Table Of ContentRahul Yadav · C. Balaji · S. P. Venkateshan
Radiative Heat
Transfer in
Participating
Media
With MATLAB Codes
Radiative Heat Transfer in Participating Media
Rahul Yadav · C. Balaji · S. P. Venkateshan
Radiative Heat Transfer
in Participating Media
With MATLAB Codes
RahulYadav C.Balaji
ApplicationEngineer—ThermoFluids DepartmentofMechanicalEngineering
Systems IndianInstituteofTechnologyMadras
AltairEngineeringInc. Chennai,TamilNadu,India
Bengaluru,Karnataka,India
S.P.Venkateshan
DepartmentofMechanicalEngineering
IndianInstituteofInformationTechnology,
DesignandManufacturing
Kancheepuram,TamilNadu,India
ISBN978-3-030-99044-2 ISBN978-3-030-99045-9 (eBook)
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Educationinitsrealsenseisthepursuitoftruth.
Itisanendlessjourneythroughknowledgeand
enlightenment.
—A.P.J.AbdulKalam
To
IITMadrasanditsvibes
Preface
This book is an outcome of continuous and ongoing research by the authors in the area
of radiative heat transfer, and gas and particle radiation applied to industrial and space
applications.Thecomputationsrelatedtoradiativeheattransferareveryrelevantinappli-
cations such as iron and steel manufacturing industries, rocket exhaust designing, fire
resistance testing, and atmospheric and solar applications.
There exist a plethora of literature and texts on heat transfer modeling in industrial
engineering applications, where the treatise has been both experimental and numerical.
Thissaid,moreaccurate,versatile,andatthesametime,computationallyefficientnumer-
ical solutions to practical heat transfer problems are critical. This is especially true in
radiative heat transfer, where the experimental approach is rather non-economic due to
large cost of fabrication, installation and operation, and severe constraint due to high
operating temperatures. This makes the use of numerical tools in radiative heat transfer
analysis of profound significance.
The contents in this text mainly outline the process of conducting numerical simula-
tions in radiative heat transfer for various industrial applications and the use of artificial
intelligence in developing and testing fast heat flux predictor tools. The numerical simu-
lations require the development of computational tools (or codes) to solve the governing
radiativeheattransferequation.Thedevelopment,testingofcodes,andresultsofnumer-
ical simulations have emerged as a part of doctoral degree research conducted at IIT
Madras during 2013–2018.
The primary motive is to help the engineers and researchers understand the governing
physics of radiative heat transfer in participating media, model a radiative heat trans-
fer system numerically, develop more physics-informed approximations, and take steps
toward developing their own codes for radiative heat transfer analysis. Keeping in view
the complexity of the subject, the discussion is very basic to help even a budding engi-
neer sail firmly through the domain of radiative heat transfer. Starting from the concepts
ofthermalradiationandreviewingtheimportantapproachestosolvetheproblem,amath-
ematical framework for the solution of governing radiative heat transfer equations with
a separate and detailed treatment of gas and particle radiation is presented. The spectral
ix
x Preface
line-basedweightedsumofgraygasesmethodalongwithitscouplingwithparticleradia-
tionframeworkusingtheMiescatteringalgorithmisanimportantinclusion.Development
of a radiative heat transfer computation tool using the finite volume method applied to a
cylindrical,conical,andrectangularcomputationaldomainwithapplicationtocombustion
cylinders,rocketexhaustplume,rocketnozzles,fireresistancetesting,andradiantfurnace
is presented. The discussion on the use of artificial intelligence concepts in speeding up
the radiative heat transfer calculations in traditional industrial setting is unique, novel,
andinlinewiththegeneration’sneed.Useofsuchpredictortoolsinapplicationsforsolv-
ing practical problems of process parameter optimization in industry is also described
in detail. The important MATLAB codes are provided in the end to help engineers begin
developingtheirownradiativeheattransfersolvers,andusetheminpracticalheattransfer
analysis.
The book, therefore, serves as a comprehensive package by taking the readers from
the basics of radiative heat transfer in participating media to equip them with their own
codes to solve practical problems of their interest. During the course of completion of
these research works, we acknowledge the kind support received from the Indian Space
Research Organization (ISRO), Trivandrum, for sharing their technical expertise on the
subject.AffectionatethanksarealsodueforourcolleaguesintheHeatTransferLab,IIT
Madras, for their continuous motivation. The constant encouragement we received from
our families has been remarkable in this journey. We wholeheartedly believe that this
text would impart comprehensive knowledge on radiative heat transfer phenomenon in
participatingmediaanditsapplication,usageofartificialintelligence,alongwithtraining
the readers on solver development for these applications.
Bengaluru, India Rahul Yadav
Chennai, India C. Balaji
Kancheepuram, India S. P. Venkateshan
Contents
1 Introduction ......................................................... 1
1.1 Thermal Radiation .............................................. 1
1.2 Importance of Radiative Heat Transfer ............................. 2
1.3 Radiative Heat Transfer in Participating Medium .................... 4
1.4 Radiative Transfer Equation ...................................... 5
1.5 Properties of a Participating Medium .............................. 8
1.6 Organization of the Book ........................................ 9
References ........................................................... 10
2 ImportantLiteraturesonRadiativeHeatTransfer ...................... 11
2.1 Introduction .................................................... 11
2.2 Background .................................................... 11
2.3 Studies on Development of RTE Solver ............................ 13
2.4 Studies on Development of Band Models .......................... 16
2.5 Studies on Inclusion of Particle Radiation .......................... 18
2.6 Application-Based Studies ........................................ 20
2.7 Relevance, Scope, and Challenges ................................. 21
References ........................................................... 22
3 MathematicalFormulation ............................................ 27
3.1 Introduction .................................................... 27
3.2 Solution Methods for RTE ....................................... 27
3.2.1 Traditional Discrete Ordinates Method ...................... 28
3.2.2 Finite Volume Method ..................................... 33
3.3 Estimation of Gas Properties ..................................... 36
3.3.1 Full Spectrum Band Models ................................ 38
3.3.2 The SLW Model .......................................... 38
3.3.3 Functional form of the ALBDF ............................. 40
3.3.4 Formulation for Non-isothermal, Non-homogeneous Media .... 42
3.3.5 SLW-Gray Approximation ................................. 43
xi
xii Contents
3.4 Estimation of Particle Properties .................................. 44
3.4.1 Scattering by a Single Particle .............................. 45
3.4.2 Scattering by a Group of Particles .......................... 45
3.4.3 Treatment of the Phase Function and Anisotropic Scattering .... 47
3.4.4 CalculationofParticlePropertiesinConjunctionwithBand
Models .................................................. 48
3.5 Modeling of Radiative Equilibrium ................................ 49
3.6 Closure ........................................................ 51
References ........................................................... 51
4 RadiativeHeatTransferinCylindricalGeometries ...................... 53
4.1 Introduction .................................................... 53
4.2 Development of the FVM-SLW Method for a Cylindrical Geometry ... 53
4.3 Solution Procedure .............................................. 57
4.4 Validation of the FVM-SLW Method for the Cylindrical Geometry .... 59
4.4.1 Validation with Experimental Results ........................ 60
4.4.2 Validation for a Non-Gray Gas-Particle Mixture .............. 61
4.4.3 Validation for Anisotropic Scattering ........................ 62
4.4.4 Decision on the Number of Gray Gases ..................... 63
4.5 Application to an Industrial Scale Delft Furnace .................... 64
4.5.1 Effect of Gas Concentration ................................ 65
4.5.2 Effect of Particle Concentration ............................ 66
4.6 Application to a Rocket Plume Base Heating Problem ............... 68
4.6.1 Effect of Gas Concentration ................................ 69
4.6.2 Effect of Particle Concentration ............................ 70
4.7 Conclusions .................................................... 71
4.8 Closure ........................................................ 73
References ........................................................... 73
5 RadiativeHeatTransferinConicalGeometries ......................... 75
5.1 Introduction .................................................... 75
5.2 FVM-SLW Formulations for Body-Fitted Conical Geometries ........ 75
5.3 Validation ...................................................... 79
5.3.1 Validation with Absorbing Emitting and Scattering Medium .... 79
5.4 Application to a Conical Diffuser ................................. 79
5.4.1 Decision on the Minimum Number of Gray Gases ............ 81
5.4.2 Effect of Gas Concentration ................................ 82
5.4.3 Effect of Particle Concentration ............................ 83
5.4.4 Effect of Cone Angle ..................................... 84
5.4.5 Effect of Anisotropic Scattering ............................ 85
5.4.6 Effect of Wall Emission ................................... 87
