Heat Transfer Modelling Using COMSOL [1 ed.] 9781683922872

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H eat Tr ansfer M odelling Using COMSOL

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®

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By purchasing or using this book and companion files (the “Work”), you agree that this license grants permission to use the contents contained herein, but does not give you the right of ownership to any of the textual content in the book or ownership to any of the information, files, or products contained in it. This license does not permit uploading of the Work onto the Internet or on a network (of any kind) without the written consent of the Publisher. Duplication or dissemination of any text, code, simulations, images, etc. contained herein is limited to and subject to licensing terms for the respective products, and permission must be obtained from the Publisher or the owner of the content, etc., in order to reproduce or network any portion of the textual material (in any media) that is contained in the Work. Mercury Learning and Information (“MLI” or “the Publisher”) and anyone involved in the creation, writing, or production of the companion disc, accompanying algorithms, code, or computer programs (“the software”), and any accompanying Web site or software of the Work, cannot and do not warrant the performance or results that might be obtained by using the contents of the Work. The author, developers, and the Publisher have used their best efforts to insure the accuracy and functionality of the textual material and/or programs contained in this package; we, however, make no warranty of any kind, express or implied, regarding the performance of these contents or programs. The Work is sold “as is” without warranty (except for defective materials used in manufacturing the book or due to faulty workmanship). The author, developers, and the publisher of any accompanying content, and anyone involved in the composition, production, and manufacturing of this work will not be liable for damages of any kind arising out of the use of (or the inability to use) the algorithms, source code, computer programs, or textual material contained in this publication. This includes, but is not limited to, loss of revenue or profit, or other incidental, physical, or consequential damages arising out of the use of this Work. The sole remedy in the event of a claim of any kind is expressly limited to replacement of the book and disc, and only at the discretion of the Publisher. The use of “implied warranty” and certain “exclusions” vary from state to state, and might not apply to the purchaser of this product. Companion files are also available by writing to the publisher at [email protected].

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H eat Tr ansfer M odelling Using COMSOL® Slab to Radial Fin

L. S. Mayboudi, PhD

MERCURY LEARNING AND INFORMATION Dulles, Virginia Boston, Massachusetts New Delhi

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Copyright ©2019 by Mercury Learning and Information LLC. All rights ­reserved. This publication, portions of it, or any accompanying software may not be reproduced in any way, stored in a retrieval system of any type, or transmitted by any means, media, electronic display or mechanical display, including, but not limited to, photocopy, recording, Internet postings, or scanning, without prior permission in writing from the publisher. Publisher: David Pallai Mercury Learning and Information 22841 Quicksilver Drive Dulles, VA 20166 [email protected] www.merclearning.com 1-800-758-3756 L. Mayboudi. Heat Transfer Modelling Using COMSOL®: Slab to Radial Fin. ISBN: 978-1-683921-72-1 The publisher recognizes and respects all marks used by companies, manufacturers, and developers as a means to distinguish their products. All brand names and product names mentioned in this book are trademarks or service marks of their respective companies. Any omission or misuse (of any kind) of service marks or trademarks, etc. is not an attempt to infringe on the property of others. Library of Congress Control Number: 2018943740 181920321  Printed on acid-free paper in the United States of America Our titles are available for adoption, license, or bulk purchase by institutions, corporations, etc. For additional information, please contact the Customer Service Dept. at 800-232-0223(toll free). All of our titles are available in digital format at authorcloudware.com and other digital vendors. The sole obligation of Mercury Learning and Information to the purchaser is to replace the book or disc, based on defective materials or faulty workmanship, but not based on the operation or functionality of the product.

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To the astronauts and pilots (real and virtual) who keep cool when heated; to their perseverance, patience, prudence, and proven character that endures with persistence to accomplish the purpose to an eternal perspective.

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CONTENTS Nomenclaturexi Glossary of Acronyms xvii Prefacexix Acknowledgmentsxxiii Introductionxxv Chapter 1: Heat Transfer through the Ages

1

Chapter 2: Extended Surfaces 2.1  Fins in Nature 2.2  Fins in Industry 2.3  Heat Transfer in Fins

7 7 8 9

Chapter 3: Conservation of Energy 3.1  Thermo-physical Properties 3.2  Heat Transfer 3.3  Governing Equations

13 14 16 19

Chapter 4: Finite Element Analysis 4.1  Material Properties 4.2 Geometry 4.3  Analysis Types 4.4  Boundary Conditions 4.5  Solution Control and Convergence 4.6  Mesh Size and Time Step

23 25 27 28 30 31 34

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viii  • Contents

Chapter 5: Sensitivity Analysis 5.1  Parametric Sweep 5.2  Function Sweep 5.3  Material Sweep 5.4  Auxiliary Sweep 5.5  Verification and Validation

37 38 40 40 41 42

Chapter 6: Fin Geometries

47

Chapter 7: COMSOL Multiphysics®-Extended Surfaces Thermal Case Study 7.1  Assumptions: From Slab to Radial Fins

49 51

Chapter 8: Fin with Rectangular Cross Section 8.1 One-Dimensional Analysis (1D) 8.2 Thermal Model for the 2D Slab 8.3 Thermal Model for the 3D Slab 8.4 Thermal Model for the 3D Slab with Central Cooling Channel

53 54 72 80

Chapter 9: Fin with Circular Cross Section 9.1 Fin with Circular Cross Section and a Central Cooling Channel 9.2 Fin with Circular Cross Section and Finned Central Cooling Channel

97

89

102 106

Chapter 10: Side-Rectangular Fin with Triangular Cross Section

115

Chapter 11: Side-Triangular Fin with Rectangular Cross Section

121

Chapter 12: Side-Concave Fin with Rectangular Cross Section

127

Chapter 13: Side-Convex Fin with Rectangular Cross Section

133

Chapter 14: Side-Concave-Trapezoidal Fin with Rectangular Cross Section

139

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Contents  •  ix

Chapter 15: Pin Fin with Circular Cross Section

145

Chapter 16: Radial Fin with Hyperbolic Profile

151

Chapter 17: Webbed Radial Fin with Hyperbolic Profile (Duckling* Radial Fin)

157

Chapter 18: Forced Convective Webbed Radial Fin with Hyperbolic Profile (Not the Ugly Duckling)

163

Chapter 19: Mathematical Methods to Solve Heat and Wave Problems 171 19.1  General Analytical Approaches 171 19.2 Analytical Approaches to Solve Heat Equations178 19.3  Fin Efficiency and Optimum Length 179 19.4  Hands-On Exercise 180 Chapter 20: Lean Six Sigma Implementation

181

Chapter 21: Good Practices

187

Chapter 22: Case Studies FEM Models and Applications 191 22.1 Example 1—1D Straight Line Fin with Three Boundary Conditions192 22.2 Example 2—2D Rectangular Fin with Three Boundary Conditions194 22.3 Example 3—3D Rectangular Cross Section Fin without Cooling Channel 196 22.4 Example 4—3D Rectangular Cross Section Fin with Cooling Channel 198 22.5 Examples 5, 6, and 7—3D Cylindrical Fin with Cooling Channel200 22.6 Examples 8—3D Side-Rectangular Fin 204 22.7 Examples 9—3D Side-Triangular Fin with Rectangular Cross Section 205 22.8 Examples 10—3D Side-Concave Fin with Rectangular Cross Section 206 22.9 Examples 11—3D Side-Convex Fin with Rectangular Cross Section 207

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x  • Contents

22.10 Examples 12—3D Side-Concave-Trapezoidal Fin with Rectangular Cross Section 22.11 Examples 13—3D Pin Fin with Circular Cross Section 22.12 Examples 14—3D Radial Fin with Hyperbolic Profile 22.13 Examples 15—3D Webbed Radial Fin with Hyperbolic Profile

208 209 210 211

Conclusion213 Appendix A217 Appendix B239 Index245

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NOMENCLATURE VARIABLES a fit parameter

aF mean free path of the particles (m)

b fit parameter dx finite difference along the x-coordinate (m) dy finite difference along the y-coordinate (m) dz finite difference along the z-coordinate (m) f non-homogeneity function for boundary condition

f v volume fraction (%)

f1 ( x ) area function: cross-sectional area f2 ( x ) width function

g gravitational constant (m/s2), non-homogeneity ­function for boundary condition h non-homogeneity function for boundary condition hc convective heat transfer coefficient (W/m2K), Planck constant (Js) hr radiative heat transfer coefficient (W/m2K) k thermal conductivity (W/mK), Boltzmann constant (J/K)

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xii  • Nomenclature

k x thermal conductivity along the x-coordinate (W/mK) ky thermal conductivity along the y-coordinate (W/mK) kz thermal conductivity along the z-coordinate (W/mK)

l characteristic length (m)

m mass (kg), constant (m-2) m f constant (m-2) m o constant (m-2)

n multiplier

p homogeneity response function, dependent variable, response function q homogeneity response function, dependent variable, response function

q0 initial heat generation (W/m3) qx heat flux along the x-coordinate (W/m2) qy heat flux along the y-coordinate (W/m2) qz heat flux along the z-coordinate (W/m2) q gen heat generation (W/m3) r radius (m), radius ratio

ri internal radius (m) r0 external radius (m) t time (s)

th fin height, thickness (m) u dependent variable, response function, axis along the x-coordinate ux change of dependent variable along the x-coordinate uxx derivative of change of dependent variable along the x-coordinate

uq angular change of dependent variable v laser beam scanning speed (m/s), dependent variable, response function, axis along the y-coordinate

vi dependent variable component along the y-coordinate

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Nomenclature  •  xiii

w dependent variable, response function, axis along the z-coordinate

wi dependent variable component along the z-coordinate x coordinate, axis along the slab length

xi coordinate y coordinate, axis along the slab height

yi coordinate z coordinate, axis along the slab depth

zi coordinate

A area (m2) Ab fin base area (m2)

A1 area (m2) A2 area (m2)

A fin fin surface area (m2) C heat capacity (J/kgK) Cp heat capacity at constant pressure (J/kgK) Cv heat capacity at constant volume (J/kgK)

Egen internal energy (J/kgK) Ein internal energy at input (J/kgK)

Eout internal energy at output (J/kgK) Est internal energy storage (J/kgK)

F non-homogeneity function L fin length (m)

M constant (m-3) P laser beam power (W), pressure (Pa), perimeter S shape factor

T temperature (ºC), absolute temperature of the body (K) Tamb ambient temperature (ºC, K) Tb fluid bulk temperature (ºC, K) Ti initial temperature (ºC, K)

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xiv  • Nomenclature

Tm melting temperature (ºC, K) Ts element surface temperature (ºC, K)

T¥ surroundings temperature (ºC, K) V volume (m3)

GREEK SYMBOLS a absorptivity, thermal diffusivity (m2/s) b coefficient of thermal expansion (1/K)

b t constant

d constant length (m) e emissivity h f fin efficiency

l wavelength (nm) J kinematic viscosity (m2/s) r density (kg/m3), reflectivity s Stefan Boltzmann constant (W/m2K4), standard deviation t transmissivity

Dt time step (s) Dx step size along the x-coordinate (m) Dy step size along the y-coordinate (m)

Dz step size along the z-coordinate (m)

DT temperature change (°C, K) q dimensionless fluid bulk temperature q amb dimensionless ambient temperature q b dimensionless fluid bulk temperature q i dimensionless initial temperature

q m dimensionless melting temperature q s dimensionless element surface temperature

q ¥ dimensionless surroundings temperature

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Nomenclature  •  xv

SUBSCRIPTS a absorption amb ambient

g glass transition point m melting point ¥ surroundings

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GLOSSARY OF ACRONYMS Bi

Biot Number ( hl / k )

CAD

Computer-Aided Design

CDIO

Conceive, Design, Implement, and Operate

CFD

Computational Fluid Dynamics

DOF

Degree of Freedom

DVD

Digital Video Disc

EES

Engineering Equation Solver

FDM

Finite Difference Method

FEM

Finite Element Method

FO

Fourier Number (at / l2)

EVA

Extravehicular Activity

GAMS

General Algebraic Modelling System

Gr

Grashof Number (gb DTl 3 / n 2)

HG

Heat Generation (W/m3)

ISS

International Space Station

JCG

Jacobi Conjugate Gradient

Kn

Knudsen Number (l / l)

LAB Laboratory

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xviii  •  Glossary of Acronyms

Laser Light Amplification by Stimulated Emission of Radiation LED

Light Emitting Diode

LEED Leadership in Energy and Environmental Design

LTW MUDA

Laser Transmission Welding Futility, Uselessness, and Wastefulness -- 無駄

NASA National Aeronautics and Space Administration Nu

Nusselt Number ( hl / k )

OR

Operational Research

PA Polyamide PC Polycarbonate PS Polystyrene

RAM

Random Access Memory

Re

Reynolds Number (r ul / m)

SAFER

Simplified Aid for EVA Rescue

TCS

Thermal Control System

TIMWOODS Transportation, Inventory, Material, Waiting, Over-production, Over-processing, Defects, and Skills 1D One-Dimensional 2D Two-Dimensional 3D Three-Dimensional

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PREFACE This book introduces main fin designs and investigates heat transfer regimes applicable to them. The information is provided with increasing level of complexity and in most cases, it is assumed that the earlier sections have been reviewed in advance. Having completed the main sections (Chapters 1 to 9), one may read the remaining chapters selectively. The book starts with an introduction of heat transfer along with some historical information on the concept of temperature and heat and the path along which heat transfer and thermodynamics developed (Chapters 1). It then progresses to introduce the use of extended surfaces—also referred to as fins—as an effective thermal management technique with examples in nature and industry (Chapter 2). Heat transfer modes in fins are also discussed in this section. Chapter 3 focuses on energy conservation and in particular on energy balance applicable to fins. In this section, thermo-physical properties, modes of heat transfer, and governing equations are discussed. Chapter 4 is an introduction to the finite element analysis and the steps required to set up a physical model and complete the solution: preprocessing, post-processing, and analysis. This section comprises eight topics: material properties, geometry, analysis type, boundary conditions, solution control, convergence, mesh size, and time step.

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xx  • Preface

Chapter 5 focuses on the sensitivity analyses and approaches that may be taken when using a commercial software package such as COMSOL Multiphysics®. Although this section may be treated as part of the finite element analysis, it is given extra emphasis because of its importance and so is introduced as a standalone chapter. Chapter 6 presents the fin geometries used in thermal management applications. Chapter 7 describes how COMSOL Multiphysics® in combination with the Heat Transfer module were used to set up and carry out the numerical analysis for case studies presented in this publication, including all the steps from the preprocessing to post-processing. The subsequent chapters deal with main fin designs, geometry creation or import, setting up material properties, defining physics, assigning initial and boundary conditions, setting up the solution, solving the problem, performing parametric studies, and eventually extracting data in the form of diagrams, contour plots, and analyzing the solutions. Chapter 8 focuses on the slab fin with side-rectangular profile and rectangular cross section. This problem is solved using one, two, and three-dimensional models. Three boundary conditions that form the foundation of heat transfer problems are addressed in these scenarios: 1) semi-infinite fin, 2) insulated tip, and 3) convective surfaces. A majority of real problems either use these conditions individually or in a combined form. At the end of this chapter, a fin that includes a rectangular central cooling channel is presented whose physics are treated as either a heat transfer in solid with housing a liquid coolant or conjugate heat transfer, where the solid heat transfer model interacts with the flow model (laminar or turbulent) through a multiphysics node. A comparison between the results for the three boundary condition scenarios is also provided. Chapter 9 focuses on the cylindrical slab or fin with circular cross section. This problem is solved as a three-dimensional case study, with three geometries with similar profiles: 1) without a cooling channel, 2) with a central cooling channel, and 3) with a finned central cooling channel. The physics are treated as either a solid heat transfer with embedded fluid or conjugate heat transfer, where the solid heat transfer model interacts with the flow model (laminar or

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Preface  •  xxi

turbulent) through a multiphysics node. The material switch feature is employed to investigate the effect of different liquid coolants on the heat transfer process. The following chapters represent fins with different geometries such as a side-rectangular fin with triangular cross section (Chapter 10), side-triangular fin with rectangular cross section (Chapter 11), side-concave fin with rectangular cross section (Chapter 12), sideconvex fin with rectangular cross section (Chapter13), side-concavetrapezoidal fin with rectangular cross section (Chapter 14), pin fin with circular cross section (Chapter 15), radial fin with hyperbolic profile (Chapter 16), and webbed radial fin with hyperbolic profile— the author calls it a “Duckling” Radial Fin (Chapter 17). Chapter 18 presents the webbed radial fin earlier attempted in Chapter 17 interacting in a conjugate heat transfer model with its surroundings fluid. The author calls it, “Not the Ugly Duckling,” for it is the ultimate progression of a simple one-dimensional fin represented by a straight line, advancing to a two-dimensional rectangular fin, and then developing into a three-dimensional cuboid fin, and eventually, progressing through more complex designs, it becomes the most complex fin structure investigated in this work. Chapter 19 presents the mathematical methods that are used for numerical analysis in the form of second-order wave or first-order heat equations. The following methods are introduced to analytically solve these equations: 1) separation of variables, 2) variation of parameters, 3) Duhamel’s theorem, 4) complex combinations, 5) superposition, 6) Laplace transform, 7) integral method, and 8) perturbation method. A subsection is dedicated to the analytical approaches used for heat equations with a definition for fin’s efficiency and optimization, an approach adopted from the field of operational research. Chapter 20 provides an overview of Lean Six Sigma and its implementation when making engineering design decisions. Chapter 21 shares some of the personal experiences when attempting numerical analysis in general and heat transfer specifically, perhaps not commonly discussed.

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xxii  • Preface

Chapter 22 is dedicated to the case studies used in this book, whose electronic files as well as the applications, are provided. The purpose is to select the case study, review the file and related application, and attempt the problem by implementing various input data as the boundary or initial conditions or introduce new ones as desired. The Conclusion emphasizes the importance of the responsible use of energy and how this may be achieved by implementing effective thermal management methods such as employing extended surfaces. The Appendix A lists the analytical solutions for the numerical problems presented in the case studies. The author encourages the reader to attempt the mathematical models by means of the tools provided in Chapter 19 and obtain the approximate or exact analytical solutions provided for each design. The Appendix B is a list of references used in this publication. There are a number of references that are not directly used, but have been provided as a source for the readers to enrich their experience in the field of heat transfer.

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ACKNOWLEDGMENTS The world of scientific publication entails challenges but also rewards, especially at the time of completion. This book, from conception to publication, underwent a long journey, which has been intertwined with a personal one. Although I recall the individuals and instances that have helped me accomplish this work, from the janitor of my schools to the professors, the Publisher, and family, I cannot fully express the depth of my appreciation toward these individuals. So below is what I feel when I think about all of you: There are some who raise you up to be a better person you have aspired to be… They are some who put fire in your heart to create and to succeed… There are some who bring love of life back to the tired and hopeless… There are some who make positive impacts in your life… There are some who nurture and give wings to fly to deep blue yonder and to continue flying… There are some who walk the earth and spread love and respect… There are some who keep giving with no expectation in return or anybody knowing they ever existed… Either it is about you or not, it is in your head or you have been exposed to its merciless shock waves, it is about something close to your heart, your experiences and that makes it personal…

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INTRODUCTION Heat transfer is a complex engineering field with numerous ­references and textbooks that are a click of a mouse away. This book takes an avant-garde approach to the heat transfer field by incorporating science, engineering, gastronomy, literature, art, culture, and fashion. The purpose is to make a connection between heat transfer and everyday activities in order to make this complex scientific subject more understandable. Heat transfer applies to a variety of fields—literally from the Earth’s core to the exosphere and beyond. Understanding, monitoring, and optimizing it therefore are of great importance, knowing that the energy for running any business is directly related to the bottom line. Conservation of energy states that energy in the ­universe remains the same but it transforms from one form to another; for instance, electrical energy transforms to mechanical energy. This process is not 100 percent efficient, meaning that in this scenario not all the electrical energy is transformed to mechanical energy. The wasted energy is associated with the process efficiency. Keeping in mind that majority of the industrial processes involve change from one form of energy to another, the importance of designing high-efficiency processes cannot be overemphasized. Heat is the by-product of the majority of these processes. Most of the time this heat is wasted; occasionally, this heat is recycled through processes such as cogeneration and trigeneration. In the former case, in applications such as combined steam or gas power plants, the byproduct of the generated work is heat that is discharged to water or

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xxvi  • Introduction

space. In trigeneration, the generated heat may be used for both heating and cooling, namely, in an absorption refrigerator. Because the total energy is constant, generating unused heat as a by-product equals a waste of resources. Think of converting the lost energy in a ­combustion process to dollar values, assuming that clean energy sources such as hydrogen are not the source for power generation; other common sources of fuel such as Avgas or Mogas are the crude oil by-products that are connected with environmental, economical, and geopolitical implications. Change of internal temperature does not necessarily constitute the value of a system but for any system the determining factor is what temperature-dependent characteristics are of interest and contribute to the energy transformed or lost during an energy conversion process. Let us look at some examples. An aircraft needs fuel to taxi, take off, cruise, and land. Looking at aircraft performance charts, you will see that the rate of energy consumption per distance required for take-off is higher than that of the cruise. In fact, the fuel-to-air ratio needed to make the combustion possible greatly depends on the aircraft performance, in addition to the air density. The engine cowl and particularly the engine are warm to the touch after landing. This heat affects the performance of engine components if it does not receive the due attention. The most common problem will be excessive oil temperature and pressure so that the oil cannot properly circulate and cool the engine, which can lead to engine damage. Aircraft engine fire is another scenario. This could occur because of multiple reasons such as wires that short-circuited or connections that frayed. You know that an irregular electric current might cause overheating and ultimately lead to engine fire. You then realize that designing a mechanism in order to detect such current irregularity early enough is vital. Another overheating reason could be that the installation or the connection is not made properly, which might cause ohmic heating, producing a similar result. If any of these scenarios occur when you are in the air, you may need to declare a “mayday”, but even then, you must thermally manage the engine, so you introduce more air into the engine in order to cool it down as much as possible and land in a safe place. Perhaps you would want to stay away from the areas that are a fire hazard or conducive to fire due to surface friction after a potential semi-soft landing.

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Introduction  •  xxvii

When you exercise, you burn calories; you know it because your cardio activities increase, you feel the blood flow under your skin, and your temperature increases. Receiving a soggy lukewarm pizza on Super Bowl night is another reminder that heat transfer from the pizza box is to be managed in an effective way. There are a number of similar inventive methods that target pizza boxes such as building in either ventilating radial holes or slits into the top of the pizza box to let the steaming pizza breathe. The purpose of this work is to explore the possibilities when designing an extended surface, while doing some math in order to compare the efficiency and effectiveness of different designs, examine the complexities of each design, nourish the imagination, experiment with the curves and lengths of the geometry, and examine the added value in the form of improved heat dissipation while also reducing the manufacturing costs. It should be used as a guide for further studies and to provoke thoughts about how to improve designs at the next iteration for the product platform.

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CHAPTER

1

HEAT TRANSFER THROUGH THE AGES Heat transfer, thermodynamics, and fluid mechanics, commonly known as thermo-fluid sciences, share an interesting historical background. Heat transfer involves change of internal or external ­temperatures, depending on the closed or open system, over time or space. You may expect a steady-state temperature in which temperature remains constant but heat continues flowing; for example, when a phase change occurs. The development must have started when the need for measuring temperature was recognized and attempted to be addressed. The idea of heat dates back to prehistoric era, mainly associated with fire as part of the simplistic ­categorization of elements into four—air, fire, water, and Earth. It has been ­referenced in a Greek text called the Kore Kosmou, attributed to Hermes Trismegistus. First efforts in this field are ascribed to Heron of Alexandria from the Roman province of Egypt (circa 10 AD— circa 70 AD), who noticed air contraction and expansion within a tube partially filled with air whose end was submerged inside a tub of water. In 1593, Galileo Galilei, an Italian polymath, employed this concept and invented the thermoscope, which worked based on changes in sensible heat. The air within the tube with a balloonshaped air-filled region was heated, and due to the changes in the temperature, the level of water changed as well, causing water to experience perpetual motion.

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2  •  Heat Transfer Modelling Using COMSOL Slab to Radial Fin

One may argue that Galileo set the stage for atomistic conception of heat transfer. Temperature as a term was still unknown during this time. In 1624 Leurechon, a French mathematician, first used the word “thermometer” for a similar device that was in fact a barometer. Galileo’s student Joseph Solomon Delmedigo, an Italian physician, mathematician, and music theorist, has been reported as the originator of the sealed liquid-in-glass thermometer. However, in 1654 Ferdinando II de’ Medici, an Italian musician, produced the latter instrument, which was independent of air pressure and only relied on expansion of the liquid. There have been number of thermometers experimented with using different liquids and tubes. The first standardized one that was proposed based on melting and boiling points of water was in 1694 by Carlo Renaldini, a French philosopher and experimenter. The one proposed in 1642 by Sir Isaac Newton, a British natural philosopher, was based on the scale of 12 degrees between the ice melting point and body temperature. Only in 1714, Daniel Gabriel Fahrenheit, a Dutch-German-Polish scientist, introduced the concept of the modern thermometer with the accurate scale that he developed based on mercury’s high coefficient of thermal expansion. Anders Celsius, a Swedish scientist, in 1742 proposed a 100-degree scale between the freezing point and boiling point of water. In 1848, William Thomson, a Scottish mathematician and engineer, later known as Lord Kelvin, introduced the idea of absolute zero, that was the negative reciprocal of expansion coefficient of gas (0.00366) at water freezing point per degree Celsius. One Kelvin is also defined as the fraction (1/273.16) of the triple point of water (0.01°C) which bears Celsius scale. Thermometer response time was an important factor, especially in medical applications. It is reported that early devices would have required twenty minutes to identify the correct temperature, which was reduced to five minutes in 1866 by Sir Thomas Clifford Allbutt, a British physician. Dr. Francesco Pompei, an American innovator and entrepreneur, proposed the precise two-second-temporal artery thermometer in 1999. Francis Bacon, a British scientist, philosopher, and statesman, distinguished between heat and temperature in 1620[4]; however, Joseph Black, a Scottish physician and chemist, quantitatively distinguished between the two. He applied the conservation of mass—earlier

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Heat Transfer through the Ages  •  3

suggested by Newton and adopted by Antoine-Laurent Lavoisier, a French chemist, for all chemical reactions—and d ­ eveloped calorimetry and latent heat. Specific heat that a­ ccompanies melting or vaporization was introduced by Joseph Black (1763), who also articulated the theory of latent heat. Black’s work was later p ­ ublished by [5] John Robinson, a Scottish scientist, in 1803 . Theory of heat was introduced in 1798 by Sir Benjamin Thompson, an American-born British physicist and innovator, which was expanded upon by Sadi Carnot, a French physicist, in 1824, who proposed equivalency of heat and mechanical work[6,7,8]. Thompson made no attempt to further quantify the heat generated or to measure the mechanical equivalent of heat; however, Carnot measured the equivalent work done by a gas expanded under isothermal conditions. William Thompson, a Scottish mathematician, in 1851 re-emphasized this equivalency, that heat is not a substance and presented Dynamical Theory of Heat that developed into the science of thermodynamics, which is concerned with heat generation due to position and motion. Terms such as specific heat, calorimetry, and combustion, developed in the nineteenth century, were rooted in this theory. Joseph Fourier, a French mathematician and physician, worked on the analytical theory of heat. His work in 1807, which introduced Fourier series, transcendental functions, Fourier integral, and operator calculus methods, was eventually published in 1872. His masterpiece on Analytical Theory of Heat was published in 1822 and translated in 1878. His derivation of transient heat transfer equation took similar macroscopic approach as that of Euler’s derivation of continuity of mass and motion for a frictionless fluid in 1761. Leonhard Euler, a Swiss mathematician, physicist, astronomer logician, and engineer, is well known for his work in fluid dynamics, mechanics, and music theory in addition to his modern mathematical terminology and notation. He concluded that bodies do not possess the same capability to contain heat, receive, or transmit it across their surfaces or through the interior of their masses. Pierre-Simon, marquis de Laplace, a French scholar, made an effort to present a mathematical model for heat diffusion for the experiments he conducted in 1804. Jean-Baptiste Biot, a French physicist, astronomer,

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and mathematician, employed Newton’s Law of Cooling in 1804 and introduced a distinction between the conduction of heat within the body and into the surrounding atmosphere. Fourier setup his convective boundary condition employing Biot number. Fourier’s law of heat conduction is similar to that of Newton’s law of viscosity, with temperature difference substituted for velocity difference. Ohm’s Law, presented by Georg Simon Ohm, a German physicist and mathematician, and Fick’s law, presented by Adolf Fick, a German physician and physiologist, later used the Fourier’s law of heat conduction for their analogy. Fourier also studied the radiation heat transfer theory later on, as well as dimensional analysis pertinent in parametric studies. His energy equation in convective heat transfer differs from that of the Navier-Stokes since the latter addresses the viscous flow. Thermal management, either in the form of heating or c­ ooling, dates back centuries ago. In fact, industrial revolution is not ­responsible for invention of heating, ventilation, and cooling in buildings, nor for invention of refrigerators and freezers. Persian (the residents of the old Iran) engineers used Yakhchal, meaning an “ice container” in Farsi, as early as 400 BCE, to store during the summer months the ice created in winter. Iranian aqueduct, also known as Qanat, was responsible for transfer of water to Yakhchal’s tall conical-shaped structure, where it was cooled to the freezing point by its surroundings. To make the heat transfer more efficient, the flow was directed through the northern wall to use its shadow and keep the water cool and an additional eastern-western wall was made to protect the wall from Sun’s radiation. The same word is still used for refrigerator and freezer in modern-day Iran. Abanbar, a Persian cistern, is another thermal engineering example where a water reservoir with an insulating structure built below ground level was able to manage water temperature and the ventilating effect due to installation of windcatchers while avoiding creation of mold and mildew due to stagnation. Windcatchers were the traditional ventilating system old Iran used, which is still in use; it can be unidirectional, bidirectional, and multi-directional. The structure is even resistant to earthquakes, which are common in this area. If you wonder what material was used to make such insulating

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Heat Transfer through the Ages  •  5

structure, think of baking a low-fat sand cubed cake with two-meter sides: egg white, sand, clay, lime, goat hair, and ash—a recipe rooted in vernacular architecture. One wonders if Marco Polo truly appreciated this innovation as he followed the silk road, taking refuge in caravanserais at the end of each leg of his journey.

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CHAPTER

2

EXTENDED SURFACES Extended surfaces, also known as fins, are one of the primary thermal management measures. Fins have been well integrated into everyday engineering problems to the point that their first use has not been given well-deserved credit.

2.1  FINS IN NATURE Looking at nature, one can recognize the use of “fins” in f­oliage (e.g., conifers), reptiles (e.g., crocodiles and lizards), birds (e.g., ­altricial birds and chickens), marine mammals (e.g., whales), and other types of mammals (e.g., deer and rabbits). In these cases, metabolic heat is transferred to the appendage by the blood flow. The appendage then transfers this heat to the environment mainly by natural convection and to some extent radiation. Human hair and hair in animal fur are extended surfaces in which thermal management from the skin of the living creature is achieved. Could you possibly consider a snake as a giant moving fin? That may be the case; due to its large body surface, it loses a lot of heat and that encourages the male snake to pretend to be female so that as a “she-male” it takes advantage of the body warmth of competing males at the time of mating, not to mention saving the concerned female snakes from aggressive males. In the world of dogs, the heat is transferred through their tongue, an extended surface, to

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the environment by evaporation of their saliva so that they keep cool. So now you would be able to identify the heat transfer modes when your old panting bulldog, Robert, extends his tongue fully to lick the honey off the vertical edge of your neighbor’s fence. A simple way to experience different modes of heat transfer for yourself is to take a shower and then cover your body except for your hands with a towel and let the rest get exposed to the environment. You will feel the evaporation of the water and as the result its cooling effect on your bare hands, while the rest of the body covered with towel remains warm. Keeping towel for an extended period of time, you will sense your body temperature rising, which is due to the heat that your body is generating (i.e., sensible heat). Your body is an endothermic bio-mechanism, generating heat on its own. Your bare hands on the other hand get colder. You can easily test it by bringing the area close to your covered skin and feeling the sensation of warmth on your bare skin. Soaking your feet in the ocean while looking at a crab generates the same effect, resulting in cold feet. Your crab friend, on the other hand, is an ectotherm, also known as a cold-blooded animal, meaning their bodies generate very little or no heat, making them interested in basking in the warmth of their environment to regulate their metabolic functions. Let us assume you are making a sand castle and as part of the creation process you extend your fingers inside the sand. The sand that has been heated by the Sun’s radiation transfers the heat immediately to your fingertips by conduction. Your palm will feel the warmth with a considerable delay. Would you say your arms, fingers, and legs function as extended surfaces?

2.2  FINS IN INDUSTRY Let us return now from our expedition into nature and enter the world of industry. Extended surfaces have been used for ­thermal m ­ anagement in applications such as light emitting diode (LED) lamps, soldering, and microprocessors. For industrial applications, fins may be designed in a variety of shapes and therefore may be easily tailored to the application. They can geometrically differ in cross sections and side profiles. Rectangular, circular, triangular,

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Extended Surfaces  •  9

tapered, concave, convex, radial, and pin fin shapes may be used. Fins may be an integrated part of the design or assembled to the main structure by methods such as welding, clipping, wiring, and using standoffs and pushpins. The latter design, where fins are assembled into the main structure, addresses thermal management in heated areas after manufacturing has been completed, namely as a retrofit solution. An example of integrated structure is oil-cooling system on a Lycoming O-320-E2D engine such as that of a Cessna 172 ­aircraft. Another example is use of extended surfaces in heritage homes on the floor, walls, and ceilings; examples include radiators. There are also cases in which fins are combined with other thermal management solutions such as using extended surfaces on heat pipes in thermal energy storage units or space applications or fan-cooled fins attached to personal computer processors. Additional measures such as thermal tape or thermal grease can be used to improve thermal contact between the extended surface and the rest of the structure. Since the purpose of employing a fin is to manage thermal energy to meet the design specifications, its performance is evaluated based on the amount of energy that it dissipates. This is known as a fin’s thermal efficiency. Although increasing the fin’s surface area; for example, by enlarging the cross section or extending the length will improve the heat transfer to the surroundings, the heat transfer plateaus at some point, or the improvement does not justify the added manufacturing cost or the extra space requirement. Optimization of the fin is therefore a compromise between the heat transferred and the size of the fin, which is a function of its cross-sectional area and length. This is an important factor, especially in limited spaces and where arrays of fins are to be employed.

2.3  HEAT TRANSFER IN FINS Let us revisit the example of cooking on a skillet and identify the process: ●●

The decision as to what to cook is made,

●●

The skillet is selected,

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●●

The skillet is put on top of the gas stove,

●●

The gas is turned on,

●●

●● ●●

●●

The fire comes in direct contact with the bottom surface of the skillet, The skillet heats up, The content inside the skillet (e.g., oil, water, food ingredients including spices) heat up, Chemical reactions between the ingredients happen at the right temperature and after appropriate time,

●●

Dinner is ready,

●●

Lamb bourguignon is served.

The stages indicated above are thermally and chemically induced and temperature is playing a major role in all of them. Let us ignore the chemical reaction of mixing ingredients for now and focus on the thermal aspects of the problem. The main source of heat is the burning gas. In an endothermic reaction, heat is absorbed and the surrounding area is cooled as the result. In an exothermic reaction, heat is released and causes the temperature of the surrounding area to increase. One way to gain a better understanding of an endothermic versus an exothermic reaction is to consider the example of cooking pierogis or pasta. Water is heated. The formation of bubbles suggests that boiling process has started. Pasta is then submerged inside the boiling water. The bubbles disappear, meaning that pasta has absorbed some heat upon coming in contact with water. The same is valid for stirring sugar in your morning cappuccino. Initially, the drink is hot; then the temperature reduces as soon as you add the sugar and stir it; especially, if you were to use a metal spoon, which is a good heat conductor. An example of an exothermic reaction is the food that is heated inside a special self-heating packaging hikers use, or the care bag you pack in rover for the next Mars expedition. The package consists of multiple layers; the food or drink can be located anywhere inside the interior or exterior of the package. The heat is generated after the membrane between the heating substance (e.g., calcium oxide) and water is removed and the two are mixed.

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Extended Surfaces  •  11

If you were to isolate the skillet handle from the rest and include the contact with the skin, the following terms may be identified: ●●

The energy conducted from the skillet body to the handle,

●●

The energy conducted from the handle to the skin,

●●

●●

The energy accumulated inside the handle as a function of time, The energy that is convected to the surrounding area.

Note that there is no energy generated inside the skillet handle in this case.

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CHAPTER

3

CONSERVATION OF ENERGY In order to apply the conservation of energy, you need to b ­ alance the energy for a small and identifiable portion of the material that could exist in the form of a continuum. This continuum can adopt any shape. Energy can enter and exit the box; however, the boundaries remain constant and so it forms a closed system. The box is identified by its size, mass, and thermo-physical properties. Thermo-physical properties may be either temperature-dependent or constant. They can also vary in different directions along the length of the continuum or remain the same throughout the box. The variation of spatial distribution of molecules within the continuum is associated with homogeneity of the molecules inside the box. Assuming that the box is a small representative of the rest of the environment, the homogeneity concept may be expanded to the rest. Energy is defined in different forms inside this environment. It is either in the form of heat conduction entering the box, heat generation inside the box, changes of internal energy, or heat conduction leaving the box. The heat leaving the box by conduction is the same as the heat entering the box by conduction plus the spatial variations due to the length of the travel, and is time-independent. The internal energy on the other hand is time-dependent; it focuses on the variation of internal energy, which is presented in the form of thermal capacity of the box.

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14  •  Heat Transfer Modelling Using COMSOL Slab to Radial Fin

Conservation of energy stipulates that the total energy inputted to the system is the same as that of the total outputted from the system. Figure 1 shows the schematic of general form of energy balance for a continuum—equation (1).

FIGURE 1:  Energy balance diagram.



E in + E gen = E out + E st 

(1)

3.1  THERMO-PHYSICAL PROPERTIES Temperature is an inherent property of a molecule and is the driving force for heat transfer. Heat is transferred from the entity with higher temperature to the one with the lower one. Heat transfer continues until the two entities reach an equilibrium ­ state, meaning their temperature equalizes. Molecules consist of atoms that are harmonic oscillators vibrating in their equilibrium ­positions as dependent, semi-dependent, or independent entities. The dependent entities are easily excited by the energy of the surrounding neighbors. Nevertheless, statistical thermodynamics assumptions rely heavily on the semi-independence or independence of the particles. Mode of heat transfer depends on the media in which energy is transmitted. On the microscopic level, matter is made of molecules and atoms. In normal states, the molecules carry on their usual task of defining the state of material: gas, liquid, or solid. The main difference between the gas and solid states is the proximity of the molecules. Mean free path is the factor that defines what state the material is in. Knudsen number (Kn), defined as the ratio of the mean free path to the characteristic length, a dimension that defines the scale of a physical system, describes the level of freedom enjoyed by the molecules. This freedom is more available for gaseous or low viscosity molecules; therefore, it can be associated

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Conservation of Energy   •  15

with gaseous and liquid state of the material. Knudsen number smaller than one identifies if there is a continuum flow, values equal to one are associated with slip flow, and values greater than one define free molecular flow. Assuming a continuum flow passes over a wall, the magnitude of the flow velocity at the vicinity of the wall will be zero, while for a slip flow this value is not zero since the flow slides over the wall. The parabolic velocity profile associated with the flow passing over the wall changes to a linear profile in a free molecular flow, meaning that the flow, starting at a non-zero velocity magnitude at the wall, changes linearly with increasing distance from the wall. The heat transfer is achieved primarily by the mechanisms of conduction and radiation. For conduction to happen, either in its pure or subsidiary forms (such as convection), molecules need to be present. While in the radiation form, electromagnetic waves are the energy transmitting agents and no intervening molecules are needed. It is because of this mode of energy transfer that Soyuz (Сою́з) spacecraft is covered by a thermal shield to remain cool when exposed to the Sun. When fluid of any kind (liquid or gas) comes in contact with a solid surface, the heat transfer mechanism takes a hybrid approach. It combines the conduction mode in solid and liquid. Additionally, the momentum of the bulk of the fluid contributes to transferring some of the heat. You wake up in the morning and decide to make tomato and cheese omelette mixed with the last night leftover salad. You put the copper skillet on fire, add the extra virgin olive oil and get distracted by the squirrel running in your backyard, digging and nibbling your red tulip bulbs. You touch the hot skillet and you feel a burning sensation. The hurt you feel from the juicy blister that is quickly forming on your skin and what to do about it; your ego is bruised in the next aftershock. On the macroscopic level, agglomeration of molecules forms a material that is identified by the three main properties, known as mass [kg], length [m], and time [s]. All other material properties are derived from these main properties. Some properties are related to physical nature of the material and therefore are called physical properties; such as density, a term that represents the ratio of weight to the volume occupied by the material.

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Energy is what excites—for example, in a diatomic molecule that has six degrees of freedom: translation, vibration, and rotation. Temperature is the by-product of such excitement and is the constituent part of thermal properties: Heat capacity is the amount of energy that is needed to increase the temperature of molecule by one degree centigrade [J/kg°C]. Thermal conductivity describes how fast the energy can travel through the unit length of material and causes a temperature rise of one degree centigrade [W/m°C]. The main thermo-physical properties described above-namely density, thermal conductivity, and heat capacity-form the foundation of all heat transfer problems. Any type of conservation either in the form of mass, energy, or momentum employs one or more of these properties. Temperature is the driving force in many heat transfer problems either directly or indirectly by converting other types of energy to heat.

3.2  HEAT TRANSFER Let us review the example discussed earlier, the case of burning finger in contact with the handle of a hot skillet. If you consider amount of energy generated per hour by standard unsealed burner that is 9,000 Btu of heat per hr (2638 W), it will not take long for the skillet handle to reach 70°C. If you then accidentally touch the handle, you will feel the agonizing burning sensation within a second. There are physical measures; however, that could influence that painful contact: factors such as gas top structure, surface area, temperature of the fire, structure and thermo-physical properties of the skillet, environmental factors such as temperature and atmospheric pressure, and external factors such as the cooling mechanism. To have a better idea, you may be interested in comparing the rate of heat transfer from the objects (i.e., skillet to skin) as well as capacity in which the heat can be stored within the objects. This will introduce a new concept known as diffusion of heat. In other words,

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Conservation of Energy   •  17

diffusion of heat is an indication of the ratio of the heat conducted to the capacity of heat absorbed. Employing the new concept, you learn that copper heat of diffusion is 802 times that of skin. The main question is how to manage the heat and the damage it can cause when such contact happens. There are number of ways to achieve this; for example, the surface area of the skillet can be increased and that simply means that due to the larger mass of the skillet body, more time is needed for the heat to transfer from the center of the skillet directly exposed to the fire to the handle. Another method would be to construct the skillet of a material that is less conducive to transferring heat such as aluminum or stainless steel (e.g., thermal conductivity of stainless steel is 24 times less than that of copper). The heat capacity of material, meaning how much heat can be stored is an important aspect in cases where time is an important factor. Employing diffusion of heat, that shows the power of the heat conductivity to the heat capacity, is a more representative indication in this case (e.g., stainless steel has a diffusivity that is 34 times smaller than that of copper). Assuming any of these changes have been made, one may still compromise on the efficiency of the cooking process and heat interaction with food ingredients. This is where the purpose of thermal management becomes clear. Thermal management is the science that takes into account all the factors that are either influenced by or influence the thermal performance of a system— anything at any stage from the design conception to implementation, and execution. The purpose is to design a product or system that is efficient (to function and to operate) and is ergonomically and aesthetically pleasing. Thermal management is not always about dissipating heat but also ensuring certain materials operate at their optimum temperature. An example is ferrite in a linear induction motor, where magnetism of the material increases at higher temperatures; in other words, its coercivity, which is its resistance to demagnetization, improves. Another example is to use coloredpigmented plastics in ­ exterior surfaces of vehicles to emit the radiated heat back to the environment and help with cooler interior surfaces. One cannot ignore the aesthetic effect of the product when surrounded by many examples such as smart tablets.

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As for the skillet example, nowadays modern cookware uses multiple layers of materials with different conductivity and diffusivity to improve the efficiency of cooking and also the interaction of heat and cooking ingredients. The layers may vary from three to five, incorporating the three aforementioned materials. For example, a five-ply cookware may consist of internal and external layers of stainless steel, as well as internal aluminum layers with a copper central core. Contrary to what the know-it-all at a breakfast buffet may say, that you need to have a non-stick pan to make a perfect omelette, all you need to do is to ensure you expose the ingredients to the right amount of heat for the right amount of time and watch them dance together, diffusing the tastes in this thermally provoked chemical reaction. Imagine the case where the skillet did not have a handle, and you were to hold it by the external surface without using a glove. That would cause even more injury to your hand, since the contact surface would have increased considerably and so did the temperature of the skillet. Additionally, since the skillet in this case has more energy per unit mass compared to the case where it has handles, more energy in the form of heat is distributed per unit mass of the material, causing the temperature of the skillet to rise even further. Due to the actual physical contact with the molecules and their proximity, heat is transferred by the mode of conduction to your skin. In the case of the environment where the widelyspread molecules are adjacent to the tightly-spread ones (i.e., the solid in contact with fluid), convection heat transfer modes are dominant. “Fluid” here means both liquids and gases; however, assume for now the skillet is being used on Earth (versus Mars; for example, where ambient conditions are not known), submerged in air versus water (when soaked to be washed). Convection is the combination of conduction and advection, where the solid as well as bulk liquid characteristics dominate. During convection mode the molecules are not in contact, so they work in a stochastic tag team, transferring the bundles of energy from one to the other. This random interaction does not encourage an efficient communication between the members, and also does not help the process efficiency.

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Conservation of Energy   •  19

Depending on the suppleness of the molecules expressed by convective heat transfer coefficient, the heat transfer can be expedited or impeded. There are ways to improve this agility. One way is to use external sources such as fans. Using fans encourages the fluid molecules in the vicinity of the solid ones to move faster, to run versus walk to dissipate heat from the solid surfaces. Nevertheless, there are limited options in selecting the fluids in which the solid is submerged, given that you are Earth-bound. Use of extended surfaces, also known as fins, is the next best option where the interaction is increased by increasing the surface area. In this case the solid molecules will have higher chance of meeting with their fluid counterparts and therefore will be capable of transferring energy bundles more efficiently. By introducing fins, you are extending the surface in contact with the fluid so that it may alleviate the undesirable effects due to the high temperature. Had the skillet not had a handle, the burning sensation would have been stronger, which is the direct result of high temperature due to decreased rate of thermal dissipation. In this example, the skillet handle functioned as an extended surface. Had the skillet handle had extra surfaces attached to it, the heat would have been transferred to the surroundings more effectively and you would not have been burned as much.

3.3  GOVERNING EQUATIONS Experimental correlations have been the basis of many thermofluids formulae. In this approach, tests are carried out to investigate the influence of change of a single parameter or number of them on a control volume or system. The parameters can either be thermophysical properties of the material such as heat capacity and thermal conductivity or temperature induced ones such as stress, creep and oxidation lives, magnetic fields, and phase change, to name a few. In a complex system such as a heat exchanger, water temperature, pressure, and velocity are the determining factors for heat transfer mechanisms, and its efficiency as well as flow regimes.

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Experiments are to be conducted in order for the mathematical correlations representing the physical phenomena to be derived. Conducting experiments imposes challenges and involves distinct stages: 1) conception of ideas, 2) setting objectives, 3) identifying design approaches, 4) establishing pros and cons of each technique, 5) selecting the approach that is best suited to the objective of the tests given the resources, 6) carrying on with the design of experiments, 7) conducting the tests, 8) collecting relevant and critical data, 9) analyzing the data objectively, 10) ­reporting analysis results, 11) presenting analysis results, 12) publishing analyses to share with peers, 13) presenting a technical version of research findings, and 14) presenting a mass-consumable version of research findings. When analyzing data, the relationship between the parameters is identified to the best knowledge of the experimenter. Prejudices play a major role in this stage, as they can haze the better judgement of the person by questioning just about everything; for example, what is the nature of that unknown and unwelcome shocking electric discharge. The fundamental theories, which were held true and honoured for years and were the source of peace of mind, vanish in a blink of an eye and new ones are formed only to be questioned again in a period not so long from the last one that posed the question. Examples include the discovery of element Polonium by Marie (and Pierre) Curie, and Stephen Hawking’s ­theory of a black hole’s emission of radiation; they either challenged previous findings or set the platform for future challenges. Theoretical relations derived from experiments show an approximate relationship between the two or more parameters; for example, they identify that these parameters are directly or inversely related. The correlating factor which enforces the exactness of the theoretical relationship can be material-dependent, such as conductivity in Fourier’s law, or not material-dependent, such as the convective heat transfer coefficient in Newton’s law of cooling. The correlated value obtained from the former case defines a thermo-physical property of the material while the experimental setup or processes influence that of the latter. An example is convective heat transfer coefficient employed in Newton’s law of cooling. In some cases, the correlation value is a constant and bears its status in general form. An example is Stephen-Boltzmann’s constant that correlates the radiated electromagnetic energy to object’s temperature in Stephen-Boltzmann’s law. There are cases in which no exactness can be achieved by fitting an

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Conservation of Energy   •  21

experimental relationship into a theory; this is the definition for an empirical relationship. Examples include release of magnetic energy during a solar flare, heat transfer in external flows, and shear stress in non-Newtonian fluids. Figure 2 shows the general form of conservation of energy, including all modes of heat transfer. The radiation and convection terms shown are applied in the form of boundary condition. Like any other physics phenomenon, such as solid mechanics, where balance of forces is evaluated along similar axes, balance of energy is approached the same way. In other words, to apply the energy conservation law to the box above, the relationships along the x, y, and z directions are to be assessed separately—equations (2) to (8). A new matrix can be created that is a linear combination of the conservation in three dimensions in addition to the time component. T, q, k, x, y, z, and t are temperature, heat flux, thermal conductivity, coordinates in the x, y, and z directions, and time, respectively. The heat flux presented in the forms of equations (2) to (7) is proportional to the temperature gradient, where the conductivity (k) is the proportionality constant. Equation (8) shows the rate of energy as a function of change of internal energy over time and energy generated inside the material due to any heat source or sink.

FIGURE 2:  General form of energy conservation diagram.

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22  •  Heat Transfer Modelling Using COMSOL Slab to Radial Fin



 dT  qx = − k x   (2)  dx 



 dT  qy = − ky   (3)  dy 



 dT  qz = − kz    dz 

(4)



 dq qx + dx = qx +  x  dx

  dx  

(5)



 dqy  qy+ dy = qy +   dy   dy 

(6)



 dq  qz+ dz = qz +  z  dz   dz 

(7)

Einternal = m Cp



dT ( x, y, z ) dt

+ E gen 

(8)

Substituting aforementioned relations into the energy balance in equation (1) results in equations (9) and (10), where spatial and temporal temperature profiles are associated with the change of internal energy and heat generation within the material.

dqy dT ( x, y, z )  dqx dq + Egen = 0 (9) dx + dy + z dz + m Cv dx dy dz dt



dT  dT  dT  dT  dT  dT  dT + q gen (10)  ky +  kz  = ρ Cv +  kx dx  dx  dy  dy  dz  dz  dt

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CHAPTER

4

FINITE ELEMENT ANALYSIS Finite Element Method (FEM) is a numerical technique in which geometry is divided into finite number of small pieces, called elements. One advantage of defining such elements is that it enables division of regions into smaller portions that can more accurately represent the associated physics. Element size and shape may vary for regions, depending on the physics they represent. Each element carries its own properties. Elements are in contact with the surrounding elements. Solving an FEM problem consists of solving “n” number of conservation equations in which “n” is the number of elements. Each equation requires its own boundary and initial condition. From algebra, you may recall that solving an equation that consists of two independent variables, and results in a unique solution is only possible if it is done in combination with another linearly independent equation that includes at least one of the independent variables. Expanding the equation to “n” state variables requires defining “n” equations that each include at least one independent variable. The same concept applies to the finite element technique. The boundary condition is usually a known relationship for the elements situated at the boundary. The interior elements do not have conditions of their own unless in exceptional cases where specific contact conditions such as contact resistance, internal heat generation, and heat sink or source are to be introduced to interior boundaries of assembled parts. You will find that even in those cases there

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is a third relationship to be considered to make it meaningful. When solving a single equation consisting of two independent variables, an unlimited number of solutions can be obtained. Any assumption for one variable will result in a solution for another one. Let us discuss some examples in which governing equations are derived from experimental observations. If you were to optimize the problem, you would look at the regions in which the desired results are to be found. For example, if you are interested in preparing a balanced diet for yourself in which your required nutritional values and calorie intake are met in order to function as efficiently as possible, you may need to first define an objective function. Let us assume you are interested in optimizing your calorie intake. The total calorie is the cumulative calories from each individual meal ingredients. The next step is to setup the required nutrition (e.g., required amount of daily intake for riboflavin or protein). This is usually the foundation for optimization discussed in operations research and there are tools such as General Algebraic Modelling System (GAMS) that can help you achieve the joy of optimizing every piece of your everyday life problems, from optimized number of hours to sleep to number of person-hours tasks to number of exercise hours you need to achieve a healthy lifestyle, to number of climaxes you can take when riding a roller coaster, given your heart condition. In the finite element analysis, you simply do not have the luxury of selecting number of scenarios and working your way up to identify what combination works out (also known as bottom-up approach). A top-down approach is usually a better technique, where you have the problem defined and steps taken to identify the solution. The missing link in this case is the continuity between the elements. Elements possess the same dependent properties as the geometry as a whole (e.g., temperature). These properties are to be the same at ­element interfaces to enforce continuity from an element to the adjoining element. Imagine you and your neighbor having a fence of your own or a common fence separating your ­properties. The advantage of having your own fence is to ensure your privacy is respected, opting out of common short fences by building taller fence; for example, if you are planning on sunbathing on your concrete patio. You will see that this is an identifier for unions

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versus assemblies; this is not to be confused with the concept of assemblies and unions in COMSOL if you are familiar with it but let us assume a subsection versus the entire geometry. Assemblies keep their independence; however, their interface satisfies certain boundary conditions such as thermal contacts. Though unions do care to follow through from a sub-component to the adjoining subcomponent, which is the neighboring element in this case.

4.1  MATERIAL PROPERTIES Material properties are essential constituents of any modelling. However, some properties may be more dominant than others, depending on the thermo-fluids regime, mode of heat transfer, or the analysis type. These properties may vary in space (spatial), time (temporal), or under environmental conditions (environmental). Non-constant properties will introduce nonlinearities and non-homogeneities to the physics that make the problem more challenging to tackle. Materials can generally be divided into homogeneous, nonhomogeneous, and heterogeneous families, depending on their molecular structure. Nonhomogeneous materials consist of molecules that are different but are coexisting in a mixture. Let us discuss some examples of material properties. An example will be making your favorite mayonnaise recipe; even when wearing your reading glasses, you are not able to distinguish the ingredients while looking at the mixture. When zoomed in sufficiently, you should be able to see the different ingredients under a microscope. If you decide to make a different sauce such as vinegar and olive oil; however, the story is different; they simply do not mix although they maintain a peaceful and independent coexistence. These are examples of heterogeneous mixtures. Next time you make your Sunday French toast, pay careful attention when adding the vanilla extract to the liquid (i.e., egg and milk); at first, they do not mix; the burnt sienna fluid moves freely inside the creamy iridescent liquid as you move the bowl around, creating the most astonishing patterns of forms and colors. It takes some effort to mix the two; only after adding fresh eggs they are mixed, and a solution is formed. Another

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example is adding sugar or milk to your English breakfast tea. You are not able to identify either of the components after stirring the mixture and a homogeneous mixture is created. It does not take an alchemist to know that gold carat is a determining factor in the manipulation process. This rare precious metal, the most malleable and nonreactive of all, both in pure form and as a solution, has a wide range of applications, from your mother’s gold jewelry set given to you as a wedding gift, to a manned satellite—also known as International Space Station (ISS). It is an excellent thermo-electrical conductor, making it a perfect candidate for circuitry, an excellent coating for polyester films used as thermal shields against space radiation, and an effective lubricant between mechanical parts, due to its low shear strength. The purity of your favorite 24-carat heart is 100% compared to that of the 10-carat one that is in the order of 42%, but also requires more care when handling. The carat of the gold used in satellites is almost always higher than that of the one in an 18-carat Welsh gold band. Spatial properties change with geometry. Examples include non-homogeneities in physical properties. In most cases, variations in physical properties such as density will result in changes in thermo-mechanical properties such as thermal conductivity or elasticity. So far, you developed an understanding of the thermophysical properties. The elasticity on the other hand, is a mechanical property which may also vary with temperature; this includes non-homogeneities in physical properties. Recall Elastigirl in The Incredibles[9] by Disney PIXAR Animation Studios[10]. She is Mr. Incredible’s wife, and a former jet pilot feminist who is made of a material that can be stretched to one-mm thin and up to 100-ft tall in a variety of shapes from a parachute to a rubber boat, and resume its original shape. There are also cases in which one property may be nonhomogeneous while others may be semi-homogeneous, homogeneous, or approximated as such for simplicity of calculations. The change can be either spatial, such as varying thermal conductivity in different directions, or thermal, such as change of heat capacity with temperature. There are numerous cases in nature in which properties change as time passes by, in addition to age-related items, such as olive oil and wine.

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4.2 GEOMETRY Selecting the dimensions in which your physics is setup depends on the size of the model, its degree of complexity, time constraints, and computational capability limitations. The first question to ask is, what kind of information a three-dimensional (3D) model can produce that a two-dimensional (2D) model cannot, or if the physics captured by a 2D model are comprehensive enough to produce meaningful results. A 3D model may seem a sophisticated choice, but you may be able to start with a simpler approach to work out the physics and validate the results more efficiently before moving on to more complex problems. Processing a 3D model is more complex than that of a 2D. For example, it may take two minutes to mesh and solve a 3D model using an HPCVL machine, while similar activities may take under a minute for a 2D model. You may suggest there is not much of a difference between two minutes and 30 seconds. But you should realize that to fully develop an accurate working numerical model, multiple solutions will likely be needed. The size of the model directly affects the consequent stages such as number of elements and boundary conditions. Additionally, the possibility for introducing singularities increases. Those factors directly affect the random-access memory (RAM) your computer needs and utilizes when solving a problem. The process of writing the analysis report and extracting related diagrams for the presentation you need to make urgently to the customer to earn you the brownie points you need to secure that contract will be delayed as a result. So, you need to give a lot of consideration when selecting the number of dimensions in which the physics are to be investigated. In some cases, you do not even need to solve for the entire geometry. You may be able to take advantage of geometry’s physical characteristics such as symmetry features to reduce the size of the model by half, quarter, or even more. The initial time spent to identify those features is effort well spent, since it forms the foundation for the following steps. The geometry may be created either externally using a dedicated CAD tool such as SolidWorks®, Solid Edge®, Autodesk®, or

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Inventor™ or internally using COMSOL Multiphysics® built-in ­features. To utilize external CAD tools, you will need to have either a CAD Import Module or a LiveLink™ module, each associated with a specific CAD tool. Any of the LiveLink™ modules include the CAD Import Module functionality. The LiveLink™ modules enable you to have a live interface with the specific third-party CAD software; the CAD Import Module allows you to import the previously created geometries in supported format (e.g., *.stp files). Another method is to take advantage of the available COMSOL Multiphysics® built-in part library to import geometry as desired. Using the LiveLink™ module allows you to update your geometry in COMSOL as soon as you make changes in the CAD software. In addition, if you employed variables to define your CAD model (such as lengths or angles), you will be able to access these in COMSOL and use them as part of parametric solutions, or in any internal calculations. As these variables are updated in CAD, they will be synchronized in COMSOL. Using external CAD tools may become necessary as your model complexity increases. Another way to deal with more complex modeling tasks is to use the COMSOL Design Module, which adds expanded modelling functionality. On the other hand, the benefit of using COMSOL internal geometry generation tools is that all creation steps are accessible and can be modified by adjusting internal COMSOL variables. Sensitivity to dimensions using Parametric Sweep may also be employed.

4.3  ANALYSIS TYPES Analysis type identifies if the problem solution is time-­ independent (i.e., stationary) or time-dependent (i.e., transient). The former is appropriate in cases where the solution does not change over time. It does not mean that the solution has never changed over time; it simply means that you are interested in the part of the solution that has reached the steady state. The latter however is

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associated with cases in which solution varies with time. When solving transient solutions, one may come across the cases in which the solution has become steady. For example, temperature does not rise any further when heat input has levelled off in a thermal problem, meaning it has achieved the steady state. If the geometry is exposed to a heat input that increases linearly over time, you will not expect the temperature to reach a steady state. Selecting the analysis type depends on the objective of analyses. If you are interested in studying thermal response of a train underframe to fire to make sure it complies with the ASTM 2061 standard for fire test code for rail transportation vehicles, you need to study the time response for the first 15-minute exposure. Knowing that the structure is made of a grade of steel with a softening and melting temperature of approximately 400°C and 1300°C, and that the fire temperature-time curve is an increasing concave that shows steep variation with time for the first 15 minutes, you need to perform a transient analysis in which you obtain the temperature-time history of the structure exposed to fire. Temperature profile as a function of time provides interesting information in this case; it not only provides more detailed information on the transient structural integrity of the part when exposed to fire, but also provides an insight to the areas that need to be structurally strengthened. Such thermal reinforcements can be achieved by introducing additional materials in areas with minimum limitations for space and weight to thermally insulate the part, or applying intumescent coatings as fire retardant in areas that are less accessible or have limited space available. Knowing transient spatial temperature profile inside the train when exposed to fire will ensure passenger safety by assessing evacuation time at different locations. If you are interested in evaluating the thermal performance of a heat exchanger, you may not be interested in plotting the transient temperature but instead want to identify the temperature profile along a desired path, such as the liquid cooling channel for stationary case where temperature stabilizes.

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4.4  BOUNDARY CONDITIONS When defining boundary conditions, the main objective is to expose the surfaces of a fin to conditions that are common in fin applications. The intent of the upcoming sections is to investigate three main scenarios: Case 1) very long fin, Case 2) insulated fin, and Case 3) fin exposed to either air or radiation or both. In Case 1, where the fin is very long, it can be treated as a semi-infinite object, meaning that temperature at the tip of the fin equalizes with that of the ambient. In Case 2, insulated tip suggests that there is no heat transfer from the tip of the fin and temperature has equalized on both sides of the tip of the fin. The main difference between this case and the previous one is the length of the fin, meaning it can accommodate the shorter lengths. This boundary condition does not require for the fin to be too long. Case 3 is the most comprehensive condition and incorporates both convection and radiation from the entire fin including the tip. The objective of the following sections is to investigate the sensitivities of the results to boundary conditions. Solving any problem with complex physics in full three dimensions is a challenging task both numerically and analytically. When attempting such problems using analytical techniques, the full range of mathematical tools is required. Even the most advanced theorems may not be able to solve the cases in which non-homogeneity and nonlinearity are present. These nonlinearities can be spatial, temporal, or environmental. The modelling starts with zero dimension and progressively becomes more complex moving through one (1D), two (2D), and three (3D) dimensions. The zero-dimension approach, also known as lumped capacity technique, assumes that temperature is spatially uniform throughout the material but varies exponentially with time. In a 1D numerical analysis, one coordinate is required to identify the position of a point and heat is transferred in only one direction (e.g., x-coordinate), meaning that heat transfer along the remaining coordinates, which form a plane, is ignored or heat is integrated over the remaining plane. One advantage of 1D numerical analyses is that they enable comparison with simplified analytical solutions that are

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derived for number of physics-related phenomena. The comparison provides a tool to validate the numerical analysis. In a 2D numerical analysis, two coordinates are needed to identify the position of a point and heat is transferred in two directions (e.g., xy-coordinates). In other words, the heat transfer transverse to the active work-plane is ignored or the heat is integrated over the third dimension of the geometry. In a 3D numerical analysis, the most comprehensive of the previous approaches, three coordinates are needed to represent the position of a point within the geometry (xyz-coordinates) and heat is transferred in three directions. In any of the said cases, heat input and the resulting heat transfer may change with time. The coordinate system presented above is Cartesian (x-y-z). Depending on the model geometry shape, it may be more appropriate to choose one of either Cartesian, cylindrical, or spherical coordinates. For example, a cylindrical or a wedge shape would be more easily represented by a cylindrical coordinate system. This would make meshing easier; for example, in the sharp corner of the wedge.

4.5  SOLUTION CONTROL AND CONVERGENCE Conservation laws are satisfied when solving physics for the heat transfer of any type. Dependent variables (e.g., Temperature) are to be calculated using independent variables (e.g., process parameters), as well as initial values. They are inputted to the model. The equation is solved and the residuals are obtained. The ideal is to obtain zero residuals; however, this is normally not possible, and so a tolerance value is set. If the residuals are within the tolerance level, it means you have achieved a “good enough” solution. This is where the iteration is complete and the step solution has converged to an acceptable value that can then be used as an input for the next step for a transient analysis or suffices on its own, as in the solution for a steady-state analysis. For instance, if you set 10-5 tolerance value for a solid heat transfer analysis problem, most probably you will be happy with the results: the solution should converge to reasonable values and in

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reasonable time, assuming that the physics, boundary conditions, and material properties are properly defined. However, if you were to use the same tolerance for a flow problem, there is a good chance that analysis may require excessive number of iterations, leading to very long solution (convergence) times or in some cases to not converging at all; ill-conditioned solutions may be another consequence of incorrectly setting the tolerance. You may then decide to relax the tolerance to, say, 10-3. Thus, tolerance value setting should be chosen with care. Figure 3 is an example for a convergence plot for a singleparameter analysis. It shows reciprocal of step size versus time for a transient analysis using a logarithmic scale. Step size identifies the time in between the analysis steps required for accurate computation of the solution. A new data point is added to the plot at every iteration while the analysis progresses. You are also able to view the message console under the convergence plot providing information related to the previous steps taken before running the analysis, such as number of degrees of freedom, number of domains and boundaries, number of domain elements, the name under which the file was saved, and solution time in seconds.

FIGURE 3:  Example of a convergence plot for a 3D analysis for a heat transfer model.

Figure 4 is an example for a convergence plot for a parametric analysis. This type of analysis which investigates the sensitivity of the model to different parameters will be discussed in later sections.

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Each plot represents data (reciprocal of step size versus time for a transient analysis) associated with a new set of parameters that are added to the diagram as analysis progresses with every iteration.

FIGURE 4:  Example of a convergence plot for a parametric study of a 3D analysis for a solid heat transfer model.

Figure 5 is an example for a convergence plot for a parametric analysis. Data (error versus iteration number for a steady analysis) has been presented in a linear scale. This is a multiphysics analysis including solid heat transfer and non-isothermal physics. Depending on the type of flow analysis, you may see additional diagrams such as a diagram associated with a segregated solver to represent dissipated turbulent energy that is supposed to converge to the equivalent temperature-related one for acceptability of the results. The plot is filled with new data with each iteration. The message console is similar to the above example and includes information such as number of degrees of freedom, number of domains and boundaries, number of domain elements, file name, and solution time in seconds. The latter is particularly informative in flow applications where computational resource allocation is paramount. Figure 5 is not a parametric representation, but merely an iteration of the solution to the converging status.

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FIGURE 5:  Example of a convergence plot for a 3D analysis for a conjugate heat transfer model.

4.6  MESH SIZE AND TIME STEP Finite element technique involves dividing the geometry into small elements and solving the energy and mass governing equations for each element and for whatever number of time steps is required in order to reach the total analysis time (for transient problems) or steady state (for stationary problems). The number of iterations required for a solution to converge depends on the initial conditions that the solver employs to start the solution and it may increase or decrease depending on the residuals, i.e., the difference between the calculated and desired values, and also the proximity of the initial conditions to the final results. You can control the temporal and spatial steps when setting up the analyses. Spatial step is defined by the mesh size that may vary within the geometry and so is the temporal (time) step. Think of a fishing net and Formula One car; the purpose is to catch fish and win that race; however, the process is just as important if not the same. Lave, lift, and plankton nets are all fishing nets designed to catch different sea creatures; however, they are

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employed differently. You cannot catch plankton using a lave net. An element size performs similar function. If you have large temperature gradients in certain areas, expect to observe large errors if you are not prepared to reduce the element size in those areas. It is like zooming in to see the details. How accurately you would be able to describe the details of a rug texture (e.g., rug knot density and material) if you are standing 6 ft away? You need to zoom in on the desired areas to give an accurate account of the scene. Most intense heating processes such as laser welding require zooming in at the exposed regions where temperature is changing rapidly in space and time, and so requires reduction of the element size and time step. There are engineering tools such as Engineering Equation Solver (EES) that may be employed to estimate optimized time step for a certain element size and vice versa—given physics by manipulating the engineering equations and minimizing the residues; you may even make a parametric table in which you can decide which element size to use with which time step to generate the most accurate results (i.e., by minimizing an error function). A similar approach may be taken by using Microsoft Excel, or MATLAB. Assume that you decided on the optimized mesh size using one of the said techniques. The next step is to make sure the element size produces the converging results that are reasonable. One way to achieve this is to change the element size from larger to smaller values (or vice versa depending on the solution convergence) and review the variation of the numerical results (i.e., sensitivity ­analyses); keeping in mind that the objective is not to be perfect but to be good enough for the application under study. The sensitivity analysis is usually conducted for the dependent variable whose numerical analysis has been performed. If heat transfer analysis is the main focus of the investigation, temperature is most probably the dominant dependent variable (unless you are interested in the pressure gradient in a non-isothermal analysis). Temperature gradient alone, in the form of temperature difference from a reference value, does not suffice. You need to also take into account the range in which the changes occur. For example, if the total temperature change in a heat transfer problem is 10°C and you are off by 1°C, that is already a 10 percent error. However, if the former is 100°C and the latter

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remains the same, it becomes a one percent error. So be prepared to defend your “good enough” numerical results. On the other hand, there is no need to select fine meshes over the coarse ones if your objective is purely to provide methodology examples for educational purposes, or where the mesh sensitivity analysis is telling you that a finer mesh is not necessary. Note that there is no point in refining the solution to one percent accuracy if there is much higher uncertainty in other model data, such as material properties or contact resistance between the geometry sub-components. Recall that there are two types of errors: 1) round-off and 2) truncation errors. The former occurs when you decide to round the number to the closest value, taking into account the desired decimals. Meaning you are compromising on some accuracy. The latter on the other hand is when you decide to show only specific number of decimals. A simple example will be to show 45.657123 either as 45.66, 45.65, or 45.657. The first two examples show the same number when it is either rounded off or truncated with two figures after the decimal; the third example could be either rounded off or truncated to the same number when three decimals are employed. There is a balance between the two errors; especially where they are accumulated due to the increased number of numerical equations. They usually show an opposite trend—i.e., decreasing versus increasing concave temperature profiles for the round-off and truncation errors, respectively. A time step-mesh sensitivity analysis provides a good compromise. So, it is due to the accumulation of the above computational errors that with decreasing element size, after converging to the most accurate solution, the solution may start diverging (i.e., getting less accurate).

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CHAPTER

5

SENSITIVITY ANALYSIS There are a number of methods that can be employed in order to calculate the influence of different parameters on the numerical results in COMSOL Multiphysics® Heat Transfer module. These parameters may be material, geometry, and mesh characteristics, as well as boundary conditions including coefficients, fluxes, and temperatures. Either a constant value or a function of any type may define these conditions that impose constraints on the solution outcome. You may investigate the analysis response in order to study its sensitivity to any of the said parameters. There may be limitations as to how this method may be implemented when modeling experimental processes given the limitations imposed by process parameters such as the power setting limits for a laser. There are number of methods to employ when carrying out these sensitivity analyses, which may be used in a hybrid ­arrangement, in combination with other methods, depending on the analysis type. The purpose of this section is to provide an overview of the ­available tools. The following techniques (Parametric Sweep, Function Sweep, Material Sweep, and Auxiliary Sweep) are recommended for carrying out the sensitivity analysis studies (Figure 6). Note that the analyses may be activated, deactivated, or moved up or down to specify the order in which the actions are to be carried out. For the ­remainder of this work, pop-up menus are used extensively. The author recommends reviewing the pop-up options to become

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familiar with the available options when setting up the physics or boundary conditions.

FIGURE 6:  Example of sweep types to perform sensitivity analyses.

5.1  PARAMETRIC SWEEP You may use parametric sweep to carry out sensitivity analysis of the results with respect to material properties, initial conditions, geometry features, mesh characteristics, and boundary conditions including coefficients, fluxes, and temperatures. The example shown

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herein depicts the steps taken in order to setup such an analysis. Assume that you are interested in varying convective heat transfer coefficients to investigate the environmental cooling. You should be following good modelling practices and define parameters (or variables) by giving descriptive names. Under the Study branch, pick Parametric Sweep (Figure 6). Parametric sweep is added as an independent item under the analysis node with a settings window in which you may input desired parameters. In the example shown, the multiplier “n” is selected as the parameter whose sensitivity analysis is to be performed. Note that this parameter is a dimensionless multiplier for the convection coefficient (i.e., hc in Figure 7). Care is to be taken when defining the parameter unit; if you are to choose hc instead of “n”; for instance, you should make sure the dimensions are repeated in this window for whatever number of scenarios (i.e., number of analysis) the hc is to vary. You may also plot the results for the analyses as they run or select a probe (e.g., temperature) to collect data for a desired location. The approach of using “n” is recommended since it enables you to scale any given value without changing the parameter value directly. So, instead of using [0.0025 m, 0.005 m , 0.0075 m], you may simply convert it to [0.05 m × 0.05 m, 0.05 m × 0.1 m or 0.05 m × 0.15 m] and use [0.05,0.1,0.15] for “n” values.

FIGURE 7:  Sensitivity analysis by a Parametric Sweep.

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5.2  FUNCTION SWEEP You may employ functions (e.g., analytical, interpolation, random, step, and waveform) when defining material properties, initial conditions, geometry features, mesh characteristics, and boundary conditions including coefficients, fluxes, and temperatures. Note that the function definition can be made either under the main tree or for each component (if you are working with multiple components). To perform a Function Sweep, you need to have defined the Function Sweep under the component for which you wish the analysis to be performed (Figure 6). Your functions of any type are grouped under the pre-defined Function Switch. The next step is to communicate your intention to the analysis node (e.g., steady or transient) that you are interested in performing the Function Sweep by adding the capability under your analysis tree. You may then add a switch by selecting its name in the study settings window (Figure 8).

FIGURE 8:  Sensitivity analysis by a Function Sweep.

5.3  MATERIAL SWEEP Let us say that you are interested in finding out how varying a component material (e.g., liquid cooling agent) will affect your results. You can use a Material Sweep to investigate sensitivity of the results with respect to material (Figure 6). For this example, you have selected six possible cooling agents: ethylene glycol, water, mercury, air, transformer oil, and engine oil. Recall that you may set material on the global or local levels. When setting up materials

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locally, it is possible to add materials as separate nodes or to define them in clusters under the material switch node where you can add any blank or pre-defined material. These materials are ordered as added and form a family. Let us say the family name is Switch 1, consisting of ethylene glycol (member 1), water (­member 2), and so forth. In the next step, you add a material sweep node to your analysis and add Switch 1 family to this node (Figure 9). You are then able to pick whatever number of the sensitivity scenarios you are interested in performing. For example, for your family of six, you may select any family members in any order (i.e., using their sequence numbers such as 1, 3, 6, 4 associated with ethylene glycol, mercury, engine oil, and air, respectively). You may specify the selected materials in whatever order you wish (Figure 9). One reason to pick different orders is to facilitate solution convergence in applications such as turbulent flow analysis. Figure 9 presents range (1,1,3) when defining Material Sweep. It means materials are to be used in the order of material 1, and incremented by unit of 1, being solved for materials 2 and 3, respectively.

FIGURE 9:  Sensitivity analysis by a Material Sweep.

5.4  AUXILIARY SWEEP Auxiliary Sweep is available for stationary, transient, and frequency domain studies. You may use Auxiliary Sweep to carry out sensitivity analysis of the results with respect to material properties,

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initial conditions, geometry features, mesh characteristics, and boundary conditions including coefficients, fluxes, and temperatures. Let us consider the same example discussed earlier where you were interested in varying convective heat transfer coefficient. Under the study branch, you may define as many steps as you wish, either steady or transient—independently, or as the input to the next analysis step. Under the specific step in which you wish the sensitivity analysis to be performed, you would activate the Auxiliary Sweep feature and select the parameter names along with their dimensions in whatever order you wish. In this example, the dimensionless multiplier “n” for convective heat transfer coefficient is selected as the parameter whose sensitivity analysis is to be performed and values 1 and 100 are solved for (Figure 10).

FIGURE 10:  Auxiliary sweep to perform sensitivity analysis.

5.5  VERIFICATION AND VALIDATION Let us assume that a numerical model explaining a physical phenomenon has been created using a commercial finite element software to output certain information for given inputs. The inputs to

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such model may be material properties or manufacturing process parameters. They are incorporated in the form of boundary conditions or any other terms defined in the energy balance equations. How do you know if the numerical results obtained after solving the equations are valid? While you may say in general “the outputs are only as good as the inputs,” there are number of other points to consider. The following are the factors which will likely enhance the validity of the solutions: 1) setting up the physics as accurately as possible—meaning introducing a comprehensive set of equations that precisely represent the physical phenomena, 2) making simplifying assumptions that do not jeopardize the integrity of the problem, 3) creating geometries that accurately represent the real system, including the correct interfaces and boundaries, 4) defining model inputs correctly, 5) solving the model using proper techniques, and 6) selecting outputs that closely match the questions sought when setting up the physical system. Nevertheless, the solution is to be validated, and there are techniques that can facilitate this process. One of the most powerful methods is to employ an analytical solution. Analytical methods may be employed to produce analytical solutions for simplified numerical models. You can find a variety of these models in heat transfer textbooks. These are scenarios to be used as the bases for validating numerical results. To make the comparison relevant, the numerical results are to simulate the conditions which the analytical results have been derived from. Another approach to validate the numerical analysis is to use experimental results for a system that closely simulates the conditions for which the mathematical model has been defined and solved for. One may argue if it would be more practicable to simulate the experiments numerically as the reverse process may be more challenging. The answer is not as obvious as it seems, and depends on the complexities surrounding setting up experiments and models. There are examples in which tests encounter limitations and interpolating or extrapolating the results is the only way moving forward. The limitations may be due to any variables such as limitations facing process parameters for a multiphysics modelling. It is certain that unlimited deductions cannot be made from one set of data and at least three

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sets of data are required to make such interpolations or extrapolations. Carrying out physical experiments is generally expensive and time-consuming. One may only be able to carry them out for a few conditions. With a numerical model, you may run it for any conditions, even those that may not be possible to achieve with available equipment. However, you may validate your model by comparing with experimental results at least for a few selected settings. For example, heat transfer modelling in laser welding is a challenging task. Selecting the combination of process parameters to make the joining happen can be time-consuming. In this case, demanding the experiments to follow a low power setting is not always reasonable since laser capability may not allow for small power variations. On the other hand, increasing the power level to very large values may degrade the material. Introducing process parameters for the numerical models therefore requires careful design of experiments along with setting up upper and lower process limits. Thermal imaging is a technique that may be used to validate the numerical results from heat transfer modelling of the said laser welding process. The model should be setup in the exact way or as closely to the experiments as possible so that the problem is faithfully represented. For example, if you are interested in measuring the temperature at the surface facing the thermal imager, you need to make sure the camera lens is parallel to the surface. Furthermore, the front surface is covered by a coating of known emissivity so that temperature is measured accurately, taking into account that imager object signals are calibrated correctly to the temperature. Although analytical solutions may be developed for variety of physical phenomena, they all share the same limitations. They represent a simplified form of a more complex physical system. Any attempt to solve most complex problems may be neither effective nor efficient given the magnitude of the effort that may be required. Unless you are interested in investing hours to develop complex mathematical models for the pure joy of challenging yourself, you may wish to employ numerical coding or commercial software as a more efficient and comprehensive approach to setting up and analyzing physical phenomena.

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Sensitivity Analysis  •  45

If you were to utilize numerical coding in your approach, meaning you are using a suitable programming language such as Fortran or C++ to setup and solve a physical phenomenon, there are extra challenges of verifying the solution in addition to validating the solution. Validating your model requires the steps described at the beginning of this section. Verifying, on the other hand, requires that the physical equations be carefully investigated for precision and accuracy. If you are wondering about the correctness and accuracy of commercial software packages, you may review the verification documentations that they release with their products and the methods through which they have accomplished this verification. Some commercial codes (e.g., Code-Aster a commercial FEM code designed for solving structural mechanics) are accompanied by an ISO-9001 qualification certificate. On the other hand, some commercial software may not allow in their End-User-Agreement to carry out direct comparison of their numerical results with those of the competitor’s software.

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CHAPTER

6

FIN GEOMETRIES Extended surfaces (Fins) can be as simple as slabs or as complex as radial fins. Heat transfer mechanism varies depending on the ­profile of the fin, including its cross-sectional shape and area. Although a fin with larger surface area increases the amount of heat transferred, space and weight limitations are important factors to be considered. It is possible to use other thermal management ­techniques such as fans in combination with fins. Note that maximizing the heat dissipation is not always the goal when dealing with thermal management problems. There are cases where maintaining the temperature at a certain value is required for optimized performance of the system, such as the ­ferrite temperature in a linear induction motor. The purpose of the following sections is to investigate the effect of a fin’s varying cross section and profile on heat dissipation. This provides a tool for analysts, as well as engineers, when selecting the type of fin during preliminary stages of design and analysis. The sections have been organized to include the stages required to construct the thermal models consisting of geometry creation, materials definition, setting up physics, including initial and boundary conditions, identifying analysis type, meshing, defining the solutions, and post-processing. Each analysis includes number of assumptions varying from the selection of dimension to material properties to the methodology in which inputs interact with the physics, and boundary conditions. The main objective of this presentation is to establish a

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baseline to evaluate fin performance, and therefore temperature at the base of the fin is assumed to be known. Fin geometries are summarized in Appendix A along with the applied physics and mathematics. Fin’s performance characteristics (e.g., efficiency) are presented. The author recommends the reader to attempt some of the solutions to better understand the mathematical and physical concepts presented in this publication.

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CHAPTER

7

COMSOL MULTIPHYSICS®EXTENDED SURFACES THERMAL CASE STUDY The purpose of this section is to become familiar with the process of setting up a thermal model in general and using a commercial software package specifically. Most numerical software packages follow the same logic when setting up such models in any of the applied science fields such as mechanical, electrical, chemical, or biomedical. Organization of the model into accessible compartments is more viable when using a commercial software package than when writing your own code. To develop a model, in general, requires three phases: 1) preprocessing, 2) solution, and 3) post-processing. Phase 1 consists of setting up the model such as deciding which physics to apply, defining the geometry, assigning material properties, setting up boundary and initial conditions, and meshing the part. Phase 2 consists of setting up the analysis type and solving the model for the parameters that control the output. In Phase 3, you review the numerical results either visually in the form of contour plots or in the form of tables, and diagrams. Think of a numerical model as an artwork, in the form of visualization of behavioural representation of a physical system, showing

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a tree with branches, leaves, and fruits. Your model as a whole is the tree trunk. You first need to decide how realistically you would like to represent this tree. The most realistic way would be to sculpt it (i.e., a 3D model). The next simpler way would be to have it painted on a blank canvass (i.e., a 2D model), and the most basic way is to associate a line with the whole thing, similar to your kindergarten drawings (i.e., a 1D model). Did you ever try assigning a dot to a tree or planet as part of the entire milky way, which would constitute a 0D model? You then need to decide on the shape of the tree (i.e., geometry). Do you wish to depict a straight or a curved tree? Different parts of the tree such as main and side branches, and leaves require ­different materials. The environmental conditions which contribute to the tree growth and its livelihood are the boundary conditions. Was it exposed to the sunshine radiation in the early morning or cold wintery weather? Did it act as a heat sink when you sat under its shade reading your favorite novel? Looking at the final product of your imagination, you view sections of it piece by piece using your main or peripheral visions. If you stitch the segments (i.e., meshes) together, you are able to see your design in a complete form. The first thing to keep in mind when solving the tree puzzle is to figure out how the tree comes to existence. Are you interested in observing the status of the tree five years from now when it is all mature and fruit bearing (i.e., steady state at a certain age) or would you like to continuously monitor its growth up to a said point (i.e., transient)? Whatever the purpose of the exercise is, either to enjoy the shade or harvesting the fruits of your work, you may wish to obtain variety of information on the tree any time during this journey (post-processing). You may wish to collect data such as the physical properties of the trunk (e.g., mass and width), the thermallydependent characteristics of the leaves (e.g., temperature), the energy generated by photosynthesis when exposed to the sunshine (e.g., heat flux), the carbon monoxide or heat absorbed from the environment (e.g., mass or heat sink). You may wish to exercise your artistic nature by associating colors with the tree parts; for example, temperature visualization in the form of diagrams or contours. This is when the visualization tools such as plots and diagrams are employed.

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COMSOL Multiphysics®-Extended Surfaces Thermal Case Study  •  51

Choice of mesh shape and element size depends on the geometry to a large extent. It is desirable to introduce meshes that are as uniform as possible with aspect ratios, which is defined as the ratio of the element’s smallest-to-largest dimensions, that honour the integrity of the numerical solution. This is usually achieved by means of sensitivity analysis where the influence of various element characteristics such as size is investigated for the given geometry and physics. Note that the same geometry with different physics may require elements of different characteristics or size, such as cases where conduction is mainly the focus versus cases in which conduction and convection are dominant in a fluid dynamic analysis. It is to be noted that there are other techniques that make this comparison possible. For example, you may export the data from different analyses to an Microsoft Excel sheet or MATLAB subroutine and decide to present it in the form of a table where the minimum and maximum independent values such as temperature is desired or Microsoft Excel/MATLAB generated diagrams. There are also other regression analysis tools such as Minitab® or JMP® Statistical Software that can carry out additional analysis in order to find the relation between varying parameters and the main function. This type of study, called sensitivity analysis, is very important when setting up mathematical models and where optimization is of interest.

7.1  ASSUMPTIONS: FROM SLAB TO RADIAL FINS Note that the initial conditions are assumed to be that of the ambient, which can be modified to the desired values, and the surfaces to which boundary conditions are to be applied are assumed insulated unless otherwise specified. It is assumed that the skillet over the stove top has reached the steady state temperature and therefore the temperature at the outer surface of the skillet, is the same as the temperature at the base also known as the skillet handle. For the upcoming analyses, the skillet handle base is assumed to have a constant temperature of 100°C and the initial temperature is 20°C. Note that the proceeding boundary conditions override

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their counterpart initial conditions. The slab is connected to a block throughout the case studies; this “block” is a visual representation of the skillet’s main body and is not included in the solution. In other words, the fin is an extended surface that transfers heat from an object. In the following sections, the order of figures are from the left to the right with “a” representing the far-left figure.

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CHAPTER

8

FIN WITH RECTANGULAR CROSS SECTION This section focuses on the simplest form of the extended surfaces also known as fins- a slab with a rectangular cross section (Figure 11). The analyses level of complexity increases as the section unfolds by introducing one-dimensional (1D), two-dimensional (2D), and three-dimensional (3D) models of the same problem. Steady state as well as transient analyses are investigated and the results are compared for different scenarios.

FIGURE 11:  Fin with side-rectangular profile and rectangular cross section.

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8.1 ONE-DIMENSIONAL ANALYSIS (1D) To setup a new model, go to File > New, where either the Model Wizard or Blank Model is available to you. The Blank Model is usually used to setup highly specialized models (e.g., mathematical models). The focus of this section is on the Model Wizard (Figure 12a). The next step is to select the space dimension (Figure 12b). Physics and study selections are the next stages. To setup a heat transfer model for slab, related physics is applied (heat transfer in solids). The dependent variable in this case is temperature, whose name can be assigned to any desired letter or combined with numerical subscripts (Figure 13a). The next step is to select the study type that is either stationary (steady state) or time-dependent (transient) (Figure 13b).

FIGURE 12:  a) Setting up a new model, b) Selecting space dimension.

FIGURE 13:  a) Selecting physics, b) Selecting study.

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Adding materials to the model tree is the next step, which can be achieved either by searching the built-in library or adding a blank material and filling out the property fields. Materials can be either added directly under the sub-component (e.g., tree leaf: local) or to the upper level component (e.g., tree trunk: global). Material thermo-physical properties can be defined or modified (Figure 14). In the former case, a link is to be created to associate the subcomponent material to its global parent (Figure 15). Note that the material section has its own place within the tree either in the global or local levels.

FIGURE 14:  Defining material properties at a global level.

FIGURE 15:  Defining material properties at a local level by linkage to the global parent.

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Creating a 1D geometry is either to be done using the points or intervals. The 1D slab in this case is created by using interval feature and its length is set to 0.1 m. Metric or Imperial unit systems can be employed when creating the geometry (Figure 16). Use of ­parameters is recommended when setting up the geometry to provide flexibility when revising dimensions (Figure 17 and Figure 18).

FIGURE 16:  Creating slab geometry within the component.

FIGURE 17:  Setting up global parameters.

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FIGURE 18:  Creating sub-component geometry by means of parameters.

Selection of the model was done earlier when selecting the physics for the model. Switching to a different physics or adding new physics to the model is to be performed by selecting the component model and adding new physics (Figure 19).

FIGURE 19:  Adding physics to the model tree.

Either way you need to assign sub-physics under the main physics to compensate for the missing information; think of filling the blanks in a technical document related to an aircraft landing gear, where information such as tire pressure, lubricant type, shock absorber, and oleo mechanism is to be found out. Figure 20 shows the sub-physics

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under the Heat Transfer in Solids. Note that by default, the physics such as Solid, Initial Values and Thermal Insulation are defined. These are the bases for setting up a FEM model that defines state equations as well as boundary and initial conditions.

FIGURE 20:  Defining boundary and initial conditions for the component sub-physics.

8.1.1 Case 1 Boundary Condition: Semi-Infinite Fin The semi-infinite fin assumes that the fin is very long; therefore, the temperature of its tip equalizes with that of the environment (Tamb  = 20°C) (Figure 21). Base temperature is assumed constant (Tbase = 100°C) (Figure 22). Creating a mesh is the next step: You may either choose a physics-controlled mesh or a user-controlled mesh option. When selecting each option, you may choose the element

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size, from Extra Coarse to Extra Fine (Figure 23). In this case, the maximum element size of 0.0053 m has been set (Size). This resulted in a mesh of 0.005 m and 20 elements. Mesh definitions such as general physics, fluid dynamics options, and other details may be accessed on the mesh tab (Figure 24). You are able to define maximum element size, maximum growth rate, and resolution of narrow regions within this tab. Element size will be limited to the maximum element size. The growth rate, which is the allowed relative increase of element size as the mesh is automatically generated, will be limited to the maximum growth rate. The maximum mesh resolution for the narrow regions is also set here. These last two settings become relevant for 2D and 3D problems. For example, as the software is meshing an area, a narrow region will require a fine mesh. When mesh generation extends to a less restricted area, it will start “growing” the elements, and this growth will be capped by the growth rate setting. Thus, a smaller setting will extend the finely meshed region. A higher setting will result in faster transition to a coarser mesh.

FIGURE 21:  Assigning tip temperature for boundary condition.

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FIGURE 22:  Assigning base temperature for boundary condition.

FIGURE 23:  Creating mesh for the 1D component.

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FIGURE 24:  Defining mesh size for the 1D component.

The last stage is to setup the analysis (Figure 25). Note that the model is to be identified within this section as well as the related mesh. If there are multiple models within a single file, the program selects all of the previous ones when setting up the solution. More analyses may be added to the same physics through the top model by selecting Add Study feature (Figure 36). For a thermal model consisting of multiple analyses scenarios, the new analysis does not supersede the previous one, but will be cumulative to the previous analyses, meaning that they will be executed in the order of creation if desired. Therefore, it is paramount to double check the solution, physics and mesh in which the analysis is setup to run.

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FIGURE 25:  Setting up transient analysis for the 1D component.

Figure 26 is the convergence diagram generated after carrying out the Compute command. The solution tolerance may be setup within the solver configuration and is usually on the order of 10-5. You may decide to tighten or loosen the tolerance, depending on the physics and analysis type. For example, in the fluid flow analysis, you may wish to relax the tolerance. Beware of the compromise made when making such decisions. The solution is obtained by computing the analysis and then the default temperature profiles or diagrams are created (Figure 27). You may then select the time(s) for which the results are to be presented through Time Selection menu from a list or by interpolation technique. There are a number of visual options you may choose from when plotting a diagram, such

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as legend display, axis labels and range, and grid options (Figure 28). Temperature profiles for different times along the fin length (x-­coordinate) are shown in Figure 28. Note the decreasing convex profile at the beginning of the heating process that develops into a linear profile as time progresses. The set temperatures of 100°C and 20°C at the base and tip of the 1D fin are maintained. You may add plots to either depict the temperature profile along a coordinate (spatial) as shown in Figure 28 in the form of a predefined line or as a function of time (temporal) for a predefined point. The latter is achieved by adding a new group of plots (i.e., 1D, 2D, and 3D line plots) that represents the cross section in which the data set belongs (Figure 29a and Figure 29b). The data set definition can be found under the Solution node, as it represents a subset of the results as a function of time or location. The same methodology is used to define a Cut-Point 1D as a subset of the numerical results (Figure 30). Figure 31 shows the transient results for the defined point. The temperature at midpoint starts from the initial value of 20°C and stabilizes at 60°C (steady state) at around 40 s.

FIGURE 26:  Convergence plot and message console.

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FIGURE 27:  Plotting transient analysis solution.

FIGURE 28:  Setting up temperature profile a) Selecting axis labels, b) Displaying legend, c) Plotting temperature profiles along the fin length (1D) for different times.

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FIGURE 29:  a) Defining data set for a plot, b) Defining sub-plots within the main plot.

FIGURE 30:  Defining a point data set for transient results.

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FIGURE 31:  Transient temperature profile at the base (0 m), middle (0.05 m), and tip (0.1 m) for a semi-infinite fin.

Figure 32 shows the steady state results for the slab along the fin length (x-coordinate). Figure 33 compares the steady state results with the transient results after the solution has reached steady state (t = 60 s) and demonstrates that the two plots are nearly identical.

FIGURE 32:  Temperature profile along the fin length in stationary analysis for a semi-infinite fin (1D).

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FIGURE 33:  Comparing temperature profiles along the fin length for steady-state (1) and transient (60 s) analyses.

8.1.2 Case 2 Boundary Condition: Insulated Tip This case assumes the slab tip is insulated, meaning that there is no temperature gradient at the tip. The easiest way to setup this new case would be to modify the existing model (Case 1). However, it is recommended to create a new setup (Case 2) by adding new physics to the same component (Figure 34).

FIGURE 34:  Adding new physics to the model tree.

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The main difference between setting up the model for the Case 1 and Case 2 boundary conditions is the physics to define the tip conditions (i.e., fixed temperature versus insulated for the fin tip) (Figure 35).

FIGURE 35:  Assigning insulated tip for boundary condition.

The study in which the analysis is performed is then added to the main level (Figure 36). Ensure that it represents Case 2 physics and mesh by carefully checking the tick marks on the selection panel within the study settings. Analysis is then computed (Figure 37). Before computing the results, you can choose to generate the default and convergence plots. The default plots are usually the temperature distribution and basic contour plots that can be modified as desired. Selecting to activate the intermediate solution storage option will result in greater memory usage and data storage requirement, which may lead to issues with largescale solutions. Temperature results for transient analysis are shown in Figure 38. At the beginning of the process (t = 1 s), temperature shows a nonlinear behaviour throughout the slab along the fin length (x-coordinate) where the heat transfer takes place. As time progresses, the steeply decreasing convex temperature curve progressively decreases its slope, becoming nearly linear and reaching over 80°C at the right boundary after one minute.

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FIGURE 36:  Adding study to the model tree.

FIGURE 37:  Setting up stationary analysis for the insulated tip.

FIGURE 38:  Temperature along the fin length in transient analysis for the insulated tip (1D).

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8.1.3 Case 3 Boundary Condition: Convective Surfaces This case assumes convective elements for the slab, meaning that change of temperature at the right boundary is proportional to the convective heat transfer coefficient. This coefficient is assumed to equal 5 W/m2K (a typical value for the horizontal surfaces). For this case, as well as the upcoming cases, two values for this ­coefficient have been adopted and defined by a multiplier “n”; n = 1 is associated with the natural convection case (base value), whereas n = 100 is associated with the forced convection case, where it is expected the temperature of the slab approaches closer to that of the environment and also within a shorter time. Temperature results for transient analysis are shown in Figure 39. At the beginning of the process (t = 1 s), temperature shows a ­nonlinear behaviour throughout the slab. As time progresses, the steeply decreasing convex temperature curve progressively decreases its slope; temperature at the right boundary reaches over 81°C after a minute of exposure for the free convection and over 72°C for the forced convection.

FIGURE 39:  Temperature along the fin length in transient analysis for the convective tip (1D).

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8.1.4 Comparing the 1D Analyses for Three Boundary Conditions (Cases 1, 2, and 3) Comparison between the transient analysis results for a midpoint location within the 1D slab for the three-boundary condition cases is shown in Figure 40. It is seen that the insulated tip as well as the convective tip almost follow the same trend, while the forced convective conditions reduce the temperature rise within the slab by about 5°C. The semi-infinite slab constrained to ambient temperature (Case 1) at the slab tip shows the smallest temperature rise and reaches the steady state sooner (after 40 s). Transient temperature distributions for the three boundary conditions along the fin length (x-coordinate) after a minute of exposure are compared in Figure 41. The semi-infinite slab with a fixed tip temperature shows a linear trend for the temperature distribution while the insulated case shows a nonlinear trend of decreasing convex profile that develops to a more linear form for a slab with forced-convective elements for the boundary conditions. If the slab were long enough for the fixed tip temperature assumption to be valid, you would obtain similar results between the temperature distribution of the insulated cases.

FIGURE 40:  Comparing the 1D transient results for the three cases of boundary conditions (x = 0.05 m).

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FIGURE 41:  Comparing the 1D transient profiles for the three cases of boundary conditions along the fin length (t = 60 s).

8.2 THERMAL MODEL FOR THE 2D SLAB This section focuses on a 2D slab that is exposed to the previously defined boundary conditions. Heat transfer fundamentals remain the same except for the element shapes and types (i.e., plane elements). To build this 2D model, one could create a new model by selecting File > New and following the steps presented in section Thermal Model for the 1D Slab. Another method would be to build upon the previous example by adding a new 2D component (Figure 42). The advantage of the latter technique is that it makes it easier to compare the 1D and 2D results in one plot, which can be useful for more complex problems.

FIGURE 42:  Adding component to the model tree.

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FIGURE 43:  Creating the 2D slab geometry by using parameters.

The next step is to create the geometry. After making decision as to how to create the 2D slab, either as an independent component or in a parallel setting (i.e., equal in existence and independent or inter-dependent in operation), you need to define the geometry. In general, the geometry may be created either in a dedicated CAD tool such as SolidWorks® and imported into the COMSOL Multiphysics® or created using COMSOL Multiphysics® ­depending on the software modules available to you. Simple shapes may be created within the software with relative ease. In this example, the geometry has been created using rectangle and polygon commands, which are part of the core package of COMSOL Multiphysics® (Figure 43). One of the places in which “less is more”—cited from Robert Browning’s “Faultless Painter”[11]—is applicable, is when setting up a numerical model in order to save resources such as computational time and memory[12]. Simplicity in addition to clarity helps understanding, definition, development, and presentation of the analysis results. Taking advantage of symmetry features is a key technique one should use to simplify the model. Care is to be taken to ensure that the boundary conditions on the symmetrical features are identical in addition to the geometry symmetry. Examining the 2D model in Figure 43, note the symmetry about the horizontal

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centerline (Figure 45). Figure 44 shows the meshed model taking advantage of the symmetry. Note that only the highlighted meshed area constitutes the physical domain included in the model.

FIGURE 44:  Assigning mesh size for the 2D slab.

8.2.1 Case 1 Boundary Condition: Semi-Infinite Fin The contour plot in Figure 45 shows the temperature variation within the geometry after one-minute exposure for Case 1 boundary condition where the semi-infinite slab is investigated, meaning that the temperature at the tip of the slab equalizes to that of the environment. Note that the slab base is kept at a constant temperature (100°C), and therefore the results of the extended surface (fin) are the focus throughout the investigation. The semi-infinite 2D fin assumes that the fin is very long; therefore, the temperature of its tip equalizes with that of the environment (20°C) (Figure 45). The 2D transient analysis results showing temperature for a midpoint on the slab versus time are compared to those for the 1D case in Figure 46. It is seen that the two temperature curves are nearly identical.

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FIGURE 45:  Assigning tip temperature for boundary condition (t = 60 s).

FIGURE 46:  Transient temperature profiles at the center for the 2D slab compared to the 1D slab for the semi-infinite slab.

8.2.2 Case 2 Boundary Condition: Insulated Tip Figure 47 shows the boundary conditions for the 2D fin with insulated tip. The temperature gradient at the tip is zero, meaning that the temperature at the node adjacent to the tip equalizes with that of its vicinity, which is not necessarily the same as that of the environment (Figure 47). Base temperature remains constant

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at 100°C. Setting up the mesh is identical to Case 1 (Figure 48). Figure 49 compares transient temperature profiles at a midpoint location for the insulated tip for 2D analysis to that of the 1D analysis. It is observed that the profiles are closely matching, testifying to the accuracy of the 1D model for this type of analysis.

FIGURE 47:  Assigning insulation boundary conditions.

FIGURE 48:  Assigning mesh size for the 2D slab with insulated tip.

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FIGURE 49:  Transient temperature profiles at the center for the 2D slab compared to the 1D slab for the insulated tip.

8.2.3 Case 3 Boundary Condition: Convective Surfaces Figure 50 shows the boundary conditions for the 2D fin with convective surfaces (highlighted by blue lines). Temperature contours within the geometry are also shown in the same figure. Color distribution shows the temperature after one-minute. Convective coefficient of 5 W/m2K has been assumed for the surfaces exposed to the ambient. Base temperature remains constant at 100°C. Transient temperature results are presented in Figure 51. Note that the temperature for the 2D case when the cooling is done by means of natural convection is comparable with that of the 1D. However, when the cooling is achieved by means of forced convection, the temperature at the midpoint location is reduced more in the case of 2D versus the 1D. This is expected since the surface area for the 2D analysis has increased and that, in addition to the increased convection coefficient, contributes to increased heat dissipated in the case of 2D analysis.

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FIGURE 50:  Setting up convective boundary conditions for the 2D slab (t = 60 s).

FIGURE 51:  Transient temperature profiles at the center for the 2D slab compared with the 1D slab for the convective tip (t = 60 s).

8.2.4 Comparing the 2D Analyses for Three Boundary Conditions (Cases 1, 2, and 3) Figure  52 compares transient temperature results at a ­centerpoint for the three boundary conditions. Temperature profiles are of

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increasing convex nature and temperature reaches approximately 47°C, 60°C, and over 85°C for forced convective (n = 100) after exposure time for fixed tip temperature, insulated, and natural convection (n = 1) cases, respectively. Figure 53 compares transient temperature along the fin length (x-coordinate) after one minute of exposure time for the three boundary conditions. It is seen that the semi-infinite boundary condition introduces a linear temperature profile that decreases from the base where the temperature is the highest (100°C) to the tip (20°C). The convective boundary condition introduces a nonlinearity term to the heat transfer equation with a decreasing convex profile. With increasing convective coefficient, temperature within the slab equalizes more uniformly from the base to the tip, since the heat transfer is done more effectively. The cases of the insulated tip and natural convection display very similar profiles. Note that it is helpful to select the midpoint for observing temperature profiles that are least affected by the boundaries.

FIGURE 52:  Comparison of the 2D transient temperature profiles for the semi-infinite slab at a midpoint location for the three boundary conditions.

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FIGURE 53:  Comparison of the 2D transient temperature profiles along the fin length for the semi-infinite slab for the three boundary conditions (t = 60 s).

8.3 THERMAL MODEL FOR THE 3D SLAB This section presents the natural progression of the 2D slab to the 3D slab which is exposed to the previously defined boundary conditions. Heat transfer concept is the same as those of the previous cases except for the element shapes and types (i.e., volume elements). Similar to the 2D analysis, the 3D model can be created either by making a new model (File >New) and following the steps to define the physics and study type, or building upon the 2D study by adding a new 3D component (Figure 54). Using the latter approach makes possible to compare the results of 3D and other types of analysis in one plot. The 3D geometry can either be created by importing a solid geometry that is generated in a CAD software such as CATIATM or made directly inside COMSOL. For the example presented, the 3D geometry has been created using blocks, a feature available in the COMSOL core module (Figure 54). Note that to create any geometry in 3D or other dimensions, you need to input the dimensions along the fin length (x-coordinate) individually and also identify the center or corner of the 3D shape. There are also other characteristics such as the angle of rotation, work-plane and so forth, which may be included to accurately represent the geometry.

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Similar to the previous scenario for the 2D analysis, one can use symmetry to simplify the solid geometry. In order to achieve this, a cutting plane parallel to the x-z plane is defined along the y axis at the midpoint of the fin base. Another symmetry plane would be a cutting plane parallel to the x-y plane that is defined along the z axis. The intention is to present the geometry scenario that is similar to that of the 2D. Either of the sliced volumes (left or right of the cutting plane) may be employed. For this example, the left portion of the solid body has been selected for further analysis.

FIGURE 54:  Creating the 3D slab geometry with rectangular cross section.

8.3.1 Case 1 Boundary Condition: Semi-Infinite Fin Figure 55 presents the semi-infinite solid where temperature at the tip equals that of the ambient. Temperature contours after one minute are shown as well. Symmetry plane is presented in Figure 56. By default, the remaining surfaces where boundary conditions have not been defined are assumed insulated. Comparison of the transient temperature profiles of the 1D, 2D, and 3D semi-infinite slabs is shown in Figure 57. The plot shows three very similar curves, attesting to the accuracy of the lowerdimension analyses. Note the temperature scale set from 20°C to 100°C. You may wish to zoom in to show different analysis time regions and make more accurate comparison. For the purpose of

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this investigation, the displayed accuracy suffices. Figure 58 compares the transient temperature profiles along the main axis of the semi-infinite slab for the 1D, 2D, and 3D cases after one minute. It is seen that a linear profile may be assumed for the stabilized transient temperature. A good technique to keep in mind is to vary the frequency of the markers representing different curves and also to cycle them in terms of design and color to ensure correct interpretation, especially if their curves are similar.

FIGURE 55:  Assigning tip boundary conditions for the 3D semi-infinite slab with rectangular cross section (t = 60 s).

FIGURE 56:  Assigning symmetry plane for the 3D slab with rectangular cross section.

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FIGURE 57:  Comparison of the 1D, 2D, and 3D transient temperature profiles for the semi-infinite slab at a midpoint location for the fixed tip temperature.

FIGURE 58:  Comparison of the 1D, 2D, and 3D transient temperature profiles along the fin length for the semi-infinite slab for fixed tip temperature (t = 60 s).

8.3.2 Case 2 Boundary Condition: Insulated Tip Figure 59 presents the insulated boundaries when Case 2 boundary condition is valid. Temperature contours within the 3D slab after one minute are given as well. Symmetry plane is the same as that presented in Figure 56. By default, the remaining surfaces where boundary conditions have not been defined are assumed insulated. Comparison of the transient temperature profiles of the 1D, 2D, and 3D fins with insulated tip is presented in Figure 60. Maximum

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temperature of over 85°C is observed after one minute. Figure 61 compares the transient temperature profiles along the main axis of the semi-infinite slab for the 1D, 2D, and 3D cases after one minute. The profiles are now nonlinear and are almost identical between the base and the midpoint; however, approaching the tip of the slab, the temperatures deviate slightly (about 1°C).

FIGURE 59:  Assigning insulated planes for the 3D slab with rectangular cross section (t = 60 s).

FIGURE 60:  Comparison of the 1D, 2D, and 3D transient temperature profiles for the semi-infinite slab at a midpoint location for the insulated tip.

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FIGURE 61:  Comparison of the 1D, 2D, and 3D transient temperature profiles along the fin length for the semi-infinite slab for the insulated tip (t = 60 s).

8.3.3 Case 3 Boundary Condition: Convective Surfaces Figure 62 presents the convective boundaries. The convective heat transfer equation shown in Heat Flux section indicates that the heat transfer at convective surfaces is proportional to the temperature difference between the ambient temperature and temperature at the surface of the slab adjoining to the ambient. T9 is the independent variable that is defined when model physics are set. External temperature type may be identified by means of the dropdown menu, either as a user-defined, by choosing; for example, the ambient temperature, or by linking it to any of the physics equations defined for the model. Contour profiles of the similar cases have been presented in Figure 63. It is observed that the case in which n = 100 (forced convection) shows the highest heat transfer to the surroundings. In other words, you may say that more heat is removed in the right plot as it shows lower temperatures on the cooler (blue) end. Temperature of the surroundings and ambient are assumed to be the same.

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FIGURE 62:  Assigning convective planes for the 3D slab with rectangular cross section.

FIGURE 63:  Comparison between the 3D transient analysis for the midpoint of the slab at t = 60 s:  a) Moderately convective surfaces, b) Highly convective surfaces.

Comparison of the transient temperature profiles of the 1D, 2D, and 3D fins with convective surfaces is presented in Figure 64. For all cases, heat transfer increases with increasing the convective coefficient from natural to forced convection, resulting in decrease of temperature values. Thus, for the 1D case, the convective heat transfer coefficient would only apply to the exterior farthest (tip) boundary point in Figure 37. The interval between the left and right points cannot have the boundary conditions as it is “inside” the 1D model. For 2D model, the line boundaries (Figure 45) identify the convective heat transfer boundary conditions. For 3D model three surfaces present the convective boundary condition (Figure 62). In all cases, you are increasing the area over which the heat is transferred. This illustrates the point that if you have significant heat transfer through the boundary, a 1D analysis may not be appropriate.

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Temperature decrease is greatest for the 3D analysis, followed by the 2D analysis. For the 2D and 3D models, where forced convection applies, temperatures stabilize at values over 45°C and 35°C, respectively. Figure 65 displays a decreasing convex profile for the temperature distribution along the fin length (x-coordinate) of the slab at a midpoint location and after one-minute exposure for both natural convection (n = 1) and forced convection (n = 100) cases.

FIGURE 64:  Comparison of the 1D, 2D, and 3D transient temperature profiles for the semi-infinite slab at a midpoint location for the insulated tip.

FIGURE 65:  Comparison of the 1D, 2D, and 3D transient temperature profiles along the fin length for the semi-infinite slab for the insulated tip (t = 60 s).

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8.3.4 Comparing the 3D Analyses for Three Boundary Conditions (Cases 1, 2, and 3) Figure 66 presents transient temperature at the slab midpoint for the three boundary conditions (i.e., fixed, insulated, and convective surfaces). Figure 67 compares the 3D transient temperature results along the fin length (x-coordinate) for the three boundary conditions (i.e., fixed, insulated, and convective surfaces). The semi-infinite slab (fixed tip temperature) diagram shows a linearly decreasing temperature from the fin base at 100°C to the tip at 20°C. The insulated case shows a decreasing convex diagram where the temperature difference between the base and tip is in the range of 20°C and is similar to the profile presented for that of the convective surfaces where natural convection case has been assumed. The forced convection case; however, shows approximately a 75°C temperature decrease after one-minute exposure.

FIGURE 66:  Comparison of the 3D transient temperature profiles for the semi-infinite slab at a midpoint location for the three boundary conditions.

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FIGURE 67:  Comparison of the 3D transient temperature profiles along the fin length for the semi-infinite slab for the three boundary conditions (t = 60 s).

8.4 THERMAL MODEL FOR THE 3D SLAB WITH CENTRAL COOLING CHANNEL Two approaches are taken when setting up this analysis: 1) adopting the same physics presented in the previous section where a 3D slab with rectangular cross section was created and solved for the Case 3 boundary condition and then adding a cooling channel, 2) conjugate heat transfer model, where physics related to thermally active components made of solids and fluids are presented individually and linked by a multiphysics node when solution is performed using non-isothermal flow physics. In conjugate heat transfer model, two individual physics are defined: one for solid and one for the liquid. The liquid flow is now modelled and heat is transferred by conduction and convection in the media. In other words, two sets of physics are solved and then responses are combined in a multiphysics node, meaning that the dependent variables such as temperature of the fluid nodes in the vicinity of the solid nodes affect the solid physics and vice versa. It is a two-way communication. Approach 1 is presented in Figure 69 where a solid model for the 3D slab with rectangular cross section including a central cooling

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channel is introduced. This model is built upon the previous slab solid model with convective boundary surfaces where initial as well as boundary conditions such as insulated surfaces, heat flux, fixed temperatures, and symmetry planes are defined. A liquid cooling agent (e.g., water) has been introduced as a solid sub-component with fluid properties within the solid heat transfer physics. Initial temperature of the still cooling liquid is set at 50°C. Additionally, it is assumed that a thin conductive layer with known thermo-physical properties (e.g., copper) separates the cooling channel from its surrounding metal (e.g., aluminium) so that it introduces some thermal resistance at the contact surface between the liquid and solid. The cooling channel inside is similar to a heat pipe. Heat pipes have applications in aerospace, cooling of the spacecraft system, and even cooking large pieces of meat. When making Thanksgiving turkey, heat can be transferred to the center of the roast by inserting one end of the heat pipe into it. The other side then receives the heat by convection and radiation from the oven. Cooling liquid is water in this case. The pipe consists of an interior copper wrapped in stainless steel and therefore is highly conductive at the center, allowing for heat concentration at desired location. Since simultaneous heating is done from both inside and outside, the cooking process is more efficient.

FIGURE 68:  Adding physics for the 3D slab with rectangular cross section and central cooling channel (approach 1).

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FIGURE 69:  Assigning boundary conditions for the 3D slab with rectangular cross section and central cooling channel (approach 1).

Approach 2 is presented in Figure 70, where a conjugate heat transfer model has been selected for the physics of a new 3D model with a transient analysis type. This approach defines two separate sub-physics, one for the liquid and one for the solid phases. Figure 71 shows the solid sub-physics data, which takes into account the fluid as well as a medium of heat transfer, meaning that convection in fluid is not taken into account. Inflow (defined by Inlet node in Laminar Flow in Figure 71) may be specified by flow velocity, mass flow rate, laminar inflow, fully developed flow, or pressure. The velocity may be either defined by its magnitude that is generally normal to the inlet plate or as a vector, which is applicable to the cases in which the flow enters the inlet at an angle, such as air duct inlets. Outflow (defined by Outlet node in Laminar Flow in Figure 71, or outflow node in heat transfer conjugate physics node) may be specified by the pressure, velocity, or laminar outflow. For the case of pressure outflow condition, the pressure difference from that of the surroundings is inputted to the physics, taking into account that the backflow may be suppressed and the flow may be chosen to be normal to the outlet plane. Figure 72 presents physics related to the fluid such as its regime (i.e., laminar in this case versus turbulent) and incorporates the ambient conditions, inflow, outflow,

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initial temperature, and symmetry planes. Note that the remaining surfaces that border the liquid and have undefined boundary conditions are treated as walls in fluid problems.

FIGURE 70:  Adding physics for the 3D slab with rectangular cross section and central cooling channel (approach 2).

FIGURE 71:  Assigning boundary conditions for the 3D slab with rectangular cross section and central cooling channel (approach 2).

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FIGURE 72:  Assigning cooling channel properties for the 3D non-isothermal flow within the rectangular cooling channel (approach 2).

Results of the analyses are shown in Figure 73 to Figure 75. Approach 1 results are identified as “Channel Temp” in the legend and are presented in Figure 73 to Figure 75, while Approach 2 results are identified as “Conjugate Tip” and are presented in Figure 75 to Figure 78. Figure 73a shows transient temperature profiles for a midpoint at the symmetry plane shown with a red dot. Temperature sensitivity to different liquid coolants (i.e., ethylene glycol, water, and mercury) has been investigated. Ethylene glycol, water, and mercury as cooling agents moderate the temperature in the range of 77°C to 82°C, with ethylene glycol the most efficient liquid coolant among the three. Figure 73b shows transient temperature profiles along the fin length (x-coordinate) after one-minute exposure along the line passing through the midpoint of the symmetry plane. The convective tip data is related to the case where no cooling channel existed. Forced convection (n = 100) still produces the greatest reduction of temperature of all cases shown in both Figure 73 and Figure 75, where temperature stabilizes at about 37°C (Figure 73) and 51°C (Figure 75).

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FIGURE 73:  Transient temperature profiles for the 3D slab with rectangular cross section and central cooling channel:  a) Temperature profiles versus time, b) Temperature profiles along the fin length (t = 60 s).

Figure 74 presents the contour plots for the slab with embedded cooling channel for three cooling liquids (i.e., ethylene glycol, water, and mercury). The base of the slab has the highest temperature (100°C); however, the liquid cooling agent enters the channel at 50°C and heats up as it passes through the channel; as the liquid approaches the end of the channel it heats up to its maximum temperature. Figure 75a shows transient temperature profiles for a midpoint at a distance away from the symmetry plane by half of cooling channel width shown with a red dot. Choice of this location is to facilitate the comparison between the uncooled and liquidcooled slabs. Temperature sensitivity to different liquid coolants (i.e., ethylene glycol, water, and mercury) has been investigated. Ethylene glycol, water, and mercury as cooling agents moderate the temperature in the range of 52°C to 92°C, with ethylene glycol the most efficient liquid coolant among the three. Figure 75b shows transient temperature profiles along the fin length (x-coordinate) at a distance of half of the cooling channel width away from the symmetry plane after one-minute exposure. The two “Convective Tip” plots (n = 1 and n = 100) compare the results for the case where no cooling channel existed.

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FIGURE 74:  Transient temperature contours for the 3D slab with rectangular cross section and central cooling channel at t = 60 s:  a) Ethylene Glycol, b) Water, c) Mercury.

FIGURE 75:  Transient temperature profiles for the 3D slab with rectangular cross section and central cooling channel: a) Temperature profiles versus time, b) Temperature profiles along the fin length (t = 60 s).

Figure 76, Figure 77, and Figure 78 present the contour plots for the slab with embedded cooling channel for the three cooling liquids. Temperature contours, shown in plots on the left of each figure, depict the distribution of the temperature within the depth of the slab along the fin length (x-coordinate). Plots on the right of each figure show the cross-section contour plots of the velocity magnitude profile for the flow inside the cooling channel. These plots show that water introduces the least resistance to the flow moment due to its low viscosity. Mercury introduces the most resistance to the flow due to its high viscosity. Initial conditions are the same as the ambient temperature (20°C) and water temperature at inlet is 50°C. This may be achieved by adjusting the flow or by using a cooling mechanism at the inlet before water flows into the cooling channel.

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FIGURE 76:  Transient analysis for the 3D slab with rectangular cross section and central cooling channel filled with mercury at t = 60 s:  a) Transient temperature contours, b) Cross sectional channel velocity profiles.

FIGURE 77:  Transient analysis for the 3D slab with rectangular cross section and central cooling channel filled with water at t = 60 s:  a) Transient temperature contours, b) Cross sectional channel velocity profiles.

FIGURE 78:  Transient analysis for the 3D slab with rectangular cross section and central cooling channel filled with ethylene glycol at t = 60 s:  a) Transient temperature contours, b) Cross sectional channel velocity profiles.

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CHAPTER

9

FIN WITH CIRCULAR CROSS SECTION This section focuses on another form of the extended surfaces (or fins)—a cylinder (Figure 79). First, a basic design is presented. Next, a more complex design is introduced that embeds a rectangular cooling channel into the cylindrical fin. A cylindrical fin with a finned cooling channel (which may be employed in aerospace applications) concludes the section.

FIGURE 79:  Fin with circular cross section.

Figure 79 shows the energy balance for an element within the fin. Equations for three boundary condition cases are displayed in the figure. For the upcoming analyses, the most comprehensive

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scenario (Case 3) is investigated, in which the surfaces adjoining to the environment are cooled by convection. The effects of forced or natural convection are represented by a multiplier that changes the base condition from the natural to forced convection scenario. Steps taken to setup the 3D model are similar to those of the previous analyses. You need to create a new model or component on the upper tree, add a 3D component, define physics (heat transfer in solids), and decide which analysis type you are interested in performing (transient analysis). You may either import the 3D geometry using a dedicated CAD tool or create it using the built-in geometry-creation features within the core COMSOL Multiphysics® application. The geometry for this analysis is simple enough to have been created using the latter approach. Figure 80 shows a block—representing the surface whose heat is to be transferred to the surrounding—and a cylinder of 10 mm diameter and 100 mm length comprising the model geometry. To take advantage of the symmetry, the model may be cut in half by employing a work-plane. Thus, a work-plane ­parallel to x-z plane is defined that passes through the model center. This plane is then used to partition the model into two halves ­(indicated by purple line in the figure). The left half is kept, and the right half is discarded to create the final geometry, which takes advantage of symmetry to save on computational time and memory resources. There are a variety of methods to mesh a cylindrical geometry similar to the one presented in this example; one method is to use the mapped mesh feature by selecting a (source) surface, creating the mapped mesh, and finally sweeping the results to another ­(destination) surface. Thus, one end surface was meshed first and then used as the source, and the opposite face was used as the destination to perform mapped meshing. One advantage of mapped meshing is that it introduces uniformly distributed surface or volume elements in a geometry, which results in more accurate and efficient solutions. Note that some surfaces need to have the least number of surrounding constraints in order for the mapped mesh to be swept, meaning that the surfaces used as the

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source or destination to perform mapped meshing cannot be over-­ constrained (i.e., h ­ aving too many adjoining surfaces). Figure 81 shows the meshed cylindrical fin; you may start with 0.75 mm mesh size. In this example, the cylindrical curved surface and the ­symmetry plane is selected as the surfaces to be mapped and the volume as the domain to be swept. Figure 82 shows constant temperature (100°C) applied to the fin’s base by using a fixed-temperature boundary condition. The symmetry plane BC is set to insulated. The rest of the fin’s exposed surfaces transfer heat by convection to the surrounding ambient. Radiation heat transfer effect is negligible compared to that of the heat convection effect. Note that this term can either be defined as a relationship similar to that of the convection equation in which the fourth power temperature difference can be expressed in terms of difference in squares. The multiplier of the first order temperature difference is the equivalent heat convection coefficient. This equivalent coefficient takes into account the radiation constants in addition to the ambient and material emissivity. Radiation boundary conditions may also be applied to the physical system.

FIGURE 80:  Creating geometry and adding physics for the 3D slab with circular cross section.

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FIGURE 81:  Meshing the 3D slab with circular cross section.

Temporal and spatial analysis results are given in Figure 83, which also include similar results for the rectangular cross section case for comparison. Figure 83a shows temperature as a function of time for a midpoint location indicated by a red dot in the cross-sectional sketch. The four curves appear as two groups: One group corresponds to natural and the other to forced convection cases. The latter case is clearly the most efficient cooling method, moderating the maximum temperature to over 35°C (versus 85°C for the natural convection). Circular fin curves are very close to those of the rectangular fin. Decreasing convex temperature profiles along the fin length (x-coordinate shown in Figure 83b as red line in the 3D sketch) are similar to those previously presented for the rectangular fin. For the forced convection case, the fin’s tip temperature reaches about 25°C, while in the case of natural convection, the fin’s tip temperature reaches 80°C, attesting to the effectiveness of the forced convection technique.

FIGURE 82:  Assigning base boundary conditions for the 3D slab with circular cross section.

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FIGURE 83:  Transient temperature profiles for the 3D slab with circular cross section: a) Temperature profiles versus time, b) Temperature profiles along the fin length (t = 60 s).

Temperature contour plots are shown in Figure 84 for the cases where n = 1 (natural convection) and n = 100 (forced convection) after one-minute exposure time. Temperature gradient for the natural convection is less compared to that of the forced convection, visible through transition of colors from the coldest point (indigo) to the warmest point (crimson). Figure 85 shows similar contour plots where temperature contours have been sliced in equal intervals. A sliced contour plot helps with identifying temperature distribution within the depth of the solid model. No variation through the depth can be observed in these plots.

FIGURE 84:  Transient temperature contours for the 3D slab with circular cross section at t = 60 s: a) n = 1, b) n = 100.

FIGURE 85:  Transient temperature contour slices for the 3D slab with circular cross section at t = 60 s: a) n = 1, b) n = 100.

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9.1 FIN WITH CIRCULAR CROSS SECTION AND A CENTRAL COOLING CHANNEL To generate this model, you can either create a new model or add a new component to the upper tree, select a 3D geometry component, define physics (conjugate heat transfer), and select the analysis type you would like to perform (transient analysis). The geometry for this analysis is similar to that of the previous one—a fin with circular cross section—with an additional central rectangular cooling channel (2.5 mm × 5 mm) created using COMSOL core application geometry creation capability (Figure  86). You may employ the work-plane generated in the previous example to partition the central cooling channel. Meshing this solid model can be achieved by either meshing the source and sweeping the plane elements to the destination surface by predefined element depth or by selecting individual geometry components and meshing them in multiple steps. You may choose tetrahedral over brick elements when there is space limitation or irregularities such as curvatures (Figure 87). Initial temperatures for both the solid and fluid have been defined. Base boundary conditions differ from the initial conditions, where a fixed or varying temperature (as a function of time or location) is applied to the nodes throughout the geometry (Figure 88). Figure 88 shows a constant temperature (100°C) applied to the fin’s base (a fixed-temperature boundary condition). The symmetry plane is insulated. The rest of the fin’s exposed surfaces transfer heat by convection to the surrounding ambient.

FIGURE 86:  Setting up the 3D slab with circular cross section and central cooling channel.

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FIGURE 87:  Meshing the 3D slab with circular cross section and central cooling channel.

FIGURE 88:  Assigning base boundary conditions for the 3D slab with circular cross section and central cooling channel.

Temporal and spatial analysis results are given in Figure 89. Figure 89 a and b show temperature profiles as a function of time and along the fin length at a midpoint location at the ­distance half width of the cooling channel away from the symmetry plane, ­respectively, with a red dot or line in the embedded sketch. The three groups of the curves represent the natural and forced convection

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for cylinder with no channel, as well as with embedded cooling channel and three different coolants. The cooling channel quickly moderates the temperature to a maximum of 50°C. Introducing a forced convection technique with a high convective coefficient is not always feasible (e.g., due to the space limitations); therefore, implementing liquid cooling technique appears to be an effective approach. Additionally, temperature in the vicinity to the cooling channel decreases at a faster rate compared to that of the forced convection, which is beneficial. This can be seen by variation of temperature along the fin length. Water is the most effective coolant, followed by ethylene glycol and transformer oil, moderating a fin’s tip temperature to 57°C, 62°C, and 66°C, respectively. Oil can dissipate heat for temperatures above 100°C due to its higher vaporization temperature compared to water. That makes oil a good candidate for applications where temperature exceeds water’s boiling point (at atmospheric pressure). Different types of oil are used in cooling applications; for example, in aerospace and electronics fields. Oil type may even vary in one application depending on the life cycle of the component; for example, the oil used for operating a new aircraft engine is a mineral oil that is a replacement for the oil mixed with an anticorrosive agent when the engine was first manufactured and stored. The oil that is used after 25-50 hours of engine operation is an ashless dispersant oil. Each have their own thermo-physical properties as well as dynamic viscosity that represents the dependency of shear stress on the gradient of the velocity of the coolant moving tangentially with respect to the part to be cooled. Engine oil fulfils multiple functional needs: cooling, lubricating, sealing, and cleaning. Transformer oil thermally manages the excess heat, and also provides an insulating function. Temperature along the height of the slab is shown in Figure 90 for natural (n = 1) and forced (n = 100) boundary conditions. It is seen that the temperature of the cooling water that is originally at 50°C, decreases to about 34°C for the forced case. This further confirms the effectiveness of forced versus natural convection cooling techniques. The sharp temperature gradient at the interface

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between the fluid and solids is due to the change of thermophysical properties from one medium to the other, keeping in mind that the thermal conductivity of copper slab is over 600 times that of the liquid cooling agents flowing inside the copper slab. Note that a thin conductive layer was defined for the solid-fluid interface to introduce the resistance that this layer may cause to the heat flow. This layer may be particularly useful if the solid surrounding the cooling channel has very small thickness, or if the interface is coated with different material; for example, a wick structure in a heat pipe scenario. Temperature contour plots are shown in Figure 91 for different liquid coolants (i.e., transformer oil, ethylene glycol, and water) after one-minute exposure time. Looking at the color distribution, noting that contour scales are similar, you may see that water as a liquid cooling agent is the most effective. You can observe that by extension of the blue color representing lower temperatures through the cylinder. Figure 92 shows plots where temperature contours have been sliced at equal intervals. It is seen that the temperature distribution within the slice is relatively constant, with the exception of the boundary around the cooling channel, where temperature gradient varies from zero in the mid-region between the external surface and cooling channel wall to its maximum value at the cooling channel wall.

FIGURE 89:  Transient temperature profiles for the 3D slab with circular cross section and central cooling channel: a) Temperature profiles versus time, b) Temperature profiles along the fin length (t = 60 s).

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FIGURE 90:  Transient temperature profiles along the height of the part for the 3D slab with circular cross section and central cooling channel (t =0 s, 60 s), forced convection (n = 100).

FIGURE 91:  Transient temperature contours for the 3D slab with circular cross section and central cooling channel at = 60 s: a) Transformer oil, b) Ethylene glycol, and c) Water.

FIGURE 92:  Transient temperature contour slices for the 3D slab with circular cross section and central cooling channel at t = 60 s: a) Transformer oil, b) Ethylene glycol, and c) Water.

9.2 FIN WITH CIRCULAR CROSS SECTION AND FINNED CENTRAL COOLING CHANNEL Use of extended surfaces as an efficient thermal management method in conjunction with other cooling methods is an accepted practice. In a majority of these cases, choice of material is as important as the geometry itself. Heat pipe is a cooling mechanism that has been effectively employed in space applications. Heat pipes consist of a wick structure that directs the heated fluid

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from the warmer side to the cooler side by means of capillary forces. The heated molecules that are in gaseous state (vapor) travel upstream to the cool side (condenser) by means of capillary forces, lose the excess heat, and form liquid, which then travels to the warm end (evaporator) of the pipe and the cooling cycle continues. In some aerospace applications aluminum sheets were used as the heat pipe envelope[13]. The application in this case was to ensure that the space nuclear systems are operating within the recommended temperature range of 130°C to 280°C. Although aluminum is easily machinable, allowing for manufacture of interior longitudinal grooves to increase surface area of the heat pipe envelope and consequently achieve a more efficient heat transfer, operating conditions within the specified temperature range are not as desirable. Therefore, titanium has been suggested as an alternative that has high strength-to-weight ratio. Those characteristics, in addition to its anticorrosive properties, make it a desirable material in aerospace applications. The main challenge in using the titanium is its machinability. The following case study presents a modified heat pipe that consists of a cylindrical fin which embeds a finned cooling channel (Figure 93).

FIGURE 93:  Setting up the 3D slab with circular cross section and finned central cooling channel.

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The geometry employed in the previous example is adapted to incorporate the abovementioned longitudinal grooves, as well as water as the liquid cooling agent. The purpose is not to incorporate separate physics to model the heat gain or release due to vaporization and condensation as seen in a heat pipe, but to simplify the spacecraft cooling mechanism to better understand the importance of extended surfaces for such applications.

FIGURE 94:  Meshing the 3D slab with circular cross section and finned central cooling channel: a) Fin full length, b) Fin close up.

To generate this model, you can either create a new model or add a component to the upper tree, choose a 3D geometry component, define physics (conjugate heat transfer), and select the analysis type (transient analysis). Note that for this scenario, conjugate heat transfer modelling is employed where the physics for the solid and fluid are joined using a multiphysics node. The geometry for this analysis is cylindrical, as in the previous case, but with a round finned cooling channel. Use of a circular pattern when creating this channel is recommended. Using COMSOL Multiphysics® equipped with the geometry importer module, you are able to create rectangular patterns, which may

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be done in COMSOL Multiphysics®. The geometry in this case study has been created using Solid Edge® (a mid-level CAD tool by Siemens PLM). This method is recommended when creating complex geometries. Note that when importing geometry into the model, you need to ensure that volumes are imported intact (meaning that the surfaces forming them are closed entities) so that they are distinguishable when setting up the components and associating material groups with them. Meshing is similar to the previous case. Care is to be taken to ensure the minimum element size does not exceed the minimum side length of the interior fin’s edges (Figure 94). If you choose the free mesh option, the geometry will be filled with tetrahedral elements—these are higher order elements (i.e., brick elements with central nodes). This method may be recommended in cases such as this, although the geometry may be divided into regions and map and sweep features may be employed. Different base temperatures for the boundary conditions are defined for the solid (100°C) and the liquid (50°C) components (Figure 95). Temperature for initial conditions is assumed to equal that of the ambient. The symmetry plane is set as insulated. The rest of the fin’s exposed surfaces transfer heat by convection to the surrounding ambient.

FIGURE 95:  Assigning base boundary conditions for the 3D slab with circular cross section and finned central cooling channel: a) Fin full length, b) Fin close up.

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110  •  Heat Transfer Modelling Using COMSOL Slab to Radial Fin

Temporal and spatial analysis results are given in Figure 96 and Figure 97. Figure 96 and Figure 97 show temperature profiles as a function of time and along the fin length at a midpoint location presented respectively with a red dot or a line in the cross-sectional sketch. The three main groups of the diagrams seen herein represent the natural and forced convection for the fin with circular cross section, cylindrical fin with embedded unfinned cooling channel, and cylindrical fin with embedded finned cooling channel. For the liquid cooled cases, the effect of liquid cooling agents (i.e., water, ethylene glycol, and transformer oil) has been examined as well. Assuming that the interior diameter of the finned channel equals the diameter of the unfinned channel (10 mm) (due to the increased finned surface area, you expect a more moderate temperature distribution along the fin compared to that of the unfinned case). In this case study; however, the outer diameter of the finned cooling channel equals the diameter of the unfinned surface. Nevertheless, given the fin’s thickness (0.875 mm – 1.75 mm), the surface areas are similar. This explains the small temperature difference (under 5°C) observed between these two cases. The cooling channel moderates the temperature to 50°C on average. Introducing a forced convection technique with a convective coefficient of 100 times than that of the original one (natural convection) introduces a more effective heat transfer mechanism than the liquid cooling.

FIGURE 96:  Transient temperature profiles for the 3D slab with circular cross section and finned central cooling channel.

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Fin with Circular Cross Section  •  111

FIGURE 97:  Transient temperature profiles along the fin length for the 3D slab with circular cross section and finned central cooling channel (t = 60 s).

Figure 98 shows temporal and spatial temperature profiles for water as the cooling agent for locations shown in Figure 99 for three scenarios: without the cooling channel, with rectangular cooling channel, and with finned cooling channels. It is seen that the finned channel is more effective in terms of heat transfer than the rest of the methods. Figure 100 shows temperature along a line perpendicular to the fin length in the middle of the fin along the lines shown in Figure 101. It is seen that temperature distribution is even for the natural (n = 1) and forced (n = 100) convection cases; however, it shows a variation for the fins with embedded cooling channels. At the symmetry plane of the fin where liquid is flowing, temperature remains at the liquid cooling temperature (approximately 50°C) and as the liquid finds more chance to transfer heat to its surroundings, the temperature cools down.

FIGURE 98:  Transient temperature profiles for the 3D slab with circular cross section and finned central cooling channel for water: a) Temperature profiles versus time, b) Temperature profiles along the fin length (t = 60 s).

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112  •  Heat Transfer Modelling Using COMSOL Slab to Radial Fin

FIGURE 99:  Location of points at which transient temperature profiles for the 3D slab with circular cross section has been shown: a) Circular cross section, b) Circular cross section with rectangular channel, c) Finned circular cross section.

FIGURE 100:  Transient temperature profiles for the 3D slab with circular cross section and finned central cooling channel along the fin depth (t = 60 s).

FIGURE 101:  Location of lines transverse to the cooling channel for which transient temperature profiles for the 3D slab with circular cross section have been shown: a) Circular cross section, b) Circular cross section with rectangular channel, c) Finned circular cross section.

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Fin with Circular Cross Section  •  113

Temperature contour plots are shown in Figure 102 for water as the liquid cooling agent after one-minute exposure time. Looking at the color distribution, you see that water (indigo color) is cooler than the surrounding metal. Figure 103 shows similar contour plots where temperature contours have been sliced at equal intervals. It is seen that the temperature distribution within the metal area of slice is relatively even, with the exception of the boundary around the cooling channel, where there is a greater temperature gradient than that of the unfinned cooling channel scenario.

FIGURE 102:  Transient temperature contours for the 3D slab with circular cross section and finned central cooling channel at t = 60 s: a) Fin close up, b) Fin full length.

FIGURE 103:  Transient temperature contour slices for the 3D slab with circular cross section and finned central cooling channel at t = 60 s: a) Fin close up, b) Fin full length.

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SIDE-RECTANGULAR FIN WITH TRIANGULAR CROSS SECTION To generate this model, you can either create a new model or add a new component to the upper tree, select a 3D geometry component, define physics (solid heat transfer), and select analysis type (transient analysis). Figure 104 shows the applicable physics and the three possible boundary condition cases; Case 3, where surfaces transfer heat by convection, has been considered here.

FIGURE 104:  Side-rectangular fin with triangular cross section.

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The geometry is created using COMSOL Multiphysics® core application (Figure 105). One way to create this geometry is to 1) create a block, 2) define an xz work-plane that is rotated 45 degrees about the y-coordinate, and 3) partition the block volume and delete the top half (delete entity). Another method is to draw the triangular profile on the yz work-plane and then extrude it along the x-coordinate by the length of the fin. Meshing is achieved by first meshing the source face (the triangle at the fin tip) and sweeping the plane elements to the destination surface (opposite triangular face) by predefined element depth. Another technique is to select individual geometry components and mesh them in multiple steps. Tetrahedral elements may be by-products of the latter approach to compensate for the non-rectangular parts. The transition to the tetrahedral elements occurs gradually from elements with aspect ratios closer to those of the brick to the tetrahedral elements (Figure 105). One base temperature for the boundary conditions associated with the base is defined herein; it belongs to the main body of the skillet part that is partially represented by a vertical slab adjoining the solid fin (Figure 106). It is only included for a visual effect and is not part of the FEA model. The initial condition temperature is the ambient (20°C) and it is applied uniformly to the nodes throughout the geometry. Figure 106 presents a constant temperature (100°C) applied to the fin’s base. No symmetry plane has been defined for this scenario due to its asymmetrical geometry. The rest of the fin’s exposed surfaces transfer heat by convection to the surrounding ambient.

FIGURE 105:  Meshing the 3D rectangular slab with triangular cross section.

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Side-Rectangular Fin with Triangular Cross Section  •  117

FIGURE 106:  Assigning base boundary conditions for the 3D rectangular slab with triangular cross section.

Temporal and spatial analysis results are given in Figure 107 and Figure 108. Figure 107a shows temperature as a function of time for a midpoint location indicated by a red dot in the crosssectional sketch. The three main groups of the curves seen herein represent the natural (n = 1) and forced (n = 100) convection for the rectangular, circular, and triangular cross sections. It is seen that the forced convection grants the most efficient cooling mechanism moderating the peak temperatures to over 37°C for rectangular and circular cross sections and 31°C for the triangular one. When natural convection cooling is in place, maximum temperature is 86°C for rectangular and circular cases and 83°C for the triangular case. Figure 107b shows transient temperature profiles versus the fin length (red line along the x-coordinate in the 3D sketch). The profile is a decreasing convex curve that eventually moderates to 27°C for rectangular and circular cases and 23°C for the triangular case at the tip of the fin for the forced convection boundary condition. For the natural convection case, the maximum temperatures are 79°C (rectangular/circular) and 77°C (triangular) at the tip of the fin. Figure 108 focuses on the comparison between the temporal and spatial temperature profiles for the rectangular and triangular cross

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sections. Note that the fin with rectangular cross section is slightly less effective than the fin with triangular cross section due to the more heat to be dissipated from the surface area for the case of natural convection (n = 1). In forced convection scenario (n = 100), this difference becomes more pronounced.

FIGURE 107:  Transient temperature profiles for the 3D rectangular slab with triangular cross section:  a) Temperature profiles versus time, b) Temperature profiles along the fin length (t = 60 s).

FIGURE 108:  Comparison between transient temperature profiles for the 3D rectangular slabs with triangular and rectangular cross sections:  a) Temperature profiles versus time, b) Temperature profiles along the fin length (t = 60 s).

Temperature contour plots are shown in Figure 109 for the cases where n = 1 (natural convection) and n = 100 (forced convection) after one-minute exposure time. Temperature gradient for the natural convection is less compared to that of the forced convection, as the fin is more uniformly hot from the base to tip. Figure 110 shows similar contour plots where temperature contours have been sliced at equal intervals. A sliced contour plot helps with identifying temperature distribution within the depth of the solid model—along the y-coordinate, which shows a uniform temperature in this case.

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FIGURE 109:  Transient temperature contours for the 3D rectangular slab with triangular cross section at t = 60 s:  a) n = 1, b) n = 100.

FIGURE 110:  Transient temperature contour slices for the 3D rectangular slab with triangular cross section at t = 60 s:  a) n = 1, b) n = 100.

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SIDE-TRIANGULAR FIN WITH RECTANGULAR CROSS SECTION To generate this model, you can either create a new model or add a new component to the upper tree, select a 3D geometry component, define physics (solid heat transfer), and select analysis type (transient analysis). Figure 111 illustrates the applicable physics and the three boundary condition cases; in this study surfaces transfer heat by convection (Case 3).

FIGURE 111:  Fin with side-triangular profile and rectangular cross section.

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122  •  Heat Transfer Modelling Using COMSOL Slab to Radial Fin

The geometry is created using COMSOL Multiphysics® core ­application tools (Figure 112). One method to create the geometry is by starting from a rectangular slab and then define an xy work-plane that is rotated 22.5 degrees about the y-coordinate in order to partition the volume and delete the top half of the 3D fin. Another method to define the two cutting planes is by three points (vertices). Make two points at the triangle tip and one point at the upper and lower corners each where the base meets the “skillet”. Another method is to draw the triangular profile on the xz work-plane and then extrude it along the y-coordinate by the depth of the fin. Meshing is accomplished by meshing the source surface and sweeping the plane elements to the destination surface by predefined element depth or by selecting individual geometry components and meshing them in multiple steps. The upper and lower boundary planes may be selected as the mapped surfaces, and the slab body is the domain in which sweeping is to be performed for. Mapped meshing and sweeping afterwards make it possible for uniformly-distributed brick elements to be formed as the primary element shapes and thus rectangular cross section is maintained even after tapering edges (Figure 112). Base boundary condition temperature is defined herein, which belongs to the main body of the skillet part partially represented by a vertical slab adjoining the solid fin (Figure 113). The initial condition is assumed the same as that of the ambient (20°C) and is applied uniformly to the nodes throughout the geometry. Figure 113 shows a constant temperature (100°C) applied to the fin’s base. A symmetry plane dividing the solid part into half along the y-coordinate has been implemented by means of a third (xz) work-plane. The rest of the fin’s exposed surfaces transfer heat by convection to the surrounding ambient.

FIGURE 112:  Meshing the 3D side-triangular slab with rectangular cross section.

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Side-Triangular Fin with Rectangular Cross Section  •  123

FIGURE 113:  Assigning base boundary conditions for the 3D side-triangular slab with rectangular cross section.

Temporal and spatial analysis results are given in Figure 114 and Figure 115. Figure 114a shows temperature as a function of time for a midpoint location indicated by a red dot in the cross-sectional sketch. The four main groups of the curves seen herein represent the natural (n = 1) and forced (n = 100) convection for the ­rectangular, circular, triangular cross section, and side-triangular fins. It is seen that the natural convection (n = 1) results in the temperature profiles peaking at about 96°C, 86°C, and 83°C for side-triangular, ­rectangular/circular and triangular cross section fins, respectively. At the tip of the slab, temperature profiles stabilize at 93°C, 79°C, and 77°C, respectively, for the aforementioned geometries. Forced convection (n = 100) generates the most efficient heat transfer from the slab generating peak temperatures to over 39°C, 37°C and 31°C for side-triangular, rectangular/circular and triangular cross section fins. The tip of the slab for this case reaches 23°C, 27°C and 22°C for the side-triangular, rectangular, and triangular cross section fins, respectively. The slab with triangular sides has higher surface-to-volume ratio near the tip compared to that of the rectangular side, and that explains the faster temperature drop for the fins with triangular sides for the forced convection case. Figure 115 zooms on the comparison between the temporal and spatial temperature profiles for the sidetriangular and rectangular cross section fins. Note that the fin with rectangular cross section is more effective than the side-triangular

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124  •  Heat Transfer Modelling Using COMSOL Slab to Radial Fin

fin due to its surface area for both free (n = 1) and forced (n = 100) convection scenarios. The difference is more significant in natural convection (n = 1), while in forced convection (n = 100), temperature profiles converge to similar values after one-minute exposure.

FIGURE 114:  Transient temperature profiles for the 3D side-triangular slab with rectangular cross section: a) Temperature profiles versus time, b) Temperature profiles along the fin length (t = 60 s).

FIGURE 115:  Comparison between transient temperature profiles for the 3D side-triangular and rectangular cross section slabs: a) Temperature profiles versus time, b) Temperature profiles along the fin length (t = 60 s).

Temperature contour plots are shown in Figure 116 for the cases where n = 1 (natural convection) and n = 100 (forced convection) after a one-minute exposure time. Temperature gradient for the natural convection is less steep compared to that of the forced convection. Figure 117 shows similar contour plots where temperature contours have been sliced at equal intervals. The sliced contour plots show nearly constant temperature in this case.

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Side-Triangular Fin with Rectangular Cross Section  •  125

FIGURE 116:  Transient temperature contours for the 3D side-triangular slab with rectangular cross section at t = 60 s: a) n = 1, b) n = 100.

FIGURE 117:  Transient temperature contour slices for the 3D side-triangular slab with rectangular cross section at t = 60 s: a) n = 1, b) n = 100.

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SIDE-CONCAVE FIN WITH RECTANGULAR CROSS SECTION Figure 118 presents the applicable physics and the three boundary condition cases; in this study surfaces transfer heat by convection (Case 3). You can either create a new model or add a new 3D component to the upper tree, define physics (solid heat transfer), and select the analysis type (transient analysis). The geometry for this analysis is similar to that of the side-triangle fin with rectangular cross section and modified work-planes presented in Figure 111.

FIGURE 118:  Fin with side-concave profile and rectangular cross section.

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128  •  Heat Transfer Modelling Using COMSOL Slab to Radial Fin

You may either create the 3D geometry by importing a solid geometry that is generated in a CAD module such as SolidWorks® or generate the geometry using COMSOL. For the following example, the 3D geometry has been created in COMSOL Multiphysics® using blocks to represent the elongated rectangular fin and the skillet body. Parametric surfaces then have been introduced to represent the upper and lower concave surfaces, which have been used as partitioning work-planes as well. The upper and lower solid entities remaining after partition have then been deleted. Think of creating geometry in COMSOL Multiphysics® as moulding a cake using a Pumice paste, starting with a block and then carving out the upper and lower parts in a concave curve pattern. You may choose any planes to setup the parametric curved surfaces and rotate them over the third coordinate to define your work-planes. Ensure that you pick the correct axes when defining the work-planes. Use of variables in addition to parameters is recommended when setting up the geometry to provide flexibility when revising dimensions or for parametric solution sweeps. To mesh this geometry, you may choose the front surface as the source and the back one as the destination. It is also possible to choose the upper and lower surfaces; however, due to tapering the fin in a curved shape, there may be some challenges when setting up the element size (Figure 119). Tetrahedral elements primarily fill the solid geometry (Figure 119). Fixed temperature has been assumed for the skillet main body in the vicinity of the fin at 100°C (Figure 120). The initial condition is set at the uniformly applied ambient (20°C). A symmetry plane dividing the solid part into half along the y-coordinate has been implemented by means of a xz work-plane. The rest of the fin’s exposed surfaces transfer heat by convection to the surrounding ambient.

FIGURE 119:  Meshing the 3D side-concave slab with rectangular cross section.

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Side-Concave Fin with Rectangular Cross Section  •  129

FIGURE 120:  Assigning base boundary conditions for the 3D side-concave slab with rectangular cross section.

Temporal and spatial analysis results are given in Figure 121 and Figure 122. Figure 121a presents temperature as a function of time for a midpoint location indicated by a red dot in the cross-­sectional sketch. The five main groups of the curves represent the natural (n = 1) and forced (n = 100) convection for the side-­concave, ­rectangular, circular, triangular cross section, and side-­triangular fins. It is seen that the natural convection (n = 1) results in the temperature peaking at about 98°C, 96°C, 86°C, and 83°C for side-concave, side-triangular, rectangular/circular and triangular cross section fins, respectively. At the tip of the slab, temperature reaches 79°C, 93°C, 79°C, and 77°C, respectively, for the aforementioned geometries. Forced convection (n = 100) generates the most efficient heat transfer from the slab with peak temperatures 36°C, 39°C, 37°C and 31°C for side-concave, side-triangular, and rectangular/circular and triangular cross section fins. The tip of the slab for this case reaches 20°C, 23°C, 27°C, and 22°C for the side-­concave, side-triangular, and rectangular/circular and ­triangular cross section fins, respectively. The slab with concave sides has higher surface-to-volume ratio towards the tip compared to the rectangular side fin, which explains the faster temperature drop for the fins with concave side and rectangular cross section for the forced convection case. Figure 122 zooms in on the comparison between the temporal and spatial temperature profiles for the sideconcave and rectangular cross section fins. Note that the fin with

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130  •  Heat Transfer Modelling Using COMSOL Slab to Radial Fin

rectangular cross section is more effective than the side-concave fin due to its surface area for both free (n = 1) and forced (n = 100) convection scenarios. The difference is more significant in natural convection (n = 1), while in forced convection (n = 100), temperature profiles converge to similar values after one-minute exposure.

FIGURE 121:  Transient temperature profiles for the 3D side-concave slab with rectangular cross section: a) Temperature profiles versus time, b) Temperature profiles along the fin length (t = 60 s).

FIGURE 122:  Comparison between transient temperature profiles for the 3D side-concave and rectangular cross section slabs: a) Temperature profiles versus time, b) Temperature profiles along the fin length (t = 60 s).

Temperature contour plots are shown in Figure 123 for the cases where n = 1 (natural convection) and n = 100 (forced convection) after a one-minute exposure time. Temperature gradient for the natural convection is much less than those for the previous geometries exposed to similar boundary conditions. Forced convection introduces a smooth temperature transition from the base to the tip of the fin. Figure 124 shows similar contour plots where temperature contours have been sliced at equal intervals. It is seen that temperature distribution is relatively uniform along the fin depth.

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Side-Concave Fin with Rectangular Cross Section  •  131

FIGURE 123:  Transient temperature contours for the 3D side-concave slab with rectangular cross section at t = 60 s: a) n = 1, b) n = 100.

FIGURE 124:  Transient temperature contour slices for the 3D side-concave slab with rectangular cross section at t = 60 s: a) n = 1, b) n = 100.

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SIDE-CONVEX FIN WITH RECTANGULAR CROSS SECTION Figure 125 presents the applicable physics as well as the three-boundary condition cases, with Case 3 being assumed in this study. You can either create a new model or add a new 3D component to the upper tree, define physics (solid heat transfer), and select the analysis type (transient analysis). The geometry for this analysis is similar to that of the side-concave fin with rectangular cross section and modified work-planes presented in Figure 118.

FIGURE 125:  Fin with side-convex profile and rectangular cross section.

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134  •  Heat Transfer Modelling Using COMSOL Slab to Radial Fin

You may either create the 3D geometry by importing a CAD generated solid geometry, or produce the geometry in COMSOL internally. For the following example, the 3D geometry has been created in COMSOL Multiphysics® using blocks to represent the elongated rectangular fin and the skillet body. Convex parametric surfaces then have been used to represent the upper and lower surfaces. These surfaces are used as work-planes that partition the block shape so that it is divided into entities from which the extraneous ones are removed (Figure 125). It is recommended to use variables whenever possible when setting up the geometries. .

Similar to the previous case, to mesh this geometry, you may choose the front surface as the source and the back one as the destination (Figure 126). Tetrahedral elements primarily fill the solid geometry (Figure 126). The skillet main body is at the fixed temperature of 100°C (Figure 127). The initial condition is assumed to be the ambient (20°C) applied throughout the geometry. An xz symmetry plane divides the solid component along the y-coordinate. The rest of the fin’s exposed surfaces transfer heat by convection to the surrounding ambient. The purpose is to compare the results of this analysis with those of the previous one; therefore, surfaces have similar curvatures.

FIGURE 126:  Meshing the 3D side-convex slab with rectangular cross section.

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Side-Convex Fin with Rectangular Cross Section  •  135

FIGURE 127:  Assigning base boundary conditions for the 3D side-convex slab with rectangular cross section.

Temporal and spatial analysis results are given in Figure 128 and Figure 129. Figure 128a presents temperature as a function of time for a midpoint location indicated by a red dot in the cross-sectional sketch. The six main groups of the curves represent the natural (n = 1) and forced (n = 100) convection for the side-convex, sideconcave, rectangular/circular, side-triangular, and triangular cross section fins. It is seen that the natural convection (n = 1) results in the temperature profiles of about 92°C, 98°C, 96°C, 86°C, and 83°C for the above sequence of geometries. At the tip of the slab, temperature profiles stabilize at 89°C, 79°C, 93°C, 79°C, and 77°C for these geometries. Forced convection (n = 100) generates the most efficient heat transfer from the slab generating peak temperatures of 38°C, 36°C, 39°C, 37°C, and 31°C. The tip of the slab for this case reaches temperatures of 26°C, 20°C, 23°C, 27°C, and 22°C for the same geometry set. Figure 129 focuses on the comparison between the temporal and spatial temperature profiles for the side-convex and rectangular cross section. It is seen that the fin with rectangular cross section is more thermally effective than the side-convex fin due to its surface area for both free (n = 1) and forced (n = 100) convection scenarios. The difference is more significant in natural convection (n = 1) while in forced convection (n = 100), temperature profiles converge to similar values after one-minute exposure. Figure 130 shows the comparison between the side-concave and side-convex fins. It is seen that the side-convex fin is more thermally effective than the side-concave fin due to the larger surface area. Transient

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temperature distributions at a midpoint location along the length of the slab for the case of free convection (n = 1) is deviating between the two geometries while for the forced convection (n = 100), temperature distributions converge to similar values.

FIGURE 128:  Transient temperature profiles for the 3D side-convex slab with rectangular cross section: a) Temperature profiles versus time, b) Temperature profiles along the fin length (t = 60 s).

FIGURE 129:  Comparison between transient temperature profiles for the side-convex and rectangular cross section slabs: a) Temperature profiles versus time, b) Temperature profiles along the fin length (t = 60 s).

FIGURE 130:  Comparison between transient temperature profiles for the 3D side-convex and side-convex slabs: a) Temperature profiles versus time, b) Temperature profiles along the fin length (t = 60 s).

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Side-Convex Fin with Rectangular Cross Section  •  137

Temperature contour plots are shown in Figure 131 for the cases where n = 1 (natural convection) and n = 100 (forced convection) after a one-minute exposure time. Temperature gradient for the natural convection is more than that of the side-concave presented for the previous geometry exposed to similar boundary conditions. This is associated with the surface areas and volume. The latter is larger in this case, while the former remains very similar to that of the previous geometry. Care was taken to introduce a curved surface that mirrored that of the previous geometry for a fair comparison of the results. Forced convection introduces a uniform temperature transition from the base to the tip of the fin. Figure 132 presents temperature contours sliced at equal intervals; they indicate that temperature distribution is relatively uniform along the fin depth. Slab with convex sides has larger volume and similar surface area compared to those of the concave sides, which explains the reason for the slower temperature drop for the fins with convex side for forced convection scenario.

FIGURE 131:  Transient temperature contours for the 3D side-convex slab with rectangular cross section at t = 60 s: a) n = 1, b) n = 100.

FIGURE 132:  Transient temperature contour slices for the 3D side-convex slab with rectangular cross section at t = 60 s: a) n = 1, b) n = 100.

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CHAPTER

14

SIDE-CONCAVETRAPEZOIDAL FIN WITH RECTANGULAR CROSS SECTION Figure 133 presents the applicable physics, as well as the three boundary condition cases (Case 3, convective, assumed here). Similar to the previous cases, you may either generate a new model within a new file or add a new 3D component under

FIGURE 133:  Fin with side-concave-trapezoidal profile and rectangular cross section.

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the main tree, define physics (solid heat transfer), and select the analysis type (transient analysis). The geometry for this analysis is a modified version of the side-concave fin with rectangular circular cross section and modified work-planes presented in Figure 118. You may either ­create the 3D geometry by importing a CAD solid geometry or generate the geometry in COMSOL internally. For the following example, the 3D geometry has been created in COMSOL Multiphysics® using blocks to represent the elongated rectangular fin and the skillet body. Concave parametric surfaces have then been used to represent the upper and lower surfaces. These surfaces are work-planes that partition the original block shape into entities, with unneeded ones then deleted to obtain the geometry presented in Figure 134. Variables were used to define this shape so that it can be changed easily in the future, if needed. The geometry is meshed using the sweep feature where either upper or front surfaces may be selected as the source or primary plane boundaries and the remaining volume (i.e., the slab) may be selected as sweeping domain. You may also select front and back boundary planes as the mapped sources to sweep the desired domain (Figure 134). The tip of the fin is a result of a gradual decreasedconcave profile (a similar slope to that of the previous cases for comparison purposes, resulting in larger surface area) and works well with brick elements (Figure 134). The skillet main body is assumed to have a fixed temperature of 100°C (Figure 135). The initial condition is set to the ambient (20°C) throughout the geometry. Symmetry conditions are applied using an xz symmetry plane. Convection is the mode of heat transfer for the remaining exposed surfaces.

FIGURE 134:  Meshing the 3D side-concave-trapezoidal slab with rectangular cross section.

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Side-Concave-Trapezoidal Fin with Rectangular Cross Section  •  141

FIGURE 135:  Assigning base boundary conditions for the 3D side-concave-trapezoidal slab with rectangular cross section.

Transient temperature as a function of time and location along the fin length (x-coordinate) are presented in Figure 136 to Figure 139. Figure 136a shows temperature as a function of time for a midpoint location illustrated by a red dot in the cross-­sectional sketch. The seven main groups of the diagrams seen herein represent the natural (n = 1) and forced (n = 100) ­convection for the side-convex, sideconcave, rectangular, circular, side-triangular, and triangular cross section fins. It is seen that the natural convection (n = 1) results in the temperature profiles of about 92°C, 92°C, 98°C, 96°C, 86°C, and 83°C for the side-concave-trapezoid, side-convex, sideconcave, side-triangular, rectangular and triangular cross section fins, respectively. A similar profile to that of the rectangular cross section is seen for the circular cross section. At the tip of the slab, temperature profiles stabilize at 87°C, 89°C, 79°C, 93°C, 79°C, and 77°C, respectively, for the aforementioned geometries. Forced convection (n = 100) generates the most efficient heat transfer from the slab generating peak temperatures to over 36°C, 38°C, 36°C, 39°C, 37°C and 31°C for side-concave-trapezoid, side-convex, side-concave, side-triangular, and rectangular and triangular cross

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section fins. Circular cross section stabilizes at a peak temperature similar to that of the rectangular cross section (37°C). The tip of the slab for this case reaches 25°C, 26°C, 20°C, 23°C, 27°C, and 22°C for the said geometries, respectively, with the circular cross section demonstrating a similar trend to that of the rectangular cross section. The slab with side-concave-trapezoidal profile has larger volume and larger surface area compared to that of the side-concave, and for the same reason the temperature drop is more visible. Figure 137 shows the comparison between the temporal and spatial temperature profiles for the side-concave-trapezoid and rectangular cross section fins. Note that the fin with rectangular cross section is more effective than the side-convex fin due to its surface area for both free (n = 1) and forced (n = 100) convection scenarios. The difference is more significant in natural convection (n = 1) while in forced convection (n = 100), temperature profiles converge to similar values after one-minute exposure. Figure 138 and Figure 139 show a comparison between temporal and spatial temperature profiles for natural (n = 1) and forced (n = 100) convection scenarios for sideconcave, side-convex, and side-concave-trapezoidal (flat-tip) fins. It is seen the side-convex fin is the most effective to thermally manage the heat from the slab, the side-concave fin is the least effective, and the side-concave-trapezoidal (flat-tip) demonstrates closer temperature distributions to that of the side-convex fin in natural convection (n = 1) case. The three profiles show similar temperature distributions for the forced convection scenario.

FIGURE 136:  Transient temperature profiles for the 3D side-concave-trapezoidal slab with rectangular cross section: a) Temperature profiles versus time, b) Temperature profiles along the fin length (t = 60 s).

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Side-Concave-Trapezoidal Fin with Rectangular Cross Section  •  143

FIGURE 137:  Comparison between transient temperature profiles for the 3D side-concave-trapezoid (flat-tip) and rectangular cross section slabs: a) Temperature profiles versus time, b) Temperature profiles along the fin length (t = 60 s).

FIGURE 138:  Comparison between transient temperature profiles versus time for the 3D side-concave, side-convex, and side-concave-trapezoidal (flat-tip) slabs: a) Natural convection (n = 1), b) Forced convection (n = 100).

FIGURE 139:  Comparison between transient temperature profiles along the fin length for the 3D side-concave, side-convex, and side-concave-trapezoidal (flat-tip) slabs at t = 60 s: a) Natural convection (n = 1), b) Forced convection (n = 100).

Temperature contour plots are shown in Figure 140 for the cases where n = 1 (natural convection) and n = 100 (forced convection) after one-minute exposure time. Temperature gradient for the natural convection is considerably more than that of the side-concave

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presented for the previous geometry exposed to similar boundary conditions; however, they present similar trend for the case of forced convection. This gradient is associated with the surface areas and volume, as discussed. The side-concave-trapezoidal surfaces have identical curvatures with those of the side-concave surfaces for comparison purposes. Both natural and forced convection scenarios introduce a smooth temperature transition from the base to the tip of the fin, varying from the minimum value (indigo) to the maximum value (crimson). Figure 141 presents similar contour plots where temperature contours have been sliced at equal intervals. It is shown that temperature distribution is relatively uniform along the y-coordinate.

FIGURE 140:  Transient temperature contours for the 3D side-concave-trapezoidal slab with rectangular cross section at t = 60 s: a) n = 1, b) n = 100.

FIGURE 141:  Transient temperature contour slices for the 3D side-concave-trapezoidal slab with rectangular cross section at t = 60 s: a) n = 1, b) n = 100.

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CHAPTER

15

PIN FIN WITH CIRCULAR CROSS SECTION Figure 142 presents the applicable physics, as well as three cases for boundary conditions. It is assumed that exposed surfaces transfer heat to the surroundings by means of convection. Similar to the previous cases, you may either generate a new model within a new file or add a new 3D component under the main tree, define physics (solid heat transfer), and select the analysis type (transient analysis).

FIGURE 142:  Fin with pin profile and circular cross section.

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You may either create the 3D geometry by importing a solid geometry that is generated in a CAD module such as SolidWorks® or generate the geometry internally in COMSOL. For the following example, the 3D geometry has been created in COMSOL Multiphysics® using block and cone features. An xz work-plane partitions the solid fin at the y-coordinate midpoint to take advantage of the symmetry (Figure 143). Thereafter, the left half of the geometry is deleted. The geometry may be meshed using the mapped mesh feature, where the pin fin surface is selected as the surface where mapped mesh is to be applied, and the volume is meshed based on the extension of the pattern generated on its surface (Figure 143). Decreasing the element size may introduce a challenge when meshing the tip of the fin. In order to address this, one method will be to divide the surface or volume by partitioning the geometry into multiple regions and treat each region as a separate entity. Care is to be taken in such cases to ensure continuity of the elements. Brick elements are mainly expected when employing this technique (Figure 143). The skillet main body is assumed to have fixed temperature of 100°C (Figure 144). The initial condition is the same as that of the ambient (20°C) and is applied constantly throughout the geometry. Symmetry conditions are applied using the predefined xz-symmetry plane. Convection is the main mode of heat transfer for the remaining exposed surfaces.

FIGURE 143:  Meshing the 3D pin slab with circular cross section.

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Pin Fin with Circular Cross Section  •  147

FIGURE 144:  Assigning base boundary conditions for the 3D pin slab with circular cross section.

Transient temperature as a function of time and location along the fin length (x-coordinate) are presented in Figure 145 and Figure 146. Figure 145a presents temperature as a function of time for a midpoint location illustrated by a red dot in the cross-sectional sketch. The four main groups of the diagrams seen herein represent the natural (n = 1) and forced (n = 100) convection for the pin fin, and the fins with rectangular, circular, and triangular cross sections. It is seen that the natural convection (n = 1) results in the temperature profiles of about 97°C, 86°C, and 83°C for pin fin, and those with rectangular and triangular cross sections, respectively. Similar profile to that of the rectangular cross section is seen for the circular cross section. At the tip of the slab, temperature profiles stabilize at about 96°C, 79°C, and 77°C, respectively, for the aforementioned geometries. Forced convection (n = 100) generates the most efficient

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148  •  Heat Transfer Modelling Using COMSOL Slab to Radial Fin

heat transfer from the slab generating peak temperatures over 42°C, 37°C, and 31°C for pin fin and those with rectangular and triangular cross sections. Circular cross section stabilizes at a peak temperature similar to that of the rectangular cross section (37°C). The tip of the slab for this case reaches 23°C, 27°C, and 22°C for the said geometries, respectively, with circular cross section demonstrating similar trend to that of the rectangular cross section. The slab with pin profile and circular cross section has smaller volume and surface area than those of the cylindrical one with circular cross section with identical length. This explains the slightly lower temperatures for the forced convection case. On the contrary, for the same reason, temperature within the solid increases when the heat transfer to the surroundings diminishes due to the natural convection regime, meaning that the accumulated heat will not have a chance to escape the volume as efficiently as it would have if it were forced-cooled. This is the main reason temperature increase is pronounced for the natural convection scenario (Figure 145). Figure 146 shows the comparison between the temporal and spatial temperature profiles for the pin and rectangular cross section fins. Note that pin fin is less effective in heat transfer compared to the rectangular fin due to the smaller surface area. This temperature deviation is more significant for the case of natural convection (n = 1). For the forced convection (n = 100) scenario, the two profiles demonstrate similar trends and also values varying in the range of 5°C.

FIGURE 145:  Transient temperature profiles for the 3D pin slab with circular cross section: a) Temperature profiles versus time, b) Temperature profiles along the fin length (t = 60 s).

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Pin Fin with Circular Cross Section  •  149

FIGURE 146:  Comparison between transient temperature profiles for the 3D pin and rectangular cross section slabs: a) Temperature profiles versus time, b) Temperature profiles along the fin length (t = 60 s).

Temperature contour plots are presented in Figure 147 for the cases where n = 1 (natural convection) and n = 100 (forced convection) after one-minute exposure time. Pin fin temperature gradient for the natural convection is similar to that of the cylindrical one with circular cross section exposed to similar boundary conditions. Similar trend is seen for the case of forced convection. Both natural and forced convection scenarios introduce a smooth temperature transition from the base to the tip of the fin, varying from the minimum value (indigo) to the maximum value (crimson). Figure 148 shows similar contour plots where temperature contours have been sliced at equal intervals. It is seen that temperature distribution is relatively uniform along the y-coordinate.

FIGURE 147:  Transient temperature contours for the 3D pin slab with circular cross section at t = 60 s: a) n = 1, b) n = 100.

FIGURE 148:  Transient temperature contour slices for the 3D pin slab with circular cross section at t = 60 s: a) n = 1, b) n = 100.

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CHAPTER

16

RADIAL FIN WITH HYPERBOLIC PROFILE Figure 149 presents the applicable physics as well as three cases for boundary conditions, with heat transfer by convection (Case 3) applying here. In this case, the 3D geometry has been created in COMSOL Multiphysics® using blocks, spheres, and parametric ­surfaces to model the main body of the skillet, and the hyperbolic radial fin. Two xy and xz work-planes partition the solid fin at the y and z-coordinates at the centerpoint to take advantage of the s­ ymmetry. Thus, only one quadrant of the radial fin is modelled.

FIGURE 149:  Radial fin with hyperbolic profile.

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The geometry was meshed by sweeping, with the radial fin’s upper surface selected as the source boundary and its lower surface as the destination (Figure 150). This method introduces trapezoidal elements of brick family, meaning that there should be fewer elements than if they were to be of tetrahedral family. The aspect ratio, the ratio of the element’s longest to shortest dimensions, remains similar for the majority of surface mapped mesh (Figure 150). The skillet main body is assumed to have fixed temperature of 100°C. This is where the fin is connected to the main block (Figure 151). The initial condition is the same as that of the ambient (20°C) and is applied to all the geometry. Symmetry conditions are applied using the predefined xy symmetry plane. Convection is the main mode of heat transfer for the remaining exposed surfaces.

FIGURE 150:  Meshing the 3D radial slab with hyperbolic profile.

FIGURE 151:  Assigning boundary conditions for the 3D radial slab with hyperbolic profile: a) Base temperature, b) Symmetry plane.

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Radial Fin with Hyperbolic Profile  •  153

Transient temperature as a function of time and location along the fin length (x-coordinate) is presented in Figure 152 and Figure 153. Figure 152a shows temperature as a function of time for a ­midpoint location indicated by a red dot in the cross-sectional sketch. The three main groups of the diagrams seen herein represent the natural (n = 1) and forced (n = 100) convection boundary conditions for the radial fin as well as fins with side-concave and side-concavetrapezoidal profiles with rectangular cross sections. It is seen that the natural convection (n = 1) results in the temperature reaching about 60°C, 92°C, and 98°C for radial fins, and the fins with side-concavetrapezoidal and side-concave cross sections, respectively. At the tip of the slab, temperature profiles stabilize at 54°C, 87°C, and 79°C, respectively, for the aforementioned geometries.

FIGURE 152:  Transient temperature profiles for the 3D radial slab with hyperbolic profile: a) Temperature profiles versus time, b) Temperature profiles along the fin length (t = 60 s).

Forced convection (n = 100) results in the most efficient heat transfer from the slab, generating peak temperatures of over 30°C, 36°C, and 36°C for radial fin and fins with side-concavetrapezoidal and side-concave with rectangular cross sections. The tip of the slab for this case reaches 23°C, 25°C, and 20°C for the said geometries, respectively. The slab with radial fin has larger volume and larger surface area compared to the previously modelled geometries; this leads to a more moderate temperature rise for this scenario. Figure 153 shows the comparison between the temporal and spatial temperature profiles for the radialconcave and side-concave-trapezoidal (concave-flat-tip) fins. Note that a side-concave-trapezoidal (concave-flat-tip) fin is thermally less effective in heat transfer compared to the radial-concave fin

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due to the smaller surface area. This temperature deviation is more significant for the case of natural (n = 1) convection. For the forced convection case (n = 100), the difference between the two temperature distributions is less significant and is under 10°C.

FIGURE 153:  Comparison between transient temperature profiles for the 3D radial-concave and side-concave-trapezoidal (concave-flat-tip) slabs: a) Temperature profiles versus time, b) Temperature profiles along the fin length (t = 60 s).

Temperature contour plots are shown in Figure 154 for the cases where n = 1 (natural convection) and n = 100 (forced convection) after one-minute exposure time. Temperature gradient for the natural and forced boundary conditions follow the same contour shapes; however, temperature drop for the forced convection is larger than that for the natural convection. Figure 155 presents similar contour plots where temperature contours have been sliced at equal intervals along the x-coordinate. It is seen that the temperature distribution is varying, presenting its highest value at the base in contact with the skillet to the lowest value at the tip of the radial fin.

FIGURE 154:  Transient temperature contours for the 3D radial slab with hyperbolic profile at t = 60 s: a) n = 1, b) n = 100.

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Radial Fin with Hyperbolic Profile  •  155

FIGURE 155:  Transient temperature contour slices for the 3D radial slab with hyperbolic profile at t = 60 s: a) n = 1, b) n = 100.

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CHAPTER

17

WEBBED RADIAL FIN WITH HYPERBOLIC PROFILE (DUCKLING* RADIAL FIN) In this model, cut-out slots are introduced to the radial fin. Note that this does not necessarily increase the surface-to-volume ratio, shown for this scenario later on (Figure 156).

FIGURE 156:  Webbed radial fin with hyperbolic profile.

*  The name “Duckling Fin” has been coined by the author; it reflects the progression of the slab fin from the conception to the realization - the simplest to the most complex case.

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158  •  Heat Transfer Modelling Using COMSOL Slab to Radial Fin

Figure 156 illustrates the applicable physics as well as three cases for boundary conditions (Case 3, heat transfer by c­ onvection, ­applying here). For the following example, the 3D geometry has been created in COMSOL Multiphysics® using blocks, spheres, and parametric surfaces to introduce the main body of the fin and skillet, hyperbolic radial fin, and webs. Two xy and xz work-planes partition the solid fin at the y and z-coordinates at the centerpoint to take advantage of the geometry symmetrical features. Thus, only the second quadrant of the radial fin is modelled. The geometry poses the most complex case given the webs. The number of work-planes passing through the centerline have been defined in order to introduce the cutting planes to carve the webbed parts. The geometry was meshed by sweeping, with the radial fin’s upper surface selected as the source and the bottom surface as the destination (Figure 157). The skillet main body is assumed to have a fixed temperature of 100°C, shown by the two corner surfaces at the boundary between the block and the fin (Figure 158). The initial condition is the ambient temperature (20°C) and it is applied to all the geometry. Symmetry conditions are applied using the predefined xy and xz-symmetry planes. Convection is the main mode of heat transfer for the remaining exposed surfaces.

FIGURE 157:  Meshing the 3D webbed radial slab with hyperbolic profile.

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Webbed Radial Fin with Hyperbolic Profile  •  159

FIGURE 158:  Setting up boundary conditions for the 3D webbed radial slab with hyperbolic profile.

Transient temperature as a function of time and location along the fin length (x-coordinate) is presented in Figure 159. Figure 159a presents temperature as a function of time for a midpoint location indicated by a red dot in the cross-sectional sketch. The two main groups of the diagrams seen herein represent the natural (n = 1) and forced (n = 100) convection boundary conditions for the webbedradial and radial fins with hyperbolic profiles. It is seen that the natural convection (n = 1) results in the temperature reaching about 70°C and 60°C, respectively. At the tip of the slab, temperature profiles stabilize at 59°C and 54°C, respectively, for the aforementioned geometries. Forced convection (n = 100) results in the most efficient heat transfer from the slab, generating peak temperatures of over 37°C and 30°C for webbed radial and radial fins with hyperbolic profiles. The tip of the slab for this case reaches 23°C for both webbed radial and radial fins. The slab with webbed radial fin has larger volume and larger cooling surface areas compared to those of the radial fin, and for

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that reason the maximum temperature profiles are higher for the webbed radial fin in both cases of forced and natural convections (Figure 159). Another factor to keep in mind is that the fin-block ­contact surfaces are also larger for the case of the webbed radial fin compared to that of the radial fin, and that is related to higher heat transfer to the webbed radial fin and therefore higher temperature compared to that of the radial fin (Figure 159). Figure 160 demonstrates a method in which surface and volume can be measured for webbed-radial (5.50×10-5 m2 and 0.020 m3) and radial (5.02×10-5 m2 and 0.018 m3) fins. To achieve this, one may select the geometry whose boundaries or domains are to be measured and access available features (right-click), select the features to be measured from the finalized geometry, and measure single or multiple surfaces or domains. Figure 161 shows the contact surfaces where the base temperature has been identified for webbed-radial (8.34×10-5 m2) and radial (5.00×10-5 m2) fins, respectively.

FIGURE 159:  Transient temperature profiles for the 3D webbed radial slab with hyperbolic profile:  a) Temperature profiles versus time, b) Temperature profiles along the fin length (t = 60 s).

FIGURE 160:  Fin volume and cooling area measured for the 3D: a) Webbed radial slab with hyperbolic profile, b) Radial slab with hyperbolic profile.

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Webbed Radial Fin with Hyperbolic Profile  •  161

FIGURE 161:  Fin-Block contact areas for the 3D: a) Webbed radial slab with hyperbolic profile, b) Radial slab with hyperbolic profile.

Temperature contour plots are shown in Figure 162 for the cases where n = 1 (natural convection) and n = 100 (forced convection) after one-minute exposure time. Temperature gradients for the natural and forced convection boundary conditions follow the same contour shapes; however, temperature drop for the forced convection is larger than that of the natural convection. Figure 163 shows similar contour plots where temperature contours have been sliced at equal intervals along the x-coordinate. It is seen that temperature distribution is varying, presenting its highest value at the base in contact with the skillet and the lowest value at the tip of the radial fin. Figure 164 shows isothermal profiles for the natural (n = 1) and forced (n = 100) convection scenarios. It is seen that temperature is uniform at equal radial distances from the central block.

FIGURE 162:  Transient temperature contours for the 3D webbed radial slab with hyperbolic profile at t = 60 s:  a) n = 1, b) n = 100.

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162  •  Heat Transfer Modelling Using COMSOL Slab to Radial Fin

FIGURE 163:  Transient temperature contour slices for the 3D webbed radial slab with hyperbolic profile at t = 60 s: a) n = 1, b) n = 100.

FIGURE 164:  Transient temperature radial isothermal contours for the 3D webbed radial slab with hyperbolic profile at t = 60 s: a) n = 1, b) n = 100.

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CHAPTER

18

FORCED CONVECTIVE WEBBED RADIAL FIN WITH HYPERBOLIC PROFILE (NOT THE UGLY DUCKLING) This section presents the concluding example. The case study is similar to the previous one, where a webbed radial fin with hyperbolic profile and convective boundary conditions were analyzed (Figure 165).

FIGURE 165:  Webbed radial fin with hyperbolic profile inside a box of cooling liquid.

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164  •  Heat Transfer Modelling Using COMSOL Slab to Radial Fin

However, in this scenario a flow passes by the solid components, acting as a cooling agent. The geometry employs conjugate heat transfer methodology where the model is subdivided into two main physics, solid heat transfer and non-isothermal flow, which are coupled by a multiphysics interface. In addition, sensitivity of the analysis results to the liquid cooling agent (i.e., transformer oil, water, or air) has been studied. Figure 165 illustrates the applicable physics, as well as three possible scenarios for surfaces exposed to the surroundings. The symmetry planes are insulated. Similar to the previous cases, to generate this model, you define physics (conjugate heat transfer, heat transfer in solids, non-­isothermal flow) and select the analysis type (stationary analysis). This geometry was generated using the built-in tools by following the steps discussed earlier. The additional geometry feature in this case study is the cooling liquid box surrounding the solid components; it should be large enough to comfortably contain the entire geometry with additional space to minimize the interference by fluid boundaries. A method to mesh this geometry is to mesh the interior solid parts using mapped and sweep techniques and carefully extend the surface elements to the surrounding liquid. This could be laborious to ensure results will converge. The solid surfaces will act as walls in contact with the adjacent fluid, and this is where use of boundary elements is recommended. In this case number of layers of thin elements surround the solid surfaces in contact with the liquid vary to five layers. This is the area where transition between the physics takes place. The boundary elements assist to better represent flow characteristics in proximity to the surfaces, such as pressure gradient to find drag or lift forces in applications such as aerodynamic studies for vehicles moving through air. You may expect formation of tetrahedral as well as brick elements exceeding a million depending on the size of your elements. The author recommends employing the physics-­controlled meshing in similar situations as a comparison with your choice of custom-made meshing. In this case, selection of coarse meshing resulted in 403,000 elements, using element sizes larger than 14 mm and smaller than 75 mm. The skillet main body, also known as the block, is assumed to have fixed temperature of 100°C, which is applied to the entire

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skillet volume. Since the block and fin are part of the air-cooled geometry, they both have been included in the geometry. The initial condition is the same as that of the ambient (20°C) and is applied to the whole fin. Symmetry conditions are applied using the xy and xz symmetry planes similar to those conditions defined in the previous case study, where no box surrounded the geometry. Convection is the main mode of heat transfer for the remaining exposed surfaces. However, in this simulation, convection is modelled by representing the actual fluid flow, as opposed to using a heat transfer coefficient. This is expected to give more accurate results as the convection heat transfer coefficient will not accurately reflect the intricacies of the fluid flow around the solid. Inflow and outflow from the box are defined (Figure 165). Inflow characteristics such as velocity, mass flow rate, volume flow rate, pressure, or temperature may be set as input conditions. Note that some inflow settings may only be available in COMSOL Multiphysics® CFD module; however, it is possible to calculate these values manually if needed. For instance, you may calculate the mass flow rate knowing the flow velocity, density, and area. A useful feature available in Heat Transfer module is the capability to define the velocity profile for the input flow, meaning you are able to identify the velocity components along the three-major axis (i.e., vx, vy, vz). This is especially useful if you were to setup the fan conditions such as performance curves. Selection of flow physics depends on the flow velocity; faster flow may result in a turbulent boundary regime, while the slower one may result in a laminar regime. Note that the combination of both regimes is also possible where the flow path is long enough to introduce a characteristic length that transitions to the next phase. Recall Knudsen number where the ratio of the mean free path to that of the characteristic length defines the flow type. The Reynolds number, which shows the ratio of inertia forces to viscous forces, is a determining factor when deciding if flow is laminar, turbulent, or in transition. Flow characteristics such as density, velocity, characteristic length, and viscosity have been incorporated into the Reynolds number, which make it a descriptive dimensionless number when determining flow regime in a non-vacuum environment. In this study, flow velocity defines the inlet condition and atmospheric pressure is defined as the outlet.

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166  •  Heat Transfer Modelling Using COMSOL Slab to Radial Fin

In this model, outflow conditions are associated with the flow characteristics when leaving the solid part. It could be atmospheric conditions, for example. Note that Bernoulli’s law for two points within the flow path may be tested for the flow model to ensure the physics are consistent, where the combination of dynamic pressure and hydraulic pressure (i.e., pressure head plus elevation) are constant throughout an incompressible, steady flow with ignorable viscous forces. This suggests that the best location to test the Bernoulli’s theory is away from the solid boundary surfaces, where viscous forces are minimum. Recall for energy to flow, a gradient is required that acts as the driving force. Passage of the flow over the heated parts depends on the pressure gradient of the liquid cooling agent—liquid flows from the higher-pressure point to the lower one. Initial conditions (e.g., temperature) for the liquid cooling agent are the same as those of the desired environment in which the test is conducted. Stationary temperature as a function of location along the fin length (x-coordinate) is presented in Figure 166 for the webbed radial fin without and with surrounding box. The two main groups of the d ­ iagrams seen herein represent the natural (n = 1) and forced (n = 100) convection boundary conditions as well as the sensitivity

FIGURE 166:  Stationary temperature profiles for the 3D webbed radial slab with hyperbolic profile with and without surrounding fluid box along the fin length.

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of the temperature profiles to the cooling agent liquid. It is seen that the natural (n = 1) and forced (n = 100) convection boundary conditions result in the temperature profiles of about 95°C and 37°C, respectively, for the midpoint webbed radial. Temperature reaches 93°C and 24°C for the said scenarios at the tip of the webbed radial fin after one-minute exposure time. Coolant fluid sensitivity shows a temperature variation that is more significant for water, compared to transformer oil, and lastly air, meaning that water is the most effective coolant of the three. It is seen that the transformer oil, water, and air cooling agents each result in 23°C, 22°C, and 85°C, respectively, at a midpoint location. Temperature reaches 20°C and 77°C for the water and air cooling agents at the tip of the slab. The case involving transformer oil shows similar trend to that of the water at the tip of the slab. This example clearly demonstrates the effectiveness of using water and transformer liquid coolants over air for thermal management applications. Recall that in this section, the natural and forced convective conditions were simulated by combining a solid heat transfer analysis with a non-isothermal flow analysis in the form of a conjugate heat transfer analysis. Material switch feature has been employed to depict the sensitivity of the temperature results to the cooling fluid choice. Temperature contour plots are shown in Figure 167 for the case where transformer oil is employed as the cooling agent. Note the size

FIGURE 167:  Stationary temperature contours for the 3D webbed radial slab with hyperbolic profile and transformer oil as the cooling agent.

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of the surrounding box with respect to the solid parts. Figure 168 presents sensitivity of the temperature contour plots to the three cooling agents (i.e., transformer oil, water, and air). Note that viscosity is highest for transformer oil (0.016 Pa.s at 25°C), less for water (0.00084 Pa.s at 25°C), and least for air (1.85×10-5 Pa.s at 25°C), affecting the flow regime. Liquid cooling (especially water) is generally more efficient than air cooling. This model is a good example for submerged surfaces, where the temperature of a hot surface is managed by submerging the surface into a liquid. Note that skillet temperature is assumed constant at 100°C for the entire cooling process. Figure 169 and Figure 170 present equally-spaced sliced velocity contour plots along the x-coordinate. It is seen that velocity profile is varying, presenting its maximum value within the volume of air, followed by transformer oil and water.

FIGURE 168:  Stationary temperature contours for the 3D webbed radial slab with hyperbolic profile: a) Water, b) Transformer oil, c) Air.

FIGURE 169:  Velocity contour plot slices for the 3D webbed radial slab with hyperbolic profile: a) Transformer oil, b) Water, c) Air.

FIGURE 170:  Velocity contour plot slices for the 3D webbed radial slab with hyperbolic profile: a) Transformer oil, b) Water, c) Air.

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An example for the pressure contour plot is shown in Figure 171 for the case where transformer oil is employed as the cooling agent. Figure 172 shows an example of the sensitivity analysis of the pressure contour plots to the three cooling agents (i.e., transformer oil, water, and air). Note that highest pressure gradient magnitude is observed where the viscosity is highest and in the vicinity of the solid surfaces for transformer oil, water, and air, in that order. That is due to the stickiness of the liquid molecules to the solid surface as they sweep over it. Note that in order to obtain accurate data on different drag coefficients on surfaces in contact with the air, detailed investigative studies are to be performed where the effect of the solid-fluid interactions is fully explored.

FIGURE 171:  Pressure contours for the 3D webbed radial slab with hyperbolic profile and transformer oil as the cooling agent.

FIGURE 172:  Pressure contours for the 3D webbed radial slab with hyperbolic profile: a) Transformer oil, b) Water, c) Air.

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CHAPTER

19

MATHEMATICAL METHODS TO SOLVE HEAT AND WAVE PROBLEMS There is a variety of analytical techniques that savvy mathematicians may employ to solve physics equations. Contrary to what you may think, special cases of 2D and 3D problems may also be solved by means of specialized analytical practices. The first part of this s­ ection summarizes some of the methods that you will find ­interesting, if you have a flair for math sciences. Knowing them proves to be useful; especially in cases such as sitting for your doctoral comprehensive examination—assuming that you pick advanced heat transfer topics or maths as your specialized subject.

19.1  GENERAL ANALYTICAL APPROACHES This section focuses on the analytical techniques that may be employed to solve heat transfer equations. The most general form is a wave equation that consists of a second-order linear p ­ artial differential equation for the description of waves with respect

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to time and space. The application of wave equations varies from sound waves, light waves, and water waves, and is important in fields such as acoustics, electromagnetics, and fluid dynamics. The purpose is to equip the reader with primarily practical tools to evaluate and solve their engineering problems.

19.1.1  Separation of Variables By separating variables, you define a dependent variable (e.g., temperature in heat transfer problems) and treat it as an individual or independent variable; in other words, their combined effect has been divided to show their unique impact. For the 3D case studies discussed earlier, this involves defining an energy equation such as the one defined in equation (11)—each function is a function of a single variable that may be space or time, equation (12). There are “m” number of linearly independent boundary conditions matching the number of the highest derivatives times the number of independent variables in a differential equation. For instance, for a second order three-dimensional steady heat transfer, six boundary conditions are required, representing the conditions for each side of the brick. To facilitate solving these problems, change of variables to the ones that result in homogeneous differential equation or boundary conditions is recommended, which in a majority of cases results in dimensionless equations. For example, assuming that a boundary is kept at the surrounding temperature (e.g., Tx = L = T∞ ), difference between the main dependent variable (T) and ambient temperature (T∞) may be defined as a new variable (θ = T − T∞), substituting its counterpart in the heat transfer equation. Note that in this case, the derivatives are to be revised for the new variable to implement this dT dT dθ change ( e.g., = ). A number of these mathematical reladx dθ dx tions can be solved using Fourier transform, which represents a complex function for the real dependent variable.

 



dT  d dT d dT d dT  dT ( k ) + (k ) + (k ) = ρ Cp  + vi  + q ⋅  (11) dx dx dx  g en dy dy dz dz  dt T ( x, y, z, t ) = X ( x ) Y ( y ) Z ( z )τ ( t ) 

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(12)

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As a practice example, you may attempt the following 2D ­problem, equation (13), that simulates the conditions presented by boundary conditions, equations (14). The solution is provided by equation (15). The first step is to define a new dependent variable for temperature (q), where θ = T − T∞. d 2T d 2T + =0 dx2 dy2



 T ( 0, y ) x = 0 = Fy  T ( L, y ) x = L = T∞





θ ( x, y ) θ0

(13)

 T ( x, 0 )y= 0 = T∞   T x, L )y= L = T∞  (

 − ( 2 k +1)π x ( 2k + 1)π 4 ∞ e l = ∑ sin π k=0 ( 2 k + 1) l 

(14)  y  

(15)

19.1.2  Variation of Parameters The concept for the partial solutions and variation of parameters is similar to the separation of variables. These are the steps to be taken to solve such problems: 1) setup a problem that represents the homogeneous case for ( x,θ ), where x and q are dimension and time, 2) determine the eigenfunctions, 3) construct a solution using a function such as u ( x,θ ) = ∑ m Am (θ )φm ( x ), 4) evaluate Am (θ ) by orthog-

onality of φm ( x ) , 5) setup an ordinary equation, 6) solve for Am (θ ), and 7) complete the solution. This method is particularly useful if you are performing a transient analysis. Attempt to solve equation (16), given boundary conditions presented by equations (17). Note that the second order equation with respect to space (x) and first order with respect to time (t) require three boundary conditions.

uxx = uθ 

(16)



 u ( 0,θ ) x = 0 = 1   u (1,θ ) x =1 = 0   u ( x, 0 ) = 0 θ =0 

(17)

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− ( nπ ) θ 2 ∞  sin ( nπ x )  2 ∞  sin ( nπ x ) e  u ( x,θ ) = ∑  − ∑  π n=1  nπ n  π n=1  2



  

(18)

19.1.3  Duhamel’s Theorem The problems in this category are essentially similar to the previous scenarios except that u ( x,θ ) is the response to a boundary condition that is initially zero and then progresses to a constant value or the problem is non-homogeneous in general terms, equation (19). Attempt problem presented by equation (20) with the boundary conditions presented by equations (21). The solution is given by equation (22).



u ( x,θ ) =

θ

∫ u ( x,θ − τ ) F′ (τ ) dτ + ∑

τ =0

N i =1

u ( x,θ − τ i ) ∆Fi 

(19)



uxx = uθ (20)



 u ( 0,θ ) x = 0 = F (θ ) = cos ( wθ )  u (1,θ ) x =1 = 0 (21)   u ( x, 0 )θ = 0 = 0   ∞  nsin ( nπ x ) ( nπ )2 cos ( wθ ) + wsin ( wθ )   u ( x,θ ) = 2π ∑ n=1  4 2    ( nπ ) + w   (22) 3 ∞  n sin ( nπ x ) − ( nπ )2 θ  3 e − 2π ∑ n=1   4 2  ( nπ ) + w 

19.1.4  Complex Combinations The following steps may be adopted when solving such problems: 1) define a new variable that is 90° out of phase with that of the main dependent variable (n)—this variable is the imaginary component of the ultimate solution, 2) present a new variable that is the conjugate of the real and imaginary parts w = ( u + iv), this variable is the ultimate solution, 3) define the ultimate solution as w = X ( x ) e± iwθ,

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using Euler’s formulae, where e ix = ( cosx + i sinx ), 4) solve for X ( x ), and 5) obtain the final complex variable ( w). Equation (23) along with boundary conditions given by (24) is a relatively complex problem that may be solved using this technique. Equation (25) is the solution expressed as a function of the complex variable w.

uxx + cos ( wθ ) = uθ (23)



 u ( 0,θ ) x = 0 = 0  u ( ∞,θ ) x =∞ = 0 (24)   u ( x, 0 ) = T sin ( wθ ) 0 θ =0  u ( x,θ ) =

1  −x e w 

w 2

  w sin  x  cos ( wθ )  2  

 −x + 1 − e 



w 2

  w cos  x  sin ( wθ )  2  

(25)

19.1.5 Superposition There are scenarios where you may superimpose multiple solutions you have attempted using different techniques. The boundary conditions may be either homogeneous, constant, or periodic. Attempt equation (26) given boundary conditions presented with equations (27). Note that you may convert equation (26) to four components consisting of cases where one non-homogeneity is taken into consideration at a time, equations (28) and (29).

uxx + F ( x,θ ) = uθ (26)



 u ( 0,θ ) x = 0 = g (θ )   u (1,θ ) x =1 = h (θ ) (27)  u ( x, 0 ) = f ( x ) θ =0 



u ( x,θ ) = v ( x,θ ) + w ( x,θ ) + p ( x,θ ) + q ( x,θ ) (28)

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Partial solution Partial boundary conditions

vxx + F ( x,θ ) = vθ

wxx = wθ

pxx = pθ

q xx = qθ

 v ( 0,θ ) x = 0 = 0   v (1,θ ) x =1 = 0  v ( x, 0 ) = 0 θ =0 

 w ( 0,θ ) x = 0 = g (θ )   w (1,θ ) x =1 = 0  w ( x, 0 ) = 0 θ =0 

 p ( 0,θ ) x = 0 = 0   p (1,θ ) x =1 = h (θ )  p ( x, 0 ) = 0 θ =0 

 q ( 0,θ ) x = 0 = 0   q (1,θ ) x =1 = 0  q ( x, 0 ) = f ( x ) θ =0 



(29)

19.1.6  Laplace Transform This transformation is very similar to the Fourier transformation; however, it is more comprehensive in the sense that both function and variable (i.e., frequency) are complex. The inverse transformation is also possible where a complex variable such as frequency is transformed to a real variable (i.e., time). Attempt equation (30) with the boundary conditions presented by equation (31), resulting in the solution presented by equation (32). dt hA + ( T − T0 ) = 0  dθ ρ Cv



 t∞ = t1 + T0 cos ( wθ ) = Fy (31)  t (0 ) = ti 





(30)

1  −θ  T (θ ) =  T ( 0 ) − e 1 + w2   1 cos ( wθ ) + w sin ( wθ )   +  1 + w2 

(32)

19.1.7  Integral Method This method is an approximate solution to relatively complicated problems and may be attempted by taking the following steps: 1) estimate a temperature profile as a function of the dependent variables where one variable is incorporated as a multiplier and the other one as the variable in the polynomial relationship, T ( x, t ) = a ( t ) + b ( t ) x + c ( t ) x2 + d ( t ) x3 , 2) define a penetration depth as a function of the non-polynomial dependent variable,

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which satisfies the initial condition x = ρ ( t ) , 3) calculate the multipliers using the variable defined in step 2—taking into account the boundary and initial conditions, and 4) obtain the final solution by integrating from the main equation, taking into account the ρ (t)

variable in step 2 as the boundary limits, θ ( t ) = ∫ T ( x, t ) dx. You 0 may attempt equation (33) keeping in mind group of boundary conditions presented in equation (34). The solution is presented by equation (35). d 2 T ( x, t ) dT ( x, t ) =k 0< x