decrease the value of id to increase or decrease the amount of isotropic stabilization. References for the Heat Transfer Interfaces 1. F.P. Incropera and D.P. DeWitt, Fundamentals of Heat and Mass Transfer, 4th ed., John Wiley & Sons, R. Codina, Comparison of Some Finite Element Methods for Solving the Diffusion-Convection-Reaction Equation, Comp. Meth.Appl. Mech. Engrg, vol. 156, pp , A. Bejan, Heat Transfer, Wiley, G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, R.L. Panton, Incompressible Flow, 2nd ed., John Wiley & Sons, M. Kaviany, Principles of Convective Heat Transfer, 2nd ed., Springer, CHAPTER 2: HEAT TRANSFER THEORY

49 7. T. Poinsot and D. Veynante, Theoretical and Numerical Combustion, Second Edition, Edwards, THEORY FOR THE HEAT TRANSFER INTERFACES 49

50 About Infinite Elements In this section: Modeling Unbounded Domains Known Issues When Modeling Using Infinite Elements Note For more information about this feature, see About Infinite Element Domains and Perfectly Matched Layers in the COMSOL Multiphysics User s Guide. Modeling Unbounded Domains Many environments modeled with finite elements are unbounded or open, meaning that the fields extend toward infinity. The easiest approach to modeling an unbounded domain is to extend the simulation domain far enough that the influence of the terminating boundary conditions at the far end becomes negligible. This approach can create unnecessary mesh elements and make the geometry difficult to mesh due to large differences between the largest and smallest object. Another approach is to use infinite elements. There are many implementations of infinite elements available, and the elements used in this module are often referred to as mapped infinite elements (see Ref. 1). This implementation maps the model coordinates from the local, finite-sized domain to a stretched domain. The inner boundary of this stretched domain coincides with the local domain, but at the exterior boundary the coordinates are scaled toward infinity. The principle can be explained in a one-coordinate system, where this coordinate represents Cartesian, cylindrical, or spherical coordinates. Mapping multiple 50 CHAPTER 2: HEAT TRANSFER THEORY

51 coordinate directions (for Cartesian and cylindrical systems only) is just the sum of the individual coordinate mappings. t r 0 t unscaled region t p unscaled region w scaled region Figure 2-2: The coordinate transform used for the mapped infinite element technique. The meaning of the different variables are explained in the text. Figure 2-2 shows a simple view of an arbitrary coordinate system. The coordinate r is the unscaled coordinate that COMSOL Multiphysics draw the geometry in (reference system). The position r 0 is the new origin from where the coordinates are scaled, t p is the coordinate from this new origin to the beginning of the scaled region also called the pole distance, and w is the unscaled length of the scaled region. The scaled coordinate, t, approaches infinity when t approaches t p w. To avoid solver issues with near infinite values, it is possible to change the infinite physical width of the scaled region to a finite large value, pw. The true coordinate that the PDEs are formulated in is given by where t comes from the formula r' = r 0 + t w t' = t p p t t p = pw The following figures show typical examples of infinite element regions that work nicely for each of the infinite element types. These types are: Stretching in Cartesian coordinate directions, labeled Cartesian Stretching in cylindrical directions, labeled Cylindrical Stretching in spherical direction, labeled Spherical User-defined coordinate transform for general infinite elements, labeled General t p t p ABOUT INFINITE ELEMENTS 51

52 Figure 2-3: A cube surrounded by typical infinite-element regions of Cartesian type. Figure 2-4: A cylinder surrounded by typical cylindrical infinite-element regions. Figure 2-5: A sphere surrounded by a typical spherical infinite-element region. If other shapes are used for the infinite element regions not similar to the shapes shown in the previous figures, it might be necessary to define the infinite element parameters manually. 52 CHAPTER 2: HEAT TRANSFER THEORY

53 The poor element quality causes poor or slow convergence for iterative solvers and make the problem ill-conditioned in general. For this reason it is strongly recommended to use swept meshing in the infinite element domains. The sweep direction should be selected the same as the direction of scaling. For Cartesian infinite elements in regions with more than one direction of scaling it is recommended to first sweep the mesh in the domains with only one direction of scaling, then sweep the domains with scaling in two directions, and finish by sweeping the mesh in the domains with infinite element scaling in all three direction. GENERAL STRETCHING With manual control of the stretching, the geometrical parameters that defines the stretching are added as Manual Scaling subnodes. These subnodes have no effect unless the type of the Infinite Elements node is set to General. Each Manual Scaling subnode has three parameters: Scaling direction, which sets the direction from the interface to the outer boundary. Geometric width, which sets the width of the region. Coordinate at interface, which sets an arbitrary coordinate at the interface. When going from any of the other types to the General type, subnodes that represent stretching of the previous type are added automatically. Known Issues When Modeling Using Infinite Elements Be aware of the following when modeling with infinite elements: Use of One Single Infinite Elements Node Use a separate Infinite Elements node for each isolated infinite element domain. That is, to use one and the same Infinite Elements node, all infinite element domains must be in contact with each other. Otherwise the infinite elements do not work properly. Element Quality The coordinate scaling resulting from infinite elements also yields an equivalent stretching or scaling of the mesh that effectively results in a poor element quality. (The element quality displayed by the mesh statistics feature does not account for this effect.) The poor element quality causes poor or slow convergence for iterative solvers and make the problem ill-conditioned in general. For this reason, it is strongly recommended to use swept meshing in the infinite element domains. The sweep direction should be selected the same as the direction of scaling. For Cartesian infinite ABOUT INFINITE ELEMENTS 53

54 elements in regions with more than one direction of scaling it is recommended to first sweep the mesh in the domains with only one direction of scaling, then sweep the domains with scaling in two directions, and finish by sweeping the mesh in the domains with infinite element scaling in all three direction. Complicated Expressions The expressions resulting from the stretching get quite complicated for spherical infinite elements in 3D. This increases the time for the assembly stage in the solution process. After the assembly, the computation time and memory consumption is comparable to a problem without infinite elements. The number of iterations for iterative solvers might increase if the infinite element regions have a coarse mesh. Erroneous Results Infinite element regions deviating significantly from the typical configurations shown in the beginning of this section can cause the automatic calculation of the infinite element parameter to give erroneous result. Enter the parameter values manually if this is the case. See General Stretching. Use the Same Material Parameters or Boundary Conditions The infinite element region is designed to model uniform regions extended toward infinity. Avoid using objects with different material parameters or boundary conditions that influence the solution inside an infinite element region. REFERENCE FOR INFINITE ELEMENTS 1. O.C. Zienkiewicz, C. Emson, and P. Bettess, A Novel Boundary Infinite Element, International Journal for Numerical Methods in Engineering, vol. 19, no. 3, pp , CHAPTER 2: HEAT TRANSFER THEORY

55 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 a surface by natural or forced convection. In principle, it is possible to model this process in two ways: Use a heat transfer coefficient on the convection-cooled surfaces Extend the model to describe the flow and heat transfer in the cooling fluid The second approach is the correct approach if the geometry or the external flow is complicated. The Heat Transfer Module includes the Conjugate Heat Transfer interface for this purpose. However, such a simulations can become costly, both in terms of computational time and memory requirement. The first method is simple, yet powerful and efficient. Convection cooling is then modeled by specifying the heat flux on the boundaries that interface with the cooling fluid as being proportional to the temperature difference across a fictitious thermal boundary layer. Mathematically, the heat flux is described by the equation where h is a heat transfer coefficient and T inf the temperature of the external fluid far from the boundary. The main difficulty in using heat transfer coefficients is in calculating or specifying the appropriate value of the h coefficient. That coefficient depends on the cooling fluid, the fluid s material properties, and the surface temperature and, for forced-convection cooling, also on the fluid s flow rate. In addition, the geometrical configuration affects the coefficient. The Heat Transfer interface provides built-in functions for heat transfer coefficients. For most engineering purposes, the use of these coefficients is an accurate and numerically efficient modeling approach. In this section: n kt = ht inf T Heat Transfer Coefficient Theory Nature of the Flow the Grashof Number Available Heat Transfer Coefficients References for the Heat Transfer Coefficients ABOUT THE HEAT TRANSFER COEFFICIENTS 55

56 Heat Transfer Coefficient Theory It is possible to divide convection cooling into four main categories depending on the type of convection conditions (natural or forced) and on the type of geometry (internal or external convection flow). In addition, these four cases can all experience either laminar or turbulent flow conditions, resulting in a total of eight types of convection, as in Figure 2-6. Natural Forced External Internal Laminar Flow Turbulent Flow Figure 2-6: The eight possible categories of convective cooling. The difference between natural and forced convection is that in the latter case an external force such as a fan creates the flow. In natural convection, buoyancy forces induced by temperature differences and the thermal expansion of the fluid drive the flow. Heat transfer handbooks generally contain a large set of empirical and theoretical correlations for h coefficients. The Heat Transfer Module includes a subset of them. The expressions are based on the following set of dimensionless numbers: The Nusselt number, Nu L (Re, Pr, Ra)hL/k The Reynolds number, Re L U L/ The Prandtl number, PrC p /k The Rayleigh number, RaGr Pr 2 gc p T L 3 /(k) 56 CHAPTER 2: HEAT TRANSFER THEORY

57 where h is the heat transfer coefficient (SI unit: W/(m 2 K)). L is the characteristic length (SI unit: m). T is the temperature difference between surface and cooling fluid bulk (SI unit: K). g is the acceleration of gravity (SI unit: m/s 2 ). k is the thermal conductivity of the fluid (SI unit: W/(m K)). is the fluid density (SI unit: kg/m 3 ). U is the bulk velocity (SI unit: m/s). is the dynamic viscosity (SI unit: Pa s). C p equals the heat capacity of the fluid (SI unit: J/(kg K)). is the thermal expansivity (SI unit: 1/K) Further, Gr refers to the Grashof number, which is defined as the ratio between the buoyancy force and the viscous force. Nature of the Flow the Grashof Number In cases of externally driven flow, such as forced convection, the flow s nature is characterized by the Reynolds number, Re, which describes the ratio of the inertial to viscous forces. However, the velocity is largely unknown for internally driven flows such as natural convection. In such cases the Grashof number, Gr, characterizes the flow. It describes the ratio of the internal driving force (buoyancy force) to a viscous force acting on the fluid. Similar to the Reynolds number it requires the definition of a length scale, the fluid s physical properties, and the temperature scale (temperature difference). The Grashof number is defined as: gt s T 0 L 3 Gr L = where g is the acceleration of gravity, is the fluid s coefficient of volumetric thermal expansion, T s denotes the temperature of the hot surface, T 0 equals the temperature of the surrounding air, L is the length scale, represents the fluid s dynamic viscosity, and is the density. In general, the coefficient of volumetric thermal expansion is given by ABOUT THE HEAT TRANSFER COEFFICIENTS 57

58 = T p which for an ideal gas reduces to = 1 T The transition from laminar to turbulent flow occurs at a Gr value of 10 9 ; the flow is turbulent for larger values. Available Heat Transfer Coefficients EXTERNAL NATURAL CONVECTION Vertical Wall The correlations are equations 9.26 and 9.27 in Ref. 1: h = --- k 0.67Ra1/ 4 L L 0.492k / 16 4/ 9 C p Ra L k 0.387Ra1/ L L 0.492k / 16 8/ 27 Ra L 10 9 C p (2-17) where L, the height of the wall, is a correlation input and g T p C p T T ext L 3 Ra L = k (2-18) where in turn g is the acceleration of gravity equal to 9.81 m/s 2. All material properties are evaluated at TT ext 2. Inclined Wall The correlations are equations 9.26 and 9.27 in Ref. 1 (same as for vertical wall): 58 CHAPTER 2: HEAT TRANSFER THEORY

59 h = --- k 0.67cosRaL 1/ L 0.492k / 16 4/ 9 C p Ra L k 0.387Ra1/ 6 L L 0.492k / 16 8/ 27 Ra L 10 9 C p (2-19) where L, the height of the wall, is a correlation input and is the tilt angle (angle between the wall and the vertical direction, =0 for vertical walls). These correlations are valid for -60 < < 60. The definition of Raleigh number, Ra L, 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 (2-20) where in turn g denotes the gravitational acceleration, equal to 9.81 m/s 2. For turbulent flow, 1 is used instead of cos( ) in the expression for h, since this gives better accuracy (see Ref. 2). Note According to Ref. 1., correlations for inclined walls are only satisfactory for the top side of a cold plate or the down face of a hot plate. Hence, these correlations are not recommended for the bottom side of a cold face and for the top side of a hot plate. The laminar-turbulent transition depends on (see Ref. 2). Unfortunately, few data is available about this transition. There is some data available in Ref. 2 but this data gives only approximations of this transition, according to the authors. In addition, data is only provided for water (Pr around 6). For this reasons, we define a flow as turbulent, independently of value, when g T p C p T T ext L k 9 All material properties are evaluated at TT ext 2. ABOUT THE HEAT TRANSFER COEFFICIENTS 59

60 Horizontal Plate, Upside The correlations are equations in Ref. 1 but can also be found as equations 7.77 and 7.78 in Ref. 2. If TT ext, then h k Ra1/ 4 L L Ra L 10 7 = k Ra1/ 3 L L Ra L 10 7 (2-21) while if T T ext, then h = k Ra1/ 4 L L (2-22) Ra L is given by Equation 2-18, and L, the plate diameter (defined as area/perimeter, see Ref. 2) is a correlation input. The material data are evaluated at TT ext 2. Horizontal Plate, Downside Equation 2-21 is used when T T ext and Equation 2-22 is used when T T ext. Otherwise it is the same implementation as for Horizontal plate, upside. INTERNAL NATURAL CONVECTION Narrow Chimney, Parallel Plates If Ra L HL and T T ext, then h = ---- k 1 H Ra L (2-23) where L, the plate distance, and H, the chimney height, are correlation inputs (equation 7.96 in Ref. 2). Ra L is given by Equation The material data are evaluated at TT ext 2. Narrow Chimney, Circular Tube If Ra D HD, then k h = Ra 1 H128 D where D, the tube diameter, and H, the chimney height, are correlation inputs (table 7.2 in Ref. 2 with D h D). Ra D is given by Equation 2-18 with L replaced by D. The material data are evaluated at TT ext CHAPTER 2: HEAT TRANSFER THEORY

61 EXTERNAL FORCED CONVECTION Plate, Averaged Transfer Coefficient This correlation is an assembly of equations 7.34 and 7.41 in Ref. 1: h = 2 k L Pr 1/ 3 Re 1/ L Pr 2/ 3 1/ 4 Re L k L ---Pr1/ Re4/ 5 L 871 Re L (2-24) where Prc p k and Re L U ext L. L, the plate length and U ext, the exterior velocity are correlation inputs. The material data are evaluated at TT ext 2. Plate, Local Transfer Coefficient This correlation corresponds to equations 5.79b and in Ref. 2: h = k Pr 1 3 maxx eps k Pr 1 3 maxx eps / Re x 1 2 / Re x 4 5 / Re 5 10 x 5 / Re 5 10 x 5 (2-25) where Prc p k and Re x U ext x. x, the position along the plate, and U ext, the exterior velocity are correlation inputs. The material data are evaluated at TT ext 2. INTERNAL FORCED CONVECTION Isothermal Tube This correlation corresponds to equations 8.55 and 8.61 in Ref. 1: h = --- k 0.027Re4 5 D D k Re D D 2500 / Pr n T Re D 2500 (2-26) where Prc p k, Re D U ext D and n0.3 if TT ext and n0.4 if T T ext. D, the tube diameter and U ext, the exterior velocity, are correlation inputs. All material data are evaluated at T ext except T which is evaluated at the wall temperature, T. ABOUT THE HEAT TRANSFER COEFFICIENTS 61

62 References for the Heat Transfer Coefficients 1. F.P. Incropera and D.P. DeWitt, Fundamentals of Heat and Mass Transfer, Fifth ed. John Wiley & Sons, A. Bejan, Heat Transfer, John Wiley & Sons, CHAPTER 2: HEAT TRANSFER THEORY

63 About Highly Conductive Layers 2D 2D Axi This module supports heat transfer in highly conductive layers in 2D, 2D axisymmetry, and 3D. 3D The highly conductive layer feature is efficient for modeling heat transfer in thin layers without the need to create a fine mesh for them. The material in the thin layer must be a good thermal conductor. A good example is a copper trace on a printed circuit board, where the traces are good thermal conductors compared to the board s substrate material. More generally, the highly conductive layer feature can be applied in a part of a geometry with the following properties: The part is a thin layer compared to the thickness of the adjacent geometry The part is a good thermal conductor compared to the adjacent geometry Because the layer is very thin and has a high thermal conductivity, you can assume that no variations in temperature and in-plane heat flux exist along the layer s thickness. Furthermore, think of the difference in heat flux in the layer s normal direction between its upper and lower face as a heat source or sink that is smeared out along the layer thickness. A significant benefit is that a layer can be represented as a boundary instead of a domain, which simplifies the geometry and reduces the required number of mesh ABOUT HIGHLY CONDUCTIVE LAYERS 63

64 elements. Figure 2-7 shows an example where a highly conductive layer reduces the mesh density significantly. Copper wire modeled with a mesh Copper wire represented as a highly conductive layer Figure 2-7: Modeling a copper wire as a domain (top) requires a denser mesh compared to modeling it as a boundary with a highly conductive layer (bottom). To describe heat transfer in highly conductive layers, the Highly Conductive Layer feature uses a variant of the heat equation that describes the in-plane heat flux in the layer: T d s s C s t t d s k s t T = q q + d s Q S = q s (2-27) Here the operator t denotes the del or nabla operator projected onto the plane of the highly conductive layer. The properties in the equation are: s is the layer density (kgm 3 ) C s is the layer heat capacity (J(kg K)) k s is the layer thermal conductivity at constant pressure (W(m K)) d s is the layer thickness (m) q is the heat flux from the surroundings into the layer (Wm 2 ) q is the heat flux from the layer into the domain (Wm 2 ) Q s represents internal heat sources within the conductive layer (Wm 3 ) q s is the net outflux of heat through the top and bottom faces of the layer (Wm 2 ) With the above boundary equation inserted, the general heat flux boundary condition becomes 64 CHAPTER 2: HEAT TRANSFER THEORY

65 n q = T d s s C p s t t d s k s t T on See Also Highly Conductive Layer Features ABOUT HIGHLY CONDUCTIVE LAYERS 65

66 Theory of Out-of-Plane Heat Transfer See Also Out-of-Plane Heat Transfer Features When the object to model in COMSOL Multiphysics is thin or slender enough along one of its geometry dimensions, there is usually only a small variation in temperature along the object s thickness or cross section. For such objects, it is efficient to reduce the model geometry to 2D or even 1D and use the out-of-plane heat transfer mechanism. Figure 2-8 shows examples of likely situations where this type of geometry reduction can be applied. q q up q down Figure 2-8: Geometry reduction from 3D to 1D (top) and from 3D to 2D (bottom). The reduced geometry does not include all the boundaries of the original 3D geometry. For example, the reduced geometry does not represent the upside and downside surfaces of the plate in Figure 2-8 as boundaries. Instead, heat transfer through these boundaries appears as sources or sinks in the thickness-integrated version of the heat equation used when out-of-plane heat transfer is active. Equation Formulation When out-of-plane heat transfer is enabled, the equation for heat transfer in solids, Equation 3-1 is replaced by T d z C p d t z kt = d z Q (2-28) 66 CHAPTER 2: HEAT TRANSFER THEORY

67 where d z is the thickness of the domain in the out-of-plane direction. The equation for heat transfer in fluids, Equation 3-2, is replaced by C p d T z u T = d t z kt + d z Q (2-29) The Pressure Work attribute on Solids and Fluids and the Viscous Heating attribute on Fluids are not available when out-of-plane heat transfer is activated. Note Heat Source features that are added to a model with out-of-plane heat transfer enabled are multiplied by the thickness, d z. Boundary conditions are also adjusted. Activating Out-of-Plane Heat Transfer and Thickness Using a 1D or 2D model, activate the features for out-of-plane heat transfer and the thickness property by clicking the main Heat Transfer feature and selecting the Out-of-plane heat transfer check box under Physical Model. THEORY OF OUT-OF-PLANE HEAT TRANSFER 67

68 Theory for the Bioheat Transfer Interface The Bioheat Transfer Interface uses the bioheat equation and the corresponding features in the Heat Transfer interface, This is used to model heat transfer within biological tissue. This feature uses Pennes approximation to represent heat sources from metabolism and blood perfusion. The equation for conductive heat transfer using this approximation: C T p + k T = b C t b b T b T+ Q met (2-30) The density, heat capacity C p, and thermal conductivity k are the thermal properties of the tissue. For a steady-state problem the temperature does not change with time and the first term disappears. To model Equation 2-30 add the Biological Tissue model equation, with a Bioheat feature. The Biological Tissue model provides the left-hand side of Equation 2-30 while the Bioheat feature provides the two source terms on the right-hand side of Equation Reference for the Bioheat Interface 1. A. Bejan, Heat Transfer, Wiley, CHAPTER 2: HEAT TRANSFER THEORY

69 Theory for the Heat Transfer in Porous Media 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: C p C p u T = k eq T+ Q eq T t (2-31) with the following material properties: is the fluid density. C p is the fluid heat capacity at constant pressure. (C p ) eq is the equivalent volumetric heat capacity at constant pressure. k eq is the equivalent thermal conductivity (a scalar or a tensor if the thermal conductivities are anisotropic). u is the fluid velocity field, either an analytic expression or a velocity field from a fluid-flow interface. u should be interpreted as the Darcy velocity, that is, the volume flow rate per unit cross-sectional area. The average linear velocity (the velocity within the pores) can be calculated as u L u L, where L is the fluid s volume fraction, or equivalently the porosity. Q is the heat source (or sink). Add one or several heat sources as separate features. The equivalent thermal conductivity of the solid-fluid system, k eq, is related to the conductivity of the solid k p and to the conductive of the fluid, 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, which is related to the volume fraction of the liquid L (or porosity) by L + = 1 For a steady-state problem the temperature does not change with time, and the first term in the left-hand side of Equation 2-31 disappears. p THEORY FOR THE HEAT TRANSFER IN POROUS MEDIA INTERFACE 69

70 70 CHAPTER 2: HEAT TRANSFER THEORY

71 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 the The Radiation Heat Transfer Branch. In this chapter: About the Heat Transfer Interfaces The Heat Transfer Interface Heat Transfer Interface Advanced Features Highly Conductive Layer Features Out-of-Plane Heat Transfer Features The Bioheat Transfer Interface The Heat Transfer in Porous Media Interface 71

72 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 suitable for modeling heat transfer in solids and fluids, porous media, and biological tissue. The interfaces are available in 1D, 2D, and 3D and for axisymmetric models with cylindrical coordinates in 1D and 2D. The default dependent variable is the temperature, T. There are Heat Transfer interfaces displayed in the Model Builder with the same name but with different icons and default models. After selecting a Heat Transfer interface in the Model Wizard, default settings are added under the main node. For example, if Heat Transfer in Solids ( ) is selected, a Heat Transfer node is added with a default Heat Transfer in Solids model. If Heat Transfer in Fluids ( ) is selected, a Heat Transfer in Fluids model is added instead, but the parent nodes are both called Heat Transfer. Any interface based on the main Heat Transfer feature has the suffix ht. Tip To locate and search all the documentation, in COMSOL, select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree. Heat Transfer in Solids and Heat Transfer in Fluids Use the Heat Transfer in Solids (ht) ( ) to model mainly heat transfer in solid materials. A default Heat Transfer in Solids model is added, but all functionality for including fluid domains is also available. This interface has Surface-to-surface radiation and radiation in participating media check boxes to model radiation. Use the Heat Transfer in Fluids (ht) ( ) to model mainly heat transfer in fluid materials. A default Heat Transfer in Fluids model is added, but all functionality for including solid domains is also available. This interface has Surface-to-surface radiation and radiation in participating media check boxes to model radiation. 72 CHAPTER 3: THE HEAT TRANSFER BRANCH

73 Heat Transfer in Porous Media Use the Heat Transfer in Porous Media (ht) ( ) to model mainly heat transfer in porous materials. The Porous Matrix and Heat Transfer in Fluids models are added, but all functionality for including solid domains is also available. Heat Transfer with Surface-to-Surface Radiation Use the Heat Transfer with Surface-to-Surface Radiation (ht) ( ), found under the Radiation branch ( ), to model heat transfer that includes surface-to-surface radiation. It is a Heat Transfer interface with the Physical Model>Surface-to-surface radiation check box selected, which enables the Radiation Settings section. A Heat Transfer in Solids default model with surface-to-surface radiation is added and available as a boundary condition, but all functionality to include both solid and fluid domains is also available. All available features are as described in Domain, Boundary, Edge, Point, and Pair Features for the Heat Transfer Interfaces. Model Radiative Heat Transfer in a Utility Boiler: Model Library path Heat_Transfer_Module/Process_and_Manufacturing/boiler Bioheat Transfer Use the Bioheat Transfer (ht) ( ) to model heat transfer in biological tissue. A Biological Tissue default model is added, but all functionality to include both solid and fluid domains is also available. See The Bioheat Transfer Interface. Joule Heating Select Joule Heating (jh) ( ), found under the Electromagnetic Heating subbranch ( ), to combine all features from the Electric Currents interface with the Heat Transfer interface for modeling Joule heating (also called resistive heating or ohmic heating). See The Joule Heating Interface in the COMSOL Multiphysics User s Guide. Conjugate Heat Transfer Select the Conjugate Heat Transfer (nitf) ( ), Laminar Flow ( ) or Conjugate Heat Transfer, Turbulent Flow ( ) interfaces to use a predefined multiphysics coupling ABOUT THE HEAT TRANSFER INTERFACES 73

74 consisting of a Single-Phase Flow interface, using a compressible formulation, in combination with a Heat Transfer interface. See The Conjugate Heat Transfer Branch. Note The Non-Isothermal Flow (nitf) ( ), Laminar Flow ( ) and Non-Isothermal Flow, Turbulent Flow ( ) interfaces, found under the Fluid Flow branch, are identical to the Conjugate Heat Transfer interfaces. The only difference is that Fluid is selected as the default model. If Heat transfer in solids is selected as the default model, the interface changes to a Conjugate Heat Transfer interface. To change the default model, select the Heat Transfer interface node and locate the Physical Model section in the settings window. Surface-to-Surface Radiation Select Surface-to-Surface Radiation (rad) ( ), found under the Radiation branch ( ), to use a model that treats heat transfer by surface-to-surface radiation as a process that transfers energy directly between boundaries. The radiation therefore contributes to the boundary conditions rather than to the heat equation itself. See The Surface-To-Surface Radiation Interface. This physics interface solves only for the radiation variable. To solve for radiation and temperature, use a Heat Transfer interface instead. Radiation in Participating Media Select Radiation in Participating Media (rpm) ( ), found under the Radiation branch ( ), to model radiative heat transfer inside a participating medium. This physics interface solves for radiative intensity field. See The Radiation in Participating Media Interface. This physics interface solves only for radiation variables. In order to solve for radiation and temperature, use a Heat Transfer interface. Heat Transfer in Thin Shells Select Heat Transfer in Thin Shells (htsh) ( ) to model conductive heat transfer in thin thermally conducting shells. The Thin Conductive Shell interface is added when this is selected. A Thin Conductive Layer default model, and all functionality for modeling heat conduction and out-of-plane radiation and convective cooling is added. See The Heat Transfer in Thin Shells Interface. 74 CHAPTER 3: THE HEAT TRANSFER BRANCH

75 The Heat Transfer Interface The Heat Transfer (ht) interface is available in many forms and each one has the equations, boundary conditions, and sources for modeling conductive and convective heat transfer, and solving for the temperature. When this interface is added, default nodes are added to the Model Builder based on the selection made in the Model Wizard Heat Transfer in Solids or Heat Transfer in Fluids, Thermal Insulation (the default boundary condition), and Initial Values.Right-click the Heat Transfer node to add other features that implement, for example, boundary conditions and sources. Note Depending on the version of the Heat Transfer interface selected, these default nodes may be different. INTERFACE IDENTIFIER The interface identifier is a text string that can be used to reference the respective physics interface if appropriate. Such situations could occur when coupling this interface to another physics interface, or when trying to identify and use variables defined by this physics interface, which is used to reach the fields and variables in expressions, for example. It can be changed to any unique string in the Identifier field. The default identifier (for the first interface in the model) is ht. See Also About the Heat Transfer Interfaces DOMAIN SELECTION The default setting is to include All domains in the model to define heat transfer and a temperature field. To choose specific domains, select Manual from the Selection list. THE HEAT TRANSFER INTERFACE 75

76 PHYSICAL MODEL Select a Default model Heat transfer in solids or Heat transfer in fluids. 1D 2D If required for 1D or 2D models, select the Out-of-plane heat transfer check box and then enter the Thickness of the plane (d z ). The default is 1 m and applies to the entire geometry. If another thickness is specified for some of the domains, use the Change Thickness feature. Note If Heat Transfer with Surface-to-Surface Radiation (ht) is selected from the Model Wizard, the Surface-to-surface radiation check box selected by default. This enables the Radiation Settings section as described. All other available features are the same as described for the Heat Transfer interface. Thermo-Photo-Voltaic Cell: Model Library path Heat_Transfer_Module/Electronics_and_Power_Systems/tpv_cell Model Cavity Radiation: Model Library path Heat_Transfer_Module/ Verification_Models/cavity_radiation Select the Surface-to-surface radiation check box to add a Radiation Settings section on the Heat Transfer interfaces. Select the Radiation in participating media check box to add a Participating Media Settings section. Select the Heat transfer in porous media check box to add Porous Matrix nodes for modeling the thermal properties of the immobile solids in the porous matrix. Select the Heat Transfer in biological tissue check box to make Biological Tissue the default model. See the Physical Model section in The Bioheat Transfer Interface for details. RADIATION SETTINGS To display this section select the Surface-to-surface radiation check box under Physical Model on any version of the Heat Transfer interface settings window. 76 CHAPTER 3: THE HEAT TRANSFER BRANCH

77 Select a Surface-to-surface radiation method Hemicube (the default) or Direct area integration. See below for descriptions of each method. If Hemicube is selected, select a Radiation resolution 256 is the default. If Direct area integration is selected, select a Radiation integration order 4 is the default. For either method, also select the Use radiation groups check box to enable the ability to define radiation groups, which can, in many cases, speed up the radiation calculations. Hemicube Hemicube is the default method for the heat transfer interfaces. The more sophisticated and general hemicube method uses a z-buffered projection on the sides of a hemicube (with generalizations to 2D and 1D) to account for shadowing effects. Think of it as rendering digital images of the geometry in five different directions (in 3D; in 2D only three directions are needed), and counting the pixels in each mesh element to evaluate its view factor. Its accuracy can be influenced by setting the Radiation resolution of the virtual snapshots. The number of z-buffer pixels on each side of the 3D hemicube equals the specified resolution squared. Thus the time required to evaluate the irradiation increases quadratically with resolution. In 2D, the number of z-buffer pixels is proportional to the resolution property, and thus the time is, as well. For an axisymmetric geometry, G m and F amb must be evaluated in a corresponding 3D geometry obtained by revolving the 2D boundaries about the axis. COMSOL Multiphysics creates this virtual 3D geometry by revolving the 2D boundary mesh into a 3D mesh. The resolution can be controlled in the azimuthal direction by setting the number of azimuthal sectors, which is the same as the number of elements to a full revolution. Try to balance this number against the mesh resolution in the rz-plane. Direct Area Integration COMSOL Multiphysics evaluates the integrals in Equation 2-10 and Equation 2-11 directly, without considering which face elements are obstructed by others. This means that shadowing effects (that is, surface elements being obstructed in nonconvex cases) are not taken into account. Elements facing away from each other are, however, excluded from the integrals. THE HEAT TRANSFER INTERFACE 77

78 Direct area integration is fast and accurate for simple geometries with no shadowing, or where the shadowing can be handled by manually assigning boundaries to different groups. Note If shadowing is ignored, global energy is not conserved. Control the accuracy by specifying a Radiation integration order. Sharp angles and small gaps between surfaces may require a higher integration order for accuracy but also more time to evaluate the irradiation. PARTICIPATING MEDIA SETTINGS To display this section select the Radiation in participating media check box under Physical Model on any version of the Heat Transfer interface settings window. Define the Refractive index of the participating media, n. Note The same refractive index is used for the whole model. Select the Discrete ordinates method order from the list. This order defines the discretization of the radiative intensity direction. 3D In 3D, S2, S4, S6, and S8 generate 8, 24, 48, and 80 directions, respectively. 2D In 2D, S2, S4, S6, and S8 generate 4, 12, 24, and 40 directions, respectively. 78 CHAPTER 3: THE HEAT TRANSFER BRANCH

79 Select Linear (the default), Quadratic, Cubic, Quartic, or Quintic to define the discretization level of the Radiative intensity fields. DEPENDENT VARIABLES The Heat Transfer interface has a dependent variable for the Temperature T. For surface-to-surface radiation, there is a dependent variable for the Surface radiosity J. The dependent variable names can be changed but the names of fields and dependent variables must be unique within a model. ADVANCED SETTINGS To display this section, click the Show button ( ) and select Advanced Physics Options. Normally these settings do not need to be changed. Performance Index Select a Performance index P index from the list. Select a value between 0 and 1 that modifies the strategy used to define automatic solver settings. The default is 0.5. With small values, a robust setting for the solver is expected. With large values (up to 1), less memory is needed to solve the model. DISCRETIZATION To display this section, click the Show button ( ) and select Discretization. Select a Temperature Quadratic (the default), Linear, Cubic, or Quartic. Select an element order for the Surface Radiosity Linear (the default), Quadratic, Cubic, or Quartic. Specify the Value type when using splitting of complex variables Real (the default) or Complex. CONSISTENT AND INCONSISTENT STABILIZATION To display this section, click the Show button ( ) and select Stabilization. The Streamline diffusion check box is selected by default and should remain selected for optimal performance for heat transfer in fluids or other applications that include a convective or translational term. Crosswind diffusion provides extra diffusion in the region of sharp gradients. The added diffusion is orthogonal to the streamline diffusion, so streamline diffusion and crosswind diffusion can be used simultaneously. THE HEAT TRANSFER INTERFACE 79

80 If Crosswind diffusion is selected, enter a Lower gradient limit g lim (SI unit: K/m). The default is 0.01[K]/ht.helem. See Also Show More Physics Options Domain, Boundary, Edge, Point, and Pair Features for the Heat Transfer Interfaces Consistent and Inconsistent Stabilization Methods for the Heat Transfer Interfaces Theory for the Heat Transfer Interfaces Show Stabilization in the COMSOL Multiphysics User s Guide Domain, Boundary, Edge, Point, and Pair Features for the Heat Transfer Interfaces The Heat Transfer Interface has these domain, boundary, edge, point, and pair conditions available (listed in alphabetical order): Boundary Heat Source Continuity on Interior Boundary Convective Cooling Continuity Heat Flux Heat Source Heat Transfer in Fluids Heat Transfer in Solids Highly Conductive Layer Initial Values Inflow Heat Flux Line Heat Source Open Boundary Outflow Pair Boundary Heat Source Pair Thin Thermally Resistive Layer Periodic Heat Condition 80 CHAPTER 3: THE HEAT TRANSFER BRANCH

81 Point Heat Source Pressure Work Reaction Heat Flux Surface-to-Ambient Radiation Symmetry Temperature Thermal Insulation (the default boundary condition) Thin Thermally Resistive Layer Translational Motion Viscous Heating 1D Axi 2D Axi For axisymmetric models, COMSOL Multiphysics takes the axial symmetry boundaries into account and automatically adds an Axial Symmetry node to the model that is valid on the axial symmetry boundaries only. Tip To locate and search all the documentation, in COMSOL, select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree. Heat Transfer in Solids The Heat Transfer in Solids model uses the heat equation version in Equation 3-1 as the mathematical model for heat transfer in solids: T C p kt = Q t (3-1) For a steady-state problem the temperature does not change with time and the first term disappears. It has these material properties: density, heat capacity C p, thermal conductivity k (a scalar or a tensor if the thermal conductivity is anisotropic), and Q, which is the heat source (or sink) one or more heat sources can be added separately. THE HEAT TRANSFER INTERFACE 81

82 When parts of the model (for example, a heat source) are moving, right-click the Heat Transfer in Solids node to add a Translational Motion feature to take this into account. Add Pressure Work or Opaque to the Heat Transfer in Solids node as required. Model 2D Heat Transfer Benchmark with Convective Cooling: Model Library path COMSOL_Multiphysics/Heat_Transfer/heat_convection_2d DOMAIN SELECTION From the Selection list, choose the domains to define the heat transfer. The default is to use all domains. 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. Initially, this section is empty. COORDINATE SYSTEM SELECTION The Global coordinate system is selected by default. The Coordinate system list contains any additional coordinate systems that the model includes (except boundary coordinate systems). The coordinate system is used for interpreting directions of orthotropic and anisotropic thermal conductivity. HEAT CONDUCTION The default setting is to use the Thermal conductivity k (SI unit: W/(m K)) From material. If User defined is selected, choose Isotropic, Diagonal, Symmetric, or Anisotropic based on the characteristics of the thermal conductivity, and enter another value or expression. The thermal conductivity describes the relationship between the heat flux vector q and the temperature gradient T as in q = kt, which is Fourier s law of heat conduction. Enter this quantity as power per length and temperature. The components of a thermal conductivity k in the case that it is a tensor (k xx, k yy, and so on) are available as ht.kxx, ht.kyy, and so on (using the default Heat Transfer interface identifier ht). The single scalar mean effective thermal conductivity ht.kmean is the mean value of the diagonal elements k xx, k yy, and k zz. 82 CHAPTER 3: THE HEAT TRANSFER BRANCH

83 THERMODYNAMICS The default Density (SI unit: kg/m 3 ) and Heat capacity at constant pressure C p (SI unit: J/(kg K)) use values From material. Select User defined to enter other values or expressions. The heat capacity at constant pressure describes the amount of heat energy required to produce a unit temperature change in a unit mass. Model Axisymmetric Transient Heat Transfer: Model Library path COMSOL_Multiphysics/Heat_Transfer/heat_transient_axi Translational Motion The Translational Motion node provides movement by translation to model heat transfer in solids. It adds the following contribution to the right-hand side of Equation 3-1: C p u T The contribution describes the effect of a moving coordinate system that is required to model, for example, a moving heat source. Caution Special care must be taken at boundaries where n u0. The Heat Flux boundary condition does not, for example, work at boundaries where n u0. Model Heat Generation in a Disc Brake: Model Library path Heat_Transfer_Module/Tutorial_Models/brake_disc THE HEAT TRANSFER INTERFACE 83

84 DOMAIN SELECTION From the Selection list, choose the domains to prescribe a translational motion. Note By default, the selection is the same as for the Heat Transfer in Solids node that it is attached to, but it is possible to use more than one Heat Translation subnode, each covering a subset of the Heat Transfer in Solids node s selection. TRANSLATIONAL MOTION Enter component values for x, y, and z (in 3D) for the Velocity field u trans (SI unit: m/s). Heat Transfer in Fluids The Heat Transfer in Fluids model uses the following version of the heat equation as the mathematical model for heat transfer in fluids: T C p C t p u T = kt+ Q (3-2) For a steady-state problem the temperature does not change with time and the first term disappears. It has these material properties: The density () The fluid heat capacity at constant pressure (C p ) describes the amount of heat energy required to produce a unit temperature change in a unit mass The fluid thermal conductivity (k) a scalar or a tensor if the thermal conductivity is anisotropic The fluid velocity field (u) can be an analytic expression or a velocity field from a fluid-flow interface The heat source (or sink) (Q) one or more heat sources can be added separately Note Right-click to add Viscous Heating (for heat generated by viscous friction), Opaque, or Pressure Work nodes to the Heat Transfer in Fluids feature. Also, the ratio of specific heats is defined. It is the ratio of heat capacity at constant pressure, C p, to heat capacity at constant volume, C v. When using the ideal gas law to 84 CHAPTER 3: THE HEAT TRANSFER BRANCH

85 describe a fluid, specifying is enough to evaluate C p. For common diatomic gases such as air, 1.4 is the standard value. Most liquids have 1.1 while water has 1.0. is used in the streamline stabilization and in the variables for heat fluxes and total energy fluxes. It is also used if the ideal gas law is applied. See Thermodynamics. Model Heat Transfer by Free Convection: Model Library path COMSOL_Multiphysics/Multiphysics/free_convection DOMAIN SELECTION From the Selection list, choose the domains to define the heat transfer. MODEL INPUTS This section has fields and values that are inputs to expressions that define material properties. If such user-defined property groups are added, the model inputs appear here. There are also two standard model inputs Absolute pressure and Velocity field. The absolute pressure is used in some predefined quantities that include the enthalpy (the energy flux, for example). Note Absolute pressure is also used if the ideal gas law is applied. See Thermodynamics. Absolute Pressure Enter the Absolute pressure p A (SI unit: Pa). The default is atmosphere pressure, 1 atm (101,325 Pa). This section controls both the variable as well as any property value (reference pressures) used when solving for pressure. There are usually two ways of calculating the pressure when describing fluid flow, and mass and heat transfer. Solve for the absolute pressure or a pressure (often denoted gauge pressure) that relates back to the absolute pressure through a reference pressure. Using one or the other usually depends on the system and the equations being solved for. For example, in a straight incompressible flow problem, the pressure drop over the modeled domain is probably many orders of magnitude less than atmospheric pressure, THE HEAT TRANSFER INTERFACE 85

86 which, if included, reduces the chances for stability and convergence during the solving process for this variable. In other cases, the absolute pressure may be required to be solved for, such as where pressure is a part of an expression for gas volume or diffusion coefficients. The absolute pressure model input is controlled by both a drop-down list and a check box within this section. Use the User defined option to manually define the absolute pressure in a system. This is the default setting. The pressure variables solved for by a fluid-flow interface can also be used, which is selected from the list as, for example, Pressure spf/fp. Selecting a pressure variable also activates a check box for defining the reference pressure, where 1[atm] (1 atmosphere) is the default value. This makes it possible to use a system-based (gauge) pressure as the pressure variable while automatically including the reference pressure in places where it is required, such as for gas flow governed by the gas law. While this check box maintains control over the pressure variable and instances where absolute pressure is required within this respective physics interface, it may not with physics interfaces that being coupled to. In such models, check the coupling between any interfaces using the same variable. Velocity Field From the Velocity field list, select an existing velocity field in the model (for example, Velocity field (spf/fp1) from a Laminar Flow interface) or select User defined to enter values or expressions for the components of the Velocity field (SI unit: m/s). COORDINATE SYSTEM SELECTION The Global coordinate system is selected by default. The Coordinate system list contains any additional coordinate systems that the model includes (except boundary coordinate systems). The coordinate system is used for interpreting directions of orthotropic and anisotropic thermal conductivity. HEAT CONDUCTION The default Thermal conductivity k (SI unit: W/(m K)) is taken From material. If User defined is selected, choose Isotropic, Diagonal, Symmetric, or Anisotropic based on the characteristics of the thermal conductivity, and enter another value or expression. The thermal conductivity describes the relationship between the heat flux vector q and the temperature gradient T as in q = kt which is Fourier s law of heat conduction. Enter this quantity as power per length and temperature. 86 CHAPTER 3: THE HEAT TRANSFER BRANCH

87 THERMODYNAMICS The default Density (SI unit: kg/m 3 ), Heat capacity at constant pressure C P (SI unit: J/(kg K)), and Ratio of specific heats (unitless) for a general gas or liquid use values From material. Select User defined to enter other values or expressions. Select a Fluid type: Select Gas/liquid to specify the density, the heat capacity at constant pressure, and the ratio of specific heats for a general gas or liquid. The default settings are to use data from the material. Select User defined to enter another value for the density, heat capacity, or ratio of specific heats. Select Ideal gas to use the ideal gas law to describe the fluid. In this case, specify the thermodynamics properties by selecting a gas constant type and selecting between entering the heat capacity at constant pressure or the ration of specific heats: From the list under Gas constant type, select Specific gas constant to specify the specific gas constant R s, or select Mean molar mass to specify the mean molar mass M n. If Mean molar mass is selected, the software uses the universal gas constant R J/(mol K), which is a built-in physical constant. For both properties, the default setting is to use the property value from the material. Select User defined to define another value for either of these material properties. From the list under Specify Cp or, select Heat capacity at constant pressure to specify the heat capacity at constant pressure C p, or select Ratio of specific heats to specify the ratio of specific heats. For both properties, the default setting is to use the property value from the material. Select User defined to define another value for either of these material properties. Tip For an ideal gas, specify either C p or the ratio of specific heats,, but not both since these, in that case, are dependent. Heat Source The Heat Source describes heat generation within the domain. Express heating and cooling with positive and negative values, respectively. Add one or more nodes as required all heat sources within a domain contribute to the total heat source. Specify the heat source as the heat per volume in the domain, as linear heat source, or as a total heat source (power). THE HEAT TRANSFER INTERFACE 87

88 DOMAIN SELECTION From the Selection list, choose the domains to add the heat source. HEAT SOURCE Click the General source, Linear source, or Total power button. Note For the Heat Transfer in Porous Media interface, and for the Batteries & Fuel Cells Module, Corrosion Module, or Electrodeposition Module, when General Source is selected, heat sources from the electrochemical current distribution interfaces are also made available in the list under Heat Source. Choose the appropriate one or User defined to defined your own. If General source is selected, enter a value for the distributed heat source Q (SI unit: W/m 3 ). If Total power is selected, enter the total power (total heat source) P tot (SI unit: W). If Linear source (Qq s T) is selected, enter the Production/absorption coefficient q s (SI unit: W/(m 3 K)). Tip The advantage of writing the source in this form is that it can be stabilized by the streamline diffusion. The theory covers q s that is independent of the temperature, but some stability can be gained as long as q s is only weakly dependent on the temperature. See Also Stabilization Techniques in the COMSOL Multiphysics Reference Guide Initial Values The Initial Values node adds an initial value for the temperature that can serve as an initial condition for a transient simulation or as an initial guess for a nonlinear solver. Right-click to add additional Initial Values nodes. DOMAIN SELECTION From the Selection list, choose the domains to define an initial value. 88 CHAPTER 3: THE HEAT TRANSFER BRANCH

89 INITIAL VALUES Enter a value or expression for the initial value of the Temperature T (SI unit: K). The default value is approximately room temperature, K (20 ºC). Temperature Use the Temperature node to specify the temperature somewhere in the geometry; for example, on boundaries. BOUNDARY SELECTION From the Selection list, choose the boundaries to apply a temperature. TEMPERATURE The equation for this condition is T = T 0 where T 0 is the prescribed temperature on the boundary. Enter the value or expression for the Temperature T 0 (SI unit: K). The default is K. CONSTRAINT SETTINGS To display this section, click the Show button ( ) and select Advanced Physics Options. By default Classic constraints is selected. If required, select the Use weak constraints check box. Use Discontinuous Galerkin constraints as an alternative to the Classic constraints when it does not work satisfactorily. This option is especially useful to prevent oscillations on inlet boundaries where convection dominates. Unlike the Classic constraints, these constraints do not enforce the temperature on the boundary extremities. This is relevant on fluid inlets where the temperature may not be enforced on the walls at the inlet extremities. Thermal Insulation The Thermal Insulation node is the default boundary condition for all heat transfer interfaces. This boundary condition means that there is no heat flux across the boundary: n kt = 0 This condition specifies where the domain is well insulated. Intuitively this equation says that the temperature gradient across the boundary must be zero. For this to be true, the temperature on one side of the boundary must equal the temperature on the THE HEAT TRANSFER INTERFACE 89

90 other side. Because there is no temperature difference across the boundary, heat cannot transfer across it. An interesting numerical check for convergence is the numerical evaluation of the thermal insulation condition along the boundary. Another check is to plot the temperature field as a contour plot. Ideally the contour lines are perpendicular to any insulated boundary. BOUNDARY SELECTION Note The default Thermal Insulation feature does not require any user input. If required, add more features by right-clicking the Heat Transfer node and selecting the boundaries to apply the thermal insulation. Outflow The Outflow node provides a suitable boundary condition for convection-dominated heat transfer at outlet boundaries. In a model with convective heat transfer, this condition states that the only heat transfer over a boundary is by convection. The temperature gradient in the normal direction is zero, and there is no radiation. This is usually a good approximation of the conditions at an outlet boundary in a heat transfer model with fluid flow. BOUNDARY SELECTION The Outflow node does not usually require any user input. If required, select the boundaries that are convection-dominated outlet boundaries. Symmetry The Symmetry node provides a boundary condition for symmetry boundaries. This boundary condition is similar to an insulation condition, and it means that there is no heat flux across the boundary. Note The symmetry condition only applies for the temperature field. It has no effect on the radiosity (surface-to-surface radiation) and on the radiative intensity (radiation in participating media). 90 CHAPTER 3: THE HEAT TRANSFER BRANCH

91 BOUNDARY SELECTION Note In most cases, the Symmetry node does not require any user input. If required, define the symmetry boundaries. Heat Flux Use the Heat Flux node to add heat flux across boundaries. A positive heat flux adds heat to the domain. This feature is not applicable to inlet boundaries. Tip For inlet boundaries, use the Inflow Heat Flux condition instead. For the Thin Conductive Shell interface, the Heat Flux feature adds a heat source (or sink) to edges. It adds a heat flux qd e q 0. BOUNDARY OR EDGE SELECTION From the Selection list, choose the boundaries or edges to add the heat flux contribution. PAIR SELECTION If Heat Flux is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect. HEAT FLUX Click one of the General inward heat flux, Inward heat flux, or Total heat flux buttons. If General inward heat flux q 0 (SI unit: W/m 2 ) is selected, it adds to the total flux across the selected boundaries. Enter a value for q 0 to represent a heat flux that enters the domain. For example, any electric heater is well represented by this condition, and its geometry can be omitted. If Inward heat flux is selected, enter the Heat transfer coefficient h (SI unit: W/ (m 2 K)). The default is 0. Also enter an External temperature T ext (SI unit: K). The default is K. The value depends on the geometry and the ambient flow conditions. Inward heat flux is defined by this equation: THE HEAT TRANSFER INTERFACE 91

92 q 0 = ht ext T For a thorough introduction about how to calculate heat transfer coefficients, see Incropera and DeWitt in Ref. 1. If Total heat flux is selected, enter the total heat flux q tot (SI unit: W) for the total heat flux across the boundaries where the Heat Flux node is active. Surface-to-Ambient Radiation Use the Surface-to-Ambient Radiation boundary condition to add surface-to-ambient radiation to boundaries. The net inward heat flux from surface-to-ambient radiation is q = 4 T amb T 4 where is the surface emissivity, is the Stefan-Boltzmann constant (a predefined physical constant), and T amb is the ambient temperature. Model Continuous Casting: Model Library path Heat_Transfer_Module/ Process_and_Manufacturing/continuous_casting BOUNDARY SELECTION From the Selection list, choose the boundaries to add surface-to-ambient radiation contribution. SURFACE-TO-AMBIENT RADIATION The default Surface emissivity e (a dimensionless number between 0 and 1) is taken From material. An emissivity of 0 means that the surface emits no radiation at all and an emissivity of 1 means that it is a perfect blackbody. Enter an Ambient temperature T amb (SI unit: K). The default is K. Periodic Heat Condition Use the Periodic Heat Condition node to add a periodic heat condition to boundaries. Right-click the node to add a Destination Selection feature. 92 CHAPTER 3: THE HEAT TRANSFER BRANCH

93 BOUNDARY SELECTION From the Selection list, choose the boundaries to add a periodic heat condition. Periodic Condition is also described in the COMSOL Multiphysics User s Guide: Periodic Condition See Also Destination Selection Using Periodic Boundary Conditions Periodic Boundary Condition Example Boundary Heat Source The Boundary Heat Source node models a heat source (or heat sink) that is embedded in the boundary. BOUNDARY SELECTION From the Selection list, choose the boundaries to apply the heat source. BOUNDARY HEAT SOURCE Click the General source or Total boundary power button. If General source is selected, enter the boundary heat source Q b (SI unit: W/m 2 ). A positive Q b is heating and a negative Q b is cooling. The default is 0. If Total boundary power is selected, enter the total power (total heat source) P b, tot (SI unit: W). Pair Boundary Heat Source The Pair Boundary Heat Source node models a heat source (or heat sink) that is embedded in the boundary. It also prescribes that the temperature field is continuous across the pair. BOUNDARY SELECTION From the Selection list, choose the boundaries to apply the heat source. THE HEAT TRANSFER INTERFACE 93

94 PAIR SELECTION When Pair Boundary Heat Source is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect. BOUNDARY HEAT SOURCE Click the General source or Total boundary power button. If General source is selected, enter the boundary heat source Q b (SI unit: W/m 2 ). A positive Q b is heating and a negative Q b is cooling. The default is 0. If Total boundary power is selected, enter the total power (total heat source) P b, tot (SI unit: W). Continuity The Continuity node can be added to pairs. It prescribes that the temperature field is continuous across the pair. Continuity is only suitable for pairs where the boundaries match. BOUNDARY SELECTION The selection list in this section shows the boundaries for the selected pairs. PAIR SELECTION When Continuity is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect. In the COMSOL Multiphysics User s Guide: See Also Identity and Contact Pairs Specifying Boundary Conditions for Identity Pairs Pair Thin Thermally Resistive Layer Use the Pair Thin Thermally Resistive Layer node to define the thickness and thermal conductivity of a resistive material located on boundaries. This material can be formed of one or more layers. It can be added to pairs. The heat flux across the Pair Thin Thermally Resistive Layer is defined by 94 CHAPTER 3: THE HEAT TRANSFER BRANCH

95 T u T d n d k d T d = k s d s T d T u n u k u T u = k s where u and d subscript refer respectively to the upside and the downside of the pair. When the material has a multi-layer structure k s and d s in the expressions above are replaced by d tot and k tot which are defined according to following expressions: d s d tot = nl j = 1 d sj (3-3) k tot = d tot nl d sj k sj j = 1 (3-4) where nl is the number of layers. BOUNDARY SELECTION The selection list in this section shows the boundaries for the selected pairs. PAIR SELECTION When Pair Thin Thermally Resistive Layer is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect. PAIR THIN THERMALLY RESISTIVE LAYER The Multiple layers check box enables the definition of multiple sandwiched thin layers with different thermal conductivities. Note By default Multiple layers is not selected. Define the properties of a single layer. Enter a value or expression for the Layer thickness d s (SI unit: m). The default is to use the Thermal conductivity k s (SI unit: W/(m K)) From material. Select User defined to enter another value or expression. THE HEAT TRANSFER INTERFACE 95

96 When Multiple layers is selected, define the properties of a multiple layers structure. Select the number of layer to define (1 to 5) and set following properties for each: Assign a material to each layer by selecting the appropriate material in the Solid material i list. Enter a value or expression for the Layer thickness d si (SI unit: m). The default is to use the Thermal conductivity k si (SI unit: W/(m K)) From material. This gets the thermal conductivity from the material selected in Solid material i. Select User defined to enter another value or expression. Thin Thermally Resistive Layer Use the Thin Thermally Resistive Layer node to define the thickness and thermal conductivity of a resistive material located on boundaries. This material can be formed of one or more layers. The heat flux across the Thin Thermally Resistive Layer is defined by T u T d n d k d T d = k s d s T d T u n u k u T u = k s where u and d subscript refer respectively to the upside and the downside of the slit. When the material has a multi-layer structure k s and d s in the expressions above are replaced by d tot and k tot which are defined according to following expressions: d s d tot = nl j = 1 d sj (3-5) k tot d tot = nl d sj k sj j = 1 (3-6) where nl is the number of layers. 96 CHAPTER 3: THE HEAT TRANSFER BRANCH

97 BOUNDARY SELECTION From the Selection list, choose the boundaries to add a thermally resistive layer. MODEL INPUTS The temperature model input used to evaluate the material properties is equal to the mean temperature on interior boundaries. THIN THERMALLY RESISTIVE LAYER The Multiple layers check box enables the definition of multiple sandwiched thin layers with different thermal conductivities. Note By default Multiple layers is not selected. Then define the properties of a single layer. Enter a value or expression for the Layer thickness d s (SI unit: m). The default is to use the Thermal conductivity k s (SI unit: W/(m K)) From material. Select User defined to enter another value or expression. When Multiple layers is selected, define the properties of a multiple layers structure. Select the number of layer to define (1 to 5) and set following properties for each: Assign a material to each layer by selecting the appropriate material in the Solid material i list. Enter a value or expression for the Layer thickness d si (SI unit: m). The default is to use the Thermal conductivity k si (SI unit: W/(m K)) From material. This gets the thermal conductivity from the material selected in Solid material i. Select User defined to enter another value or expression. THE HEAT TRANSFER INTERFACE 97

98 Line Heat Source The Line Heat Source node models a heat source (or sink) that is so thin that it has no thickness in the model geometry. Select this feature from the Edges menu. The Line Heat Source node is available in 3D only because a line in 2D is a boundary and a domain in 1D. 3D In theory, the temperature in a line source in 3D is plus or minus infinity (to compensate for the fact that the heat source does not have any volume). The finite element discretization used in COMSOL returns a finite temperature distribution along the line, but that distribution must be interpreted in a weak sense. EDGE SELECTION From the Selection list, choose the edges to apply the heat source. LINE HEAT SOURCE Click the General source or Total power button. If General source is selected, enter a value for the distributed heat source, Q l, in unit power per unit length (SI unit: W/m). Positive Q l is heating while a negative Q l is cooling. If Total power is selected, enter the total power (total heat source) P l,tot (SI unit: W) Point Heat Source The Point Heat Source node models a heat source (or sink) that is so small that it can be considered to have no spatial extension. Select this feature from the Points menu. The Point Heat Source is available in 2D and 3D. It is not available in 1D since points are boundaries (possibly internal boundaries) there. 2D 3D In theory, the temperature in a point source in 2D or 3D is plus or minus infinity (to compensate for the fact that the heat source does not have a spatial extension). The finite element discretization used in COMSOL returns a finite value, but that value must be interpreted in a weak sense. 98 CHAPTER 3: THE HEAT TRANSFER BRANCH

99 POINT SELECTION From the Selection list, choose the points to apply the heat source. POINT HEAT SOURCE Enter the quantity Q p in unit power (SI unit: W). Positive Q p is heating while a negative Q p is cooling. Continuity on Interior Boundary The Continuity on Interior Boundary node enables intensity conservation across internal boundaries. It is the default boundary condition for all internal boundaries. BOUNDARY SELECTION From the Selection list, choose the boundaries to activate the continuity on interior boundaries. PAIR SELECTION If Continuity on Interior Boundary is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect. Reaction Heat Flux Note The Reaction Heat Flux feature is available only with the Plasma Module and is documented in the Plasma Module User s Guide. THE HEAT TRANSFER INTERFACE 99

100 Heat Transfer Interface Advanced Features For the Heat Transfer Module, several advanced features are available with this interface. In addition to the nodes described in The Heat Transfer Interface, this section details these nodes and subnodes (listed in alphabetical order): Incident Intensity Infinite Elements Inflow Heat Flux Opaque Opaque Surface Open Boundary Pressure Work Radiation in Participating Media Viscous Heating See Also About the Heat Transfer Interfaces Domain, Boundary, Edge, Point, and Pair Features for the Heat Transfer Interfaces The Heat Transfer Interface Tip To locate and search all the documentation, in COMSOL, select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree. Radiation in Participating Media The Radiation in Participating Media node uses the radiative transfer equation I s = I b T I s I s 4 s CHAPTER 3: THE HEAT TRANSFER BRANCH

101 where Is) is the radiative intensity at the position s position in the direction T is the temperature,, s are absorption, extinction, and scattering coefficients I b is the blackbody radiative intensity It also adds the radiative heat source term in the heat transfer equation: Q r = qr = G 4T 4 DOMAIN SELECTION From the Selection list, choose the domains to define radiation in participating media. MODELS 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. There is one standard model input the Temperature T (SI unit: K). The default is to use the heat transfer dependant variable. ABSORPTION The default Absorption coefficient (SI unit: 1/m) uses the value From material. The absorption coefficient defines the amount of radiation, I, that is absorbed by the medium. If User defined is selected, enter another value or expression. SCATTERING The default Scattering coefficient s (SI unit: 1/m) uses the value From material. If User defined is selected, enter another value or expression. The default is 0. Select the Scattering type Isotropic, Linear anisotropic, or Nonlinear anisotropic. Isotropic (the default) and corresponds to the scattering phase function 0 = 1. If Linear anisotropic is selected, it defines the scattering phase function as 0 = 1 + a 1 0. Enter the Legendre coefficient a 1. If Nonlinear anisotropic is selected, it defines the scattering phase function HEAT TRANSFER INTERFACE ADVANCED FEATURES 101

102 0 = 1 + a m P m 0 m = 1 Enter the Legendre coefficients a 1,, a 12 as required. 12 Opaque Note The Opaque node is available for the Heat Transfer with Surface-to-Surface Radiation version of the Heat Transfer interface. It is also available for The Surface-To-Surface Radiation Interface. Right-click the Heat Transfer in Solids or Heat Transfer in Fluids node to add the Opaque feature. By default all the domains are transparent. Surface to surface boundary conditions can only be set at the interface between transparent and opaque domains. DOMAIN SELECTION From the Selection list, choose the domains to add this feature. By default, the selection is the same as for the Heat Transfer in Solids or Heat Transfer in Fluids node it is attached to. Model Cavity Radiation: Model Library path Heat_Transfer_Module/ Verification_Models/cavity_radiation Infinite Elements Note For the Infinite Elements node, see About Infinite Element Domains and Perfectly Matched Layers in the COMSOL Multiphysics User s Guide. 102 CHAPTER 3: THE HEAT TRANSFER BRANCH

103 Opaque Surface Note The Opaque Surface node is available for The Heat Transfer with Radiation in Participating Media Interface version of the Heat Transfer interface. It is also available for The Radiation in Participating Media Interface. The Opaque Surface node defines a boundary opaque to radiation. The Opaque Surface node prescribes incident intensities on a boundary and accounts for the net radiative heat flux, q w, that is absorbed by the surface. BOUNDARY SELECTION From the Selection list, choose the boundaries to define the wall condition. PAIR SELECTION If Opaque Surface is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect. MODELS 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. There is one standard model input the Temperature T (SI unit: K). The default is K and is used in the black-body radiative intensity expression. Tip The boundary temperature definition can differ from the that of the temperature in the adjacent domain. WALL SETTINGS Select a Wall type to define the behavior of the wall Gray wall or Black wall. Gray Wall If Gray wall is selected the default Surface emissivity e value is taken From material (a material defined on the boundaries). Select User defined to enter another value or expression. Enter a Diffusive reflectivity d. HEAT TRANSFER INTERFACE ADVANCED FEATURES 103

104 Both are dimensionless numbers between 0 and 1 that satisfy the relation d w 1. By default d 1 w. In this case, an emissivity of 0 means that the surface emits no radiation at all and that all outgoing radiation is diffusely reflected by this boundary. An emissivity of 1 means that the surface is a perfect black body, outgoing radiation is fully absorbed on this boundary. If d 1 w, it means that the wall is not opaque and that a part of the outgoing radiative intensity goes through the wall without being reflected nor absorbed. Radiative intensity (W/m 2 in SI units) along incoming discrete directions on this boundary is defined by I i bnd = d w I b T q out Black Wall If Black wall is selected, 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. Incident Intensity Note The Incident Intensity node is available for The Heat Transfer with Radiation in Participating Media Interface version of the Heat Transfer interface. It is also available for The Radiation in Participating Media Interface. Use an Incident Intensity node to specify the radiative intensity along incident directions on a boundary. BOUNDARY SELECTION From the Selection list, choose the boundaries to define the radiative intensity along incident directions. PAIR SELECTION If Incident Intensity is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect. 104 CHAPTER 3: THE HEAT TRANSFER BRANCH

105 INCIDENT INTENSITY Enter a Boundary radiation intensity I wall (SI unit: W/m 2 ). This represents the value of radiative intensity along incoming discrete directions. Values of radiative intensity on outgoing discrete directions are not prescribed. Tip The components of each discrete ordinate vector can be used in this expression. The syntax is interfaceidentifier.sx, interfaceidentifier.sy, interfaceidentifier.sz where interfaceidentifier is the physics interface identifier. By default, the Heat Transfer interface identifier is ht so ht.sx, ht.sy, and ht.sz correspond to the components of discrete ordinate vectors. Pressure Work Right-click the Heat Transfer in Solids or Heat Transfer in Fluids node to add this subnode. The Pressure Work node adds the following term to the right-hand side of Equation 3-7: T----S t el (3-7) where S el is the Elastic Contribution to Entropy. Compared to u + u u = pi + + F t (this is also described in the theory for Laminar Flow), the part that corresponds to sound wave propagation is neglected. The reason is the energy in the sound waves are almost always negligible compared to the contribution from Equation 3-7. The software computes the pressure work using the absolute pressure. See Also Theory for the Laminar Flow Interface in the COMSOL Multiphysics User s Guide. DOMAIN SELECTION From the Selection list, choose the domains to add pressure work. By default, the selection is the same as for the Heat Transfer in Solids or Heat Transfer in Fluids node it is attached to. HEAT TRANSFER INTERFACE ADVANCED FEATURES 105

106 MODEL INPUTS Enter a value or expression for the Elastic contribution to entropy Ent (SI unit: J m 3 K)). The default is 0. Viscous Heating The Viscous Heating node adds the following term to the right-hand side of the Heat Transfer in Fluids equation: :S (3-8) where is the viscous stress tensor and S is the strain-rate tensor. Equation 3-8 represents the heating caused by viscous friction within the fluid. DOMAIN SELECTION From the Selection list, choose the domains to add pressure work. By default, the selection is the same as for the Heat Transfer in Fluids feature that it is attached to. DYNAMIC VISCOSITY Specify the Dynamic viscosity (SI unit: Pa s). The default setting is to use the value of the viscosity from the material. Select User defined to define another value for the viscosity. COMSOL uses the dynamic viscosity together with the velocity expressions to compute the viscous stress tensor,. Inflow Heat Flux Use the Inflow Heat Flux node to model inflow of heat through a virtual domain with a heat source. The temperature at the outer boundary of the virtual domain is known. This boundary condition estimates the heat flux through the system boundary 1 n kt = q 0 u n h in h u n u n (3-9) where h in T in h = C p dt + T p A p T T dp p (3-10) 106 CHAPTER 3: THE HEAT TRANSFER BRANCH

107 A positive heat flux adds heat to the domain. The Inflow Heat Flux feature is applicable to inlet boundaries. The second integral in Equation 3-7 is neglected if the Inflow Heat Flux feature is applied to the boundary of a solid domain. BOUNDARY SELECTION From the Selection list, choose the boundaries to add the heat flux contribution. INFLOW HEAT FLUX Enter a value or expression for each of the following properties. Select the Inward heat flux or Total heat flux button. - When Inward heat flux is selected, define q 0 (SI unit: W/m 2 ) to add to the total flux across the selected boundaries. The default value is 0. - When Total heat flux is selected, define q tot. In this case q 0 q tot /A, where A is the total area of the selected boundaries. 2D Axi In 3D and 2D axial symmetry, A = 1. 3D 2D 1D Axi In 2D and 1D axial symmetry, A = dz 1, where dz is the out-of-plane thickness. If the out-of-plane property is not active, a text field is available to define dz or A c. 1D In 1D, A = A, where A c is the cross-sectional area.if the c 1 out-of-plane property is not active, a text field is available to define dz or A c. HEAT TRANSFER INTERFACE ADVANCED FEATURES 107

108 External temperature T ext (SI unit: K). The default value is K. External absolute pressure p ext (SI unit: Pa). The default value is 1 atm. Open Boundary The Open Boundary node adds a boundary condition for modeling heat flux across an open boundary: the heat can flow out of the domain or into the domain with a specified exterior temperature. Use this node to limit a modeling domain that extends in an open fashion. BOUNDARY SELECTION From the Selection list, choose the boundaries to model as open boundaries. EXTERIOR TEMPERATURE Enter the exterior Temperature T 0 (SI unit: K) outside of the open boundary. Convective Cooling The Convective Cooling node adds the following heat flux contribution to its boundaries: ht ext T where h can be defined by the user or by using a library of predefined coefficients described in the section About the Heat Transfer Coefficients. Model Power Transistor: Model Library path Heat_Transfer_Module/ Electronics_and_Power_Systems/power_transistor Free Convection in a Water Glass: Model Library path Heat_Transfer_Module/Tutorial_Models/cold_water_glass BOUNDARY SELECTION From the Selection list, choose the boundaries to add a convective cooling contribution. For information about selecting boundaries. 108 CHAPTER 3: THE HEAT TRANSFER BRANCH

109 HEAT FLUX Select a Heat transfer coefficient h (SI unit: W/(m 2 K)) to control the type of convective cooling to model User defined (the default), External natural convection, Internal natural convection, External forced convection, or Internal forced convection. For all of the options, enter an External temperature, T ext (SI unit: K). For all of the options (except User defined), follow the individual instructions below and select an External fluid Air, Transformer oil, or water. If Air is selected, also enter an Absolute pressure, p A (SI unit: Pa). The default is 1 atm. External Natural Convection Select Vertical wall, Inclined wall, Horizontal plate, upside, or Horizontal plate, downside from the list. If Vertical wall is selected, enter a Wall height L (SI unit: m). If Inclined wall is selected, enter a Wall height L (SI unit: m) and the Tilt angle (the angle between the wall and the vertical direction, for vertical walls). If Horizontal plate, upside or Horizontal plate, downside is selected, define the Plate diameter (area/perimeter) L (SI unit: m). L is approximated by the ratio between the surface area and its perimeter. The defaults are 0. Internal Natural Convection Select Narrow chimney, parallel plates or Narrow chimney, circular tubes from the list. If Narrow chimney, parallel plates is selected, enter a Plate distance L (SI unit: m) and a Chimney height H (SI unit: m). The defaults are 0. If Narrow chimney, circular tubes is selected, enter a Tube diameter D (SI unit: m) and a Chimney height H (SI unit: m). The defaults are 0. External Forced Convection Select Plate, averaged transfer coefficient or Plate, local transfer coefficient from the list. If Plate, averaged transfer coefficient is selected, enter a Plate length L (SI unit: m) and a Velocity, external fluid U ext (SI unit: m/s). The defaults are 0. If Plate, local transfer coefficient is selected, enter a Position along the plate x pl (SI unit: m) and a Velocity, external fluid U ext (SI unit: m/s). The defaults are 0. Internal Forced Convection The only option is Isothermal tube. Enter a Tube diameter D (SI unit: m) and a Velocity, external fluid U ext (SI unit: m/s). The defaults are 0. HEAT TRANSFER INTERFACE ADVANCED FEATURES 109

110 Highly Conductive Layer Features In this section: Highly Conductive Layer Layer Heat Source Edge Heat Flux Edge Temperature Edge Surface-to-Ambient Model 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 2D 2D Axi Use the Highly Conductive Layer feature to model heat transfer in thin highly conductive layers on boundaries in 2D and 3D. This feature can also be added to 2D axisymmetric models. 3D See Also About Highly Conductive Layers 110 CHAPTER 3: THE HEAT TRANSFER BRANCH

111 Right-click to add these additional features: Layer Heat Source to add an internal heat source, Q s, within the highly conductive layer. Edge (3D) or Point (2D and 2D axisymmetric) Heat Flux adds a heat flux through a specified set of boundaries of a highly conductive layer. See Edge Heat Flux and Point Heat Flux. Note If you also have the Microfluidics Module, the Edge Heat Flux and Point Heat Flux nodes are not available with the Slip Flow interface. Edge (3D) or Point (2D and 2D axisymmetric) Temperature sets a prescribed temperature condition on a specified set of boundaries of a highly conductive layer. See Edge Temperature and Point Temperature. Edge (3D) or Point (2D and 2D axisymmetric) Surface-to-Ambient Radiation adds a surface-to-ambient radiation for the highly conductive layer. See Edge Surface-to-Ambient and Point Surface-to-Ambient Radiation. BOUNDARY SELECTION From the Selection list, choose the boundaries to model as highly conductive layers. MODEL INPUTS This section contains fields and values that are inputs to expressions that define material properties. If such user-defined property groups are added, the model inputs appear here. The Layer thickness d s, displays in this section. The default value is 0.01 m. COORDINATE SYSTEM SELECTION The Global coordinate system is selected by default. The Coordinate system list contains any additional coordinate systems that the model includes (except boundary coordinate systems). The coordinate system is used for interpreting directions of orthotropic and anisotropic thermal conductivity. HEAT CONDUCTION The default Layer thermal conductivity k s (SI unit: W/(m K)) is taken From material and describes the layer s ability to conduct heat. If User defined is selected, choose HIGHLY CONDUCTIVE LAYER FEATURES 111

112 Isotropic, Diagonal, Symmetric, or Anisotropic based on the characteristics of the thermal conductivity, and enter another value or expression. THERMODYNAMICS The default Layer density s (SI unit: kg/m 3 ) and Layer heat capacity C s (SI unit: J s (SI unit: kg/m 3 )/ (kg K)) is taken From material. Select User defined to enter other values or expressions. Enter a value or expression for the Layer thickness d s (SI unit: m). The default is 0.01 m. Tip If the thickness is zero, the highly conductive layer does not take effect. See Also About Highly Conductive Layers Layer Heat Source Right-click the Highly Conductive Layer node to add this feature. Use a Layer Heat Source feature to add an internal heat source, Q s, within the highly conductive layer. Add one or more heat sources. BOUNDARY SELECTION From the Selection list, choose the boundaries to add the heat source. By default, the selection is the same as for the Highly Conductive Layer feature. LAYER HEAT SOURCE Enter a value or expression for the Layer heat source Q s (SI unit: W/m 2 ). Edge Heat Flux 3D Use the Edge Heat Flux feature for 3D models to add heat flux across boundaries of a highly conductive layer. A positive heat flux adds heat to the layer. Right-click the Highly Conductive Layer node to add this feature. 112 CHAPTER 3: THE HEAT TRANSFER BRANCH

113 Note If you also have the Microfluidics Module, the Edge Heat Flux node is not available with the Slip Flow interface. EDGE SELECTION From the Selection list, choose the edges to add the heat flux contribution. HEAT FLUX Select either the General inward heat flux or Inward heat flux buttons. If General inward heat flux q 0 (SI unit: W/m 2 ) is selected, it adds to the total flux across the selected edges. Enter a value for q 0 to represent a heat flux that enters the layer. For example, any electric heater is well represented by this condition, and its geometry can be omitted. If Inward heat flux is selected (in the form q 0 h T ext T, enter the Heat transfer coefficient h (SI unit: W/(m 2 K)). The default value is 0. Enter an External temperature T ext (SI unit: K). The default value is K. The value depends on the geometry and the ambient flow conditions. Point Heat Flux 2D 2D Axi Use the Point Heat Flux feature for 2D and 2D axisymmetric models to add heat flux across boundaries of a highly conductive layer. A positive heat flux adds heat to the layer. Right-click the Highly Conductive Layer node to add this feature. Note If you also have the Microfluidics Module, the Point Heat Flux node is not available with the Slip Flow interface. POINT SELECTION From the Selection list, choose the points to add the heat flux contribution. HIGHLY CONDUCTIVE LAYER FEATURES 113

114 HEAT FLUX Select either the General inward heat flux or Inward heat flux buttons. If General inward heat flux q 0 (SI unit: W/m 2 ) is selected, it adds to the total flux across the selected points. Enter a value for q 0 to represent a heat flux that enters the layer. For example, any electric heater is well represented by this condition, and its geometry can be omitted. If Inward heat flux is selected (in the form q 0 h T ext T, enter the Heat transfer coefficient h (SI unit: W/(m 2 K)). The default value is 0. Enter an External temperature T ext (SI unit: K). The default value is K. The value depends on the geometry and the ambient flow conditions. Edge Temperature 3D Use the Edge Temperature feature to specify the temperature on a set of edges that represent thin boundary surfaces of the highly conductive layer. Right-click the Highly Conductive Layer node to add this feature. Only edges adjacent to the boundaries can be selected in the parent node. Note EDGE SELECTION From the Selection list, choose the edges to apply an edge temperature. PAIR SELECTION If Edge Temperature is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect. TEMPERATURE Enter the value or expression for the Temperature T 0 (SI unit: K). The equation for this condition is T = T 0 where T 0 is the prescribed temperature on the edges. 114 CHAPTER 3: THE HEAT TRANSFER BRANCH

115 CONSTRAINT SETTINGS To display this section, click the Show button ( ) and select Advanced Physics Options. Select a Constraint type Bidirectional, symmetric or Unidirectional. If required, select the Use weak constraints check box. Point Temperature 2D Use the Point Temperature feature to specify the temperature on a set of points that represent thin boundary surfaces of the highly conductive layer. Right-click the Highly Conductive Layer node to add this feature. 2D Axi Only points adjacent to the boundaries can be selected in the parent node. Note POINT SELECTION From the Selection list, choose the points to apply a prescribed temperature. TEMPERATURE Enter the value or expression for the Temperature T 0 (SI unit: K). The equation for this condition is T = T 0 where T 0 is the prescribed temperature on the points. CONSTRAINT SETTINGS To display this section, click the Show button ( ) and select Advanced Physics Options. Select a Constraint type Bidirectional, symmetric or Unidirectional. If required, select the Use weak constraints check box. Edge Surface-to-Ambient 3D Use the Edge Surface-to-Ambient Radiation feature to add surface-to-ambient radiation to edges representing boundaries of a highly conductive layer. HIGHLY CONDUCTIVE LAYER FEATURES 115

116 The net inward heat flux from surface-to-ambient radiation is q = 4 T amb T 4 where is the surface emissivity, is the Stefan-Boltzmann constant (a predefined physical constant), and T amb is the ambient temperature. Right-click the Highly Conductive Layer node to add this feature. EDGE SELECTION From the Selection list, choose the edges to add surface-to-ambient radiation contribution. SURFACE-TO-AMBIENT RADIATION Enter an Ambient temperature T amb (SI unit: K). The default is K. The default Surface emissivity (a dimensionless number between 0 and 1) is taken From material. An emissivity of 0 means that the surface emits no radiation at all and an emissivity of 1 means that it is a perfect blackbody. Point Surface-to-Ambient Radiation 2D Use the Point Surface-to-Ambient Radiation feature to add surface-to-ambient radiation to points representing boundaries of a highly conductive layer. 2D Axi The net inward heat flux from surface-to-ambient radiation is q = T amb T 4 where is the surface emissivity, is the Stefan-Boltzmann constant (a predefined physical constant), and T amb is the ambient temperature. 4 POINT SELECTION From the Selection list, choose the points to add surface-to-ambient radiation contribution. 116 CHAPTER 3: THE HEAT TRANSFER BRANCH

117 SURFACE-TO-AMBIENT RADIATION Enter an Ambient temperature T amb (SI unit: K). The default is K. The default Surface emissivity (a dimensionless number between 0 and 1) is taken From material. An emissivity of 0 means that the surface emits no radiation at all and an emissivity of 1 means that it is a perfect blackbody. HIGHLY CONDUCTIVE LAYER FEATURES 117

118 Out-of-Plane Heat Transfer Features The following features are available for 1D and 2D Heat Transfer models. In this section: Out-of-Plane Convective Cooling Out-of-Plane Radiation Out-of-Plane Heat Flux Change Thickness Model Surface Resistor: Model Library path Heat_Transfer_Module/ Electronics_and_Power_Systems/surface_resistor See Also Theory of Out-of-Plane Heat Transfer Out-of-Plane Convective Cooling 1D Select the Out-of-plane heat transfer check box on the Heat Transfer interface to add this feature to 1D and 2D models. 2D Use the Out-of-Plane Convective Cooling node to model upside and downside cooling (or heating) caused by the presence of an ambient fluid. The Out-of-Plane Convective Cooling feature adds the following contribution to the right-hand side of Equation 2-28 or Equation 2-29: h u T ext,u T+ h d T ext,d T 118 CHAPTER 3: THE HEAT TRANSFER BRANCH

119 DOMAIN SELECTION Select the domains where you want to add an out-of-plane convective cooling contribution. UPSIDE HEAT FLUX See Also About the Heat Transfer Coefficients Select a Heat transfer coefficient h u (SI unit: W/(m 2 K)) to control the type of convective cooling to model User defined (the default), External natural convection, Internal natural convection, External forced convection, or Internal forced convection. If only convective flux is required on the downside, use the default, which sets h u 0. For all of the options, enter an External temperature, T ext,u (SI unit: K). For all of the options (except User defined), follow the individual instructions in the Heat Flux section described for the Convective Cooling feature, then select an External fluid Air, Transformer oil, or water. If Air is selected, also enter an Absolute pressure, p A (SI unit: Pa). The default is 1 atm. 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. Out-of-Plane Radiation 1D Select the Out-of-plane heat transfer check box on the Heat Transfer interface to add this feature to 1D and 2D models. 2D The Out-of-Plane Radiation feature models surface-to-ambient radiation on the upside and downside and adds the following contribution to the right-hand side of Equation 2-28 or Equation 2-29: OUT-OF-PLANE HEAT TRANSFER FEATURES 119

120 4 T 4 4 T 4 u T amb u + d T amb d See Also Compare to the equation in the heat theory section about Surface-to-Ambient Radiation. DOMAIN SELECTION Select the domains where you want to add an out-of-plane surface-to-ambient heat transfer contribution. UPSIDE PARAMETERS The default Surface emissivity e u, a unitless number between 0 and 1, is taken From material. Select User defined to enter another value. An emissivity of 0 means that the surface emits no radiation at all while a emissivity of 1 means that it is a perfect blackbody. Enter an Ambient temperature T amb,u (SI unit: K). The default is K. DOWNSIDE PARAMETERS Follow the instructions for the Upside Parameters for the downside parameters e d and T amb,d. Out-of-Plane Heat Flux 1D Select the Out-of-plane heat transfer check box on the Heat Transfer interface to add this feature to 1D and 2D models. 2D The Out-of-Plane Heat Flux feature adds a heat flux q 0,u as an upside heat flux and a heat flux q 0,d as a downward heat flux to the right-hand side of Equation 2-28 or Equation 2-29: 120 CHAPTER 3: THE HEAT TRANSFER BRANCH

121 d s q 0 u + d s q 0 d DOMAIN SELECTION Select the domains where you want to add an out-of-plane heat flux. UPSIDE INWARD HEAT FLUX Select between specifying the upside inward heat flux directly or as a convective term using a heat transfer coefficient. Click the General inward heat flux button to specify a value or expression for the inward (or outward, if the quantity is negative) heat flux through the upside (SI unit: W/m 2 ) in the q 0,u field. Click the Inward heat flux button to specify an inward (or outward, if the quantity is negative) heat flux through the upside (SI unit: W/m 2 ) as h u (T ext,u T). Enter a value or expression for the heat transfer coefficient in the h u field (SI unit: W/(m 2 K) and a value or expression for the external temperature in the T ext,u field (SI unit: K). The default value for the external temperature is K. 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 2D Select the Out-of-plane heat transfer check box on the Heat Transfer interface to add this feature to 2D models. The Change Thickness feature makes it possible model domains with another thickness than the overall thickness that is specified in the Heat Transfer feature s Physical Model section. DOMAIN SELECTION Select the domains where you want to use a different thickness. OUT-OF-PLANE HEAT TRANSFER FEATURES 121

122 CHANGE THICKNESS Specify a value for the Thickness d z (SI unit: m). The default value is 1 m. This value replaces the overall thickness in the domains that are selected in the Domain Selection section. 122 CHAPTER 3: THE HEAT TRANSFER BRANCH

123 The Bioheat Transfer Interface When Bioheat Transfer is selected under the Heat Transfer branch ( ) in the Model Wizard, Biological tissue is automatically selected as the Default model and a Heat Transfer ( ) interface is added to the Model Builder. When this version of the interface is added, these default nodes are added to the Model Builder Biological Tissue, Thermal Insulation (the default boundary condition), and Initial Values. All functionality to include both solid and fluid domains is also available. Right-click the Heat Transfer node to add other features that implement boundary conditions and sources. Note This interface is also added when the Heat Transfer in Biological Tissue check box is selected under Advanced Settings on the Heat Transfer interface. To display the section, click the Show button ( ) and select Advanced Physics Options. Model Hepatic Tumor Ablation: Model Library path Heat_Transfer_Module/ Medical_Technology/tumor_ablation PHYSICAL MODEL Biological tissue is automatically selected as the Default model and. If Heat transfer in solids or Heat transfer in fluids is selected as the Default model, the Biological Tissue node changes to a Heat Transfer in Solids or Heat Transfer in Fluids node. Note The interior and exterior boundary conditions are the same as for the The Heat Transfer Interface. Tip To locate and search all the documentation, in COMSOL, select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree. THE BIOHEAT TRANSFER INTERFACE 123

124 See Also The Heat Transfer Interface Theory for the Bioheat Transfer Interface Biological Tissue Bioheat Working with Geometry in the COMSOL Multiphysics User s Guide Biological Tissue The Biological Tissue feature adds the bioheat equation as the mathematical model for heat transfer in biological tissue. See Equation Right-click the node to add a Bioheat node. Tip When parts of the model (for example, a heat source) are moving, also right-click to add a Translational Motion node, which includes the effect of the movement by translation that requires a moving coordinate system. DOMAIN SELECTION From the Selection list, choose the domains to define the heat transfer. MODEL INPUTS This section has fields and values that are inputs to expressions that define material properties. If such user-defined property groups are added, the model inputs appear here. Initially, this section is empty. COORDINATE SYSTEM SELECTION The Global coordinate system is selected by default. The Coordinate system list contains any additional coordinate systems that the model includes (except boundary coordinate systems). The coordinate system is used for interpreting directions of orthotropic and anisotropic thermal conductivity. HEAT CONDUCTION The default uses Thermal conductivity k (SI unit: W/(m K)) values From material. If User defined is selected, choose Isotropic, Diagonal, Symmetric, or Anisotropic based on the characteristics of the thermal conductivity and enter another value or expression. 124 CHAPTER 3: THE HEAT TRANSFER BRANCH

125 THERMODYNAMICS The default Density (SI unit: kg/m 3 ) and Heat capacity at constant pressure C p (SI unit: J/(kg K)) are taken From material. The heat capacity describes the amount of heat energy required to produce a unit temperature change in a unit mass. If User defined is selected, enter other values or expressions. Bioheat A default Bioheat node is added to the Biological Tissue node. This feature provides the source terms that represent blood perfusion and metabolism for modeling heat transfer in biological tissue using the bioheat equation b C b b (T b T) Define the density of blood, specific heat of blood, blood perfusion rate, the arterial blood temperature and the metabolic heat source. Right-click the Biological Tissue node to add more Bioheat subnodes. DOMAIN SELECTION From the Selection list, choose the domains to add bioheat source terms. The default selection are the domains in the Biological Tissue feature, but more than one Bioheat feature can be defined with different settings for subsets of the domains where the bioheat transfer occurs. BIOHEAT Enter values or expressions for the following properties and source terms in the associated fields: Density, blood b (SI unit: kg/m 3 ), which is the mass per volume of blood. Specific heat, blood C b (SI unit: J/(kg k)), which describes the amount of heat energy required to produce a unit temperature change in a unit mass of blood. Blood perfusion rate b (SI unit: 1/s, which in this case means (m 3 /s)/m 3 ), describes the volume of blood per second that flows through a unit volume of tissue. Arterial blood temperature T b (SI unit: K), which is the temperature at which blood leaves the arterial blood veins and enters the capillaries. T is the temperature in the tissue, which is the dependent variable that is solved for and not a material property. The default tis K. Metabolic heat source Q met (SI unit: W/m 3 ), which describes heat generation from metabolism. Enter this quantity as the unit power per unit volume. THE BIOHEAT TRANSFER INTERFACE 125

126 The Heat Transfer in Porous Media Interface After selecting Heat Transfer in Porous Media, found under the Heat Transfer branch ( ) in the Model Wizard, a Heat Transfer ( ) node is added to the Model Builder with these default nodes: Porous Matrix, Heat Transfer in Fluids, Thermal Insulation, and Initial Values. Right-click the main node to open a context menu and add as many nodes as required to define the equations, properties and boundary conditions. The Heat Transfer in Porous Media interface ( ) is an extension of the generic Heat Transfer interface that includes modeling heat transfer through convection, conduction and radiation, conjugate heat transfer, and non-isothermal flow. The ability to define material properties, boundary conditions and more for porous media heat transfer is activated by selecting the Heat transfer in porous media check box on the Heat Transfer settings window (Figure 3-1). Figure 3-1: The ability to model porous media heat transfer is activated by selecting the Heat transfer in porous media check box in the Heat Transfer settings window. Tip When a different Default model is selected (for example, Heat transfer in solids), the node in the Model Builder also changes to match. This is a quick way to toggle between the Heat Transfer interfaces. 126 CHAPTER 3: THE HEAT TRANSFER BRANCH

127 See Also Show More Physics Options Domain, Boundary, Edge, Point, and Pair Features for the Heat Transfer in Porous Media Interface Theory for the Heat Transfer in Porous Media Interface Theory for the Heat Transfer Interfaces in the COMSOL Multiphysics User s Guide Domain, Boundary, Edge, Point, and Pair Features for the Heat Transfer in Porous Media Interface The Heat Transfer in Porous Media Interface has these features described in this section: Heat Transfer in Fluids Porous Matrix Thermal Dispersion These domain, boundary, edge, point, and pair conditions are described for The Heat Transfer Interface in the COMSOL Multiphysics User s Guide (listed in alphabetical order): Important The links to features described the COMSOL Multiphysics User s Guide do not work in the PDF, only from within the online help. Tip To locate and search all the documentation, in COMSOL, select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree. Boundary Heat Source Continuity Heat Flux Heat Source Heat Transfer in Solids THE HEAT TRANSFER IN POROUS MEDIA INTERFACE 127

128 Line Heat Source Outflow Periodic Heat Condition Point Heat Source Surface-to-Ambient Radiation Symmetry Temperature Thermal Insulation Thin Thermally Resistive Layer and Pair Thin Thermally Resistive Layer Heat Transfer in Fluids Note The Heat Transfer in Fluids feature is available with the basic COMSOL Multiphysics license. DOMAIN SELECTION From the Selection list, choose the domains to define the heat transfer. MODEL INPUTS If user-defined property groups are added, this section displays fields and values that are inputs to expressions that define material properties. There are also two standard model inputs Absolute pressure and Velocity field. Enter the Absolute pressure p (SI unit: Pa). The default is atmosphere pressure (101,325 Pa). From the Velocity field list, select an existing velocity field in the model (for example, Velocity field (spf/fp1) from a Laminar Flow interface) or select User defined to enter values or expressions for the components of the Velocity field (SI unit: m/s). COORDINATE SYSTEM SELECTION The Global coordinate system is selected by default. The Coordinate system list contains any additional coordinate systems that the model includes. 128 CHAPTER 3: THE HEAT TRANSFER BRANCH

129 HEAT CONDUCTION The default Thermal conductivity k (SI unit: W/(m K)) uses values From material. If User defined is selected, choose Isotropic, Diagonal, Symmetric, or Anisotropic based on the characteristics of the thermal conductivity and enter another value or expression in the field or matrix. Tip 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. THERMODYNAMICS The defaults for the Density (SI unit: kg/m 3 ), Heat capacity at constant pressure C P (SI unit: J/(kg K)), and the Ratio of specific heats (unitless) take values From material for a general gas or liquid. Select User defined to enter other values or expressions. Porous Matrix The Porous Matrix feature is used to specify the thermal properties of a porous matrix. Right-click to add a Thermal Dispersion subnode. DOMAIN SELECTION From the Selection list, choose the domains to define the heat transfer in porous media. MODEL INPUTS This section contain fields and values that are inputs to expressions that define material properties. If such user-defined materials are added, the model inputs appear here. Initially, this section is empty. IMMOBILE SOLIDS This section contains fields and values that are inputs to expressions that define material properties. The Solid material list can point to any material in the model. Enter a Volume fraction p (unitless) for the solid material. HEAT CONDUCTION The default Thermal conductivity k p (SI unit: W/(m K)) uses values From material. If User defined is selected, choose Isotropic, Diagonal, Symmetric, or Anisotropic based on THE HEAT TRANSFER IN POROUS MEDIA INTERFACE 129

130 the characteristics of the thermal conductivity and enter another value or expression in the field or matrix. Tip 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 = k p T, which is Fourier s law of heat conduction. THERMODYNAMICS The default Density p (SI unit: kg/m 3 ) uses values From material. If User defined is selected, enter another value or expression. The default Specific heat capacity C p,p (SI unit: J/(kg K)) uses values From material. If User defined is selected, enter another value or expression. Tip The specific heat capacity describes the amount of heat energy required to produce a unit temperature change in a unit mass of the solid material. The equivalent volumetric heat capacity of the solid-liquid system is calculated from C p eq = p p C p p + L C p Thermal Dispersion Right-click the Porous Matrix node to add the Thermal Dispersion feature. This adds an extra term k d T to the right-hand side of Equation 2-31 and specifies the values of the longitudinal and transverse dispersivities. DOMAIN SELECTION From the Selection list, choose the domains to activate the thermal dispersion. DISPERSIVITIES Define the Longitudinal dispersivity lo (SI unit: m) and Transverse dispersivity tr (SI unit: m). For the Transverse vertical dispersivity the Thermal Dispersion node defines the tensor of dispersive thermal conductivity 130 CHAPTER 3: THE HEAT TRANSFER BRANCH

131 kd ij = L C p L D ij where D ij is the dispersion tensor D ij = u k u l ijkl u and ijkl is the fourth order dispersivity tensor lo tr ijkl = tr ij kl ik jl + il jk THE HEAT TRANSFER IN POROUS MEDIA INTERFACE 131

132 132 CHAPTER 3: THE HEAT TRANSFER BRANCH

133 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 Interface Theory for the Heat Transfer in Thin Shells Interface 133

134 The Heat Transfer in Thin Shells Interface 3D The Thin Conductive Shell interface ( ) opens after selecting Heat Transfer in Thin Shells under the Heat Transfer branch ( ) in the Model Wizard. It is available for 3D models. The Thin Conductive Shell (htsh) interface is suitable for solving thermal-conduction problems in thin structures and has the equations, edge and point conditions, and heat sources for modeling heat transfer in thin conductive shell, solving for the temperature. The Thin Conductive Layer is the main feature. It adds the equation for the temperature and provides a setting window for defining the thermal conductivity, the heat capacity and the density (see Equation 4-1). When this interface is added, these default nodes are also added to the Model Builder Thin Conductive Layer, Insulation/Continuity (a default boundary condition), and Initial Values. Right-click the Thin Conductive Shell node to add other features that implement, for example, edge or point conditions and heat sources. Model Shell Conduction: Model Library path Heat_Transfer_Module/ Tutorial_Models/shell_conduction INTERFACE IDENTIFIER The interface identifier is a text string that can be used to reference the respective physics interface if appropriate. Such situations could occur when coupling this interface to another physics interface, or when trying to identify and use variables defined by this physics interface, which is used to reach the fields and variables in expressions, for example. It can be changed to any unique string in the Identifier field. The default identifier (for the first interface in the model) is htsh. 134 CHAPTER 4: HEAT TRANSFER IN THIN SHELLS

135 BOUNDARY SELECTION The default setting is to include All boundaries in the model to define the dependent variables and the equations. To choose specific boundaries, select Manual from the Selection list. SHELL THICKNESS Define the Shell thickness d s (SI unit: m) (see Equation 4-1). The default is 0.01 m. DEPENDENT VARIABLES The dependent variable (field variable) is for the Temperature T. The name in the corresponding field can be changed, but the names of fields and dependent variables must be unique within a model. DISCRETIZATION To display additional features for the physics interfaces and feature nodes, click the Show button ( ) on the Model Builder and then select Discretization. Select a Frame type Spatial (the default) or Material. Select Quadratic (the default), Linear, Cubic, or Quartic for the Temperature. Specify the Value type when using splitting of complex variables Real or Complex (the default). Show More Physics Options See Also Boundary, Edge, Point, and Pair Conditions for the Thin Conductive Shell Interface Theory for the Heat Transfer in Thin Shells Interface Boundary, Edge, Point, and Pair Conditions for the Thin Conductive Shell Interface The Heat Transfer in Thin Shells Interface (named Thin Conductive Shell in the Model Wizard), has the following boundary, edge, point and pair conditions described (listed in alphabetical order): Note When features are described for other interfaces, the difference for this interface is that Boundaries are selected instead of Domains. Change Effective Thickness THE HEAT TRANSFER IN THIN SHELLS INTERFACE 135

136 Change Thickness Edge Heat Source Edge Temperature (only Edge Temperature is available for this interface) Initial Values Insulation/Continuity Heat Flux Heat Source Out-of-Plane Convective Cooling Out-of-Plane Heat Flux Out-of-Plane Radiation Point Heat Source Radiation Surface-to-Ambient Radiation Thin Conductive Layer Tip To locate and search all the documentation, in COMSOL, select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree. Thin Conductive Layer The Thin Conductive Layer feature adds the heat equation for conductive heat transfer in shells (see Equation 4-1). BOUNDARY SELECTION From the Selection list, choose the boundaries to define the temperature and the heat transfer equation that defines the temperature field. MODEL INPUTS This section has fields and values that are inputs to expressions that define material properties. If such user-defined property groups have been added, the model inputs are included here. 136 CHAPTER 4: HEAT TRANSFER IN THIN SHELLS

137 HEAT CONDUCTION See Also Thermal Conductivity Tensor Components By default, the Thermal conductivity k (SI unit: W/(m K)) uses values From material. If User defined is selected, choose Isotropic, Diagonal, Symmetric, or Anisotropic based on the characteristics of the thermal conductivity and enter other values or expressions in the field or matrix. THERMODYNAMICS Specify the Heat capacity at constant pressure C p (SI unit: J/(kg K)) to describe the amount of heat energy required to produce a unit temperature change in a unit mass, and the Density (SI unit: kg/m3). The default settings use values From material. If User defined is selected, enter other values or expressions. Heat Source The Heat Source feature adds a thermal source Q. It adds the following contributions to the right-hand side of Equation 4-1: d s Q BOUNDARY SELECTION From the Selection list, choose the boundaries to define the heat source. HEAT SOURCE Enter a value or expression for the Shell heat source Q (SI unit: W/m 3 ), which describes heat generation within the shell. Express heating and cooling with positive and negative values, respectively. Initial Values The Initial Values node adds an initial value for the temperature that can serve as an initial condition for a transient simulation or as an initial guess for a nonlinear solver. If more than one set of initial values is needed, right-click the Initial Values node. THE HEAT TRANSFER IN THIN SHELLS INTERFACE 137

138 BOUNDARY SELECTION From the Selection list, choose the boundaries to define the initial value. INITIAL VALUES Enter a value or expression for the initial value of the Temperature T. The default is approximately room temperature, K (20º C). Change Thickness Use the Change Thickness feature to give parts of the shell a different thickness than that what is specified on the Thin Conductive Shell interface Shell Thickness section. BOUNDARY SELECTION From the Selection list, choose the boundaries to define the thickness. CHANGE THICKNESS Specify a value for the Shell thickness d s (SI unit: m). The default value is 0.01 m. This value replaces the overall thickness for the boundaries that are selected. Insulation/Continuity The Insulation/Continuity feature is the default edge condition. On external edges, this edge condition means that there is no heat flux across the edge: n k g T = 0 On internal edges, this edge condition means that the temperature field and its flux is continuous across the edge. EDGE SELECTION From the Selection list, choose the edges to define the insulation/continuity. PAIR SELECTION If Insulation/Continuity is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect. Radiation Use the Radiation feature to add surface-to-ambient radiation to edges. The net inward heat flux from surface-to-ambient radiation is 138 CHAPTER 4: HEAT TRANSFER IN THIN SHELLS

139 where is the Stefan Boltzmann constant (a predefined physical constant). 4 q = d e T amb T 4 EDGE SELECTION From the Selection list, choose the edges to define the radiation flux. PAIR SELECTION If Radiation is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect. SURFACE-TO-AMBIENT RADIATION The default Surface emissivity e is used From material. It is a number between 0 and 1. An emissivity of 0 means that the surface emits no radiation at all and an emissivity of 1 means that it is a perfect blackbody. Select User defined to enter another value. Enter an Ambient temperature T amb (SI unit: K). The default is K. Change Effective Thickness The Change Effective Thickness feature models edges with another thickness than the overall thickness that is specified in the Thin Conductive Shell interface Shell Thickness section. It defines the height of the part of the edge that is exposed to the ambient surroundings. EDGE SELECTION From the Selection list, choose the edges to define effective thickness. PAIR SELECTION If Change Effective Thickness is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect. CHANGE EFFECTIVE THICKNESS Enter a value for the Effective thickness d e (SI unit: m). The default is 0.01 m. This value replaces the overall thickness in the edges selected in the Edges section. Edge Heat Source The Edge Heat Source feature models a linear heat source (or sink). THE HEAT TRANSFER IN THIN SHELLS INTERFACE 139

140 EDGE SELECTION From the Selection list, choose the edges to define the heat source. PAIR SELECTION If Edge Heat Source is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect. EDGE HEAT SOURCE Enter a value or expression for the Edge heat source q e (SI unit: W/m). A positive q e means heating while a negative q e means cooling. The added heat source is equal to q e. Point Heat Source The Point Heat Source feature models a point heat source (or sink). POINT SELECTION From the Selection list, choose the point to define the heat source. POINT HEAT SOURCE Enter a value or expression for the Point heat source q p (SI unit: W). A positive q p means heating while a negative q p means cooling. The added heat source is equal to q p. 140 CHAPTER 4: HEAT TRANSFER IN THIN SHELLS

141 Theory for the Heat Transfer in Thin Shells Interface The Heat Transfer in Thin Shells Interface theory is described in this section: About Thin Conductive Shells Heat Transfer Equation in Thin Conductive Shell Thermal Conductivity Tensor Components About Thin Conductive Shells The Thin Conductive Shell (htsh) interface opens after selecting The Heat Transfer in Thin Shells Interface in the Model Wizard. This interface supports two types of heat transfer: conduction and out-of-plane heat transfer and is suitable for solving thermal-conduction problems in thin structures. Because the thermal conductivity across the shell thickness is very large or the shell is so thin, assume constant temperature through the shell thickness. The Thin Conductive Layer node is the main feature. It adds the equation for the temperature and provides a setting window for defining the thermal conductivity, the heat capacity and the density: d s C T p + t T d k T s g T = 0 (4-1) Heat Transfer Equation in Thin Conductive Shell The dependent variable is the temperature T. The interface is defined on 3D faces. The governing equation for heat transfer in thin shells is: d s C T p + t T d k T s g T = d s Q+ d s h u T ext, u T + d h T s d ext d + d s T4 u amb u + d s d T 4 amb d + d s q u + d s q d, T 4, T 4 Where T is the tangential derivative along the shell and is the density (SI unit: kg/m 3 ), T (4-2) THEORY FOR THE HEAT TRANSFER IN THIN SHELLS INTERFACE 141

142 d s is the shell thickness (SI unit: m) C p is the heat capacity (SI unit: J/(kg K) k g is the thermal conductivity (SI unit: W/(m K) Q is the heat source (SI unit: W/m 3 ) h u and h d are the out-of-plane heat transfer coefficients, upside and downside (SI unit: W/(m 2 K)) T ext, u and T ext, d are the out-of-plane external temperatures, upside and downside (SI unit: K) u and d are the out-of-plane surface emissivities, upside and downside (SI unit: 1), T amb, u and T amb, d are the out-of-plane ambient temperatures, upside and downside (SI unit: K) q u and q d are the out-of-plane inward heat fluxes, upside and downside (SI unit: W/ m 2 ) Thermal Conductivity Tensor Components The thermal conductivity k describes the relationship between the heat flux vector q and the temperature gradient T T as in q = k g T which is Fourier s law of heat conduction (see also The Heat Equation). The tensor components are specified in the shell local coordinate system, which is defined from the geometric tangent and normal vectors. The local x direction, e xl, is the surface tangent vector t 1 and the local z direction, e zl, is the normal vector n. Their cross product defines the third orthogonal direction such that: T e xl = t 1 e yl = e xl e zl = n t 1 e zl = n From this, a transformation matrix between the shell s local coordinate system and the global coordinate system can be constructed in the following way: 142 CHAPTER 4: HEAT TRANSFER IN THIN SHELLS

143 A = e xlx e ylx e zlx e xly e yly e zly e xlz e ylz e zlz The thermal conductivity tensor, k g, can be expressed as k g = AkA t THEORY FOR THE HEAT TRANSFER IN THIN SHELLS INTERFACE 143

144 144 CHAPTER 4: HEAT TRANSFER IN THIN SHELLS

145 5 The Radiation Heat Transfer Branch This chapter describes the Heat Transfer Module s interfaces for modeling radiative heat transfer, including the Theory for the Radiative Heat Transfer Interfaces. The following physics interface for modeling of radiative heat transfer are available in the Heat Transfer>Radiation branch ( ): Heat Transfer with Surface-to-Surface Radiation (ht) interface, is a Heat Transfer interface where surface-to-surface radiation is active by default, enabling the Radiation Settings section. All available features are as described in the About the Heat Transfer Interfaces section. The Surface-To-Surface Radiation Interface is used for separate radiosity computations. The Heat Transfer with Radiation in Participating Media Interface is used to model radiative heat transfer in nontransparent media in combination with the other Heat Transfer physics interfaces for heat transfer through convection, conduction, and surface-to-ambient radiation. The Radiation in Participating Media Interface is used for separate computations of radiation in participating media. 145

146 The Surface-To-Surface Radiation Interface The Surface-to-Surface Radiation interface ( ), found under the Heat Transfer>Radiation branch ( ) in the Model Wizard, treats thermal radiation as an energy transfer between boundaries and external heat sources where the medium does not participate in the radiation (radiation in transparent media). The process transfers energy directly between boundaries and external radiation sources. The radiation therefore contributes to the boundary conditions rather than to the heat equation itself. When this interface is added, the Initial Values default node is also added to the Model Builder. Right-click the node to add a Surface-to-Surface Radiation boundary condition or other feature nodes. For the Surface-to-Surface Radiation interface, select a Stationary or Time Dependent study as a preset study type. The surface-to-surface radiation is always stationary (that is, the radiation time scale is assumed to be shorter than any other time scale), but the interface is compatible with all standard study types. Important Absolute (thermodynamical) temperature units must be used. See Specifying Model Equation Settings in the COMSOL Multiphysics User s Guide. Note For this interface, COMSOL Multiphysics works under the assumption that the domain medium does not participate in the radiation process. If the media participate in the radiation, then select The Radiation in Participating Media Interface. Tip For another way to model combinations of conductive, convective, and radiative heat transfer, see the Radiation Settings section described for the Heat Transfer interface. 146 CHAPTER 5: THE RADIATION HEAT TRANSFER BRANCH

147 INTERFACE IDENTIFIER The interface identifier is a text string that can be used to reference the respective physics interface if appropriate. Such situations could occur when coupling this interface to another physics interface, or when trying to identify and use variables defined by this physics interface, which is used to reach the fields and variables in expressions, for example. It can be changed to any unique string in the Identifier field. The default identifier (for the first interface in the model) is rad. BOUNDARY SELECTION The default setting is to include All boundaries in the model to define the dependent variables and the equations. To choose specific boundaries, select Manual from the Selection list. RADIATION SETTINGS See Also See The Heat Transfer Interface for details about these settings. Select the Surface-to-surface radiation method Hemicube or Direct area integration. If Direct area integration is selected, select the Radiation integration order. Sharp angles and small gaps between surfaces may require a higher integration order for accuracy but also more time to evaluate the irradiation. If Hemicube is selected, select the Radiation resolution. Select the Use radiation groups check box to enable the ability of defining radiation groups. This can speed up the radiation calculations in many cases. DEPENDENT VARIABLES The dependent variable (field variable) is for the Surface radiosity J. The name can be changed but the names of fields and dependent variables must be unique within a model. THE SURFACE-TO-SURFACE RADIATION INTERFACE 147

148 DISCRETIZATION To display this section, click the Show button ( ) and select Discretization. Select Quadratic (the default), Linear, Cubic, or Quartic for the Surface radiosity. Specify the Value type when using splitting of complex variables Real or Complex (the default). Show More Physics Options Domain, Boundary, Edge, Point, and Pair Conditions for the Surface-to-Surface Radiation Interface See Also Radiative Heat Transfer in Transparent Media Theory for the Surface-to-Surface Radiation Interface Domain, Boundary, Edge, Point, and Pair Conditions for the Surface-to-Surface Radiation Interface The Surface-To-Surface Radiation Interface has these domain, boundary, edge, point, and pair features available (listed in alphabetical order): External Radiation Source Initial Values Opaque Prescribed Radiosity Radiation Group Reradiating Surface Surface-to-Surface Radiation (Boundary Condition) In the COMSOL Multiphysics User s Guide: Continuity on Interior Boundaries See Also Identity and Contact Pairs Specifying Boundary Conditions for Identity Pairs Important The links to the features described in the COMSOL Multiphysics User s Guide do not work in the PDF, only from within the online help. 148 CHAPTER 5: THE RADIATION HEAT TRANSFER BRANCH

149 Tip To locate and search all the documentation, in COMSOL, select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree. Surface-to-Surface Radiation (Boundary Condition) The Surface-to-Surface Radiation boundary condition feature handles radiation with view factor calculation. The feature adds a radiosity dependent variable to its selection and uses it as surface radiosity. BOUNDARY SELECTION From the Selection list, choose the boundaries to define the surface-to-surface radiation. MODEL INPUTS This section has fields and values that are inputs to expressions that define material properties. If such user-defined property groups have been added, the model inputs are included here. There is one standard model input the Temperature T (SI unit: K). The default is the temperature variable in the Heat Transfer interface or K in the Surface-to-Surface Radiation interface. This model input is used in the expression for the blackbody radiation intensity. RADIATION SETTINGS Select a Radiation Direction based on the geometric normal (nx, ny, nz): Opacity controlled is the default and requires that each boundary is adjacent to exactly one opaque domain. Opacity is controlled by the Opaque boundary condition. Select Negative normal direction to specify that the surface radiates in the negative normal direction. Select Positive normal direction if the surface radiates in the positive normal direction. Select Both sides if the surface radiates on both sides. Enter an Ambient temperature T amb (SI unit: K). The default is K. THE SURFACE-TO-SURFACE RADIATION INTERFACE 149

150 Set T amb to the far-away temperature in directions where no other boundaries obstruct the view. Inside a closed cavity, the ambient view factor, F amb, is theoretically zero and the value of T amb therefore should not matter. It is, however, good practice to set T amb to T in such cases because that minimizes errors introduced by the finite resolution of the view factor evaluation. When Both sides is selected, define the Ambient temperature T amb, u and T amb, d on the up and down side respectively. The geometric normal points from the down side to the up side. SURFACE EMISSIVITY When the Radiation Direction is set to Both sides, select an option from the Material on upside and Material on downside lists. The default uses Domain material, and the list contains other options based on the material defined in the model. Note When Both sides is selected, also define the Surface emissivity u and d on the up and down side respectively. The geometric normal points from the down side to the up side. Set the surface emissivity to a number between 0 and 1, where 0 represents diffuse mirror and 1 is appropriate for a perfect blackbody. The proper value for a physical material lies somewhere in-between and can be found from tables or measurements. By default, the Surface emissivity uses values From material, which is a property of the material surface that depends both on the material itself and the structure of the surface. Make sure that a material is defined at the boundary level (by default materials are defined at the domain level). About the Surface-to-Surface Radiation Boundary Conditions Radiation Group Boundaries See Also Domain, Boundary, Edge, Point, and Pair Conditions for the Surface-to-Surface Radiation Interface 150 CHAPTER 5: THE RADIATION HEAT TRANSFER BRANCH

151 Opaque Use the Opaque feature to assign domains to the surface-to-surface radiation when Opacity controlled is set as the Radiation Direction. DOMAIN SELECTION From the Selection list, choose the domains to define the surface-to-surface radiation opacity. Initial Values The Initial Values feature adds an initial value for the surface radiosity that can serve as an initial condition for a transient simulation or as an initial guess for a nonlinear solver. Right-click to add more than one set of initial values. BOUNDARY SELECTION From the Selection list, choose the boundaries to define the initial value of the Surface radiosity J (SI unit: W/m 2 ). INITIAL VALUES Enter a value or expression for the initial value. Reradiating Surface The Reradiating Surface feature is a variant of the surface-to-surface radiation feature with e = 0. Reradiation surfaces are common as an approximation of a surface that is well insulated on one side and for which convection effects can be neglected on the opposite (radiating) side. It resembles a mirror that absorbs all irradiation and then radiates it back in all directions: J = G. The feature adds the radiosity dependent variable to its selection and uses it as surface radiosity. Heat flux on surface-to-surface boundary is zero where q = 0. BOUNDARY SELECTION From the Selection list, choose the boundaries corresponding to reradiation surfaces. MODEL INPUTS This section has fields and values that are inputs to expressions that define material properties. If such user-defined property groups have been added, the model inputs are included here. THE SURFACE-TO-SURFACE RADIATION INTERFACE 151

152 There is one standard model input the Temperature T (SI unit: K). The default is the temperature variable in the heat transfer physic interface or K in the surface to surface physic interface. It is used in the blackbody radiation intensity expression. RADIATION SETTINGS Note See Radiation Settings for the Surface-to-Surface Radiation (Boundary Condition). Prescribed Radiosity Use the Prescribed Radiosity feature to specify radiosity on the boundary. Radiosity can be defined as graybody radiation. The radiosity expressions is then et 4. A user-defined surface radiosity expression can also be defined. BOUNDARY SELECTION From the Selection list, choose the boundaries to define the prescribed radiosity. MODEL INPUTS This section has fields and values that are inputs to expressions that define material properties. If such user-defined property groups have been added, the model inputs are included here. There is one standard model input the Temperature T (SI unit: K). The default is the temperature variable in the Heat Transfer interface or K in the Surface-to-Surface Radiation interface. It is used in the blackbody radiation intensity expression. RADIATION DIRECTION Select a Radiation Direction based on the geometric normal (nx, ny, nz): Opacity controlled is the default and requires that each boundary is adjacent to exactly one opaque domain. Opacity is controlled by the Opaque feature condition. Select Negative normal direction to specify that the surface radiates in the negative normal direction. 152 CHAPTER 5: THE RADIATION HEAT TRANSFER BRANCH

153 Select Positive normal direction if the surface radiates in the positive normal direction. Select Both sides if the surface radiates on both sides. RADIOSITY Note Radiosity does not directly affect the boundary condition on the boundary where it is specified, but rather how that boundary affects others through radiation. Select a Radiosity expression Graybody radiation, Blackbody radiation, or User defined. If Blackbody radiation is selected, it sets the surface radiosity expression J = T 4. If Graybody radiation is selected, it sets the surface radiosity expression to J = T 4. By default, the Surface emissivity is defined From material. In this case, make sure that a material is defined at the boundary level (materials are defined by default at the domain level). If User defined is selected, enter another value for. When Both sides is selected, define the Surface emissivity u and d on the up and down side respectively. The geometric normal points from the down side to the up side. If User defined is selected, it sets the surface radiosity expression to J = J 0. which specifies how the radiosity of a boundary is evaluated when that boundary is visible in the calculation of the irradiation onto another boundary in the model. Enter a Surface radiosity expression, J 0 (SI unit: W/m 2 ). The default is 0. When Both sides is selected, define the surface radiosity expression J 0, u and J 0, d on the up and down side respectively. The geometric normal points from the down side to the up side. Radiation Group Add a Radiation Group to a Surface-to-Surface Radiation interface or any version of a Heat Transfer interface where the Surface-to-surface radiation check box is selected. Note Select the Use radiation groups check box under Radiation Settings. By default the check box is not selected, which means that all radiative boundaries belong to the same radiation group. THE SURFACE-TO-SURFACE RADIATION INTERFACE 153

154 The Radiation Group feature enables you to specify radiation groups to speed up the radiation calculations and gather boundaries in a radiation problem that have a chance to see one another. When the Use radiation groups check box is selected, the feature is automatically added to the Model Builder and contains all boundaries selected in the Surface-to-Surface Radiation (Boundary Condition), a Reradiating Surface, or a Prescribed Radiosity) feature. BOUNDARY SELECTION From the Selection list, choose the boundaries that belong to the same radiation group. This section should contains any boundary that is selected in a Surface-to-Surface Radiation, a Reradiating Surface, or a Prescribed Radiosity node and that a chance to see one of the boundary that is already selected in the Radiation Group. See Also Radiation Group Boundaries External Radiation Source Note The External Radiation Source node is selected from the Global submenu and is available for 2D and 3D models in the Surface-to-Surface Radiation interface or in any version of a Heat Transfer interface where the Surface-to-surface radiation check box is selected. Add an External Radiation Source to define an external radiation source as a point or directional radiation source with view factor calculation. Each External Radiation Source feature contributes to the incident radiative heat flux, G, on all the boundaries where a Surface-to-surface or Reradiating surface boundary condition is active. The source contribution, G ext, is equal to the product of the view factor of the source by the 154 CHAPTER 5: THE RADIATION HEAT TRANSFER BRANCH

155 source radiosity. For radiation sources located on a point, G ext =F ext P s. For directional radiative source G ext = F ext q 0,s. Note The external radiation sources are ignored on the boundaries where neither Surface-to-Surface Radiation nor Reradiating Surface is active. In particular they are not contributing on boundaries where Surface-to-Surface Ambient is active. 2D 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, Infinite distance, or Solar position. Solar position is only available in 3D. If Point coordinate is selected, define the Source location x s (SI unit: m) and the Source power P s (SI unit: W, default is 0). The source radiates uniformly in all directions. Note x s should not belong to any surface where a Surface-to-surface or Reradiating surface boundary condition is active. If Infinite distance is selected, define the Incident radiation direction i s (unitless) and the Source heat flux q 0,s (SI unit: W/m 2 ). The default is 0. For 3D models, if Solar position is selected define all the required information to estimate the external radiative heat source due to the sun. Define the Latitude (decimal value, positive in the northern hemisphere; the default is Greenwich UK latitude, ), the Longitude (decimal value, positive at the East of the Prime Meridian; the default is Greenwich UK longitude, ), and the Time zone (number of hours to add to UTC to get local time; the default is Greenwich UK time zone, 0) in the Location table. THE SURFACE-TO-SURFACE RADIATION INTERFACE 155

156 Enter the Day, (default 1) the Month (default 6, June), and the Year (default 2012) in the date table. The solar position is accurate for a date between the years 2000 and Indicate the Hour (default 12), the Minute (default 0) and the Second (0) that defines the local time in the local Time table. Note The sun position is updated if the location, date, or local time changes during a simulation. In particular for transient analysis, if the unit system for the time is in seconds (default choice), the time change can be taken into account by adding t to the Second field in the Local time table. Define the incident radiative intensity coming from the sun, I s (SI unit: W/m 2, default value 1000 W/m 2 ), in the Solar energy field. I s 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. 156 CHAPTER 5: THE RADIATION HEAT TRANSFER BRANCH

157 The Radiation in Participating Media Interface The Radiation in Participating Media interface ( ), found under the Heat Transfer>Radiation branch ( ) in the Model Wizard, enables the modeling of radiative heat transfer inside a participating medium. This interface solves for radiative intensity field. When the interface is added, these default nodes are also added to the Model Builder Radiation in Participating Media, Wall, Continuity on Interior Boundary, and Initial Values. Right-click the main node to add boundary conditions or other features. Radiative Heat Transfer in Finite Cylindrical Media: Model Library path Heat_Transfer_Module/Tutorial_Models/cylinder_participating_media Model Radiative Heat Transfer in a Utility Boiler: Model Library path Heat_Transfer_Module/Process_and_Manufacturing/boiler INTERFACE IDENTIFIER The interface identifier is a text string that can be used to reference the respective physics interface if appropriate. Such situations could occur when coupling this interface to another physics interface, or when trying to identify and use variables defined by this physics interface, which is used to reach the fields and variables in expressions, for example. It can be changed to any unique string in the Identifier field. The default identifier (for the first interface in the model) is rpm. DOMAIN SELECTION The default setting is to include All domains in the model to define the dependent variables and the equations. To choose specific domains, select Manual from the Selection list. THE RADIATION IN PARTICIPATING MEDIA INTERFACE 157

158 REFRACTIVE INDEX Define the Refractive index (a dimensionless number) of the participating media, n. The default value is 1. Note The same refractive index is used for the whole model. ADVANCED SETTINGS To display this section, click the Show button ( ) and select Advanced Physics Options. Normally these settings do not need to be changed. Performance Index Select a Performance index P index from the list. Select a value between 0 and 1 that modifies the strategy used to define automatic solver settings. The default is 0.5. With small values, a robust setting for the solver is expected. With large values (up to 1), less memory is needed to solve the model. DISCRETIZATION To display this section, click the Show button ( ) and select Discretization. Select Quadratic, Linear (the default), Cubic, or Quartic for the Radiative intensity. Specify the Value type when using splitting of complex variables Real (the default) or Complex. Show More Physics Options See Also Domain, Boundary, Edge, Point, and Pair Conditions for the Radiation in Participating Media Interface DEPENDENT VARIABLES The interface includes a dependent variable for intensity along discrete directions. The number of direction depends on the Discrete ordinates method order selected. 3D In 3D, S2, S4, S6, and S8 generate 8, 24, 48, and 80 directions, respectively. 158 CHAPTER 5: THE RADIATION HEAT TRANSFER BRANCH

159 2D In 2D, S2, S4, S6, and S8 generate 4, 12, 24, and 40 directions, respectively. If required, edit the Radiative intensities default names I1... In. The name can be changed but the names of fields and dependent variables must be unique within a model. Domain, Boundary, Edge, Point, and Pair Conditions for the Radiation in Participating Media Interface The Radiation in Participating Media Interface has these domain, boundary, edge, point, and pair features available: Continuity on Interior Boundary Initial Values Opaque and Incident Intensity (described for the Heat Transfer interface) Radiation in Participating Media Tip To locate and search all the documentation, in COMSOL, select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree. Radiation in Participating Media The Radiation in Participating Media feature uses the radiative transfer equation I s = s I b T I s I s where Is is the radiative intensity at the position s position in the direction T is the temperature,, s are absorption, extinction, and scattering coefficients, respectively n 2 T 4 I b T = is the blackbody radiation intensity and n is the refractive index THE RADIATION IN PARTICIPATING MEDIA INTERFACE 159

160 DOMAIN SELECTION From the Selection list, choose the domains to define the radiation in participating media. MODELS INPUTS This section has fields and values that are inputs to expressions that define material properties. If such user-defined property groups have been added, the model inputs are included here. There is one standard model input the Temperature T (SI unit: K). The default is K and is used in the blackbody radiation intensity expression. ABSORPTION The default Absorption coefficient (SI unit: 1/m) uses the value From material. The absorption coefficient defines the amount of radiation, I, that is absorbed by the medium. If User defined is selected, enter another value or expression. SCATTERING The default Scattering coefficient s (SI unit: 1/m) uses the value From material. If User defined is selected, enter another value or expression. The default value is 0. Select the Scattering type Isotropic, Linear anisotropic, or Nonlinear anisotropic. Isotropic (the default) and corresponds to the scattering phase function 0 = 1 If Linear anisotropic is selected, it defines the scattering phase function as 0 = 1 + a 1 0 Enter the Legendre coefficient a 1. If Nonlinear anisotropic is selected, it defines the scattering phase function 0 = 1 + a m P m 0 m = 1 Enter the Legendre coefficients a 1,, a 12 as required CHAPTER 5: THE RADIATION HEAT TRANSFER BRANCH

161 Initial Values The Initial Values node defines an initial value for the discrete intensities I 1,, I n that can serve as an initial condition for a transient simulation or as an initial guess for a nonlinear solver. Right-click to add more than one set of initial values. DOMAIN SELECTION From the Selection list, choose the domains to define an initial value. INITIAL VALUES Enter a value or expression for the initial value of the Radiative intensities I 1,, I n (SI unit: W/m 2 ) in the I 1,, I n edit field. The default value is the blackbody temperature, I b. THE RADIATION IN PARTICIPATING MEDIA INTERFACE 161

162 The Heat Transfer with Radiation in Participating Media Interface The Heat Transfer with Radiation in Participating Media interface ( ), found under the Heat Transfer>Radiation branch ( ) in the Model Wizard, combines features from the Radiation in Participating Media and Heat Transfer interfaces. This enables the modeling of radiative heat transfer inside a participating medium combined with heat transfer in solids and fluids. This interface solves for radiative intensity and temperature fields. When this interface is added, the Radiation in participating media check box is selected in the Physical Model section of the main Heat Transfer node s settings window. The following default nodes are also added to the Model Builder: Heat Transfer in Solids, Thermal Insulation, Continuity on Interior Boundary, and Initial Values. Right-click the node to add other boundary conditions and features. Note Except where described below, most of the settings windows are the same as described for The Radiation in Participating Media Interface and The Heat Transfer Interface. INTERFACE IDENTIFIER The interface identifier is a text string that can be used to reference the respective physics interface if appropriate. Such situations could occur when coupling this interface to another physics interface, or when trying to identify and use variables defined by this physics interface, which is used to reach the fields and variables in expressions, for example. It can be changed to any unique string in the Identifier field. The default identifier (for the first interface in the model) is ht. DOMAIN SELECTION The default setting is to include All domains in the model to define the dependent variables and the equations. To choose specific domains, select Manual from the Selection list. 162 CHAPTER 5: THE RADIATION HEAT TRANSFER BRANCH

163 PARTICIPATING MEDIA SETTINGS To display this section select the Radiation in participating media check box under Physical Model on any version of the Heat Transfer interface settings window. Define the Refractive index of the participating media, n. Note The same refractive index is used for the whole model. Select the Discrete ordinates method order from the list. This order defines the discretization of the radiative intensity direction. 3D In 3D, S2, S4, S6, and S8 generate 8, 24, 48, and 80 directions, respectively. 2D In 2D, S2, S4, S6, and S8 generate 4, 12, 24, and 40 directions, respectively. Select Linear (the default), Quadratic, Cubic, Quartic, or Quintic to define the discretization level of the Radiative intensity fields. ADVANCED SETTINGS To display this section, click the Show button ( ) and select Advanced Physics Options. Normally these settings do not need to be changed. Performance Index Select a Performance index P index from the list. Select a value between 0 and 1 that modifies the strategy used to define automatic solver settings. The default is 0.5. With THE HEAT TRANSFER WITH RADIATION IN PARTICIPATING MEDIA INTERFACE 163

164 small values, a robust setting for the solver is expected. With large values (up to 1), less memory is needed to solve the model. See Also Show More Physics Options Theory for the Heat Transfer Interfaces Theory for the Radiative Heat Transfer Interfaces Domain, Boundary, Edge, Point, and Pair Conditions for the Heat Transfer with Radiation in Participating Media Interface Domain, Boundary, Edge, Point, and Pair Conditions for the Heat Transfer with Radiation in Participating Media Interface The Heat Transfer with Radiation in Participating Media Interface has these domain, boundary, edge, point, and pair features available and described for the Heat Transfer and Radiation in Participating Media interfaces (listed in alphabetical order): Boundary Heat Source Continuity on Interior Boundary Convective Cooling Continuity Heat Flux Heat Source Heat Transfer in Fluids Heat Transfer in Solids Highly Conductive Layer Inflow Heat Flux Incident Intensity Infinite Elements Initial Values Line Heat Source Opaque Surface Open Boundary Outflow Pair Boundary Heat Source 164 CHAPTER 5: THE RADIATION HEAT TRANSFER BRANCH

165 Pair Thin Thermally Resistive Layer Periodic Heat Condition Point Heat Source Pressure Work Radiation in Participating Media Surface-to-Ambient Radiation Symmetry Temperature Thermal Insulation Thin Thermally Resistive Layer Translational Motion Tip To locate and search all the documentation, in COMSOL, select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree. THE HEAT TRANSFER WITH RADIATION IN PARTICIPATING MEDIA INTERFACE 165

166 Theory for the Radiative Heat Transfer Interfaces The following sections provide some background information about the radiosity method and view factors and includes these topics: The Radiosity Method View Factor Evaluation Radiation and Participating Media Interactions Radiative Transfer Equation Boundary Condition for the Transfer Equation Heat Transfer Equation in Participating Media Discrete Ordinates Method The Radiosity Method The radiation interacts with convective and conductive heat transfer through the source term in the Heat Flux and Boundary Heat Source boundary conditions. By definition, this source must be the difference between incident radiation and radiation leaving the surface. According to Equation 2-12 it is given by q = G T 4 (5-1) where is the surface emissivity, a dimensionless number in the range 01. G is the incoming radiative heat flux, or irradiation (SI unit: W/m 2 ). is the Stefan-Boltzmann constant (a predefined physical constant equal to W/(m 2 T 4 )). The irradiation, G, at a point can in general be written as a sum according to: G = G m + G ext + F amb T amb CHAPTER 5: THE RADIATION HEAT TRANSFER BRANCH

167 where G m is the mutual irradiation, coming from other boundaries in the model (SI unit: W/m 2 ). G ext is the irradiation from external radiation sources (SI unit: W/m 2 ). G ext is the sum to the products, for each external source, of the external heat sources view factor by the corresponding source radiosity. For radiation sources located on a point, G _ext =F _ext(x_s) P s. For directional radiative source G _ext =F _ext(i_s) q _0,s. G ext = G ext P s + G ext q 0 s F amb is an ambient view factor whose value is equal to the fraction of the field of view that is not covered by other boundaries. Therefore, by definition, 0F amb 1 must hold at all points. T amb is the assumed far-away temperature in the directions included in F amb. The Surface-To-Surface Radiation Interface includes these radiation types: Surface-to-Surface Radiation is the default radiation type. It requires accurate evaluation of the mutual irradiation, G m. The incident radiation at one point on the boundary is a function of the exiting radiation, or radiosity, J (W/m 2 ), at every other point in view. The radiosity, in turn, is a function of G m, which leads to an implicit radiation balance: J = 1 G + T 4 = 1 G m J + G ext + F amb T amb + T 4 4 (5-2) Reradiating Surface is a variant of the Surface-to-Surface Radiation radiation type with = 0. Reradiation surfaces are common as an approximation of a surface that is well insulated on one side and for which convection effects can be neglected on the opposite (radiating) side (see Ref. 3). It resembles a mirror that absorbs all irradiation and then radiates it back in all directions. Prescribed Radiosity makes it possible to specify graybody radiation. The radiosity expressions is then T 4. A user-defined surface radiosity expression can also be defined. The Surface-to-Surface Radiation interface treats the radiosity J as a dependent variable unless J is prescribed. THEORY FOR THE RADIATIVE HEAT TRANSFER INTERFACES 167

168 View Factor Evaluation The strategy for evaluating view factors is central to any radiation simulation. Loosely speaking, a view factor is a measure of how much influence the radiosity at a given part of the boundary has on the irradiation at some other part. The quantities G m and F amb in Equation 5-2 are not strictly view factors in the traditional sense. F amb is the view factor of the ambient portion of the field of view, which is considered to be a single boundary with constant radiosity J amb = 4 T amb G m, on the other hand, is the integral over all visible points of a differential view factor times the radiosity of the corresponding source point. In the discrete model, think of it as a product of a view factor matrix and a radiosity vector. This is, however, not necessarily the way the calculation is performed. A separate evaluation is performed for each unique point where G m or F amb is requested, typically for each quadrature point during solution. Differential view factors are normally computed only once, the first time they are needed, and then stored in memory until next time the model definition or the mesh is changed. The Heat Transfer Module supports two surface-to-surface radiation methods, which are selected in the Radiation Settings section from the Heat Transfer interface. Tip View factors are always calculated directly from the mesh, which is a polygonal, flat-faceted representation of the geometry. To improve the accuracy of the radiative heat transfer simulation, the mesh must be refined rather than raising the element order. VIEW FACTOR FOR EXTERNAL RADIATION SOURCES In 3D, the view factor for a point at finite distance is given by cos 4r 2 where is the angle between the normal to the irradiated surface and the direction of the source, and r is the distance from the source. For a source at infinity, the view factor is given by cos. In 2D the view factor for a point at finite distance is given by 168 CHAPTER 5: THE RADIATION HEAT TRANSFER BRANCH

169 cos 2r and the view factor for a source at infinity is cos. SOLAR POSITION The sun is the most common example of an external radiation source. The position of the sun is necessary to determine the direction of the corresponding external radiation source. The direction of sunlight (zenith angle and the solar elevation) is automatically computed from the latitude, longitude, time zone, date, and time using similar method as described in Ref. 5. The estimated solar position is accurate for a date between year 2000 and 2199, due to an approximation used in the Julian Day calculation. The zenith angle (zen) and the azimuth (azi) angles of the sun are converted into a direction vector (is x, is y, is z ) in Cartesian coordinates assuming that the North, the West, and the up directions correspond to the x, y, and z directions, respectively, in the model. The relation between azi, zen and (is x, is y, is z ) is given by: is x = cosazisinzen is y = sinazisinzen is z = coszen RADIATION IN AXISYMMETRIC GEOMETRIES For an axisymmetric geometry, G m and F amb must be evaluated in a corresponding 3D geometry obtained by revolving the 2D boundaries about the axis. COMSOL Multiphysics creates this virtual 3D geometry by revolving the 2D boundary mesh into a 3D mesh. The resolution can be controlled in the azimuthal direction by setting the number of azimuthal sectors, which is the same as the number of elements to a full revolution. Try to balance this number against the mesh resolution in the rz-plane. This number, Azimuthal sectors, is accessible from the Radiation Settings section in physics interfaces for heat transfer. THEORY FOR THE RADIATIVE HEAT TRANSFER INTERFACES 169

170 Select between the hemicube and the direct area integration methods also in axial symmetry. Their settings work the same way as in 3D. Note While G m and F amb are in fact evaluated in a full 3D, the number of points where they are requested is limited to the quadrature points on the boundary of a 2D geometry. The savings compared to a full 3D simulation are therefore substantial despite the full 3D view factor code being used. Radiation and Participating Media Interactions Figure 5-1: Example of interactions between participating media and radiation. In some applications the medium is not completely transparent and the radiation rays interact with the medium. Let I denote the radiative intensity traveling in a given direction,. Different kinds of interactions are observed: Absorption: The medium absorbs a fraction of the incident radiation. The amount of absorbed radiation is I where is the absorption coefficient. Emission: The medium emits radiation in all directions. The amount of emitted radiative intensity is equal to I b, where I b is the blackbody radiation intensity. Scattering: A part of the radiation coming from a given direction is scattered in other directions. The scattering properties of the medium are described by the scattering 170 CHAPTER 5: THE RADIATION HEAT TRANSFER BRANCH

171 phase function i j, which gives the probability that a ray coming from one direction i is scattered into the direction j. The phase function ( i, j ) satisfies i d i = 1 4 Radiative intensity in a given direction is attenuated and augmented by scattering: - It is attenuated because a part of incident radiation in this direction is scattered into other directions. The amount of radiation attenuated by scattering is s I. - It is augmented because a part of radiative intensity coming from other directions is scattered in all direction, including the direction we are looking at. The amount of radiation augmented by scattering is obtained by integrating scattering coming form all directions i : s I 4 i i d i 4 Radiative Transfer Equation The balance of the radiative intensity including all contributions (propagation, emission, absorption, and scattering) can now be formulated. The general radiative transfer equation can be written as (see Ref. 1) I = s I b T I s I where I is the radiative intensity at a given position following the direction T is the temperature,, s are absorption, extinction, and scattering coefficients, respectively I b T = n 2 T (5-3) Equation 5-3 is the blackbody radiation intensity, and n is the refractive index of the media ' is the phase function that gives the probability that a ray from the direction is scattered into the direction. The phase function s definition is material dependent and its definition can be complicated. It is common to use approximate THEORY FOR THE RADIATIVE HEAT TRANSFER INTERFACES 171

172 scattering phase functions that are defined using the cosine of the scattering angle, 0. The current implementation handles: Isotropic phase function: Linear anisotropic phase function: ' = 0 = 1 0 = 1 + a 1 0 Nonlinear anisotropic up to the 12 th order: 0 = 1 + a n P n 0 n = 1 where P n are n th -order Legendre polynomials. Legendre polynomials can be defined by the Rodriguez formula: 12 P k x = d 2 k k! d k x k x 2 1 k A quantity of interest is the incident radiation, denoted G and defined by G = 4 0 I Boundary Condition for the Transfer Equation For gray walls, corresponding to opaque surfaces reflecting diffusively and emitting, the radiative intensity I bnd entering participating media along the direction is d I bnd = w I b T q out for all such that n 0 where I b T = n 2 T (5-4) Equation 5-4 is the blackbody radiation intensity and n is the refractive index w is the surface emissivity, which is in the range [0, 1] 172 CHAPTER 5: THE RADIATION HEAT TRANSFER BRANCH

173 d 1 w is the diffusive reflectivity n is the outward normal vector q out is the heat flux striking the wall: q out = n 0 j w j I j n j For black walls w 1 and d 0. Thus I bnd I b T. Heat Transfer Equation in Participating Media Heat flux in gray media is defined by q r n = 4 0 I n Heat flux divergence can be defined as a function of G and T (see Ref. 1): Q r = qr = G 4T 4 In order to couple radiation in participating media, radiative heat flux is taken into account in addition to conductive heat flux: qq c q r. The heat transfer equation reads T C p u T t q + q :S T p = u p c r T t + Q and is implemented using following form: p C T p u T t q c G 4T 4 :S T p = u p T t + Q p THEORY FOR THE RADIATIVE HEAT TRANSFER INTERFACES 173

174 Discrete Ordinates Method 2D The discrete ordinates method is implemented for 2D and 3D geometries. 3D Radiative intensity is defined for any direction, because the angular space is continuous. In order to treat radiative intensity equation numerically, the angular space is discretized. The SN approximation provides a discretization of angular space into nnn2 in 3D (or nnn22 in 2D) discrete directions. It consists of a set of directions and quadrature weights. Several sets are available in the literature. A set should satisfy first, second, and third moments (see Ref. 1); it is also recommended that the quadrature fulfills the half moment for vectors of Cartesian basis. Since it is not possible to fulfill exactly all these conditions, accuracy should be improved when N increases. Following the conclusion of Ref. 2, the implementation uses LSE symmetric quadrature for S2, S4, S6, and S8. LSE symmetric quadrature fulfills the half, first, second, and third moments. Thanks to angular space discretization, integrals over directions are replaced by numerical quadratures of discrete directions: Depending on the value of N, a set of n dependent variables has to be defined and solved for I 1, I 2,, I n. Each dependent variable obeys the equation i I i with the boundary condition 4 0 I n j = 1 s w j I j = I b T I i w 4 j I j j i n j = CHAPTER 5: THE RADIATION HEAT TRANSFER BRANCH

175 I i bnd = d w I b T q out for all i such that n i 0 THEORY FOR THE RADIATIVE HEAT TRANSFER INTERFACES 175

176 Theory for the Surface-to-Surface Radiation Interface The Surface-To-Surface Radiation Interface theory is described in this section: About Surface-to-Surface Radiation Solving for the Radiosity About the Surface-to-Surface Radiation Boundary Conditions Guidelines for Solving Surface-to-Surface Radiation Problems Radiation Group Boundaries References for the Radiation Interfaces About Surface-to-Surface Radiation Surface-to-surface radiation is more complex than those topics discussed in the section Radiative Heat Transfer in Transparent Media. It includes radiation from both the ambient surroundings and from other surfaces. A generalized equation for the irradiative flux is: G = G m + F amb T amb 4 (5-5) where G m is the mutual irradiation arriving from other surfaces in the modeled geometry and F amb is the ambient view factor. The latter describes the portion of the view from each point that is covered by ambient conditions. G m on the other hand is determined from the geometry and the local temperatures of the surrounding boundaries. The following sections derive the equations for G m and F amb for a general 3D case. Consider a point x on a surface as in Figure 5-2. Point x can see points on other surfaces as well as the ambient surrounding. Assume that the points on the other surfaces have a local radiosity, J', while the ambient surrounding has a constant temperature, T amb. rn' x' xj' S' S amb Figure 5-2: Example geometry for surface-to-surface radiation. The mutual irradiation at point x is given by the following surface integral: 176 CHAPTER 5: THE RADIATION HEAT TRANSFER BRANCH

177 G m = S' n' rn r J' r 4 ds The heat flux that arrives from x' depends on the local radiosity J' projected onto x. The projection is computed using the normal vectors n and n' along with the vector r, which points from x to x'. The ambient view factor, F amb, is determined from the integral of the surrounding surfaces S', here denoted as F', determined from the integral below: F amb = 1 F ' = 1 n' rn r r 4 ds The two last equations plug into Equation 5-5 to yield the final equation for irradiative flux. The equations used so far apply to the general 3D case. 2D geometries results in simpler integrals. For 2D the resulting equations for the mutual irradiation and ambient view factor are S' G m = S ' n' r n r J' ds 3 2 r (5-6) F amb = 1 S ' n' r n r ds 3 2 r where the integral over S ' geometry. denotes the line integral along the boundaries of the 2D In axisymmetric geometries, the irradiation and ambient view factor cannot be computed directly from a closed-form expression. Instead, a virtual 3D geometry must be constructed, and the view factors evaluated according to Equation 5-6. Solving for the Radiosity The previous section derived equations for the irradiation G at an arbitrary surface point x. Now recall the expression for the radiosity leaving from x J = G + T 4 (5-7) THEORY FOR THE SURFACE-TO-SURFACE RADIATION INTERFACE 177

178 Inserting the expression for G in Equation 5-7 gives the following equation for the radiosity: J = G m + F amb T amb + T 4 Assuming an ideal graybody, the last equation becomes: 4 J = 1 G m + F amb T amb + T 4 4 (5-8) This is the equation used in the Surface-to-Surface Radiation interface to solve for the radiosity, J. It applies to boundaries that participate in surface-to-surface radiation. Equation 5-8 results in a linear equation system in J that is solved in parallel with the equation for the temperature, T. About the Surface-to-Surface Radiation Boundary Conditions Heat flux on the Surface-to-Surface Radiation boundary is where e is the surface emissivity, a dimensionless number in the range 01. is the Stefan-Boltzmann constant (a predefined physical constant equal to W/(m 2 T 4 )). G is the incoming radiative heat flux, or irradiation (SI unit: W/m 2 ): where q = G T 4 G = G m + F amb T amb G m is the mutual irradiation, coming from other boundaries in the model (SI unit: W/m 2 ). F amb is an ambient view factor whose value is equal to the fraction of the field of view that is not covered by other boundaries. Therefore, by definition, 0F amb 1 must hold at all points. T amb is the assumed far-away temperature in the directions included in F amb. Surface-to-surface radiation requires accurate evaluation of the mutual irradiation, G m. The incident radiation at one point on the boundary is a function of the exiting CHAPTER 5: THE RADIATION HEAT TRANSFER BRANCH

179 radiation, or radiosity, J (W/m 2 ), at every other point in view. The radiosity, in turn, is a function of G m, which leads to an implicit radiation balance: J = 1 eg m J+ F amb T amb + e T 4 4 Guidelines for Solving Surface-to-Surface Radiation Problems The following guidelines are helpful when selecting solver settings for models that involve surface-to-surface radiation: Surface-to-surface radiation makes the Jacobian matrix of the discrete model partly filled as opposed to the usual sparse matrix. The additional nonzero elements in the matrix appear in the rows and columns corresponding to the radiosity degrees of freedom. It is therefore common practice to keep the element order of the radiosity variable, J, low. By default, linear Lagrange elements are used irrespective of the shape-function order specified for the temperature. When you need to increase the resolution of your temperature field, it might be worth considering raising the order of the temperature elements instead of refining the mesh. The Assembly block size parameter (found in the Advanced section of the solver feature) can have a major influence on memory usage during the assembly of problems where surface-to-surface radiation is enabled. It may be useful to consider a block size as small as 100. Using a smaller block size also leads to more frequent updates of the progress bar. Radiation Group Boundaries Note The Radiation Group feature is only available when the Use radiation groups check box is selected under Radiation Settings. By default this check box is not selected, which means that all radiative boundaries belong to the same radiation group. For radiation problems, a boundary grouping can be applied to save computational time. A radiation group can be defined using a Radiation Group node; see Radiation Group for details. A default group contains all boundaries selected in a Surface-to-Surface Radiation, Reradiating Surface, or Prescribed Radiosity node. When a node is added to another radiation group, it is overridden in the default group. Then this boundary can be THEORY FOR THE SURFACE-TO-SURFACE RADIATION INTERFACE 179

180 added to other radiation groups without being overridden by the manually added radiation groups. Caution 2D Axi Be careful when grouping boundaries in axisymmetric geometries. The grouping cannot be based on which boundaries have a free view toward each other in the 2D geometry. Instead, consider the full 3D geometry, obtained by revolving the model geometry about the z axis, when defining groups. For example, parallel vertical boundaries must typically belong to the same group in 2D axisymmetric models, but to different groups in a planar model using the same 2D geometry. Figure 5-3 shows four examples of possible boundary groupings. On boundaries that have no number, the user has NOT set a node among the Surface-to-Surface Radiation, Reradiating Surface, and Prescribed Radiosity nodes. These boundaries do not irradiate other boundaries, neither do other boundaries irradiate them. On boundaries that belong to one or more radiation group, the user has set a node among the Surface-to-Surface Radiation, Reradiating Surface, and Prescribed Radiosity nodes. The numbers on each boundary specify different groups to which the boundary belongs. To obtain optimal computational performance, it is good practice to specify as many groups as possible as opposed to specifying few but large groups. For example, in Figure 5-3, case (b) is more efficient than case (d). 180 CHAPTER 5: THE RADIATION HEAT TRANSFER BRANCH

181 A B C D inefficient boundary grouping Figure 5-3: Examples of radiation group boundaries. References for the Radiation Interfaces 1. M.F. Modest, Radiative heat transfer, 2nd ed., Academic Press, San Diego, California, W.A. Fiveland, The Selection of Discrete Ordinate Quadrature Sets for Anisotropic Scattering, Fundamentals of Radiation Transfer, HTD, vol. 160, ASME, F.P. Incropera and D.P. DeWitt, Fundamentals of Heat and Mass Transfer, 5th ed., John Wiley and Sons, J.R. Welty, C.E. Wicks, and R.E. Wilson, Fundamentals of Momentum, Heat, and Mass Transfer, 3rd ed., John Wiley and Sons, THEORY FOR THE SURFACE-TO-SURFACE RADIATION INTERFACE 181

182 182 CHAPTER 5: THE RADIATION HEAT TRANSFER BRANCH

183 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. This enables modeling of forced or temperature gradient-driven flows in both laminar and turbulent regimes. This chapter describes the fluid flow groups under the Single-Phase Flow branch ( ) in the Model Wizard. In this chapter: The Single-Phase Flow, Laminar Flow Interface The Single-Phase Flow, Turbulent Flow Interfaces Boundary Conditions for the Single-Phase Flow Interfaces Theory for the Laminar Flow Interface Theory for the Turbulent Flow Interfaces 183

184 The Single-Phase Flow, Laminar Flow Interface The descriptions in this section are structured based on the order displayed in the Fluid Flow>Single-Phase Flow branch ( ) of the Model Wizard. Because many of the interfaces are integrated with each other, some features described also cross reference to other interfaces. At the end of this section is a summary of the theory that goes towards deriving each physics interface. The Heat Transfer Module extends the capabilities of the basic COMSOL Laminar Flow interface. In this section: The Laminar Flow Interface Fluid Properties Volume Force Initial Values Boundary Conditions for the Single-Phase Flow Interfaces See Also Show More Physics Options 2D Axi For 2D axisymmetric models, COMSOL Multiphysics takes the axial symmetry boundaries (at r0) into account and automatically adds an Axial Symmetry feature to the model that is valid on the axial symmetry boundaries only. The Laminar Flow Interface The Laminar Flow interface ( ), found under the Single-Phase Flow branch ( ) in the Model Wizard, has the equations, boundary conditions, and volume forces for modeling freely moving fluids using the Navier-Stokes equations, solving for the velocity field and the pressure. The main feature is Fluid Properties, which adds the 184 CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

185 Navier-Stokes equations and provides an interface for defining the fluid material and its properties. Flow Past a Cylinder: Model Library path COMSOL_Multiphysics/ Fluid_Dynamics/cylinder_flow Model Terminal Falling Velocity of a Sand Grain: Model Library path COMSOL_Multiphysics/Fluid_Dynamics/falling_sand When this interface is added, these default nodes are also added to the Model Builder Fluid Properties, Wall (the default boundary condition is No slip), and Initial Values. Right-click the Laminar Flow node to add other features that implement, for example, boundary conditions and volume forces. INTERFACE IDENTIFIER The interface identifier is a text string that can be used to reference the respective physics interface if appropriate. Such situations could occur when coupling this interface to another physics interface, or when trying to identify and use variables defined by this physics interface, which is used to reach the fields and variables in expressions, for example. It can be changed to any unique string in the Identifier field. The default identifier (for the first interface in the model) is spf. DOMAIN SELECTION The default setting is to include All domains in the model to define the fluid pressure and velocity and the Navier-Stokes equations that describe those fields. To choose specific domains, select Manual from the Selection list. PHYSICAL MODEL Control the properties of the Laminar Flow interface, which control the overall type of fluid flow model. The options available in this section enables switching between other available Single-Phase Flow interfaces. For example, Tip This interface changes to The Turbulent Flow, k- Interface when the Turbulence model type selected is RANS (Reynolds-averaged Navier Stokes). THE SINGLE-PHASE FLOW, LAMINAR FLOW INTERFACE 185

186 Compressibility By default the interface uses the Compressible flow (Ma<0.3) formulation of the Navier-Stokes equations. Select Incompressible flow to use the incompressible (constant density) formulation. Turbulence Model Type Note By definition no turbulence model is needed when studying laminar flows, and no turbulence model is therefore applied in the interface. The flow state in a fluid flow model, however, is not always known beforehand. DEPENDENT VARIABLES This interface defines these dependent variables (fields). If required, edit the name, but dependent variables must be unique within a model: Velocity field u (SI unit: m/s) Pressure p (SI unit: Pa) Turbulent kinetic energy k (SI unit: m 2 /s 2 ) Turbulent dissipation rate ep (SI unit: m 2 /s 3 ) Reciprocal wall distance G (SI unit: 1/m) ADVANCED SETTINGS To display this section, click the Show button ( ) and select Advanced Physics Options. Normally these settings do not need to be changed. Select the Use pseudo time stepping for stationary equation form check box to add pseudo time derivatives to the equation when the Stationary equation form is used. When selected, also choose a CFL number expression Automatic (the default) or Manual. Automatic calculates the local CFL number (from the Courant Friedrichs Lewy condition) from a built-in expression. If Manual is selected, enter a Local CFL number CFL loc. See Also The Projection Method for the Navier-Stokes Equations and Pseudo Time Stepping for Laminar Flow Models in the COMSOL Multiphysics User s Guide 186 CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

187 DISCRETIZATION To display this section, click the Show button ( ) and select Discretization. It controls the discretization (the element types used in the finite element formulation). From the Discretization of fluids list select the element order for the velocity components and the pressure: P1+P1 (the default), P2+P1, or P3+P2. P1+P1 (the default) means linear elements for both the velocity components and the pressure field. This is the default element order for the Laminar Flow and Turbulent Flow flow interfaces. Linear elements are computationally cheaper than higher-order elements and are also less prone to introducing spurious oscillations, thereby improving the numerical robustness. P2+P1 means second-order elements for the velocity components and linear elements for the pressure field. This is the default for the Creeping Flow interface because second-order elements work well for low flow velocities. P3+P2 means third-order elements for the velocity components and second-order elements for the pressure field. This can add additional accuracy but it also adds additional degrees of freedom compared to P2+P1 elements. Specify the Value type when using splitting of complex variables Real or Complex (the default). Show More Physics Options See Also Boundary Conditions for the Single-Phase Flow Interfaces Theory for the Laminar Flow Interface Fluid Properties The Fluid Properties feature adds the momentum equations solved by the interface, except for volume forces which are added by the Volume Force feature. The node also provides an interface for defining the material properties of the fluid. Note For the Turbulent Flow interfaces, the Fluid Properties feature also adds the equations for the turbulence transport equations. DOMAIN SELECTION From the Selection list, choose the domains to apply the fluid properties. THE SINGLE-PHASE FLOW, LAMINAR FLOW INTERFACE 187

188 MODEL INPUTS Edit input variables to the fluid-flow equations if required. For fluid flow, these are typically introduced when a material requiring inputs has been applied. Tip To define the Absolute Pressure, see the settings for the Heat Transfer in Fluids node as described in the COMSOL Multiphysics User s Guide. FLUID PROPERTIES Density The default Density (SI unit: kg/m 3 ) uses the value From material. Select User defined to enter a different value or expression. Dynamic 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, water and air have a low viscosity, and substances often described as thick (such as oil) have a higher viscosity. Select User defined to define a different value or expression. Using a built-in variable for the shear rate magnitude, spf.sr, makes it possible to define arbitrary expressions of the dynamics viscosity as a function of the shear rate. DISTANCE EQUATION Important This section is only available for Turbulent Flow, Low Reynolds number k- interface since a Wall Distance interface is included. Select how the Reference length scale l ref (SI unit: m) is defined Automatic (default) or Manual: If Automatic is used, the wall distance is automatically evaluated as half the shortest side of the geometry bounding box. This is usually quite accurate but it can sometimes give too great a value if the geometry consists of several slim entities. In this case, it is recommended that it is defined manually. Select Manual to define a different value or expression for the wall distance. 188 CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

189 MIXING LENGTH LIMIT Note This section is only available for the Turbulent Flow, k- interface because an upper limit on the mixing length is required. Select how the Mixing length limit l mix,lim (SI unit: m) is defined Automatic (default) or Manual: If Automatic is used, the wall distance is automatically evaluated as the shortest side of the geometry bounding box. If the geometry is, for example, a complicated system of slim entities, this measure can give too big a result. In this case, it is recommended that it is defined manually. Select Manual to define a different value or expression. Volume Force The Volume Force feature specifies the volume force F on the right-hand side of the incompressible flow equation. Use it, for example, to incorporate the effects of gravity in a model. u T + u u = pi + u + u + F t DOMAIN SELECTION From the Selection list, choose the domains where the volume force acts on the fluid. VOLUME FORCE Enter the components of the Volume force F (SI unit: N/m 3 ). See Also The Boussinesq Approximation in the COMSOL Multiphysics User s Guide THE SINGLE-PHASE FLOW, LAMINAR FLOW INTERFACE 189

190 Initial Values The Initial Values feature adds initial values for the velocity field and the pressure that can serve as an initial condition for a transient simulation or as an initial guess for a nonlinear solver. DOMAIN SELECTION From the Selection list, choose the domains to define initial values. INITIAL VALUES Enter values or expressions for the initial value of the Velocity field u (SI unit: m/s) and for the Pressure p (SI unit: Pa). The default values are 0. Note In the Turbulent Flow interfaces, initial values for the turbulence variables are also specified. By default these are specified using the predefined variables defined by the expressions in Initial Values. 190 CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

191 The Single-Phase Flow, Turbulent Flow Interfaces The descriptions in this section are structured based on the order displayed in the Fluid Flow branch. Because many of the interfaces are integrated with each other, most of the features cross reference to other interfaces. For example, all nodes that can be added for the Turbulent Flow interfaces are described for The Single-Phase Flow, Laminar Flow Interface Click the links to go to these section topics: The Turbulent Flow, k- Interface The Turbulent Flow, Low Re k- Interface The Turbulent Flow, k- Interface The Turbulent Flow, k-interface ( ), found under the Single-Phase Flow>Turbulent Flow branch ( ) in the Model Wizard, has the equations, boundary conditions, and volume forces for modeling turbulent flow using the Reynolds averaged Navier-Stokes (RANS) equations, solving for the mean velocity field, the pressure, and the standard k- model, solving for the turbulent kinetic energy k and the rate of dissipation of turbulent kinetic energy. Model Turbulent Flow Over a Backward Facing Step: Model Library path Heat_Transfer_Module/Verification_Models/turbulent_backstep The main feature is Fluid Properties, which adds the Navier-Stokes equations and the transport equations for k and, and provides an interface for defining the fluid material and its properties. When this interface is added, these default nodes are also added to the Model Builder Fluid Properties, Wall (the default boundary condition is Wall functions), and Initial Values. THE SINGLE-PHASE FLOW, TURBULENT FLOW INTERFACES 191

192 Right-click the Turbulent Flow, k- node to add other features that implement, for example, boundary conditions and volume forces. Note Except where noted below, see The Laminar Flow Interfacefor all the other settings. PHYSICAL MODEL Turbulence Model Type Note For the Turbulent Flow, k- interface, this defaults to a RANS Turbulence model type. This enables the Turbulence Model Parameters section. Turbulence Model Tip The interface changes to the Turbulent Flow, Low Re k- interface when Low Reynolds number k- is selected. Show More Physics Options See Also Boundary Conditions for the Single-Phase Flow Interfaces Theory for the Turbulent Flow Interfaces The Turbulent Flow, Low Re k- Interface The Turbulent Flow, Low Re k-interface ( ), found under the Single-Phase Flow>Turbulent Flow branch, has the equations, boundary conditions, and volume forces for modeling turbulent flow using the Reynolds averaged Navier-Stokes (RANS) equations, solving for the mean velocity field, the pressure, and the AKN low-reynolds number k- model, solving for the turbulent kinetic energy k and the rate of dissipation of turbulent kinetic energy. The interface also includes a wall distance equation that solves for the reciprocal wall distance. 192 CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

193 The interface is the same as The Turbulent Flow, k- Interface ( Turbulence model defaults to Low Reynolds number k-. ) except the 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. Important For study information, see Stationary with Initialization, Transient with Initialization, and Wall Distance Initialization in the COMSOL Multiphysics Reference Guide Show More Physics Options See Also Boundary Conditions for the Single-Phase Flow Interfaces Theory for the Turbulent Flow Interfaces THE SINGLE-PHASE FLOW, TURBULENT FLOW INTERFACES 193

194 Boundary Conditions for the Single-Phase Flow Interfaces The boundary features in this section are for all interfaces found under the Fluid Flow>Single-Phase Flow branch ( ) in the Model Wizard. In this section: Wall Inlet Outlet Symmetry Open Boundary Boundary Stress Periodic Flow Condition Flow Continuity Pressure Point Constraint Interior Fan Fan Grille 2D Axi For 2D axisymmetric models, COMSOL Multiphysics takes the axial symmetry boundaries (at r0) into account and automatically adds an Axial Symmetry feature to the model that is valid on the axial symmetry boundaries only. Note The theory about most boundary conditions is found in P.M. Gresho and R.L. Sani, Incompressible Flow and the Finite Element Method, Volume 2: Isothermal Laminar Flow, John Wiley & Sons, CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

195 Wall The Wall feature includes a set of boundary conditions describing the fluid flow condition at a wall. No Slip (the default for laminar flow) Slip Sliding Wall Moving Wall Leaking Wall Wall Functions (the default for turbulent flow) Sliding Wall (Wall Functions) Moving Wall (Wall Functions) In the COMSOL Multiphysics User s Guide: See Also Theory for the Slip Wall Boundary Condition Theory for the Sliding Wall Boundary Condition The Moving Mesh Interface BOUNDARY SELECTION From the Selection list, choose the boundaries that represent solid walls. PAIR SELECTION If Wall is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect. BOUNDARY CONDITION Select a Boundary condition for the wall. The boundary conditions available vary by interface. No Slip No slip is the default boundary condition for a stationary solid wall. The condition prescribes u = 0, that is, that the fluid at the wall is not moving. BOUNDARY CONDITIONS FOR THE SINGLE-PHASE FLOW INTERFACES 195

196 Slip The Slip condition assumes that there are no viscous effects at the slip wall and hence, no boundary layer develops. From a modeling point of view, this may be a reasonable approximation if the important effect of the wall is to prevent fluid from leaving the domain. Sliding Wall The Sliding wall boundary condition is appropriate if the wall behaves like a conveyor belt; that is, the surface is sliding in its tangential direction. The wall does not have to actually move in the coordinate system. 2D In 2D, the tangential direction is unambiguously defined by the direction of the boundary, but the situation becomes more complicated in 3D. For this reason, this boundary condition has slightly different definitions in the different space dimensions. Enter the components of the Velocity of the tangentially moving wall U w (SI unit: m/s). In axial symmetry, if Swirl flow is selected in the interface properties, also specify a velocity, v w, in the direction. 3D Enter the components of the Velocity of the sliding wall u w (SI unit: m/s). If the velocity vector entered is not in the plane of the wall, COMSOL projects it onto the tangential direction. Its magnitude is adjusted to be the same as the magnitude of the vector entered. Moving Wall If the wall moves, so must the fluid. Hence, this boundary condition prescribes u = u w. Enter the components of the Velocity of moving wall u w (SI unit: m/s). Important 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. Leaking Wall Use this boundary condition to simulate a wall where fluid is leaking into or leaving through a perforated wall u = u l. Enter the components of the Fluid velocity u l (SI unit: m/s). 196 CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

197 MORE WALL BOUNDARY CONDITIONS FOR THE TURBULENT FLOW INTERFACES See Also Boundary Conditions for the Single-Phase Flow Interfaces Wall Functions The Wall Functions boundary condition applies wall functions to a solid walls in a turbulent flow. Wall functions are used to model the thin region near the wall with high gradients in the flow variables. Sliding Wall (Wall Functions) The Sliding Wall (Wall Functions) boundary condition applies wall functions to a wall in a turbulent flow where the velocity magnitude in the tangential direction of the wall is prescribed. The tangential direction is determined in the same manner as in the Sliding Wall feature. Enter the component values or expressions for the Velocity of sliding wall u w (SI unit: m/s). Moving Wall (Wall Functions) Important Specifying this boundary condition does not automatically cause the associated wall to move. The Moving Wall (Wall Functions) boundary condition applies wall functions to a wall in a turbulent flow with prescribed velocity u w. Enter the component values or expressions in the Velocity of moving wall fields (SI unit: m/s). CONSTRAINT SETTINGS To display this section, click the Show button ( ) and select Advanced Physics Options. Select a Constraint type Bidirectional, symmetric or Unidirectional for the following wall boundary conditions: No slip, Moving wall, and Leaking wall. The other types of wall boundary conditions with constraints use unidirectional constraints only. Select the Use weak constraints check box (available for all boundary conditions except Sliding wall, which does not add any constraints) to use weak constraints and create dependent variables for the corresponding Lagrange multipliers. BOUNDARY CONDITIONS FOR THE SINGLE-PHASE FLOW INTERFACES 197

198 Interior Wall The Interior Wall boundary condition includes a set of boundary conditions describing the fluid flow condition at an interior wall. It is similar to the Wall boundary condition available on exterior boundaries except that it applies on both sides of an internal boundary. It allows discontinuities (velocity, pressure, turbulence) across the boundary. You can use the Interior Wall boundary condition to avoid meshing thin structures by instead using no-slip conditions on interior curves and surfaces. You can also prescribe slip conditions and conditions for a moving wall. Note The Interior Wall boundary condition is only available for single-phase flow. It is compatible with laminar and turbulent flows. In the COMSOL Multiphysics User s Guide: See Also Theory for the Slip Wall Boundary Condition The Moving Mesh Interface BOUNDARY SELECTION From the Selection list, choose the boundaries that represent interior walls. BOUNDARY CONDITION Select a Boundary condition No slip (the default), Slip, or Moving wall. 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; that is, the fluid at the wall is not moving. Slip The Slip condition assumes that there are no viscous effects at both sides of the slip wall and hence, no boundary layer develops. From a modeling point of view, this can be a reasonable approximation if the important effect of the wall is to prevent fluid from leaving the domain. 198 CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

199 Moving Wall If the wall moves, so must the fluid on both sides of the wall. Hence, this boundary condition prescribes u = u w. Enter the components of the Velocity of moving wall u w (SI unit: m/s). Important 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. Inlet The Inlet node includes a set of boundary conditions describing the fluid flow condition at an inlet. The Velocity boundary condition is the default. Tip In most cases the inlet boundary conditions appear, some of them slightly modified, in the Outlet type as well. This means that there is nothing in the mathematical formulations to prevent a fluid from leaving the domain through boundaries where the Inlet type is specified. Velocity (the default) Pressure, No Viscous Stress Normal Stress Laminar Inflow Theory for the Laminar Inflow Boundary Condition In the COMSOL Multiphysics User s Guide: See Also Theory for the Pressure, No Viscous Stress Boundary Condition Theory for the Normal Stress Boundary Condition BOUNDARY CONDITIONS FOR THE SINGLE-PHASE FLOW INTERFACES 199

200 BOUNDARY SELECTION From the Selection list, choose the boundaries that represent inlets. The boundary conditional available is based on interface. BOUNDARY CONDITION Select a Boundary condition for the inlet. See Also After selecting a Boundary Condition from the list, a section with the same name displays underneath. For example, if Velocity is selected, a Velocity section displays where further settings are defined for the velocity. Important For the Velocity, Pressure, no viscous stress, and Normal stress sections, also enter the settings as described in Additional Boundary Condition Settings for Turbulent Flow Interfaces. VELOCITY Select Normal inflow velocity (the default) to specify a normal inflow velocity magnitude u = nu 0 where n is the boundary normal pointing out of the domain. Enter the velocity magnitude U 0 (SI unit: m/s). If Velocity field is selected, it sets the velocity equal to a given velocity vector u 0 when u = u 0. Enter the velocity components u 0 (SI unit: m/s) to set the velocity equal to a given velocity vector. Note This section displays when Velocity is selected as the Boundary condition. The option is available for the Inlet and Outlet boundary features. 200 CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

201 PRESSURE, NO VISCOUS STRESS The Pressure, no viscous stress boundary condition specifies vanishing viscous stress along with a Dirichlet condition on the pressure. Enter the Pressure p 0 (SI unit: Pa) at the boundary. Note This section displays when Pressure, no viscous stress is selected as the Boundary condition. The option is available for the Inlet and Outlet boundary features. Depending on the pressure field in the rest of the domain, an inlet boundary with this condition can become an outlet boundary. NORMAL STRESS Enter the magnitude of Normal stress f 0 (SI unit: N/m 2 ). Implicitly specifies that p f 0. Note This section displays when Normal Stress is selected as the Boundary condition. The option is available for the Inlet, Outlet, Open Boundary, and Boundary Stress features. LAMINAR INFLOW Select a flow quantity to specify for the inlet: If Average velocity is selected, enter an Average velocity U av (SI unit: m/s). If Flow rate is selected, enter the Flow rate V 0 (SI unit: m 3 /s). If Entrance pressure is selected, enter the Entrance pressure p entr (SI unit: Pa) at the entrance of the fictitious channel outside of the model. Then specify these parameters: Enter the Entrance length L entr (SI unit: m) to define the length of the inlet channel outside the model domain. This value must be large enough so that the flow can reach a laminar profile. For a laminar flow, L entr should be significantly greater than 0.06ReD, where Re is the Reynolds number and D is the inlet length scale (this formula is exact if D is the diameter of a cylindrical pipe and approximate for other geometries). For turbulent flow the equivalent expression is 4.4Re 1/6 D. The default is 1 m. BOUNDARY CONDITIONS FOR THE SINGLE-PHASE FLOW INTERFACES 201

202 Select the Constrain endpoints to zero check box to force the laminar profile to go to zero at the bounding points or edges of the inlet channel. Otherwise the velocity is defined by the boundary condition of the adjacent boundary in the model. For example, if one end of a boundary with a laminar inflow condition connects to a slip boundary condition, then the laminar profile has a maximum at that end. Note This section displays when Laminar inflow is selected as the Boundary condition for the Laminar Flow interface. However, it is not available when the Use memory-efficient form check box is selected from Advanced Settings on the Laminar Flow node s Settings window. ADDITIONAL BOUNDARY CONDITION SETTINGS FOR TURBULENT FLOW INTERFACES The Turbulent Flow, k- model and Turbulent Flow, low Reynolds number k- models, also require that k and are specified using one of the following: Select Specify turbulence length scale and intensity to enter values or expressions for the Turbulent intensity I T (unitless) and Turbulence length scale L T (SI unit: m). For the Turbulent Flow, k- model, also enter a value for the Reference velocity scale U ref (SI unit: m/s). The Turbulent intensity I T and Turbulence length scale L T values are related to the turbulence variables via k k3/ 2 L T 3 -- U I 2 T =, = C If Specify turbulence variables is selected, enter values or expressions for the Turbulent kinetic energy k 0 (SI unit: m 2 /s 2 ) and Turbulent dissipation rate, 0 (SI unit: m 2 / s 3 ).CONSTRAINT SETTINGS Tip For recommendations of physically sound values see Inlet Values for the Turbulence Length Scale and Intensity. To display this section, click the Show button ( ) and select Advanced Physics Options. Select a Constraint type Bidirectional, symmetric or Unidirectional. Select the Use weak constraints check box as required. 202 CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

203 Outlet The Outlet feature includes a set of boundary conditions describing fluid flow conditions at an outlet. The Pressure, no viscous stress boundary condition is the default. Other options are based on individual licenses. Selecting appropriate outlet conditions for the Navier-Stokes equations is not a trivial task. Generally, if there is something interesting happening at an outflow boundary, extend the computational domain to include this phenomenon. Tip All of the formulations for the Outlet type are also available, possibly slightly modified, in other boundary types as well. This means that there is nothing in the mathematical formulations to prevent a fluid from entering the domain through boundaries where the Outlet boundary type is specified. Pressure, No Viscous Stress (the default) Velocity Laminar Outflow Normal Stress Pressure No Viscous Stress Note The Pressure, No Viscous Stress, Velocity, and Normal Stress boundary conditions are described for the Inlet node. See Also Theory for the Laminar Outflow Boundary Condition BOUNDARY SELECTION From the Selection list, choose the boundaries that represent outlets. BOUNDARY CONDITIONS FOR THE SINGLE-PHASE FLOW INTERFACES 203

204 BOUNDARY CONDITION Select a Boundary condition for the outlet. Pressure, no viscous stress is the default. The boundary conditional available is based on interface. PRESSURE This boundary condition prescribes only a Dirichlet condition for the pressure p = p 0. Enter the Pressure p 0 (SI unit: Pa) at the boundary. While this boundary condition is flexible and seldom produces artifacts on the boundary (compared to Pressure, No Viscous Stress), it can be numerically unstable. Theoretically, the stability is guaranteed by using streamline diffusion for a flow with a cell Reynolds number Re c uh 21 (h is the local mesh element size). It does however work well in most other situations as well. LAMINAR OUTFLOW Select a flow quantity to specify for the inlet: If Average velocity is selected, enter an Average velocity U av (SI unit: m/s). If Flow rate is selected, enter the Flow rate V 0 (SI unit: m 3 /s). If Exit pressure is selected, enter the Exit pressure p exit (SI unit: Pa) at the end of the fictitious channel following the outlet. Then specify the Exit length and Constrain endpoints to zero parameters: Enter the Exit length L exit (SI unit: m) to define the length of the fictitious channel after the model domain. This value must be large enough so that the flow can reach a laminar profile. For a laminar flow, L exit should be significantly greater than 0.06ReD, where Re is the Reynolds number and D is the outlet length scale (this formula is exact if D is the diameter of a cylindrical pipe and approximate for other geometries). For turbulent flow the equivalent expression is 4.4Re 1/6 D. The default is 1. Select the Constrain endpoints to zero check box to force the laminar profile to go to zero at the bounding points or edges of the inlet channel. Otherwise the velocity is defined by the boundary condition of the adjacent boundary in the model. 204 CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

205 For example, if one end of a boundary with a Laminar inflow condition connects to a Slip boundary condition, then the laminar profile has a maximum at that end. Note This section displays when Laminar Outflow is selected as the Boundary condition for a Laminar Flow interface. However, it is not available when the Use memory-efficient form check box is selected from Advanced Settings on the Laminar Flow Settings window. NO VISCOUS STRESS The No Viscous Stress condition specifies vanishing viscous stress on the outlet. This condition does not provide sufficient information to fully specify the flow at the outlet and must be combined with pressure constraints on adjacent points. CONSTRAINT SETTINGS To display this section, click the Show button ( ) and select Advanced Physics Options. Select a Constraint type Bidirectional, symmetric or Unidirectional. Select the Use weak constraints check box to use weak constraints and create dependent variables for the corresponding Lagrange multipliers. Symmetry The Symmetry node adds a boundary condition that describes symmetry boundaries in a fluid flow simulation. The boundary condition for symmetry boundaries prescribes no penetration and vanishing shear stresses. The boundary condition is a combination of a Dirichelet condition and a Neumann condition: u n = 0, pi u + u T ui 3 n = 0 u n = 0, pi + u + u T n = 0 for the compressible and the incompressible formulation respectively. The Dirichlet condition takes precedence over the Neumann condition, and the above equations are equivalent to the following equation for both the compressible and incompressible formulation: BOUNDARY CONDITIONS FOR THE SINGLE-PHASE FLOW INTERFACES 205

206 u n = 0, K K nn = 0 K = u + u T n BOUNDARY SELECTION From the Selection list, choose the boundaries that are symmetry boundaries. 2D Axi For 2D axial symmetry, a boundary condition does not need to be defined. For the symmetry axis at r0, the software automatically provides a condition that prescribes u r 0 and vanishing stresses in the z direction and adds an Axial Symmetry feature that implements this condition on the axial symmetry boundaries only. Open Boundary The Open Boundary node adds boundary conditions that describe boundaries that are open to large volumes of fluid. Fluid can both enter and leave the domain on boundaries with this type of condition. Inlet Outlet See Also More Open Boundary Conditions for the Turbulent Flow Interfaces Pressure Point Constraint BOUNDARY SELECTION From the Selection list, choose the boundaries that are open boundaries. BOUNDARY CONDITIONS Select a Boundary condition for the open boundaries Normal stress (the default) or No viscous stress. If Normal stress f 0 (SI unit: N/m 2 ) is selected, enter a value or expression for the boundary condition. No Viscous Stress If No viscous stress is selected, which is also available for the Outlet feature, it prescribes vanishing viscous stress: 206 CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

207 u + u T 2 -- ui n = 0 3 u + u T n = 0 using the compressible and the incompressible formulation respectively. Tip This condition can be useful in some situations because it does not impose any constraint on the pressure. A typical example is a model with volume forces that give rise to pressure gradients that are hard to prescribe in advance. To make the model numerically stable, combine this boundary condition with a point constraint on the pressure. Note This boundary condition is not available if the Use memory-efficient form check box is selected (click the Show button ( ) and select Advanced Physics Options>Advanced Settings to display the section) on any single-phase flow interface Settings window. MORE OPEN BOUNDARY CONDITIONS FOR THE TURBULENT FLOW INTERFACES With any turbulent flow interface, inlet conditions for the turbulence variables also need to be specified. These conditions are used on the parts of the boundary where u n0, that is, where flow enters the computational domain. For the Turbulent Flow, k- and Turbulent Flow, low Reynolds number k-interfaces, these options are available under Exterior turbulence: Select Specify turbulent length scale and intensity to enter values or expressions for the Turbulent intensity I T (unitless), Turbulence length scale L T (SI unit: m), and Reference velocity scale U ref (SI unit: m/s). These values are related to the turbulence variables via BOUNDARY CONDITIONS FOR THE SINGLE-PHASE FLOW INTERFACES 207

208 k 3 = -- I 2 T U ref 2, = 3 4 C ITUref L T Select Specify turbulence variables to enter values or expressions for the Turbulent kinetic energy k 0 (SI unit: m 2 /s 2 ), and the Turbulent dissipation rate 0 (SI unit: m 2 /s 3 ). See Also For recommendations of physically sound values see Inlet Values for the Turbulence Length Scale and Intensity. Boundary Stress The Boundary Stress node adds a boundary condition that represents a very general class of conditions also known as traction boundary conditions. See Also InletMore Open Boundary Conditions for the Turbulent Flow Interfaces BOUNDARY SELECTION From the Selection list, choose the boundaries to apply boundary stress. BOUNDARY CONDITION Select a Boundary condition for the boundary stress General stress (the default), Normal stress, or Normal stress, normal flow. Normal Stress If Normal stress f 0 (SI unit: N/m 2 ) is selected, enter a value or expression. General Stress If General stress is selected, enter the components of the Stress F (SI unit: N/m 2 ).The total stress on the boundary is set equal to a given stress F: pi u + u T ui 3 n = F 208 CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

209 pi + u + u T n = F using the compressible and the incompressible formulation respectively. This boundary condition implicitly sets a constraint on the pressure that for 2D flows is p = 2 u n n F n (6-1) If u n n is small, Equation 6-1 states that pn F. Normal Stress, Normal Flow If Normal stress, normal flow is selected, enter the magnitude of the Normal stress f 0 (SI unit: N/m 2 ). In addition to the stress condition set in the Normal stress condition, this condition also prescribes that there must be no tangential velocities on the boundary: pi u + u T 2 -- ui + 3 n = f0 n, t u = 0 pi + u + u T n = f 0 n, t u = 0 using the compressible and the incompressible formulation respectively. This boundary condition also implicitly sets a constraint on the pressure that for 2D flows is p = 2 u n f n 0 (6-2) If u n n is small, Equation 6-2 states that pf 0. Note This boundary condition is not available if the Use memory-efficient form check box is selected (click the Show button ( ) and select Advanced Physics Options>Advanced Settings to display the section) on any single-phase flow interface Settings page. BOUNDARY CONDITIONS FOR THE SINGLE-PHASE FLOW INTERFACES 209

210 MORE BOUNDARY STRESS CONDITIONS FOR THE TURBULENT FLOW INTERFACES Turbulent Boundary Type Select a Turbulent boundary type to apply to the turbulence variables Open boundary, Inlet, or Outlet. Open Boundary If Open boundary is selected, then expect parts of the boundary to be an outlet and parts of the boundary to be an inlet. Under Exterior turbulence, enter values for the turbulence variables, which are used on the parts of the boundary where u n0, that is, where flow enters the computational domain. These settings are the same as described in More Open Boundary Conditions for the Turbulent Flow Interfaces. 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 they, for the Inlet option, apply it to the whole boundary. These settings are the same as described in More Open Boundary Conditions for the Turbulent Flow Interfaces. Outlet 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 CONSTRAINT SETTINGS To display this section, click the Show button ( Select Use weak constraints as required. ) and select Advanced Physics Options. Periodic Flow Condition The Periodic Flow Condition splits its selection in two groups: one source group and one destination group. Fluid that leaves the domain through one of the destination boundaries enters the domain over the corresponding source boundary. This corresponds to a situation where the geometry is a periodic part of a larger geometry. 210 CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

211 If the boundaries are not parallel to each other, the velocity vector is automatically transformed. Tip If the boundaries are curved, it is recommended to only include two boundaries. The Periodic Flow Condition has no input when the interface property Compressibility is set to Compressible flow (Ma<0.3). Typically when a periodic boundary condition is used with a compressible flow the pressure is the same at both boundaries and the flow is driven by a volume force. When Compressibility is set to Incompressible flow, the boundary condition contains an input field for a Pressure difference, p src p dst. This pressure difference can, for example, drive the flow in a fully developed channel flow. To set up a periodic boundary condition select both boundaries in the Periodic Flow Condition node. COMSOL automatically assigns one boundary as the source and the other as the destination. To manually set the destination selection, add a Destination Selection node to the Periodic Flow Condition node. All destination sides must be connected. CONSTRAINT SETTINGS To display this section, click the Show button ( Select Use weak constraints as required. ) and select Advanced Physics Options. Note This boundary condition is not available if the Use memory-efficient form check box is selected (click the Show button ( ) and select Advanced Physics Options>Advanced Settings to display the section) on any single-phase flow interface Settings page. In the COMSOL Multiphysics User s Guide: Destination Selection See Also Using Periodic Boundary Conditions Periodic Boundary Condition Example BOUNDARY CONDITIONS FOR THE SINGLE-PHASE FLOW INTERFACES 211

212 Flow Continuity The Flow Continuity node is suitable for pairs where the boundaries match; it prescribes that the flow field is continuous across the pair. A Wall subnode is added by default to the Flow Continuity node. The Wall feature applies to the parts of the pair boundaries where a source boundary lacks a corresponding destination boundary and vice versa. The Wall feature can be overridden by any other boundary condition that applies to exterior boundaries. Right-click the Flow Continuity node to add additional subfeatures. In the COMSOL Multiphysics User s Guide: See Also Identity and Contact Pairs Specifying Boundary Conditions for Identity Pairs BOUNDARY SELECTION From the Selection list, choose the boundaries for the selected pairs. PAIR SELECTION When Flow Continuity is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect. Note This boundary condition is not available if the Use memory-efficient form check box is selected (click the Show button ( ) and select Advanced Physics Options>Advanced Settings to display the section) on any single-phase flow interface Settings page. Pressure Point Constraint The Pressure Point Constraint feature adds a pressure constraint at a point. If it is not possible to specify the pressure level using a boundary condition, the pressure must be set in some other way, for example, by specifying a fixed pressure at a point. POINT SELECTION From the Selection list, choose the points to use a pressure constraint. PRESSURE CONSTRAINT Enter a point constraint for the Pressure p 0 (SI unit: Pa). 212 CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

213 CONSTRAINT SETTINGS To display this section, click the Show button ( ) and select Advanced Physics Options. Select the Use weak constraints check box to use weak constraints and create dependent variables for the corresponding Lagrange multipliers. Fan Use the Fan feature to define the flow direction (inlet or outlet), and the fan parameters on exterior boundaries. Use the Interior Fan node for interior boundaries. BOUNDARY SELECTION From the Selection list, choose the exterior boundaries to apply the fan. FLOW DIRECTION Select a Flow direction Inlet or Outlet. Tip After a boundary is selected, an arrow displays in the Graphics window to indicate the selected flow direction. To update the arrow if the selection changes, click any node in the Model Builder and then click the Fan node again to update the Graphics window. PARAMETERS When Inlet is selected as the Flow direction, enter the Input pressure p input (SI unit: Pa) to define the pressure at the fan input. The default is 0. When Outlet is selected as the Flow direction, enter the Exit pressure p exit (SI unit: Pa) to define the pressure at the fan outlet. The default is 0. Select a Static pressure curve to specify a fan curve Linear (the default), Static pressure curve data, or User defined. Linear For both Inlet and Outlet flow directions, if Linear is selected, enter values or expressions for the Static pressure at no flow p nf (SI unit: Pa) and the Free delivery flow rate V 0,fd (SI unit: m 3 /s). The static pressure curve is equal to the static pressure at no flow rate when V 0 0 and equal to 0 when the flow rate is larger than the free delivery flow rate. The default static pressure is 100 Pa and the default free delivery is 0.01 m 3 /s. BOUNDARY CONDITIONS FOR THE SINGLE-PHASE FLOW INTERFACES 213

214 User Defined Select User defined to enter different values or expressions. The flow rate across the selection where this boundary condition is applied is defined by phys_id.v0 where phys_id is the physics interface identifier (for example, phys_id is spf by default for laminar single-phase flow). In order to avoid unexpected behavior, the function used for the fan curve is the maximum between the user defined function and 0. Static Pressure Curve Data Select Static pressure curve data to enter or load data under the Static Pressure Curve Data section that displays. The interpolation between points given in the table is defined using the Interpolation function type list in the Static Pressure Curve Interpolation section. Then the units are specified for the flow rate and the static pressure curve in the Units section (described in the next sections). STATIC PRESSURE CURVE DATA 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). STATIC PRESSURE CURVE INTERPOLATION This section is available when Static pressure curve data is selected as the Static pressure curve. Select the Interpolation function type Linear (the default), Piecewise cubic, or Cubic spline. The extrapolation method is always a constant value. In order to avoid problems with an undefined function, the function used for the boundary condition is the maximum between the interpolated function and 0. UNITS This section is available when Static pressure curve data is selected as the Static pressure curve. Select Units for the Flow rate (the default SI unit is m 3 /s) and Static pressure curve (the default SI unit is Pa). See Also Theory for the Fan and Grille Boundary Conditions 214 CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

215 Interior Fan The Interior Fan node represents interior boundaries where a fan condition is set using the fan pressure curve to avoid an explicit representation of the fan. The Interior Fan defines a boundary condition on the slit. That means that the pressure and the velocity can be discontinuous across this boundary. One side represents a flow inlet; the other side represents the fan outlet. The fan boundary condition ensures that the mass flow rate is conserved between its inlet and outlet: inlet u n outlet + u n = 0 This boundary condition acts like a Pressure, No Viscous Stress boundary condition on each side of the fan. The pressure at the fan outlet is fixed so that the mass flow rate is conserved. On the fan inlet the pressure is set to the pressure at the fan outlet minus the pressure drop due to the fan. The pressure drop due to the fan is defined by the static pressure curve, which is usually a function of the flow rate. To define a fan boundary condition on an exterior boundary, use the Fan feature instead. BOUNDARY SELECTION From the Selection list, choose the interior boundaries to apply the fan. INTERIOR FAN Define the Flow direction by selecting Along normal vector (the default) or Opposite to normal vector. This defines which side of the boundary is considered the fan s inlet and outlet. Tip After a boundary is selected, an arrow displays in the Graphics window to indicate the selected flow direction. To update the arrow if the selection changes, click any node in the Model Builder and then click the Interior fan node again to update the Graphics window. Note The rest of the Settings for this section are the same as for the Fan feature. See Linear, Static Pressure Curve Data, and User Defined for details. BOUNDARY CONDITIONS FOR THE SINGLE-PHASE FLOW INTERFACES 215

216 See Also Theory for the Fan Defined on an Interior Boundary Grille The Grille node models the pressure drop caused by having a grille that covers the inlet or outlet. BOUNDARY SELECTION From the Selection list, choose the boundaries that are covered by the grille. FLOW DIRECTION Select a Flow direction Inlet or Outlet. Tip After a boundary is selected, an arrow displays in the Graphics window to indicate the selected flow direction. To update the arrow if the selection changes, click any node in the Model Builder and then click the Grille node again to update the Graphics window. PARAMETERS When Inlet is selected as the Flow direction, enter the Input pressure p input (SI unit: Pa) to define the pressure at the fan input. The default is 0. When Outlet is selected as the Flow direction, enter the Exit pressure p exit (SI unit: Pa) to define the pressure at the fan outlet. The default is 0. Select an option from the Static pressure curve list Linear loss (the default), Quadratic loss, Static pressure curve data, or User defined. Linear Loss Enter the Linear loss coefficient to define llc. The default is 0 Pas/m 3. llc defines the static pressure curve that is a piecewise linear function equal to 0 when flow rate is < 0, equal to V 0 llc when flow rate is > 0. Quadratic Loss Enter the Quadratic loss coefficient to define qlc. The default value is 0 Pas 2 /m 6. qlc defines the static pressure curve that is a piecewise quadratic function equal to 0 when flow rate is < 0, equal to V 0 qlc 2 when flow rate is > CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

217 User Defined Select User defined to enter different values or expressions. The flow rate across the selection where this boundary condition is applied is defined by phys_id.v0 where phys_id is the physics interface identifier (for example phys_id is spf by default for this interface). In order to avoid unexpected behavior, the function used for the grille curve is the maximum between the user-defined function and 0. Static Pressure Curve Data Select Static pressure curve data to enter or load data under the Static Pressure Curve Data section that displays. The interpolation between points given in the table is defined using the Interpolation function type list in the Static Pressure Curve Interpolation section. Then the units are specified for the flow rate and the static pressure curve in the Units section (described in the next sections). STATIC PRESSURE CURVE DATA 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). STATIC PRESSURE CURVE INTERPOLATION This section is available when Static pressure curve data is selected as the Static pressure curve. Select the Interpolation function type Linear (the default), Piecewise cubic, or Cubic spline. The extrapolation method is always a constant value. In order to avoid problems with an undefined function, the function used for the boundary condition is the maximum between the interpolated function and 0. UNITS This section is available when Static pressure curve data is selected as the Static pressure curve. Select Units for the Flow rate (the default SI unit is m 3 /s) and Static pressure curve (the default SI unit is Pa). Note Theory for the Fan and Grille Boundary Conditions BOUNDARY CONDITIONS FOR THE SINGLE-PHASE FLOW INTERFACES 217

218 Theory for the Laminar Flow Interface See Also For the basic laminar flow theory, see Theory for the Laminar Flow Interface in the COMSOL Multiphysics User s Guide. This section discusses the theory related to the enhanced features available with this module and for laminar flow. Also see Theory for the Turbulent Flow Interfaces. The Single-Phase Flow, Laminar Flow Interface theory unique to this module is described in this section: Theory for the Laminar Inflow Boundary Condition Theory for the Laminar Outflow Boundary Condition Theory for the Fan Defined on an Interior Boundary Theory for the Fan and Grille Boundary Conditions Theory for the No Viscous Stress Boundary Condition Theory for the Laminar Inflow Boundary Condition In order to prescribe an inlet velocity profile, this boundary condition adds a weak form contribution corresponding to one-dimensional Navier-Stokes equations projected on the boundary. The applied condition corresponds to the situation shown in Figure 6-1: a fictitious domain of length L entr is assumed to be attached to the inlet of the computational domain. This boundary condition uses the assumption that flow in this fictitious domain is a laminar plug flow. If the option is selected that constrains outer edges or endpoints to zero, the assumption is instead that the flow in the fictitious domain is fully developed laminar channel flow (in 2D) or fully developed 218 CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

219 laminar internal flow (in 3D). This does not affect the boundary condition in the real domain,, where the boundary conditions are always fulfilled. p entr L entr Figure 6-1: An example of the physical situation simulated when using the Laminar inflow boundary condition. is the actual computational domain while the dashed domain is a fictitious domain. If an average inlet velocity or inlet volume flow is specified instead of the pressure, COMSOL Multiphysics adds an ODE that calculates a pressure, p entr, such that the desired inlet velocity or volume flow is obtained. See Also Also see Inlet for the node Settings. Theory for the Laminar Outflow Boundary Condition In order to prescribe an outlet velocity profile, this boundary condition adds a weak form contribution corresponding to one-dimensional Navier-Stokes equations projected on the boundary. The applied condition corresponds to the situation shown in Figure 6-2: assume that a fictitious domain of length L exit is attached to the outlet of the computational domain. This boundary condition uses the assumption that the flow in this fictitious domain is laminar plug flow. If the option is selected that constrains outer edges or endpoints to zero, the assumption is instead that the flow in the fictitious domain is fully developed laminar channel flow (in 2D) or fully developed THEORY FOR THE LAMINAR FLOW INTERFACE 219

220 laminar internal flow (in 3D). This does not affect the boundary condition in the real domain,, where the boundary conditions are always fulfilled. p exit L exit Figure 6-2: An example of the physical situation simulated when using Laminar outflow boundary condition. is the actual computational domain while the dashed domain is a fictitious domain. If the average outlet velocity or outlet volume flow is specified instead of the pressure, the software adds an ODE that calculates p exit such that the desired outlet velocity or volume flow is obtained. See Also Also see Outlet for the node Settings. Theory for the Fan Defined on an Interior Boundary In this case, the inlet and outlet of the device are both interior boundaries (see Figure 6-3). The boundaries are called dev_in and dev_out. The boundary conditions are described as follows: The inlet of the device is an outlet boundary condition for the modeled domain. For this outlet boundary condition, on dev_in, the value of the pressure variable is set to the sum of the mean value of the pressure on dev_out and the pressure drop across the device. The pressure drop is calculated from a lumped curve using the flow rate evaluated on dev_in. For the inlet boundary condition, on dev_out, the pressure value is set so that the flow rate is equal on dev_in and dev_out. An ODE is added to compute the pressure value. Note In both cases, the boundary condition implementation specifies vanishing viscous stress along with a Dirichlet condition on the pressure. 220 CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

221 See Also See Interior Fan for node settings. Figure 6-3: A device between two boundaries. The red arrows represent the flow direction, the cylindrical part represents the device (that should be not be part of the model), and the two cubes are the domain that are modeled with a particular inlet boundary condition to account for the device. Theory for the Fan and Grille Boundary Conditions Fans, pumps, or grilles (devices) can be represented using lumped curves implemented as boundary conditions. These simplifications also imply some assumptions. In particular, it is assumed that a given boundary can only be either an inlet or an outlet. Such a boundary should not be a mix of inlets/outlets, nor should it change during a simulation. Manufacturers usually provide curves that describe the static pressure as a function of flow rate for a fan. See Also See Fan and Grille for node settings. THEORY FOR THE LAMINAR FLOW INTERFACE 221

222 DEFINING A DEVICE AT AN INLET In this case, the device s inlet is an external boundary, represented by the external circular boundary of the green domain on Figure 6-4. The device s outlet is an interior face situated between the green and blue domains in Figure 6-4. The lumped curve gives the flow rate as a function of the pressure difference between the external boundary and the interior face. This boundary condition implementation specifies vanishing viscous stress along with a Dirichlet condition on the pressure. The Fan boundary condition sets the following conditions: u + u T 2 -- ui 3 n = 0, p = pinput + p fan V 0 u + u T n = 0, p = p input + p fan V 0 (6-3) (6-4) The Grille boundary condition sets the following conditions: u + u T 2 -- ui n = 0, p = pinput p 3 grill V 0 u + u T n = 0, p = p input p grill V 0 (6-5) (6-6) where V 0 is the flow rate across the boundary, p input is the pressure at the device s inlet, and p fan V 0 ) and p grille (V 0 ) are the static pressure functions of flow rate for the fan and the grille. Equation 6-3 and Equation 6-5 correspond to the compressible formulation. Equation 6-4 and Equation 6-6 correspond to the incompressible formulation. 2D In 2D the thickness in the third direction, Dz, is used to define the flow rate. Fans are modeled as rectangles in this case. 222 CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

223 Figure 6-4: A device at the inlet. The arrow represents the flow direction, the green circle represents the device (that should not be part of the model), and the blue cube represents the modeled domain with an inlet boundary condition described by a lumped curve for the attached device. DEFINING A DEVICE AT AN OUTLET In this case (see Figure 6-5), the fan s inlet is the interior face situated between the blue (cube) and green (circle) domain while its outlet is an external boundary, here the circular boundary of the green domain. The lumped curve gives the flow rate as a function of the pressure difference between the interior face and the external boundary. This boundary condition implementation specifies vanishing viscous stress along with a Dirichlet condition on the pressure. The Fan boundary condition sets the following conditions: u + u T 2 -- ui n = 0, p = pext p 3 fan V 0 u + u T n = 0, p = p ext p fan V 0 (6-7) (6-8) The Grille boundary condition sets the following conditions: u + u T 2 -- ui n = 0, p = pinput + p 3 grill V 0 u + u T n = 0, p = p input + p grill V 0 (6-9) (6-10) THEORY FOR THE LAMINAR FLOW INTERFACE 223

224 where V 0 is the flow rate across the boundary, p ext is the pressure at the device outlet, and p fan (V 0 ), p vacuum pump (V 0 ), and p grille (V 0 ) are the static pressure function of flow rate for the fan, the vacuum pump, and the grille. Equation 6-7, Equation 6-8, and Equation 6-9 correspond to the compressible formulation. Equation 6-8, Equation 6-9, and Equation 6-10 correspond to the incompressible formulation. 2D In 2D the thickness in the third direction, Dz, is used to define the flow rate. Fans are modeled as rectangles in this case. Figure 6-5: A fan at the outlet. The arrow represents the flow direction, the green circle represents the fan (that should not be part of the model), and the blue cube represents the modeled domain with an outlet boundary condition described by a lumped curve for the attached fan. Theory for the No Viscous Stress Boundary Condition For this module, and in addition to the Theory for the Pressure, No Viscous Stress Boundary Condition (described in the COMSOL Multiphysics User s Guide), the viscous stress condition sets the viscous stress to zero: u + u T 2 -- ui n = CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

225 u + u T n = 0 using the compressible and the incompressible formulation, respectively. The condition is not a sufficient outlet condition since it lacks information about the outlet pressure. It must hence be combined with at pressure point constraints on one or several points or lines surrounding the outlet. This boundary condition is numerically the least stable outlet condition, but can still be beneficial if the outlet pressure is nonconstant due to, for example, a nonlinear volume force. See Also Also see Outlet for the node Settings. THEORY FOR THE LAMINAR FLOW INTERFACE 225

226 Theory for the Turbulent Flow Interfaces The Single-Phase Flow, Turbulent Flow Interfaces theory is described in this section: Turbulence Modeling The k-turbulence Model The Low Reynolds Number k- Turbulence Model Theory for the Pressure, No Viscous Stress Boundary Condition Inlet Values for the Turbulence Length Scale and Intensity Pseudo Time Stepping for Turbulent Flow Models References for the Single-Phase Flow, Turbulent Flow Interfaces See Also Theory for the Laminar Flow 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, which depend on the geometry, the smallest quickly fluctuating scales, and all the scales in between. The tendency for an isothermal flow to become turbulent is measured by the Reynolds number UL Re = (6-11) where is the dynamic viscosity, the density, and U and L are velocity and length scales of the flow, respectively. Flows with high Reynolds numbers tend to become turbulent and this is the case for most engineering applications. The Navier-Stokes equations can be used for turbulent flow simulations, although this would require a large number of elements to capture the wide range of scales in the flow. An alternative approach is to divide the flow in large resolved scales and small unresolved scales. The small scales are then modeled using a turbulence model with 226 CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

227 the goal that the model is numerically less expensive than resolving all present scales. Different turbulence models invoke different assumptions on the unresolved scales resulting in different degree of accuracy for different flow cases. This module includes Reynolds-averaged Navier-Stokes (RANS) models which is the model type most commonly used for industrial flow applications 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: u T + u u = pi + u + u + F t u = 0 (6-12) Once the flow has become turbulent, all quantities fluctuate in time and space. It is seldom worth the extreme computational cost to obtain detailed information about the fluctuations. An averaged representation often provides sufficient information about the flow. The Reynolds-averaged representation of turbulent flows divides the flow quantities into an averaged value and a fluctuating part, = + where can represent any scalar quantity of the flow. In general, the mean value can vary in space and time. This is exemplified in Figure 6-6, which shows time averaging of one component of the velocity vector for nonstationary turbulence. The unfiltered flow has a time scale t 1. After a time filter with width t 2 t 1 has been applied, there is a fluctuating part, u i, and an average part, U i. Because the flow field also varies THEORY FOR THE TURBULENT FLOW INTERFACES 227

228 on a time scale longer than t 2, U i is still time dependent but is much smoother than the unfiltered velocity u i. Figure 6-6: The unfiltered velocity component u i, with a time scale t 1, and the averaged velocity component, U i, with time scale t 2. Decomposition of flow fields into an averaged part and a fluctuating part, followed by insertion into the Navier-Stokes equation, then averaging, gives the Reynolds-averaged Navier-Stokes (RANS) equations: U T + U U + u' u' = P + U + U + F t U = 0 (6-13) where U is the averaged velocity field and is the outer vector product. A comparison with Equation 6-12 indicates that the only difference is the appearance of the last term on the left-hand side of Equation This term represents interaction between the fluctuating velocities and is called the Reynolds stress tensor. This means that to obtain the mean flow characteristics, information about the small-scale structure of the flow is needed. In this case, that information is the correlation between fluctuations in different directions. EDDY VISCOSITY The most common way to model turbulence is to assume that the turbulence is of a purely diffusive nature. The deviating part of the Reynolds stress is then expressed by u' u' --traceu' u' I = 3 T U + U T 228 CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

229 where T is the eddy viscosity, also known as the turbulent viscosity. The spherical part can be written --traceu' 3 u' I = 2 --k 3 where k is the turbulent kinetic energy. In simulations of incompressible flows, this term is included in the pressure, but when the absolute pressure level is of importance (in compressible flows, for example) this term must be explicitly included. TURBULENT COMPRESSIBLE FLOW If the Reynolds average is applied to the compressible form of the Navier-Stokes, terms of the form u appear and need to be modeled. To avoid this, a density-based average, known as the Favre average, is introduced: 1 1 = -- lim --- T T u i t + T t ( x, )u i ( x, ) d (6-14) It follows from Equation 6-14 that ũ i = u i (6-15) and a variable, u i, is decomposed in a mass-averaged component, component, u i, according to ũ i, and a fluctuating u i = ũ i + u i (6-16) Using Equation 6-15 and Equation 6-16 along with some modeling assumption for compressible flows (Ref. 7), Equation 6-14 can be written in the form ũ i ũ i + ũ t j x j ũ t x i = 0 i p x i x ũi ũ j ũ k = j x j x i 3 x ij u j u i + F i k (6-17) The Favre-averaged Reynolds stress tensor is modeled using the same argument as for incompressible flows: THEORY FOR THE TURBULENT FLOW INTERFACES 229

230 ũ i ũ u j u i T j ũ -- k = x j x i 3 T k x k where k is the turbulent kinetic energy. Comparing Equation 6-17 to its incompressible counterpart (Equation 6-13), it can be seen that except for the term 2 3k ij the compressible and incompressible formulations are exactly the same, except that the free variables are instead of ũ i ij U i = u i More information about modeling turbulent compressible flows is in Ref. 1 and Ref. 7. The turbulent transport equations are used in their fully compressible formulations (Ref. 8). The k-turbulence Model The k- model is one of the most used turbulence models for industrial applications. This module includes the standard k- model (Ref. 1). This introduces two additional transport equations and two dependent variables: the turbulent kinetic energy, k, and the dissipation rate of turbulence energy,. Turbulent viscosity is modeled by k T = C (6-18) where C is a model constant. The transport equation for k reads: k + u k T = k t + P k k (6-19) where the production term is P k T u: u+ u T 2 -- u = --k u 3 (6-20) The transport equation for reads: 230 CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

231 u T t = + C 1 --P k k C k (6-21) The model constants in Equation 6-18, Equation 6-19, and Equation 6-21 are determined from experimental data (Ref. 1) and the values are listed in Table 6-1. TABLE 6-1: MODEL CONSTANTS CONSTANT VALUE C 0.09 C C k MIXING LENGTH LIMIT Equation 6-19 and Equation 6-21 cannot be implemented directly as written. There is, for example, nothing that prevents division by zero. The equations are instead implemented as suggested in Ref. 9. The implementation includes an upper limit on the mixing length, lim : l mix k l mix = maxc 3/ llim mix (6-22) The mixing length is used to calculated the turbulent viscosity. should not be active in a converged solution but is merely a tool to obtain convergence. llim mix REALIZABILITY CONSTRAINTS The eddy-viscosity model of the Reynolds stress tensor can be written u i u j = 2 T S ij k 3 ij where ij is the Kronecker delta and S ij is the strain-rate tensor. The diagonal elements of the Reynolds stress tensor must be nonnegative, but calculating T from Equation 6-18 does not guarantee this. To assert that u i u i 0 the turbulent viscosity is subjected to a realizability constraint. The constraint for 2D and 2D axisymmetry is: i THEORY FOR THE TURBULENT FLOW INTERFACES 231

232 k 2 T S ij S ij (6-23) and for 3D and 2D axisymmetry with swirl flow it reads: T k S ij S ij (6-24) Note Swirl flow is not available with the Heat Transfer Module. Combining equation Equation 6-23 with Equation 6-18 and the definition of the mixing length gives a limit on the mixing length scale: l 2 k mix S ij S ij (6-25) Equivalently, combining Equation 6-24 with Equation 6-18 and Equation 6-22 gives: 1 l mix k S ij S ij (6-26) This means there are two limitations on l mix : the realizability constraint and the imposed limit via Equation The effect of not applying a realizability constraint is typically excessive turbulence production. The effect is most clearly visible in stagnation points. To avoid such artifacts, the realizability constraint is always applied for the RANS models. More details can be found in Ref. 4, Ref. 5, and Ref. 6. MODEL LIMITATIONS The k- turbulence model relies on several assumptions, the most important of which is that the Reynolds number is high enough. It is also important that the turbulence is in equilibrium in boundary layers, which means that production equal dissipation. These assumptions limit the accuracy of the model because they are not always true. It does not, for example, respond correctly to flows with adverse pressure gradients that can result in underpredicting the spatial extension of recirculation zones (Ref. 1). Furthermore, in the description of rotating flows, the model often shows poor 232 CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

233 agreement with experimental data (Ref. 2). In most cases, the limited accuracy is a fair trade-off for the amount of computational resources saved compared to more complicated turbulence models. WALL FUNCTIONS The flow close to a solid wall is for a turbulent flow and is very different compared to the free stream. This means that the assumptions used to derive the k- model are not valid close to walls. While it is possible to modify the k- model so that it describes the flow in wall regions (see The Low Reynolds Number k- Turbulence Model), this is not always desirable because of the very high resolution requirements that follow. Instead, analytical expressions are used to describe the flow at the walls. These expressions are known as wall functions. The wall functions in COMSOL are such that the computational domain is assumed to start a distance w from the wall (see Figure 6-7). Mesh cells w Solid wall Figure 6-7: The computational domain starts a distance w from the wall for wall functions. The distance w is automatically computed so that + w = u w where u C 1/4 k is the friction velocity, which becomes This corresponds to the distance from the wall where the logarithmic layer meets the viscous sublayer (or to some extent would meet if there was not a buffer layer in between). w is limited from below so that it never becomes smaller than half of the height of the boundary THEORY FOR THE TURBULENT FLOW INTERFACES 233

234 mesh cell. This means that coarse. w + can become higher than if the mesh is relatively Always investigate the solution to check that w is small compared to the dimension of the geometry. Also check that + w is on most of the walls. Tip If is much higher over a significant part of the walls, the accuracy might become compromised. Both the wall lift-off, w, and the wall lift-off in viscous units, + w, are available as results and analysis variables. w + The boundary conditions for the velocities are a no-penetration condition u n = 0 and a shear stress condition where is the viscous stress tensor and u n n nn = u maxc1 4 u = u + u T / ku u = u ln w + B v where in turn, v, is the von Kárman constant (default value 0.41) and B is a constant that by default is set to 5.2. The turbulent kinetic energy is subject to a homogeneous Neumann condition n k = 0 and the boundary condition for reads: See Ref. 9 and Ref. 10 for further details. C3/ 4 k 3/ 2 = v w INITIAL VALUES The default initial values for a stationary simulation are (Ref. 9), 234 CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

235 llim mix k = u = 0 p = lim where is the mixing length limit. For time dependent simulations, the initial value for k is instead l mix C k3/ 2 init = llim mix k = lim l mix The Low Reynolds Number k- Turbulence Model In some cases, the accuracy provided by wall functions is not enough. In these cases, a so called low Reynolds number model can be used. Low Reynolds number refers to the region close to the wall where the viscous effects dominate. Most low Reynolds number k- models adapt the turbulence transport equations by introducing damping functions. This module includes the AKN model (after the inventors Abe, Kondoh, and Nagano; Ref. 11). The AKN k- model for compressible flows reads (Ref. 8 and Ref. 11): k + u k T = k t + P k k u T t = + C 1 --P k k f C k (6-27) where THEORY FOR THE TURBULENT FLOW INTERFACES 235

236 P k T u: u+ u T 2 -- u = --k u 3 k T = f C f 1 e l* e R t = / R t 3 4 f 1 e l* e R t = (6-28) l * = u l w R t = k 2 u = 1/ 4 and C 1 = 1.5 C 2 = 1.9 C = 0.09 k = 1.4 = 1.4 (6-29) Also, l w is the distance to the closest wall. Realizability Constraints are applied to the low Reynolds number k- model. WALL DISTANCE The wall distance variable, l w, is provided by a mathematical Wall Distance interface that is included when using the low Reynolds number k- model. The most convenient way to handle the wall distance variable is to solve it in a separate study step. A Wall Distance Initialization study type is provided for this purpose and should be added before the actual Stationary or Transient study step. See Also The Wall Distance Interface in the COMSOL Multiphysics User s Guide Stationary with Initialization, Transient with Initialization, and Wall Distance Initialization in the COMSOL Multiphysics Reference Guide WALL BOUNDARY CONDITIONS The damping terms in the equations for k and allows a no slip condition to be applied to the velocity, that is u0. Since all velocities must disappear on the wall, so must k. Hence, k0 on the wall. The correct wall boundary condition for is 2 k n CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

237 where n is the wall normal direction. That condition is however numerically very unstable. Instead, is not solved for in the cells adjacent to a solid wall and the analytical relation = k l2 w (6-30) is prescribed in those cells. Equation 6-30 can be derived as the first term in a series expansion of 2 k n 2 For the expansion to be a valid, it is required that l c * l* c 1 is the distance, measured in viscous units, from the wall to the center of the wall adjacent cell. The boundary variable Dimensionless distance to cell center is available to ensure that the mesh is fine enough. Observe though that it is unlikely that a solution is obtained at all if l* c» 1 INLET VALUES FOR THE TURBULENCE LENGTH SCALE AND INTENSITY The guidelines given in Inlet Values for the Turbulence Length Scale and Intensity for selecting turbulence length scale, L T, and the turbulence intensity, I T, apply also to the low-reynolds number k- model. INITIAL VALUES The low-reynolds number k- model has the same default initial guess as the standard k- model (see Initial Values) but with lim replaced by l ref. l mix The default initial value for the wall distance equations (which solves for the reciprocal wall distance) 2l ref. Inlet Values for the Turbulence Length Scale and Intensity A value of 0.1% is a low turbulence intensity I T. Good wind tunnels can produce values of as low as 0.05%. Fully turbulent flows usually have intensities between five and ten percent. THEORY FOR THE TURBULENT FLOW INTERFACES 237

238 The turbulent length scale L T is a measure of the size of the eddies that are not resolved. For free-stream flows these are typically very small (in the order of centimeters). The length scale cannot be zero, however, because that would imply infinite dissipation. Use Table 6-2 as a guideline when specifying L T (Ref. 3) where l w is the wall distance, and l w + = l w l * Theory for the Pressure, No Viscous Stress Boundary Condition For this module, and in addition to the Theory for the Pressure, No Viscous Stress Boundary Condition (described in the COMSOL Multiphysics User s Guide), the turbulent intensity I T, turbulence length scale L T, and reference velocity scale U ref values are related to the turbulence variables via k 3 = -- I 2 T U ref 2, = 3 4 C ITUref L T See Also For recommendations of physically sound values see Inlet Values for the Turbulence Length Scale and Intensity. Also see Inlet and Outlet for the node Settings. is the wall distance in viscous units. TABLE 6-2: TURBULENT LENGTH SCALES FOR TWO-DIMENSIONAL FLOWS FLOW CASE L T L Mixing layer 0.07L Layer width Plane jet 0.09L Jet half width Wake 0.08L Wake width Axisymmetric jet 0.075L Jet half width Boundary layer (px0) Viscous sublayer and log-layer Outer layer + l w 1 exp l w L Boundary layer thickness Pipes and channels (fully developed flows) 0.07L Pipe diameter or channel width 238 CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

239 Pseudo Time Stepping for Turbulent Flow Models Pseudo time stepping is by default applied to the turbulence equations for stationary problems, for both 2D and 3D models. The turbulence equations use the same t as the momentum and continuity equations. If the automatic expression for CFL loc is, for 2D models: and for 3D models: 1.3 min nitercmp min nitercmp 25 9 ifnitercmp min nitercmp 50 9 ifnitercmp min nitercmp min nitercmp 30 9 ifnitercmp min nitercmp 60 9 ifnitercmp References for the Single-Phase Flow, Turbulent Flow Interfaces 1. D.C. Wilcox, Turbulence Modeling for CFD, 2nd ed., DCW Industries, D.M. Driver and H.L. Seegmiller, Features of a Reattaching Turbulent Shear Layer in Diverging Channel Flow, AIAA Journal, vol. 23, pp , H.K. Versteeg and W. Malalasekera, An Introduction to Computational Fluid Dynamics, Prentice Hall, A. Durbin, On the k- Stagnation Point Anomality, International Journal of Heat and Fluid Flow, vol. 17, pp , A, Svenningsson, Turbulence Transport Modeling in Gas Turbine Related Applications, doctoral dissertation, Department of Applied Mechanics, Chalmers University of Technology, C. H. Park and S.O. Park, On the Limiters of Two-equation Turbulence Models, International Journal of Computational Fluid Dynamics, vol. 19, No. 1, pp , J. Larsson, Numerical Simulation of Turbulent Flows for Turbine Blade Heat Transfer, doctoral dissertation, Chalmers University of Technology, Sweden, THEORY FOR THE TURBULENT FLOW INTERFACES 239

240 8. L. Ignat, D. Pelletier, and F. Ilinca, A Universal Formulation of Two-equation Models for Adaptive Computation of Turbulent Flows, Computer Methods in Applied Mechanics and Engineering, vol. 189, pp , D. Kuzmin, O. Mierka, and S. Turek, On the Implementation of the k- Turbulence Model in Incompressible Flow Solvers Based on a Finite Element Discretization, International Journal of Computing Science and Mathematics, vol. 1, no. 2 4, pp , H. Grotjans and F.R. Menter, Wall Functions for General Application CFD Codes, ECCOMAS 98, Proceedings of the Fourth European Computational Fluid Dynamics Conference, John Wiley & Sons, pp , K. Abe, T. Kondoh, and Y. Nagano, A New Turbulence Model for Predicting Fluid Flow and Heat Transfer in Separating and Reattaching Flows I. Flow field calculations, International Journal of Heat and Mass Transfer, vol. 37, no. 1, pp , CHAPTER 6: THE SINGLE-PHASE FLOW BRANCH

241 7 The Conjugate Heat Transfer Branch The Heat Transfer Module has interfaces for conjugate heat transfer, which are also under the Fluid Flow branch as Non-Isothermal Flow interfaces. The Non-Isothermal Flow Laminar Flow (nitf) and Turbulent Flow (nitf) interfaces are identical to the Conjugate Heat Transfer interfaces found under the Heat Transfer branch. This chapter discusses applications involving the Conjugate Heat Transfer branch ( ). In this chapter: About the Conjugate Heat Transfer Interfaces The Non-Isothermal Flow and Conjugate Heat Transfer, Laminar Flow Interfaces The Non-Isothermal Flow and Conjugate Heat Transfer, Turbulent Flow Interfaces Shared Feature Settings Theory for the Non-Isothermal Flow and Conjugate Heat Transfer Interfaces 241

242 About the Conjugate Heat Transfer Interfaces In this section: Selecting the Right Interface The Non-Isothermal Flow Options Conjugate Heat Transfer Options Selecting the Right Interface There are several variations of the same predefined multiphysics interface (all with the interface identifier nitf), that combine the heat equation with either laminar flow or turbulent flow. The advantage of using the multiphysics interfaces compared to adding the individual interfaces separately is that predefined couplings are available in both directions. In particular, interfaces use the same definition of the density, which can therefore be a function of both pressure and temperature. Solving this coupled system of equation usually requires numerical stabilization, which the predefined multiphysics interface also sets up. The interfaces found under the Fluid Flow>Non-Isothermal Flow ( ) and Heat Transfer>Conjugate Heat Transfer ( ) branches are multiphysics interfaces and contain the physics for modeling fluid flow, which can be laminar, turbulent, or Stokes flow, in combination with heat transfer. The settings vary only by one or two default settings (see Table 7-1), which are selected during Model Wizard selection, or from a check box or list under the Physical Model section for the interface. Figure 7-1 is an example that compares the two Settings windows. TABLE 7-1: THE NON-ISOTHERMAL FLOW AND CONJUGATE HEAT TRANSFER PHYSICAL MODEL DEFAULT SETTINGS* INTERFACE (NITF) Non-Isothermal Flow, Laminar Flow Non-Isothermal Flow, Turbulent Flow, k- TURBULENCE MODEL TYPE TURBULENCE MODEL HEAT TRANSPORT TURBULENCE MODEL DEFAULT MODEL None N/A N/A Fluid RANS k- Kays-Crawford Fluid 242 CHAPTER 7: THE CONJUGATE HEAT TRANSFER BRANCH

243 TABLE 7-1: THE NON-ISOTHERMAL FLOW AND CONJUGATE HEAT TRANSFER PHYSICAL MODEL DEFAULT SETTINGS* INTERFACE (NITF) Non-Isothermal Flow, Turbulent Flow, Low Re k- Conjugate Heat Transfer, Laminar Flow Conjugate Heat Transfer, Turbulent Flow, k- Conjugate Heat Transfer, Turbulent Flow, Low Re k- TURBULENCE MODEL TYPE RANS Low Reynolds number k- Kays-Crawford Fluid None N/A N/A Heat transfer in solids RANS k- Kays-Crawford Heat transfer in solids RANS TURBULENCE MODEL Low Reynolds number k- HEAT TRANSPORT TURBULENCE MODEL Kays-Crawford DEFAULT MODEL Heat transfer in solids *For all the interfaces, the Neglect initial term (Stokes flow) check box is not selected by default. ABOUT THE CONJUGATE HEAT TRANSFER INTERFACES 243

244 Figure 7-1: On the left is the Settings window for the Non-Isothermal Flow, Turbulent Flow interface. You can model laminar and turbulent flow, or Stokes flow, in combination with heat transfer. On the right is the Settings window for the Conjugate Heat Transfer, Turbulent Flow, Low Reynold s k- interface. Choose to model laminar and turbulent flow in combination with heat transfer. The next sections give you a brief overview of each of the interfaces to help you choose. The Non-Isothermal Flow Options Different types of flow require different equations to describe them. If the type of flow to model is known, then select it directly from the Model Wizard. However, when you are not certain of the flow type, or because it is difficult to reach a solution easily, you 244 CHAPTER 7: THE CONJUGATE HEAT TRANSFER BRANCH

245 can start instead with a simplified model and add complexity as you build the model. Usually you start with the simplest-to-set-up physics interface, which in most cases in non-isothermal flow is the Non-Isothermal Flow, Laminar Flow interface. In other cases, you may know exactly how a fluid behaves and which equations, models, or physics interfaces best describe it, but because the model is so complex it is difficult to reach an immediate solution. Simpler assumptions may need to be made to solve the problem, and other interfaces may be better to fine-tune the solution process for the more complex problem. This can be the case when you know that the flow is essentially turbulent in nature, but you would first solve it for laminar conditions in order to build knowledge of the system and provide a good initial guess for the turbulent flow simulation. The various forms of the Non-Isothermal Flow interfaces are, by default, found under the Fluid Flow branch. If a solid is chosen in the Default model list, then the interface is renamed Conjugate Heat Transfer. NON-ISOTHERMAL FLOW, LAMINAR FLOW The Non-Isothermal Flow, Laminar Flow Interface ( ) is used primarily to model slow-moving flow in environments where energy transport is also an important part of the system and application, and must coupled or connected to the fluid flow in some way. Processes where natural convection are an important component are classic areas for such modeling. The interface solves the Navier-Stokes equations together with an energy balance assuming heat flux through convection and conduction. The density term is assumed to be affected by temperature and flow is always assumed to be compressible. Stokes law (creeping flow) can be activated from the Non-Isothermal Flow, Laminar Flow interface if wanted. NON-ISOTHERMAL FLOW, TURBULENT FLOW The Turbulent Flow, k- and Turbulent Flow Low Re k- Interfaces ( ) model flows that are relatively fast-moving and geometries that change significantly to induce disorder, vortices, and eddies. Once again, the interfaces are also set up assuming that energy transport is an important part of the system and application and must be coupled or connected to the fluid flow in some way. Process or component cooling are classic examples. For this reason, the interface includes added functionality for calculating the added dispersion of heat transfer due to turbulence. This is represented by one of the Kays-Crawford or Extended Kays-Crawford Turbulence heat transport models, or by including your own turbulent Prandtl number. ABOUT THE CONJUGATE HEAT TRANSFER INTERFACES 245

246 In addition to the properties for the different turbulence models mentioned in Theory for the Non-Isothermal Flow and Conjugate Heat Transfer Interfaces, an additional important aspect is that the reward in terms of accuracy for using low-reynolds number models is even higher in non-isothermal flow simulations. The reason is that the local equilibrium assumption on which the wall functions rely is seldom fulfilled when there are temperature gradients present. This is particularly relevant for applications in non-isothermal flow where the heat flux at solid-liquid interfaces is important to the final solution. Conjugate Heat Transfer Options The various forms of the Conjugate Heat Transfer interfaces are, by default, found under the Heat Transfer branch. These are used to set up and model heat transfer throughout a fluid in collaboration with a solid where heat is transferred by conduction. If a liquid regime is chosen in the Default model list, then the interface is renamed Non-Isothermal Flow, which is the same interface as Conjugate Heat Transfer but with different default settings as in Table 7-1. CONJUGATE HEAT TRANSFER, LAMINAR FLOW The Conjugate Heat Transfer, Laminar Flow Interface ( ) is used primarily to model slow-moving flow in environments where temperature and energy transport are also an important part of the system and application, and must coupled or connected to the fluid-flow in some way. Processes where natural convection are an important component are classic areas for such modeling. The interface solves the Navier-Stokes equations together with an energy balance assuming heat flux through convection and conduction. The density term is assumed to be affected by temperature and flow is always assumed to be compressible. Stokes law (creeping flow) can be activated from the Conjugate Heat Transfer, Laminar Flow interface if required. See Table 7-1 for details. CONJUGATE HEAT TRANSFER, TURBULENT FLOW There are different versions of the Conjugate Heat Transfer, Turbulent Flow interfaces, and each use the Reynolds-Averaged Navier-Stokes (RANS) equations, solving for the mean velocity field and pressure, along with the k-e model. The Turbulent Flow, k- and Turbulent Flow Low Re k- Interfaces ( ) are used to model flows that are relatively fast-moving and/or geometries that change significantly to induce disorder, vortices, and eddies. The interfaces are set up assuming that temperature and energy transport are also an important part of the system and 246 CHAPTER 7: THE CONJUGATE HEAT TRANSFER BRANCH

247 application, and must be coupled or connected to the fluid-flow in some way. Process or component cooling are classic examples. Each interface includes added functionality for calculating the added dispersion of heat transfer due to turbulence. This is represented by one of the Kays-Crawford or Extended Kays-Crawford Turbulence heat transport models, or by including your own turbulent Prandtl number. See Also The Heat Transfer Branch ABOUT THE CONJUGATE HEAT TRANSFER INTERFACES 247

248 The Non-Isothermal Flow and Conjugate Heat Transfer, Laminar Flow Interfaces As discussed in About the Conjugate Heat Transfer Interfaces the two interfaces differ only by where they are selected in the Model Wizard and the default model selected Heat transfer in solids or Fluids. In this section: The Non-Isothermal Flow, Laminar Flow Interface The Conjugate Heat Transfer, Laminar Flow Interface Shared Feature Settings See Also The Non-Isothermal Flow and Conjugate Heat Transfer, Turbulent Flow Interfaces Theory for the Non-Isothermal Flow and Conjugate Heat Transfer Interfaces The Non-Isothermal Flow, Laminar Flow Interface The Non-Isothermal Flow version of the Laminar Flow interface ( ), found under the Fluid Flow>Non-Isothermal Flow branch ( ) of the Model Wizard, is a predefined multiphysics coupling consisting of a single-phase flow interface, using a compressible formulation, in combination with a Heat Transfer interface. When this interface is added, these default nodes are also added to the Model Builder Non-Isothermal Flow, Fluid, Wall, Thermal Insulation, and Initial Values. Right-click any node to add other features that implement, for example, boundary conditions, volume forces, or heat sources. Model Viscous Heating in a Fluid Damper: Model Library path Heat_Transfer_Module/Verification_Models/fluid_damper 248 CHAPTER 7: THE CONJUGATE HEAT TRANSFER BRANCH

249 Tip This interface changes to a Conjugate Heat Transfer interface when Heat transfer in solids is selected as the Default model. INTERFACE IDENTIFIER The interface identifier is a text string that can be used to reference the respective physics interface if appropriate. Such situations could occur when coupling this interface to another physics interface, or when trying to identify and use variables defined by this physics interface, which is used to reach the fields and variables in expressions, for example. It can be changed to any unique string in the Identifier field. The default identifier (for the first interface in the model) is nitf. DOMAIN SELECTION The default setting is to include All domains in the model to define the dependent variables and the equations. To choose specific domains, select Manual from the Selection list. PHYSICAL MODEL Define interface properties to control the overall type of model: 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. Instead use the linear Stokes equations. This flow type is referred to as creeping flow or Stokes flow and can occur in microfluidics (and MEMS devices), where the flow length scales are very small. Turbulence Model Type By definition, no turbulence model is needed when studying laminar flows. The default Turbulence model type is None. The flow state in a fluid-flow model is not, however, always known beforehand. Select RANS as the Turbulence model type and select any of k-, k-, Low Re number k- or THE NON-ISOTHERMAL FLOW AND CONJUGATE HEAT TRANSFER, LAMINAR FLOW INTERFACES 249

250 Spalart-Allmaras as Turbulence model in order to account for turbulence. This changes the interface into the turbulent version. See the Turbulent Flow Interfaces for details. Note If the default Turbulence model type selected is RANS, the additional turbulence model settings are made available. However, the node is still called Non-Isothermal Flow or Conjugate Heat Transfer with a number added at the end of the name to indicate the change. Default Model For the Non-Isothermal Flow interface the Default model is Fluid. For the Conjugate Heat Transfer interface the Default model is Heat transfer in solids. SURFACE-TO-SURFACE RADIATION Select the Surface-to-surface radiation check box to enable the Radiation Settings section. RADIATION SETTINGS Note This section is available when the Surface-to-surface radiation check box is selected. See Radiation Settings as described for The Heat Transfer Interface. DEPENDENT VARIABLES The dependent variables (field variables) are for the Velocity field, Pressure, and Temperature. The names can be changed but the names of fields and dependent variables must be unique within a model. For turbulence modeling and heat radiation, there are additional dependent variables for the turbulent dissipation rate, turbulent kinetic energy, reciprocal wall distance, and surface radiosity. DISCRETIZATION To display this section, click the Show button ( ) and select Discretization. Select a Discretization of fluids P1+P1 (the default), P2+P1, or P3+P2. The first term describes the element order for the velocity components, and the second term is the order for the pressure. The element order for the temperature is set to follow the velocity order, so the temperature order is 1 for P1+P1, 2 for P2+P1, and 3 for P3+P2. Specify the Value type when using splitting of complex variables Real or Complex (the default). 250 CHAPTER 7: THE CONJUGATE HEAT TRANSFER BRANCH

251 CONSISTENT AND INCONSISTENT STABILIZATION To display this section, click the Show button ( unique to this interface are listed below. ) and select Stabilization. Any settings The consistent stabilization methods are applicable to the Heat and flow equations. The Isotropic diffusion inconsistent stabilization method can be activated for both the Heat equation and the Navier-Stokes equations. Show More Physics Options Shared Feature Settings See Also For Settings window details for the Heat Transfer in Solids feature, see The Heat Transfer Interface The Conjugate Heat Transfer, Laminar Flow Interface The Conjugate Heat Transfer, Laminar Flow Interface The Conjugate Heat Transfer version of the Laminar Flow interface ( ), found under the Heat Transfer>Conjugate Heat Transfer branch ( ), is a predefined multiphysics coupling consisting of a single-phase flow interface, using a compressible formulation, in combination with a Heat Transfer interface. When this interface is added, these default nodes are also added to the Model Builder Conjugate Heat Transfer, Heat Transfer in Solids, Wall, Thermal Insulation, and Initial Values. Right-click any node to add other features that implement, for example, boundary conditions, volume forces, or heat sources. Model Viscous Heating in a Fluid Damper: Model Library path Heat_Transfer_Module/Verification_Models/fluid_damper Show More Physics Options The Non-Isothermal Flow, Laminar Flow Interface See Also Shared Feature Settings The Heat Transfer Interface for Settings window details for the Heat Transfer in Solids feature. THE NON-ISOTHERMAL FLOW AND CONJUGATE HEAT TRANSFER, LAMINAR FLOW INTERFACES 251

252 Tip This interface changes to a Non-Isothermal Flow interface when Fluid is selected as the Default model. 252 CHAPTER 7: THE CONJUGATE HEAT TRANSFER BRANCH

253 The Non-Isothermal Flow and Conjugate Heat Transfer, Turbulent Flow Interfaces As discussed in About the Conjugate Heat Transfer Interfaces, the Non-Isothermal Flow ( ) and Conjugate Heat Transfer ( ) branches have more than one version of the Turbulent Flow interface. All interfaces use the Reynolds-Averaged Navier-Stokes (RANS) equations as the Turbulence model type, solving for the mean velocity field and pressure. The differences are the Default model is either Heat transfer in solids or Heat transfer in fluids, and the Turbulence model can be k- or the Low Reynolds k- turbulence model. The Turbulent Flow, k- and Turbulent Flow Low Re k- Interfaces Shared Feature Settings See Also The Heat Transfer Interface Theory for the Non-Isothermal Flow and Conjugate Heat Transfer Interfaces The Turbulent Flow, k- and Turbulent Flow Low Re k- Interfaces These predefined multiphysics couplings consist of a turbulent flow interface, using a compressible formulation, in combination with a Heat Transfer interface. Model Turbulent Flow Through a Shell-and-Tube Heat Exchanger: Model Library path Heat_Transfer_Module/Process_and_Manufacturing/ turbulent_heat_exchanger THE NON-ISOTHERMAL FLOW AND CONJUGATE HEAT TRANSFER, TURBULENT FLOW INTERFACES 253

254 Most of the setting options are the same as for The Non-Isothermal Flow, Laminar Flow Interface, except where noted below. Right-click any node to add other features that implement, for example, boundary conditions, volume forces, or heat sources. Note The Neglect inertial term (Stokes flow) check box is only valid for laminar flow. Show More Physics Options See Also Shared Feature Settings Turbulent Non-Isothermal Flow Theory PHYSICAL MODEL The default Turbulence model for the Turbulent flow, k- interface is k-. For the Turbulent flow, Low Re k- interface it is Low Reynolds number k-. For all the turbulent interfaces, the default Turbulence model type is RANS and the default Heat transport turbulence model is Kays-Crawford. Other Heat transport turbulence model options are Extended Kays-Crawford or User-defined turbulent Prandtl number. The Extended Kays-Crawford model requires a Reynolds number at infinity. That input is given in the Model Inputs section of the Fluid feature node. It is always possible to specify a user-defined model for the turbulence Prandtl number. Enter the user-defined value or expression for the turbulence Prandtl number in the Model Inputs section of the Fluid feature node. TURBULENCE MODEL PARAMETERS Turbulence model parameters are optimized to fit as many flow types as possible, but for some special cases, better performance can be obtained by tuning the model parameters. DEPENDENT VARIABLES The dependent variables (field variables) are for the Velocity field, Pressure, and Temperature. The names can be changed but the names of fields and dependent variables must be unique within a model. 254 CHAPTER 7: THE CONJUGATE HEAT TRANSFER BRANCH

255 For turbulence modeling and heat radiation, there are additional dependent variables for the transported turbulence properties and also a dependent variable for Reciprocal wall distance if the Low-Reynolds number k- model or Spalart-Allmaras model is employed. CONSISTENT AND INCONSISTENT STABILIZATION To display this section, click the Show button ( ) and select Stabilization. Any settings unique to this interface are listed below. The consistent stabilization methods are applicable to the Heat and flow equations and the Turbulence Equations. When the Crosswind diffusion check box is selected, enter a Tuning parameter C k for one or both of the Heat and flow equations and Turbulence Equations. The default for the Heat and flow equations is 0.5, and 1 for the Turbulence equations. The Isotropic diffusion inconsistent stabilization method can be activated for the Heat equation, Navier-Stokes equations, and the Turbulence equations. By default there is no isotropic diffusion selected. If required, select the Isotropic diffusion check box and enter a Tuning parameter id for one or all of Heat equation, Navier-Stokes equations, or Turbulence equations. The defaults are THE NON-ISOTHERMAL FLOW AND CONJUGATE HEAT TRANSFER, TURBULENT FLOW INTERFACES 255

256 Shared Feature Settings All the versions of the Non-Isothermal Flow and Conjugate Heat Transfer interfaces have shared domain, boundary, edge, point, and pair features based on the selections made for the model. Also because these are all multiphysics interfaces, almost every feature is shared with, and described for, other interfaces. Below are links to the domain, boundary, edge, point, and pair features as indicated. In this section: Fluid Initial Values Pressure Work Viscous Heating Wall These features are described for the Laminar Flow interface (listed in alphabetical order): Boundary Stress Interior Fan Flow Continuity Inlet Interior Wall Open Boundary Outlet Periodic Flow Condition Pressure Point Constraint Symmetry Volume Force These features are described for the Heat Transfer interface (listed in alphabetical order): Boundary Heat Source 256 CHAPTER 7: THE CONJUGATE HEAT TRANSFER BRANCH

257 Convective Cooling Continuity Heat Flux Heat Source Heat Transfer in Solids Highly Conductive Layer Inflow Heat Flux Line Heat Source Outflow Pair Boundary Heat Source Pair Thin Thermally Resistive Layer Periodic Heat Condition Point Heat Source Surface-to-Ambient Radiation Symmetry Temperature Thermal Insulation Thin Thermally Resistive Layer See Also The Heat Transfer Interface The Non-Isothermal Flow and Conjugate Heat Transfer, Laminar Flow Interfaces The Non-Isothermal Flow and Conjugate Heat Transfer, Turbulent Flow Interfaces Theory for the Non-Isothermal Flow and Conjugate Heat Transfer Interfaces Tip To locate and search all the documentation, in COMSOL, select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree. SHARED FEATURE SETTINGS 257

258 Fluid The Fluid feature adds both the momentum equations and the temperature equation but without volume forces, heat sources, pressure work, or viscous heating. You can add volume forces and heat sources as separate features, and Viscous Heating and Pressure Work can be added as subnodes to the Fluid node. When the turbulence model type is set to RANS, the Fluid node also adds the equations for k and. DOMAIN SELECTION By default, All domains are selected. MODEL INPUTS Define the model inputs. If no model inputs are required, this section is empty. Tip To define the Absolute Pressure, see the settings for the Heat Transfer in Fluids node as described in the COMSOL Multiphysics User s Guide. HEAT CONDUCTION The default uses the Thermal conductivity k (SI unit: W/(m K)) From material. If User defined is selected, choose Isotropic, Diagonal, Symmetric, or Anisotropic based on the characteristics of the thermal conductivity and enter another value or expression in the field or matrix. The thermal conductivity describes the relationship between the heat flux vector q and the temperature gradient T as in q = kt which is Fourier s law of heat conduction. Enter this quantity as power per length and temperature. When the turbulence model type is set to RANS, the conductive heat flux includes the turbulent contribution: q = k+ T I)T where k is the thermal conductivity tensor, I the identity matrix and T the thermal turbulent conductivity defined by T C p T = Pr T THERMODYNAMICS Select a Fluid type Gas/Liquid or Ideal gas. The heat capacity at constant pressure C p describes the amount of heat energy required to produce a unit temperature change in a unit mass. For an ideal gas, 258 CHAPTER 7: THE CONJUGATE HEAT TRANSFER BRANCH

259 choose to specify either C p or the ratio of specific heats,, but not both since they in that case are dependent. The ratio of specific heats is the ratio of heat capacity at constant pressure, C p, to heat capacity at constant volume, C v. When using the ideal gas law to describe a fluid, specifying is enough to evaluate C p. For common diatomic gases such as air, 1.4 is the standard value. Most liquids have 1.1 while water has 1.0. is used in the streamline stabilization and in the results and analysis variables for heat fluxes and total energy fluxes. It is also used in the ideal gas law. Gas/Liquid If Gas/Liquid is selected properties of a non-ideal gas or liquid can be used. By default the Density (SI unit: kg/m 3 ), Heat capacity at constant pressure C p (SI unit: J/ (kg K)), and Ratio of specific heats (unitless) use data From material. Select User defined to enter other values or expressions. Ideal Gas If Ideal gas is selected, the ideal gas law is used to describe the fluid. In this case, specify the thermodynamics properties by selecting a gas constant type and selecting between entering the heat capacity at constant pressure or the ratio of specific heats. For an ideal gas the density is defined as M n p A p A = = RT R s T where p A is the absolute pressure, and T the temperature. Select a Gas constant type Specific gas constant R s (SI unit: J/(kg K)) or Mean molar mass M n (SI unit: kg/mol). In both cases, the default uses data From material. Select User defined to enter other values or expressions. If Mean molar mass is selected, the universal gas constant R J/(mol K), which is a built-in physical constant, is also used. From the Specify Cp or list, select Heat capacity at constant pressure C p (SI unit: J/(kg K)), and Ratio of specific heats (unitless). The default setting is to use the property value From material. Select User defined to enter another value or expression for either of material property. DYNAMIC VISCOSITY The dynamic viscosity describes the relationship between the shear rate and the shear stresses in a fluid. Intuitively, water and air have a low viscosity, and substances often described as thick, such as oil, have a higher viscosity. Non-Newtonian fluids have a SHARED FEATURE SETTINGS 259

260 shear-rate dependent viscosity. Examples of non-newtonian fluids include yoghurt, paper pulp, and polymer suspensions. Select a Dynamic viscosity (SI unit: Pa s) from the list From material (the default), Non-Newtonian power law, Non-Newtonian Carreau model, or User defined. If User defined is selected, use a built-in variable for the shear rate magnitude, spf.sr, which makes it possible to define arbitrary expressions of the dynamic viscosity. Non-Newtonian Power Law If Non-Newtonian power law is selected, enter the Power law model parameter m and Model parameter n (both unitless). This selection uses the power law as the viscosity model for a non-newtonian fluid where the following equation defines dynamic viscosity: = m n 1 Non-Newtonian Carreau Model If Non-Newtonian Carreau model is selected, enter these Carreau model parameters: The Zero shear rate viscosity 0 (SI unit: Pa s) The Infinite shear rate viscosity inf (SI unit: Pa s) The Model parameters (SI unit: s) and n (unitless) This selection uses the Carreau model as the viscosity model for a non-newtonian fluid where the following equation defines the dynamic viscosity: = + 0 inf n MIXING LENGTH LIMIT (TURBULENCE MODELS ONLY) This section is only available for the k- and k- models, which need an upper limit on the mixing length. Select a Mixing length limit Automatic (the default) or Manual. If Automatic is selected, the mixing length limit is automatically evaluated as: llim mix = 0.5l bb (7-1) where l bb is the shortest side of the geometry bounding box. If the geometry is for example a complicated system of very slender entities, Equation 7-1 tends to give a result that is too large. Then define manually. llim mix If Manual is selected, enter a value or expression for the Mixing length limit (SI unit: m). llim mix 260 CHAPTER 7: THE CONJUGATE HEAT TRANSFER BRANCH

261 DISTANCE EQUATION (TURBULENCE MODELS ONLY) This section is only available for the low-reynolds number k- model and the Spalart-Allmaras model, which need the distance to the closest wall. Select a Reference length scale Automatic (the default) or Manual. If Automatic is selected, the reference length scale is automatically evaluated as: l ref = 0.25l bb (7-2) where l bb is the shortest side of the geometry bounding box. If the geometry is for example a complicated system of very slender entities, Equation 7-1 tends to give a result that is too large. Then define l ref manually. If Manual is selected, enter a value or expression for the Reference length scale lref (SI unit: m). Wall For laminar flow, the low Reynolds number k- turbulence model and Spalart-Allmaras turbulence model, the Wall feature is identical to the single-phase flow settings (the Boundary condition defaults to No slip). In these cases, continuity of the temperature is enforced on internal walls separating a fluid and solid domain. The settings below are for when using the k- or k- turbulence model. About the Thermal Wall Function Whenever wall functions are used, there is a theoretical gap between the solid wall and the computational domain of the fluid. This gap is often ignored in so much that it is ignored when the computational geometry is drawn, but it must nevertheless be considered in the equations for the temperature field. SHARED FEATURE SETTINGS 261

262 Figure 0-1 shows the difference between internal and external walls. The approach is slightly different depending on what type of wall the condition applies to. Any wall feature that utilizes wall functions automatically detects internal and external walls. External wall Inflow Fluid Internal wall Outflow Solid Figure 7-2: A simple example that includes both an external wall and an internal wall. On internal walls, there are two temperatures, one for the solid, T s, and one for the fluid, T f. If a temperature is prescribed to an internal wall, the constraint is applied to the temperature for the solid, that is, to T s. On external walls, the temperature T is the temperature of the fluid while the wall temperature is represented by the dependent variable T w. T w is a variable that is solved for and the equation for T w is q wf = q tot where q tot is the total heat flux prescribed to the boundary. If a temperature is prescribed to an external wall, the constraint is applied to the wall temperature T w. Note Any other heat boundary condition applied to an external wall is wrong in the sense that it acts on the fluid temperature, T s, instead of the wall temperature, T w. BOUNDARY CONDITION When using the k- turbulence model or the k- turbulence model, the Boundary condition defaults to Wall functions. The other options available are Slip, Sliding wall (wall functions), and Moving wall (wall functions). 262 CHAPTER 7: THE CONJUGATE HEAT TRANSFER BRANCH

263 If any one of these options are selected Wall functions, Sliding wall (wall functions), or Moving wall (wall functions) the wall functions for the temperature field is also prescribed, which is called a thermal wall functions. If Sliding wall (wall functions) is selected, enter the coordinates for the Velocity of sliding wall u w (SI unit: m/s). If Moving wall (wall functions) is selected, enter the coordinates for the Velocity of moving wall u w (SI unit: m/s). Note Turbulence model parameters are optimized to fit as many flow types as possible, but for some special cases, better performance can be obtained by tuning the model parameters. See Also About the Thermal Wall Function Initial Values The Initial Values feature adds initial values for the velocity field, the pressure, and temperature that can serve as an initial condition for a transient simulation or as an initial guess for a nonlinear solver. For turbulent flow there are also initial values for the turbulence model variables. The surface radiosity is only applicable for surface-to-surface radiation. DOMAIN SELECTION From the Selection list, choose the domains to define initial values. INITIAL VALUES Enter values or expressions for the initial value of the Velocity field u (SI unit: m/s), the Pressure p (SI unit: Pa), and the Temperature T (SI unit: K). The default values are 0 for the velocity and the pressure, and K for the temperature. In a turbulent flow interface, initial values for the turbulence variables are also specified. By default these are specified using the predefined variables defined by the expressions in Initial Guess. SHARED FEATURE SETTINGS 263

264 The initial value for the surface radiosity J (SI unit: W/m 2 ), for surface-to-surface radiation, has a default value of 0. Pressure Work The Pressure Work feature adds the following contribution to the right-hand side of the Heat Transfer in Fluids equation: T T p p + u p t (7-3) The software computes the pressure work using the absolute pressure. DOMAIN SELECTION From the Selection list, choose the domains to add pressure work. By default, the selection is the same as for the Heat Transfer in Fluids feature that it is attached to. PRESSURE WORK FORMULATION select Full formulation or Low mach number formulation. The latter excludes the term u p from Equation 7-3, which is small for most flows with low Mach number. Viscous Heating The Viscous Heating subnode adds the following term to the right-hand side of the Heat Transfer in Fluids equation: :S (7-4) Here is the viscous stress tensor and S is the strain rate tensor. Equation 7-4 represents the heating caused by viscous friction within the fluid. DOMAIN SELECTION From the Selection list, choose the domains to add pressure work. By default, the selection is the same as for the Fluid feature that it is attached to. 264 CHAPTER 7: THE CONJUGATE HEAT TRANSFER BRANCH

265 Theory for the Non-Isothermal Flow and Conjugate Heat Transfer Interfaces In industrial applications it is common that the density of a process fluid varies. These variations can have a number of different sources but the most common one is the presence of an inhomogeneous temperature field. This module includes the Non-Isothermal Flow predefined multiphysics coupling to simulate systems where density varies with temperature. Other situations where the density might vary includes chemical reactions, for instance where reactants associate or dissociate. The Non-Isothermal Flow and Conjugate Heat Transfer interfaces contain the fully compressible formulation of the continuity equation and momentum equations: u = 0 t u + u u p u + u t T 2 = + -- ui 3 + F (7-5) where is the density (kg/m 3 ) u is the velocity vector (m/s) p is pressure (Pa) is the dynamic viscosity (Pa s) F is the body force vector (N/m 3 ) It also solves the heat equation, which for a fluid is T C p u T t q :S T p = u p + Q T t where in addition to the quantities above C p is the specific heat capacity at constant pressure (SI unit: J/(kgK)) p THEORY FOR THE NON-ISOTHERMAL FLOW AND CONJUGATE HEAT TRANSFER INTERFACES 265

266 T is absolute temperature (SI unit: K) q is the heat flux by conduction (SI unit: W/m 2 ) is the viscous stress tensor (SI unit: Pa) S is the strain-rate tensor (SI unit: 1/s) S = 1 -- u + u 2 T Q contains heat sources other than viscous heating (SI unit: W/m 3 ) The pressure work term and the viscous heating term T T are not included by default because they are commonly negligible. These can, however, be added as subnodes to the Fluid node. For a detailed discussion of the fundamentals of heat transfer in fluids, see Ref. 3. The interface also supports heat transfer in solids: p p + u p t :S T C p = q T E + Q t t where E is the elastic contribution to entropy (SI unit: J/(m 3 K)) As in the case of fluids, the pressure work term T E t is not included by default but must be added as a subfeature. See Also The Heat Equation Turbulent Non-Isothermal Flow Theory References for the Non-Isothermal Flow and Conjugate Heat Transfer Interfaces 266 CHAPTER 7: THE CONJUGATE HEAT TRANSFER BRANCH

267 Turbulent Non-Isothermal Flow Theory Turbulent energy transport is conceptually more complicated than energy transport in laminar flows since the turbulence is also a form of energy. Equations for compressible turbulence are derived using the Favre average. The Favre average of a variable T is denoted T and is defined by T = T where the bar denotes the usual Reynolds average. The full field is then decomposed as T = T + T'' With these notations the equation for total internal energy, e, becomes ũ i ũ i ẽ u i ''u i '' ũ i ũ i ũ t 2 2 x j h j 2 ũ u ''u '' + + i i j = u j ''u i ''u i '' q x j u j ''h'' + ij u i '' ũ j 2 x i ij u i ''u j '' j (7-6) where h is the enthalpy. The vector q j = T x j (7-7) is the laminar conductive heat flux and 2 ij 2S ij -- u k = x ij k is the laminar, viscous stress tensor. Notice that the thermal conductivity is denoted. The modeling assumptions are in large part analogous to incompressible turbulence modeling. The stress tensor u i ''u'' j is model with the Boussinesq approximation: THEORY FOR THE NON-ISOTHERMAL FLOW AND CONJUGATE HEAT TRANSFER INTERFACES 267

268 u i ''u'' j T 1 ij 2 T S ij -- ũ k 2 = = x ij --k k 3 ij (7-8) where k is the turbulent kinetic energy, which in turned is defined by k = 1 --u 2 i ''u i '' (7-9) The correlation between u j '' and h'' in Equation 7-6 is the turbulent transport of heat. It is model analogous to the laminar conductive heat flux u j ''h'' q T T = j = T = x j The molecular diffusion, ij u i '' and turbulent transport term, T C p T Pr T x j (7-10) 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 u j ''u i ''u i '' ij u i '' T k = k x j (7-11) Inserting Equation 7-7, Equation 7-8, Equation 7-9, Equation 7-10 and Equation 7-11 into Equation 7-6 gives ũ i ũ i ẽ k ũ i ũ i ũ t 2 + x j h k j 2 = q x j q T j T k j x j ũ x i ij + T ij j k (7-12) The Favre average can also be applied to the momentum equation, which, using Equation 7-8, can be written ũ t i p + ũ x j ũ i = j x j x ij + T ij j (7-13) 268 CHAPTER 7: THE CONJUGATE HEAT TRANSFER BRANCH

269 Taking the inner product between ũ i and Equation 7-13 results in an equation for the resolved kinetic energy, which can be subtracted from Equation 7-12 with the following result: ẽ + k ũ t x j ẽ + k p ũ j + = j x j q x j q T j T k ũ j k x j x i ij + T ij j (7-14) where the relation ẽ = h + p has been used. According to Wilcox (Ref. 1), it is usually a good approximation to neglect the contributions of k for flows with Mach numbers up to the supersonic range. This gives the following approximation of Equation 7-14 is ẽ ũ t x j ẽ p ũ j + = j x j q x j q T + j ũ j x i ij + T ij j (7-15) Larsson (Ref. 2) suggests to make the split Since for all applications of engineering interest, it will follow that and consequently ij = ij + ij '' ij ij» ij '' ij ẽ ũ t x j ẽ p ũ j + = j x j T x T ũ j x j x i ij j Tot (7-16) where THEORY FOR THE NON-ISOTHERMAL FLOW AND CONJUGATE HEAT TRANSFER INTERFACES 269

270 Tot ij = + T 2 2S ij -- ũ k x ij k Equation 7-16 is completely analogous to the laminar energy equation and can be expanded using the same theory (see for example Ref. 3): T C p u T + t j T + x j x T T = + ijs ij j x j T which is the temperature equation solved in the turbulent Non-Isothermal Flow and Conjugate Heat Transfer interfaces. p p u p + t j x j TURBULENT CONDUCTIVITY Kays-Crawford This is a relatively exact model for Pr T, still simple. In Ref. 4, it is compared to other models for Pr T and found to be good for most kind of turbulent wall bounded flows except for liquid metals. The model is given by 1 Pr T C p T C p T e 2Pr T 0.3C 1 = + p T Pr T Pr T (7-17) where the Prandtl number at infinity is Pr T 0.85 and is the conductivity. Extended Kays-Crawford Weigand and others (Ref. 5) suggested an extension of Equation 7-17 to liquid metals by introducing 100 Pr T = C p Re0.888 where Re, the Reynolds number at infinity must be provided either as a constant or as a function of the flow field. This is entered in the Model Inputs section of the Fluid feature. TEMPERATURE WALL FUNCTIONS Analogous to the single-phase flow wall functions (see Wall Functions described for the Wall boundary condition), there is a theoretical gap between the solid wall and the computational domain of the fluid also for the temperature field. This gap is often ignored in so much that it is ignored when the computational geometry is drawn. 270 CHAPTER 7: THE CONJUGATE HEAT TRANSFER BRANCH

271 The heat flux between the fluid with temperature T f and a wall with temperature T w, is: C p C1/ 4 k 1/ 2 q T T w f wf = where is the fluid density, C p is the fluid heat capacity, C is a turbulence modeling constant, and k is the turbulent kinetic energy. T is the dimensionless temperature and is given by (Ref. 6): T + T + = Pr+ w for + w + w1 15Pr 2/ w2 for w1 + + w w2 Pr ---- ln+ w + for + w2 + w where in turn w + w2 w C1/ 2 k = w1 = / Pr C p = Pr = Pr T 15Pr 2/ 3 Pr T = ln Pr T where in turn is the thermal conductivity, and is the von Karman constant equal to The computational result should be checked so that the distance between the computational fluid domain and the wall, w, is almost everywhere small compared to any geometrical quantity of interest. The distance w is available as a postprocessing variable on boundaries. References for the Non-Isothermal Flow and Conjugate Heat Transfer Interfaces 1. D.C. Wilcox, Turbulence Modeling for CFD, 2nd ed., DCW Industries, J. Larsson, Numerical Simulation of Turbulent Flows for Turbine Blade Heat Transfer, Doctoral Thesis for the Degree of Doctor of Philosophy, Chalmers University of Technology, Sweden, THEORY FOR THE NON-ISOTHERMAL FLOW AND CONJUGATE HEAT TRANSFER INTERFACES 271

272 3. R.L. Panton, Incompressible Flow, 2nd ed., John Wiley & Sons, Inc., W.M. Kays, Turbulent Prandtl Number Where Are We?, ASME Journal of Heat Transfer, 116, pp , B. Weigand, J.R. Ferguson, and M.E. Crawford, An Extended Kays and Crawford Turbulent Prandtl Number Model, International Journal of Heat and Mass Transfer, vol. 40, no. 17, pp , D. Lacasse, È. Turgeon, and D. Pelletier, On the Judicious Use of the k Model, Wall Functions and Adaptivity, International Journal of Thermal Sciences, vol. 43, pp , CHAPTER 7: THE CONJUGATE HEAT TRANSFER BRANCH

273 8 Materials The Material Browser contains libraries with an extensive set of mechanical and heat transfer properties for solid materials. In addition, it contains a limited set of fluid properties, which can be used mainly in the physics interfaces for fluid flow and heat transfer. In this chapter: Material Library and Databases Liquids and Gases Material Database 273

274 Material Library and Databases The Heat Transfer Module includes a Liquids and Gases material database. The materials include temperature-dependent fluid dynamic and thermal properties. Note For detailed information about all the other materials databases and the separately purchased Material Library, see the section Materials in the COMSOL Multiphysics User s Guide. In this section: About the Material Databases About Using Materials in COMSOL Opening the Material Browser Using Material Properties About the Material Databases Material Browser select predefined materials in all applications. Recent Materials Select from recent materials added to the model. Material Library Purchased separately. Select from over 2500 predefined materials. Built-In database Available to all users and contains common materials. Application specific material databases Available with specific modules. User-defined material database library. All COMSOL modules have predefined material data available to build models. The most extensive material data is contained in the separately purchased Material Library, 274 CHAPTER 8: MATERIALS

275 but all modules contain commonly used or module-specific materials. For example, the Built-In database is available to all users but the MEMS database is included with the MEMS Module and Structural Mechanics Module. Also create custom materials and material libraries by researching and entering material properties. All the material databases (including the Material Library) are accessed from the Material Browser. These databases are briefly described below. RECENT MATERIALS From the Recent Materials folder ( ), select from a list of recently used materials, with the most recent at the top. This folder is available after the first time a material is added to a model. MATERIAL LIBRARY An optional add-on database, the Material Library ( ), contains data for over 2500 materials and 20,000 property functions. BUILT-IN Included with COMSOL Multiphysics, the Built-In database ( ) contains common solid materials with electrical, structural, and thermal properties. See Also Predefined Built-In Materials for all COMSOL Modules in the COMSOL Multiphysics User s Guide AC/DC Included in the AC/DC Module, the AC/DC database ( some magnetic and conductive materials. ) has electric properties for BATTERIES AND FUEL CELLS Included in the Batteries & Fuel Cells Module, the Batteries and Fuel Cells database ( ) includes properties for electrolytes and electrode reactions for certain battery chemistries. LIQUIDS AND GASES Included in the Acoustics Module, CFD Module, Chemical Reaction Engineering Module, Heat Transfer Module, MEMS Module, Pipe Flow Module, and Subsurface MATERIAL LIBRARY AND DATABASES 275

276 Flow Module, the Liquids and Gases database ( ) includes transport properties and surface tension data for liquid/gas and liquid/liquid interfaces. MEMS Included in the MEMS Module and Structural Mechanics Module, the MEMS database ( ) has properties for MEMS materials metals, semiconductors, insulators, and polymers. PIEZOELECTRIC Included in the Acoustics Module, MEMS Module, and Structural Mechanics Module, the Piezoelectric database ( ) has properties for piezoelectric materials. USER-DEFINED LIBRARY The User-Defined Library folder ( ) is where user-defined materials databases (libraries) are created. When any new database is created, this also displays in the Material Browser. Important The materials databases shipped with COMSOL Multiphysics are read-only. This includes the Material Library and any materials shipped with the optional modules. See Also Creating Your Own User-Defined Libraries in the COMSOL Multiphysics User s Guide About Using Materials in COMSOL USING THE MATERIALS IN THE PHYSICS SETTINGS The physics set-up in a model is determined by a combination of settings in the Materials and physics interface nodes. When the first material is added to a model, COMSOL automatically assigns that material to the entire geometry. Different geometric entities can have different materials. The following example uses the 276 CHAPTER 8: MATERIALS

277 heat_sink.mph model file contained in the Heat Transfer Module and CFD Module Model Libraries. Figure 8-1: Assigning materials to a heat sink model. Air is assigned as the material to the box surrounding the heat sink, and aluminum to the heat sink itself. If a geometry consists of a heat sink in a container, Air can be assigned as the material in the container surrounding the heat sink and Aluminum as the heat sink material itself (see Figure 8-1). The Conjugate Heat Transfer interface, selected during model set-up, has a Fluid flow model, defined in the box surrounding the heat sink, and a Heat Transfer model, defined in both the aluminum heat sink and in the air box. The Heat Transfer in Solids 1 settings use the material properties associated to the Aluminum 3003-H18 materials node, and the Fluid 1 settings define the flow using the Air material properties. The other nodes under Conjugate Heat Transfer define the initial and boundary conditions. All physics interface properties automatically use the correct Materials properties when the default From material setting is used. This means that one node can be used to define the physics across several domains with different materials; COMSOL then uses the material properties from the different materials to define the physics in the domains. If material properties are missing, the Material Contents section on the MATERIAL LIBRARY AND DATABASES 277

278 Materials page displays a stop icon ( ) to warn about the missing properties and a warning icon ( ) if the property exists but its value is undefined. See Also The Material Page in the COMSOL Multiphysics User s Guide There are also some physics interface properties that by default define a material as the Domain material (that is, the materials defined on the same domains as the physics interface). For such material properties, select any other material that is present in the model, regardless of its selection. EVALUATING AND PLOTTING MATERIAL PROPERTIES You can access the material properties for evaluation and plotting like other variables in a model using the variable naming conventions and scoping mechanisms: To access a material property throughout the model (across several materials) and not just in a specific material, use the special material container root.material. For example, root.material.rho is the density as defined by the materials in each domain in the geometry. For plotting, you can type the expression material.rho to create a plot that shows the density of all materials. Note If you use a temperature-dependent material, each material contribution asks for a special model input. For example, rho(t) in a material mat1 asks for root.mat1.def.t, and you need to define this variable (T) manually if the temperature is not available as a dependent variable to make the density variable work. To access a material property from a specific material, you need to know the tags for the material and the property group. Typically, for the first material (Material 1) the tag is mat1 and most properties reside in the default Basic property group with the tag def. The variable names appear in the Variable column in the table under Output properties in the Settings window for the property group; for example, Cp for the heat capacity at constant pressure. The syntax for referencing the heat capacity at constant pressure in Material 1 is then mat1.def.cp. Some properties are anisotropic tensors, and each of the components can be accessed, such as mat1.def.k11, mat1.def.k12, and so on, for the thermal conductivity. For material properties that are functions, call these with input arguments such as 278 CHAPTER 8: MATERIALS

279 mat1.def.rho(pa,t) where pa and T are numerical values or variables representing the absolute pressure and the temperature, respectively. The functions can be plotted directly from the function nodes Settings window by first specifying suitable ranges for the input arguments. Many physics interfaces also define variables for the material properties that they use. For example, solid.rho is the density in the Solid Mechanics interface and is equal to the density in a material when it is used in the domains where the Solid Mechanics interface is active. If you define the density in the Solid Mechanics interface using another value, solid.rho represents that value and not the density of the material. If you use the density from the material everywhere in the model, solid.rho and material.rho are identical. Opening the Material Browser Note When using the Material Browser, the words window and page are interchangeable. For simplicity, the instructions refer only to the Material Browser. 1 Open or create a model file. 2 From the View menu choose Material Browser or right-click the Materials node and choose Open Material Browser. The Material Browser opens by default in the same position as the Settings window. 3 Under Material Selection, search or browse for materials. - Enter a Search term to find a specific material by name, UNS number (Material Library materials only), or DIN number (Material Library materials only). If the search is successful, a list of filtered databases containing that material displays under Material Selection. Tip To clear the search field and browse, delete the search term and click Search to reload all the databases. MATERIAL LIBRARY AND DATABASES 279

280 - Click to open each database and browse for a specific material by class (for example, in the Material Library) or physics module (for example, MEMS Materials). Important Always review the material properties to confirm they are applicable for the model. For example, Air provides temperature-dependent properties that are valid at pressures around 1 atm. 4 When the material is located, right-click to Add Material to Model. A node with the material name is added to the Model Builder and the Material page opens. Using Material Properties See Also For detailed instructions, see Adding Predefined Materials and Material Properties Reference in the COMSOL Multiphysics User s Guide. 280 CHAPTER 8: MATERIALS

281 Liquids and Gases Material Database In this section: Liquids and Gases Materials References for the Liquids and Gases Material Database Liquids and Gases Materials The Liquids and Gases materials database contains thermal and fluid dynamic properties for a set of common liquids. All properties are given as functions of temperature and at atmospheric pressure, except the density, which for gases is also a function of the local pressure. The database also contains surface and interface tensions LIQUIDS AND GASES MATERIAL DATABASE 281

282 for a selected set of liquid/gas and liquid/liquid systems. All functions are based on data collected from scientific publications. TABLE 8-1: LIQUIDS AND GASES MATERIALS GROUP Gases References 1, 2, 7, and 8 Liquids References 2, 3, 4, 5, 6, 7, 9, and 10 MATERIAL Air Nitrogen Oxygen Carbon dioxide Hydrogen Helium Steam Propane Ethanol vapor Diethyl ether vapor Freon12 vapor SiF4 Engine oil Ethanol Diethyl ether Ethylene glycol Gasoline Glycerol Heptane Mercury Toluene Transformer oil Water DEFAULT GRAPHICS WINDOW APPEARANCE SETTINGS 3D The MEMS Materials database has data for these materials and a default appearance for 3D models is applied to each material as indicated. 282 CHAPTER 8: MATERIALS

283 See Also See Working on the Material Page in the COMSOL Multiphysics User s Guide for more information about customizing the material s appearance in the Graphics window. TABLE 8-2: MATERIALS 3D MODEL DEFAULT APPEARANCE SETTINGS MATERIAL DEFAULT FAMILY DEFAULT LIGHTING MODEL CUSTOM DEFAULT SETTINGS All gases Air Simple All liquids (except Engine oil, Mercury, and Transformer oil) Water Cook-Torrance Engine oil Custom Cook-Torrance Mercury Custom Cook-Torrance Transformer oil Plastic Blinn-Phong NORMAL VECTOR NOISE SCALE NORMAL VECTOR NOISE FREQUENCY DIFFUSE AND AMBIENT COLOR OPACITY SPECULAR EXPONENT REFLECTANCE AT NORMAL INCIDENCE SURFACE ROUGHNESS References for the Liquids and Gases Material Database 1. ASHRAE Handbook of Fundamentals, American Society of Heating, Refrigerating and Air Conditioning Engineers, E. R. G. Eckert and M. Drake, Jr., Analysis of Heat and Mass Transfer, Hemisphere Publishing, H. Kashiwagi, T. Hashimoto, Y. Tanaka, H. Kubota, and T. Makita, Thermal Conductivity and Density of Toluene in the Temperature Range K at Pressures up to 250 MPa, Int. J. Thermophys., vol. 3, no. 3, pp , C. A. Nieto de Castro, S.F.Y. Li, A. Nagashima, R.D. Trengove, and W.A. Wakeham, Standard Reference Data for the Thermal Conductivity of Liquids, J. Phys. Chem. Ref. Data, vol. 15, no. 3, pp , LIQUIDS AND GASES MATERIAL DATABASE 283

284 5. B.E. Poling, J.M. Prausnitz, and J.P. O Connell, The Properties of Gases and Liquids, 5th ed., McGraw-Hill, C.F. Spencer and B.A. Adler, A Critical Review of Equations for Predicting Saturated Liquid Density, J. Chem. Eng. Data, vol. 23, no. 1, pp , N.B.Vargnaftik, Tables of Thermophysical Properties of Liquids and Gases, 2nd ed., Hemisphere Publishing, R.C.Weast (editor), CRC Handbook of Chemistry and Physics, 69th ed., CRC Press, M. Zabransky and V. Ruzicka, Jr., Heat Capacity of Liquid n-heptane Converted to the International Temperature Scale of 1990, Phys. Chem. Ref. Data, vol. 23, no. 1, pp , M. Zabransky, V. Ruzicka, Jr., and E.S. Domalski, Heat Capacity of Liquids: Critical Review and Recommended Values. Supplement I, J. Phys. Chem. Ref. Data, vol. 30, no. 5, pp , CHAPTER 8: MATERIALS

285 9 Glossary This Glossary of Terms contains application-specific terms used in the Heat Transfer Module software and documentation. For information about terms relating to finite element modeling, mathematics, geometry, and CAD, see the glossary in the COMSOL Multiphysics User s Guide. For references to more information about a term, see the Index in this or other manuals. 285

286 Glossary of Terms anisotropy The condition of exhibiting properties with different values when measured in different directions. bioheat equation An alternative form of the heat equation that incorporates the effects of blood perfusion, metabolism, and external heating. The equation describes heat transfer in tissue. blackbody A blackbody is a surface that absorbs all incoming radiation; that is, it does not reflect radiation. The blackbody also emits the maximum possible radiation. conduction Heat conduction takes place through different mechanisms in different media. Theoretically, conduction takes place through collisions of molecules in a gas, through oscillations of each molecule in a cage formed by its nearest neighbors in a fluid, and by the electrons carrying heat in metals or by molecular motion in other solids. Typical for heat conduction is that the heat flux is proportional to the temperature gradient. advection Heat advection takes place through the net displacement of a fluid, which translates the heat content in a fluid through the fluid's own velocity. convection The term convection is used for the heat dissipation from a solid surface to a fluid, where the heat transfer coefficient and the temperature difference across a fictitious film describes the flux. emissivity A dimensionless factor between 0 and 1 that specifies the ability of a surface to emit radiative energy. The value 1 corresponds to an ideal surface, which emits the maximum possible radiative energy. heat capacity See specific heat. highly conductive layer A highly conductive layer is a thin layer on a boundary. It has much higher thermal conductivity than the material in the adjacent domain. This allows for the assumption that the temperature is constant across the layer s thickness. The General Heat Transfer physics interface supports heat transfer in highly conductive layers. irradiation The total radiation that arrives at a surface. 286 CHAPTER 9: GLOSSARY

287 Navier-Stokes equations The equations for the momentum balances coupled to the equation of continuity for a Newtonian incompressible fluid are often referred to as the Navier-Stokes equations. The most general versions of Navier-Stokes equations do however describe fully compressible flows. opaque material An opaque body does not transmit any radiative heat flux, that is, the surface of an opaque body has a transmissivity equal to 0. radiation Heat transfer by radiation takes place through the transport of photons, which can be absorbed or reflected on solid surfaces. The Heat Transfer Module includes surface-to-surface radiation, which accounts for effects of shading and reflections between radiating surfaces. It also includes surface-to-ambient radiation where the ambient radiation can be fixed or given by an arbitrary function. participating media A media that can absorb, emit, and scatter thermal radiation. radiosity The total radiation that leaves a surface, that is, both the emitted and the reflected radiation. specific heat Refers to the quantity that represents the amount of heat required to change one unit of mass of a substance by one degree. It has units of energy per mass per degree. This quantity is also called specific heat or specific heat capacity. specific heat capacity See specific heat. thin conductive shell An physics interface for modeling heat transfer in a thin shell. Thin means that the shell is thin enough, or has high enough thermal conductivity, to allow for the assumption that the temperature is constant across the shell s thickness. See also highly conductive layer. transparent material A transparent body transmits radiative heat flux, that is, the surface of a transparent body has a transmissivity greater than 0. thermal conductivity The definition of thermal conductivity is given by Fourier s law, which relates the heat flux to the temperature gradient. In this equation, the thermal conductivity is the proportional constant. GLOSSARY OF TERMS 287

288 288 CHAPTER 9: GLOSSARY

289 Index 1D and 2D models out-of-plane heat transfer 64, 74, 131 3D models infinite elements and 52 thin conductive shells 124 A absorption coefficients 147 acceleration of gravity 55 advanced settings 18 AKN model 219 arterial blood temperature 116 axisymmetric geometries 75, 157, 166 azimuthal sectors 157 B bioheat (node) 116 bioheat transfer interface 114 theory 66 biological tissue 115 black walls 99, 149 blackbody radiation 142 blackbody radiation intensity, definition 158 blood, bioheat properties 115 boundary conditions bioheat interface 117 heat equation, and 41 heat transfer coefficients, and 53 heat transfer interfaces 88 radiation groups 166 single-phase flow interfaces 180 surface-to-surface radiation, theory 164 thin shells 127 boundary heat source (node) 94 boundary heat source variable 41 boundary stress (node) 198 built-in materials database 253 bulk velocity 55 buoyancy force 55 C Carreau model 238 Cartesian coordinates 49 cell Reynolds number 193 CFL number, pseudo time stepping, and 172 change effective thickness (node) 129 change thickness (node) out-of-plane heat transfer 112 thin conductive shell interface 127 characteristic length 55 coefficient of volumetric thermal expansion 55 conductive heat flux variable 37 conjugate heat transfer laminar flow interface 231 turbulent flow interfaces 233 conjugate heat transfer interface theory 244 consistent stabilization settings 19 constraint settings 19 contacting COMSOL 21 continuity on interior boundary (node) heat transfer interfaces 100 convection, natural and forced 54 convective cooling (node) 101 convective cooling theory 54 convective heat flux variable 37, 40 convective out-of-plane heat flux variable 38 crosswind diffusion, consistent stabilization method 46 curves, fan 207 D del operators 62 INDEX 289

290 density, blood 116 dimensionless distance to cell center variable 221 direct area integration, axisymmetric geometry and 157 direct area integration, radiation settings 74 Dirichlet condition 207 discrete ordinates method (DOM) 160 discretization settings 18 dispersivities, porous media 121 documentation, finding 20 domain heat source variable 40 domain material 256 E eddy viscosity 213 edge heat flux (node) 106 edge heat source (node) 129 edge surface-to-ambient radiation (node) 107 edge temperature (node) 106 edges heat flux 106 temperature 106 elastic contribution to entropy 79 ing COMSOL 21 emission, radiation and 158 equation view 18 evaluating view factors 156 exit length 194 expanding sections 18 External Radiation Source 143 F fan (inlet and outlet boundary conditions) 188 fan (node) single-phase flow interfaces 202 fan curves fan boundary condition 203 inlet boundary condition 189 theory 207 Favre average 213, 246 flow continuity (node) 201 fluid (node) 236 fluid flow selecting interfaces 226 turbulent flow theory 211 fluid properties (node) 173 Fourier s law 32 G Galerkin constraints 89 general stress (boundary stress condition) 198 geometry, working with 19 Grashof number 55 gravity 55 gray walls 98, 148 graybody radiation 142, 155 grill (inlet and outlet boundary conditions) 189 grouping boundaries 166 guidelines, solving surface-to-surface radiation problems 165 H heat continuity (node) 94 heat equation, highly conductive layers and 62 heat flux (node) 91 heat flux, theory 33 heat source (node) heat transfer in porous media 122 heat transfer interfaces 84 thin conductive shell interface 126 heat sources defining as total power 84, 94 edges, thin shells 129 highly conductive layers 105 line and point 100 point, thin shells 130 heat transfer coefficients INDEX

291 I out-of-plane heat transfer interfaces 110 theory 54 heat transfer in fluids (node) 80 extended features 120 heat transfer in participating media interface 151 theory 154 heat transfer in porous media interface 118 theory 67 heat transfer in solids (node) 77 heat transfer interfaces 70 selecting 226 theory 30, 154 hemicubes, axisymmetric geometry and 157 hemicubes, radiation settings 74 hide button 18 highly conductive layer (node) 103 highly conductive layers, defined 61 incident intensity (node) heat transfer interfaces 99, 149 inconsistent stabilization settings 19 infinite elements (node) 86 inflow heat flux (node) 92 initial values (node) heat transfer interface 88 non-isothermal flow/conjugate heat transfer interfaces 242 radiation in participating media interface 150 single-phase, laminar flow interface 176 thin conductive shell interface 127 inlet (boundary stress condition) 200 inlet (node) 186 insulation/continuity (node) 128 Interior wall (node) single-phase flow, turbulent flow interfaces 184 Internet resources 19 isotropic diffusion, inconsistent stabilization methods 48 K Kays-Crawford models 248 k-epsilon turbulence model 214 knowledge base, COMSOL 21 L laminar flow conjugate heat transfer interface 231 non-isothermal flow interface 228 laminar flow interface 170 turbulence model 171 turbulent flow k-epsilon 177 turbulent flow, low re k-epsilon 178 laminar inflow (inlet boundary condition) 190 laminar outflow (outlet boundary condition) 194 layer heat source (node) 105 leaking wall, wall boundary condition 183 Legendre coefficients 147 line heat source (node) 100 line heat source variable 41 liquids and gases materials 259 local CFL number 172, 222 low Reynolds number k-epsilon turbulence theory 219 neglect inertial term 229 M Mach number pressure work, and 242 manual scaling (node) 87 mapped infinite elements 49 Material Browser opening 257 Material Library 253 materials INDEX 291

292 databases 253 domain, default 256 liquids and gases 259 properties, evaluating and plotting 256 mean effective thermal conductivity 78 mechanisms of heat transfer 30 metabolic heat source 116 model builder settings 18 Model Library examples bioheat transfer interface 114 consistent stabilization 47 convective cooling 101 heat transfer in fluids 81 heat transfer in solids 77 heat transfer in thin shells interface 124 heat transfer with surface-to-surface radiation 74 highly conductive layers 103 laminar flow interface 171 non-isothermal flow interface 228, 231 out-of-plane convective cooling 109 radiation in particpating media 145 surface-to-ambient radiation 93 thermodynamics 78 translational motion 79 turbulent flow, k-epsilon interface 177, 233 Model Library, accessing in COMSOL 20 moving wall (wall functions), boundary condition 184 moving wall, wall boundary condition 183, 185 MPH-files 20 mutual irradiation 162 N nabla operators 62 natural and forced convection 54 Neumann condition 219 O no slip, interior wall boundary condition 185 no slip, wall boundary condition 182 no viscous stress (outlet boundary condition) 194 non-isothermal flow interface laminar flow 228 theory 244 turbulent flow 233 non-newtonian power law and Carreau model 238 normal conductive heat flux variable 39 normal convective heat flux variable 39 normal stress (boundary condition) 188 normal stress, normal flow (boundary stress condition) 199 normal total energy flux variable 40 normal translational heat flux variable 40 Nusselt number 54 opaque (node) heat transfer interfaces 80 surface-to-surface radiation interface 140 open boundary (boundary stress condition) 199 open boundary (node) heat transfer 93 single-phase flow interfaces 196 opening the Model Library 20 outflow (node) 90 outlet (boundary stress condition) 200 outlet (node) 192 out-of-plane convective cooling (node) 109 out-of-plane heat flux (node) 111 out-of-plane heat transfer change thickness 112 general theory INDEX

293 thin shells theory 131 out-of-plane inward heat flux variable 39 out-of-plane radiation (node) 110 override and contribution settings 18 P pair selection 19 pair thin thermally resistive layer (node) 95 parameters, infinite elements 50 participating media, radiative heat transfer 157 Pennes approximation 66 perfusion rate, blood 116 periodic flow condition (node) 200 periodic heat condition (node) 94 physics interface settings windows 18 plotting, material properties 256 point heat flux (node) 106 point heat source (node) heat transfer interfaces 101 thin conductive shell interface 130 point heat source variable 41 point surface-to-ambient radiation (node) 107 point temperature (node) 106 points heat flux 106 temperature 106 porous matrix (node) 119 power law, non-newtonian 238 Prandtl number 54, 248 prescribed radiosity (node) 141 pressure (outlet boundary condition) 193 pressure point constraint (node) 201 pressure work (node) heat transfer interfaces 79 non-isothermal flow/conjugate heat transfer interfaces 242 R pressure, no viscous stress (inlet and outlet boundary conditions) 187 pseudo time stepping advanced settings 172 turbulent flow theory 222 pumps, lumped curves and 207 radiation axisymmetric geometries, and 75, 157, 166 participating media 157 radiation (node) 128 radiation group (node) 142 radiation groups 166 radiation in participating media (node) heat transfer interfaces 85 radiation in participating media interface 147 radiation in participating media interface 145 theory 154 radiation, out-of-plane 110 radiative heat flux variable 40 radiative heat, theory 44 radiative out-of-plane heat flux variable 39 radiative transfer equation 158 radiosity expressions 142 radiosity method 154 raditation intensity, for blackbody 158 RANS theory, single-phase flow 212 ratio of specific heats 80 Rayleigh number 54 reradiating surface (node) 140 Reynolds number extended Kays-Crawford 249 low, turbulence theory 219 turbulent flow theory 211 INDEX 293

294 Reynolds stress tensor 213, 216 Reynolds-averaged Navier-Stokes. See RANS. Rodriguez formula 159 S scattering coefficients 147 scattering, radiation and 158 sectors, azimuthal 157 selecting conjugate heat transfer interfaces 226 heat transfer interfaces 70 non-isothermal flow interfaces 226 shell thickness 125 shells, conductive 124, 131 show button 18 single-phase flow turbulent flow theory 211 single-phase flow interface boundary conditions 180 laminar flow 170 turbulent flow low re k-e 178 sliding wall (wall functions), boundary condition 183 sliding wall, wall boundary condition 182 slip, wall boundary condition 182, 185 solving surface-to-surface radiation problems 165 source terms, bioheat 116 specific heat capacity, definition 32 specific heat, blood 116 spf.sr variable 174 stabilization settings 19 static pressure curves 189, 203 strain-rate tensors 83 streamline diffusion, consistent stabilization methods 46 surface-to-ambient radiation (node) 93 edges and points 107 surface-to-surface radiation (node) 138 surface-to-surface radiation interface 136 theory 162 swirl flow theory 216 symmetry (node) heat transfer interfaces 91 single-phase flow interfaces 195 T technical support, COMSOL 21 temperature (node) 89 tensors Reynolds stress 216 strain-rate 83 theory bioheat transfer interface 66 conjugate heat transfer interface 244 heat equation definition 31 heat transfer coefficients 54 heat transfer in participating media interface 154 heat transfer in porous media interface 67 heat transfer interfaces 30 non-isothermal flow interface 244 out-of-plane heat transfer 64 radiation in participating media interface 154 radiative heat transfer interfaces 154 surface-to-surface radiation interface 162 thin conductive shell interface 131 turbulent flow k-e interface 211 turbulent flow low re k-e interface 211 thermal conductivity components, thin shells 132 thermal conductivity, mean effective 78 thermal dispersion (node) 121 thermal expansivity 55 thermal insulation (node) INDEX

295 thin conductive layer (node) 125 thin conductive layers, definition 61 thin conductive shell interface 124 theory 131 thin thermally resistive layer (node) 96 total energy flux variable 38 total heat flux 92 total heat flux variable 36 total normal heat flux variable 39 total power 84, 94 traction boundary conditions 198 translational heat flux variable 38 translational motion (node) 78 turbulence models k-epsilon 214 single-phase flow 171 turbulent compressible flow 213 turbulent conjugate heat transfer interfaces theory 246 turbulent flow k-e interface 177, 233 theory 211 turbulent flow low re k-e interface 178, 233 theory 211 turbulent heat flux variable 37 turbulent kinetic energy theory k-epsilon model 215 RANS 214 turbulent length scale 222 turbulent non-isothermal flow interfaces theory 246 turbulent Prandtl number 248 typographical conventions 21 U unbounded domains, modeling 49 user community, COMSOL 21 W 221 for material properties 256 shear rate magnitude 174 velocity (inlet and outlet boundary conditions) 187 view factors 156 viscous force 55 viscous heating (node) heat transfer interfaces 83 non-isothermal flow/conjugate heat transfer interfaces 243 volume force (node) 175 wall (node) heat transfer interface 98, 148 non-isothermal flow/conjugate heat transfer interfaces 240 single-phase flow, laminar flow interfaces 181, 184 single-phase flow, turbulent flow interfaces 181 wall distance initialization study step 220 wall functions, turbulent flow 217 wall functions, wall boundary condition 183 wall types 148 weak constraint settings 19 web sites, COMSOL 21 V variables dimensionless distance to cell center INDEX 295

296 296 INDEX