Computational Analysis of Heat Transfer in Fluids and Solids (Defect and Diffusion Forum)

205 Pages·2018·38.477 MB·English

by  Oluwole Daniel Makinde

Computational Analysis of
Heat Transfer in Fluids
and Solids II

Edited by
Prof. Oluwole Daniel Makinde
Computational Analysis of
Heat Transfer in Fluids
and Solids II

Special topic volume with invited peer-reviewed papers only
Edited by
Prof. Oluwole Daniel Makinde
Copyright  2020 Trans Tech Publications Ltd, Switzerland

All rights reserved. No part of the contents of this publication may be reproduced or
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Volume 401 of
Defect and Diffusion Forum
ISSN print 1012-0386
ISSN cd 1662-9515
ISSN web 1662-9507
(Pt. A of Diffusion and Defect Data – Solid State Data ISSN 0377-6883)

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Preface

Heat transfer and fluid flows seems to pervade all aspects of our life, since almost everything
experiences heating or cooling of some kind and the entire life depends on fluid flows. Finding
appropriate solutions to heat transfer problems in fluid and solid materials will enhance functional
success of the materials and facilitate new product development in industries and engineering. This
special issue on “Computational Analysis of Heat Transfer in Fluids and Solids II” in the journal
“Defect and Diffusion Forum” addresses various nonlinear models involving heat transfer
phenomenon in fluids and solids. Computational techniques are employed to analysis the problems
and numerical results are discussed quantitatively in order to demonstrate the salient features of
practical engineering and industrial applications. The topics covered by excellence research papers in
this issue include: tribology, extended surfaces fins, reactive flow problem, Newtonian and non-
Newtonian flow, nanofluids dynamics, boundary layer flow, natural convection, hydrodynamic
stability, biomechanics, plasma physics, physics of dusty plasma, forced convection, mixed
convection, magnetohydrodynamics, thermal radiation, porous media flow, and irreversibility
analysis.
The pertinent results obtained are very staple in understanding the complex interaction of heat
transfer with fluids flow and solids mechanics. Wide range of applications of the work in this special
issue can be found in the area of materials development, hydrodynamic lubrication, thermal storage,
biomedical, solar heating, nuclear system cooling, micromixing technologies, military equipment storage,
cooling of electrical and electronics components, product management, power production, pollution
control and safety assessment.
It is our hope that this special issue will inspire and help a wide audience of researchers,
scientists, engineers and educators from various fields of human activity.
Our appreciation goes to all the participants for their excellent contribution toward the success
of this special issue. The outstanding work of the reviewers and their constructive comments are
highly appreciated.

Professor Oluwole Daniel Makinde
Editor
Table of Contents
Preface
A Study of Transient Heat Transfer through a Moving Fin with Temperature Dependent
Thermal Properties
L.P. Ndlovu and R.J. Moitsheki 1
Determination of Proper Fin Length of a Convective-Radiative Moving Fin of Functionally
Graded Material Subjected to Lorentz Force
G.A. Oguntala, G. Sobamowo, Y. Ahmed and R.A. Abd-Alhameed 14
A Note on the Similar and Non-Similar Solutions of Powell-Eyring Fluid Flow Model and
Heat Transfer over a Horizontal Stretchable Surface
R. Khan, M. Zaydan, A. Wakif, B. Ahmed, R.L. Monaledi, I.L. Animasaun and A. Ahmad 25
Biomechanics of Surface Runoff and Soil Water Percolation
J.M.M. Deng and O.D. Makinde 36
Finite Element Numerical Investigation into Unsteady MHD Radiating and Reacting Mixed
Convection Past an Impulsively Started Oscillating Plate
B.P. Reddy, P.M. Matao and J.M. Sunzu 47
Analytical and Numerical Study on Cross Diffusion Effects on Magneto-Convection of a
Chemically Reacting Fluid with Suction/Injection and Convective Boundary Condition
S. Eswaramoorthi, M. Bhuvaneswari, S. Sivasankaran and O.D. Makinde 63
Physical Aspects on MHD Micropolar Fluid Flow Past an Exponentially Stretching Curved
Surface
K.A. Kumar, V. Sugunamma, N. Sandeep and S. Sivaiah 79
MHD Flow of Non-Newtonian Molybdenum Disulfide Nanofluid in a Converging/Diverging
Channel with Rosseland Radiation
J. Raza, F. Mebarek-Oudina, P. Ram and S. Sharma 92
Buoyancy Effects on Human Skin Tissue Thermoregulation due to Environmental Influence
L.H. Adeola and O.D. Makinde 107
Turbulent Heat Transfer Characteristics of a W-Baffled Channel Flow - Heat Transfer
Aspect
Y. Menni, A.J. Chamkha and O.D. Makinde 117
MHD Boundary Layer Flow over a Cone Embedded in Porous Media with Joule Heating
and Viscous Dissipation
S. Devi and M.K. Sharma 131
Squeeze Film Lubrication on a Rigid Sphere and a Flat Porous Plate with Piezo-Viscosity
and Couple Stress Fluid
B.N. Hanumagowda, C.K. Sreekala, Noorjahan and O.D. Makinde 140
Influence of Homogeneous and Heterogeneous Chemical Reactions and Variable Thermal
Conductivity on the MHD Maxwell Fluid Flow due to a Surface of Variable Thickness
G. Sarojamma, K. Sreelakshmi, P.K. Jyothi and P.V.S. Narayana 148
Heat Transfer Analysis of Three-Dimensional Mixed Convective Flow of an Oldroyd-B
Nanoliquid over a Slippery Stretching Surface
K.V. Prasad, H. Vaidya, K. Vajravelu, G. Manjunatha, M. Rahimi-Gorji and H. Basha 164
Effects of Variable Fluid Properties on Oblique Stagnation Point Flow of a Casson
Nanofluid with Convective Boundary Conditions
H. Vaidya, K.V. Prasad, K. Vajravelu, A. Wakif, N.Z. Basha, G. Manjunatha and U.B.
Vishwanatha 183
Defect and Diffusion Forum Submitted:2019-03-29
ISSN: 1662-9507, Vol. 401, pp 1-13 Revised:2019-06-14
© 2020 Trans Tech Publications Ltd, Switzerland Accepted:2019-09-18
Online:2020-05-28
A Study of Transient Heat Transfer Through a Moving Fin with
Temperature Dependent Thermal Properties
Partner Luyanda Ndlovu1,2,a and Raseelo Joel Moitsheki1,b,∗
1School of Computer Science and Applied Mathematics, University of the Witwatersrand, Private
Bag 3, WITS 2050, Johannesburg, South Africa
2Standard Bank of South Africa, 30 Baker Street, Rosebank, Johannesburg, 2196, South Africa
[email protected], [email protected]
Keywords: Heat transfer, analytical solutions, DTM, extended surface
Abstract.Inthisarticle,heattransferthroughamovingfinwithconvectiveandradiativeheatdissipa­
tion is studied. The analytical solutions are generated using the two­dimensional Differential Trans­
form Method (2D DTM) which is an analytical solution technique that can be applied to various
types of differential equations. The accuracy of the analytical solution is validated by benchmark­
ing it against the numerical solution obtained by applying the inbuilt numerical solver in MATLAB
(pdepe). A good agreement is observed between the analytical and numerical solutions. The effects
of thermo­physical parameters, such as the Peclet number, surface emissivity coefficient, power in­
dexofheattransfercoefficient,convective­conductiveparameter,radiative­conductiveparameterand
non­dimensionalambienttemperatureonnon­dimensionaltemperatureisstudiedandexplained.Since
numerousparametersarestudied,theresultscouldbeusefulinindustrialandengineeringapplications.
Introduction
Anincreasingnumberofengineeringapplicationsareconcernedwithenergytransportrequiringrapid
heat dissipation. To increase the heat transfer rate from a heated surface, fin (or extended surface)
assembly is commonly used. The use of fins is seen in various industrial applications such as oil
carryingpipelines,spacenuclearreactorpowersystemsandmanymore.Theheattransfermechanism
of a fin is to conduct heat from a heat source by thermal conduction, and then dissipate heat to an
ambient fluid by the effect of thermal convection and radiation. An extensive review of extended
surface heat transfer is given by Kern and Kraus [1] and Kraus et al. [2]. A brief review of published
workthatisofrelevancetothisarticleispresentednext.
A detailed study of temperature distribution in a moving fin is provided for various embedding
parameters in a triangular fin [3], rectangular fin [4], an exponential fin [5] and a trapezoidal cross
section[6].SinglaandDas[7]studiedaninverseproblemtoestimatethespeedofamovingfinusing
the binary­coded generic algorithm. Sun et al. [8] applied the Spectral Collocation Method (SCM) to
derivethesolutionforaconvective­radiativeheattransferthroughamovingrodwithvariablethermal
conductivity.SunandXu[9]furtherappliedtheSCMtopredictthetemperaturedistributioninmoving
finsofcomplexcross­sections.Recently,Maetal.[10]simulatedacombinedconductive,convective
andradiativeheattransferinirregularmovingporousfinsusingSpectralElementMethod(SEM).In­
vestigation of mixed convection heat transfer along a continuously moving heated vertical plate with
suctionandblowingwasconductedbyAl­Sanea[11].Theradiationeffectsarequitesignificantinvar­
iousengineeringandindustrialprocessesespeciallyinthedesignofreliableequipments,nuclearplants
and gas turbines. Razelos and Kakatsios [12] investigated the optimum dimensions of convecting­
radiating rectangular fins. They studied the influence of all dependent parameters on optimization
and performance of rectangular fins. They also investigated the effect of the temperature­dependent
thermal conductivity and emissivity on the temperature profile. An approximate analytical solution
for convection­radiation heat transfer from a continuously moving fin with temperature­dependent
thermalconductivitywasdevelopedbyAzizandKhani[13].TorabiandZhang[14]investigatedther­
mal performance of a convective­radiative straight fin by considering different profiles containing
2 Computational Analysis of Heat Transfer in Fluids and Solids II
nonlinearities and further studied the fin efficiency. Sun et al. [15] studied the different temperature­
dependentthermalpropertiesforpredicationofheattransferinaconvective­radiativefin.Kunduand
Wongwises [16] presented a decomposition analysis of a convective­radiating fin with one side of
the primary surface being heated by a fluid with high temperature. Aziz and Makinde [17] applied
a two­dimensional heat conduction model to obtain the thermal performance and entropy generation
in an orthotropic convection pin fin used in advanced light weight heat sinks. The study illustrated
thesimultaneousrealizationoftheleastmaterialandminimumentropygenerationinpinfindesigns.
Mhlongo et al. [18] applied the local and nonlocal symmetry techniques to study transient response
ofrectangularfinstostepchangeinbasetemperatureandbaseheatflux.Theauthorsshowedthatthe
governingequationscanbereducedtothetractableErmakov­Pinneyequation.
The DTM is a semi­numerical­analytical method based on the Taylor series expansion and was
firstproposedbyZhou[19]in1986forthesolutionoflinearandnonlinearinitialvalueproblemsthat
appear in the analysis of electrical circuits. This method was subsequently used to obtain analytical
solutions of various types of higher­order differential equations. Torabi et al. [20] applied the DTM
to derive approximate explicit analytical expressions for the temperature distribution in a moving fin
withtemperature­dependentthermalconductivityandexperiencingsimultaneousconvective­radiative
surface heat loss. Moradi and Rafiee [21] extended the work of Torabi et al. to study a similar model
for various fin profiles. Ndlovu and Moitsheki [22] successfully applied the 2D DTM to a transient
heatconductionproblemforheattransferinlongitudinalrectangularandconvexparabolicfins.Some
interestingresultswereobtainedandtheeffectsoftheparametersappearinginthemodelonthetem­
perature distribution were illustrated and explained. Mosayebidorcheh et al. [23] studied transient
thermal behavior of radial fins of rectangular, triangular and hyperbolic profiles. These authors ap­
plied the Hybrid Differential Transform Method­Finite Difference Method (DTM­FDM) to generate
thenumericalsolutionstotheproblem.Falloetal.[24]appliedthe3DDTMforthefirsttimetostudy
heat transfer in a cylindrical spine fin with variable thermal properties. Hatami et al. [25] applied
the Differential Quadrature Method (DQM) together with the DTM to study magnetohydrodynamic
(MHD)two­phaseCouetteflowanalysisforfluid­particlesuspensionbetweenmovingparallelplates.
To the best our knowledge, no research to date has been performed to study transient heat transfer in
amovingfins.Numerousstudieshavebeendevotedtosteadystateanalysiswiththeassumptionthat
thetransientresponsediesoutquickly[26,27].
In this article, 2D DTM is applied to study simultaneous conductive, convective and radiative
heattransferinamovingfinofrectangularprofile.Thermalconductivity,heattransfercoefficientand
surface emissivity are all temperature dependent. The problem formulation is presented in Section 2.
Abriefdiscussiononthefundamentalsofthe2DDTMisprovidedinSection3.Thevalidationofthe
analytical solutions together with analytical results are given in Section 4. Lastly, we provide some
discussionsbasedontheresultsobtainedinSection5andtheconclusionssummarizedareSection6.
ProblemFormulation
We consider a one dimensional rectangular moving fin of length L, with cross­sectional area A ,
c
thickness δ and perimeter P while it moves horizontally with a constant velocity U as depicted in
Figure 1. The fin surface is exposed to a convective and radiative environment of temperature T
a
and the base temperature of the fin is T > T . The fluid surrounding the moving fin is heated by
b a
theconvective­radiativefinandinducesahorizontalflowfield(advection).Theradiativecomponent
wouldbemoreprominentiffinundergoesnaturalconvectionoriftheforcedconvectionisratherweak
orabsent.Theenergybalanceequationforthemovingfinlosingheatbysimultaneousconvectionand
radiationcanbeexpressedas
(cid:20) (cid:21)
∂T ∂ ∂T PH(T) Pσε(T) ∂T
ρc = K(T) − (T −T )− (T4 −T4)−ρcU , 0 ≤ X ≤ L. (1)
∂t ∂X ∂X A a A a ∂X
c c
Defect and Diffusion Forum Vol.401 3
Formostmaterials,thethermalconductivityvarieslinearlywithtemperature,thatis,
K(T) = k [1+γ(T −T )], (2)
a a
thesurfaceemissivityisalsoassumedtovarylinearlywithtemperature,
ε(T) = ε [1+η(T −T )], (3)
a a
andheattransfercoefficientmaybegivenbyapowerlawfunctionoftemperature,
(cid:18) (cid:19)
T −T n
a
h(T) = h . (4)
b T −T
b a
Fig. 1:Schematicrepresentationofarectangularmovingfin.
Here, k is the thermal conductivity of the fin at ambient temperature, γ is a measure of thermal
a
conductivityvariationwithtemperature,ε isthesurfaceemissivityofthefinatambienttemperature,
a
η is a measure of surface emissivity variation with temperature, ρ is the density of fluid; c is the
specific heat of fluid; ε and σ are the emissivity and Boltzman constant respectively, h is the heat
b
transfer coefficient at the fin base and n is a constant. The constant n may vary between ­6.6 and
5. However, in most practical applications it lies between ­3 and 3 [28]. The exponent n represents
laminar film boiling or condensation when n = −1/4, laminar natural convection when n = 1/4,
turbulent natural convection when n = 1/3, nucleate boiling when n = 2, radiation when n = 3 and
n = 0 implies a constant heat transfer coefficient. Assuming that the fin tip is adiabatic (insulated)
andthebasetemperatureiskeptconstant,thentheboundaryconditionsaregivenby[2]etal.,
T(t,L) = T and (5)
(cid:12) b
(cid:12)
∂T (cid:12)
(cid:12) = 0. (6)
∂X
X=0
Initiallythefiniskeptattheambienttemperature,
T(0,X) = T . (7)
a
4 Computational Analysis of Heat Transfer in Fluids and Solids II
Introducingthefollowingdimensionlessvariables,
X k t T T
a a
x = , τ = , θ = , θ = , ζ = ηT , β = γT ,
L ρcL2 T a T b b
b b
(8)
Ph TnL2 Pε σL2T3 k UL
N = b b , N = a b , α = a, andPe = ,
c k A (T −T )n r k A ρc α
a c b a a c
reducestheenergyequation(1)to
(cid:20) (cid:21)
∂θ ∂ ∂θ
= {1+β(θ−θ )} −N (θ−θ )n+1 −N [1+ζ(θ−θ )](θ4 −θ4)
∂τ ∂x a ∂x c a r a a
∂θ
−Pe , 0 ≤ x ≤ 1. (9)
∂x
Theprescribedboundaryconditionsaregivenby,
θ(τ,1) = 1and (10)
(cid:12)
(cid:12)
∂θ(cid:12)
(cid:12) = 0. (11)
∂x
x=0
andtheinitialconditionbecomes,
θ(0,x) = 0. (12)
Thedimensionlessvariableθrepresentsthetemperature,θ representsthedimensionlessambienttem­
a
perature, x is the dimensionless space variable, τ is the dimensionless time variable, β is the thermal
conductivitygradient,ζ isthesurfaceemissivitygradient,N istheconvection­conductionparameter,
c
N istheradiation­conductionparameter,PeisthePecletnumberwhichrepresentthedimensionless
r
speedofthemovingfin,α isthethermaldiffusivityofthefin.
FundamentalsoftheTwo­DimensionalDifferentialTransformMethod
In this section, the basic idea underlying the two­dimensional DTM is briefly introduced. If function
θ(t,x)isanalyticanddifferentiatedcontinuouslywithrespecttotimeandthespatialvariablexinthe
domainofinterest,thenwelet
(cid:20) (cid:21)
1 ∂κ+sϕ(t,x)
Φ(κ,s) = , (13)
κ!s! ∂tκ∂xs
(0,0)
where the spectrum Φ(κ,s) is the transformed function, which is also called the T­function (see [29,
30]).ThedifferentialinversetransformofΦ(κ,s)isdefinedas
X∞ X∞
ϕ(t,x) = Φ(κ,s)tκxs, (14)
κ=0 s=0
andfromequations(13)and(14)itcanbeconcludedthat
(cid:20) (cid:21)
X∞ X∞
1 ∂κ+sϕ(t,x)
ϕ(t,x) = tκxs. (15)
κ!s! ∂tκ∂xs
κ=0 s=0 (0,0)