ORIGINAL

SK Mahapatra Æ P Nanda Æ A Sarkar

Analysis of coupled conduction and radiation heat transferin presence of participating medium- using a hybrid method

Received: 15 August 2003 / Published online: 8 June 2005� Springer-Verlag 2005

Abstract The current study addresses the mathematicalmodeling aspects of coupled conductive and radiativeheat transfer in presence of absorbing, emitting andisotropic scattering gray medium within two-dimensionalsquare enclosure. The walls of the enclosure are consid-ered to be opaque, diffuse and gray. The enclosurecomprised of isothermal vertical walls and insulatedhorizontal walls. A new hybrid method where the con-cepts of modified differential approximation employedby blending discrete ordinate method and sphericalharmonics method, has been developed for modeling theradiative transport equation. The finite volume methodhas been adopted as the numerical technique. The effectof various influencing parameters i.e., radiation-con-duction parameter, surface emissivity, single scatteringalbedo and optical thickness has been illustrated. Thecompatibility of the method with regard to solving cou-pled conduction and radiation has also been addressed.

List of symbols

AW, AE, AS, AN Four face areas (West, East, South,North)

I Radiation intensity (watt/m2)Ib Black body radiation intensity

(=rT4/p)k Thermal conductivity (watt m�1 K�1)H Characteristic length (m)qR Radiation heat flux (watt/m2)QT Total heat flux

qr Radiative heat fluxqc Conductive heat fluxRC Radiation-conduction parameter

(=r TH3 H/k)

T Absolute temperature (K)TH, T_C Hot and cold wall temperaturesX, Y Dimensionless co-ordinate.wi Quadrature weight associated with in

any direction si

Greek Symbols

aa Absorption coefficient (1/m)as Scattering coefficient (1/m)b Extinction co-efficient (=as +aa)X Solid angle [sr]r Stefan Boltzman’s constant [5.67·10�8 watt

m�2 K�4]� Wall emissivityn, g X and Y direction cosinesx Single scattering albedo (as/b)q Reflectivity of the surface.s Total optical depth (=b H)h Dimensionless temperature (T/TH)

Subscripts:

c - Conduction transferR - Radiation transferH, C, L - Hot wall, Cold wall, Bottom wallw, m - wall, medium

1 Introduction

The transport of thermal radiation is an importantmechanism of energy transport in numerous engineering

S. Mahapatra Æ P. NandaMechanical Engg. Deptt., University College of Engineering,Burla, Orissa, India

S. Mahapatra (&)Burla Engineering College Campus, Qrs No. 3R/32,Professor Colony, 768018, Orissa, IndiaE-mail: swarupkumar_m@yahoo.comFax: +91663-2430204

A. SarkarMechanical Engg. Deptt.,Jadavpur University, Kolkata, India

Heat Mass Transfer (2005) 41: 890–898DOI 10.1007/s00231-004-0587-4

applications. The constitutive medium in majority ofthese systems actively participates in the radiativetransfer due to absorption, emission, and scattering ofradiation. Examples are abundant, notably fluidized bedcombustion, insulation systems, particulate solarcollectors and combustion systems such as furnacescontaining fly ash, coal particles, soot agglomerates, etc.

Due to the difficulty in finding the exact analyticalsolution to integro-differential radiative transfer equa-tion (RTE) in radiatively participating media, a diversityof numerical methods have been worked out over lastfew decades. Monte Carlo (MC) technique, zonalmethod, spherical harmonics method, discrete ordinatemethod are few established methods which are also notfree from limitations. Researchers found the possibilityof radiation modeling with Monte Carlo technique forany participating medium with any desired radiativefeature even though it is subjected to computationalexpense. In this regard, the work of Buckius [1]addresses the improvement in the computational time byusing reverse MC technique. The account of non-homogeneity, non-isothermal, reflecting boundaries, an-isotropic scattering has also been addressed in the work.Yuen and Takara [2] has laid down the concept ofsuperposition of fundamental principles in a gray-wal-led, two-dimensional rectangular enclosure with grayabsorbing, emitting, isotropic scattering medium to findthe temperature and heat flux distribution. The con-ventional zonal method has been employed in his work.The total exchange area method described by Hottel andSarofim [3], and its variant in the work of Noble [4] andLarson and [5] is another advanced superposition tech-nique, found to be valid even in multi-mode heattransfer. Direct superposition is not valid in these non-linear situations whereas the total exchange area methodfind its application with computational inefficiency.

Among the many numerical methods of solving RTE,the discrete ordinate method is considered a potentialand a very promising tool, which transforms RTE intoset of simultaneous partial differential equation. Atpresent its popularity is clearly surging. Many variationsin the DOM are suggested in the literature. In DOM, theRTE is solved only in a fixed number of directions.Neutron transport equation by Carlson and Lathrop [6]was first applied with this method. It was also applied tothree-dimensional absorbing, emitting and scatteringmedium by Fiveland [7] and therein, the stability andaccuracy of its solution is discussed as well. Baek andKim [8] analyzed the combined conduction and radia-tion heat transfer (CCR) in absorbing emitting and an-isotropic scattering media using S-N discrete ordinatemethod. Subsequent development on DOM, Cheongand Song [9] developed and critically examined varioussolution schemes for second order DOM. Koo et al. [10]presented the numerical schemes and comparison ofpredictions of radiative transfer for the first and secondorder discrete ordinate method using an interpolationscheme. It has been pointed out that the first orderDOM gives more accurate results compared to second

order DOM. Miranda [11] presented an alternativeformulation of S-N discrete ordinates for predictingradiative transfer in non gray gases. Sakami [12] treatedthe radiative equilibrium problem with modified discreteordinate method in complex enclosures with obstacles.The anomalies of the outcome due to ray effect usingdiscrete ordinate method gets minimized with modifieddiscrete ordinate method. The spherical harmonicsmethod (P1-method) known as differential approxima-tion, enjoys great popularity because of relative sim-plicity and its compatibility with standard methods forsolution of the energy equation. The fact that P1

—approximation may become very inaccurate in opti-cally thin media and thus of limited use —has promptedmany investigators to make it reasonably accurate for allconditions. In this direction, the works of Olfe [13] andGlatt and Olfe [14] developed modified differentialapproximation (MDA) and used their model to gray-walled enclosures with gray, non-scattering media. andwhich was mostly suitable to one-dimensional planeparallel media. Wu et al. [15] demonstrated for onedimensional plane parallel media, that the MDA may beextended to scattering media with reflecting boundaries.Modest [16] showed that the method can be applied tothree-dimensional linear-anisotropic scattering mediawith reflecting boundaries. Park [17] in his work men-tioned the difficulty of evaluating the integral fordetermination of surface radiosity for adjacent elementswhile solving the problem of radiative equilibrium with(MDA). The improved differential approximation wasdeveloped by Modest [18], applied to two-dimensional,non-scattering media at radiative equilibrium. Thedevelopment of discrete transfer method by Shah [19]combines the feature of discrete ordinates, zonal andMonte Carlo methods. Cumber [20] discussed thechronological development of discrete transfer methodand comments with regard to the ray tracing problemand excessive computational time. Coelho [21] has pre-sented the conservative formulation of discrete transfermethod.

Among the few earlier works on coupled conductionand radiation (CCR) problems, Grief [22] analyzed inpresence of participating medium with variable proper-ties. Yuen and Wong [23] investigated the influence ofthe anisotropic scattering on combined heat transfer inone-dimensional planar geometry. The use of finite ele-ment by Razzaque et al. [24] for solving two-dimensionalCCR problems was limited to only non-scatteringmedium with isothermal black walls. Shih [25] developeddiscretized intensity method for solving CCR problemfor two-dimensional enclosure with isothermal walls inpresence of participating medium. The product inte-gration method finds its application in the work of Tan[26] for solving CCR problem in square enclosure withisothermal walls. Collapse dimension method has beenused to investigate CCR problem by Talukdar andMishra [27]. Krishnaprakash et al. [28] analysed CCRproblem in a nonlinearly anisotropically scatteringmedium between two parallel reflecting diffusely and

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specularly surfaces. Numerov’s method has been adop-ted to solve the energy equation in addition to the RTEsolution by S32 discrete ordinate method.

It is learnt from above that till date researchers arestill in pursuit of the numerical methods for radiationmodeling for participating medium of any radiativefeature, which will be also suitable for multi-mode heattransfer in multi-dimensional geometries. The com-bined conduction and radiation problem that aresolved so far also limited to isothermal walls withsimplified assumptions. In this regard, an attempt hasbeen made in the present work i.e. coupled conductionand radiation phenomena with gray participatingmedium within square enclosure has been analyzed,developing a new method. The differential approxima-tion and discrete ordinate method are blended togetherconsidering their strength to generate a new method.The square enclosure is considered as the physicalmodel, which is assumed to comprise with verticalisothermal walls and insulated horizontal walls. Theapplicability of the blended method has been discussedwhile examining the effect of various influencingparameters such as surface emissivity, optical thicknessand scattering albedo on temperature distribution andheat flux.

2 Analysis

The physical model along with coordinate system isdepicted in Fig. 1, in which a square enclosure withoptical thickness s in either of direction is considered.

The general equation of transfer for an absorbing,emitting and isotropic scattering gray medium at anylocation and direction (r, s assuming the process to bequasi-steady, is represented as Modest [29]

srI þ b ¼ aaIb þas

4p

Z

4p

I r; sið Þ/ si; sð Þ dXi ð1Þ

In the above equation, � I refers to spatial gradient ofradiation intensity and the incoming and outgoing vec-tors are represented by si and s and the phase function/ðsi; sÞ for isotropic scattering is taken as unity. Theright part of the Eq 1 represents the source function forthe radiation intensity. For the diffusely reflecting walls,the above Eq 1 subjected to the following boundarycondition

Iðrw; sÞ ¼JwðrwÞ

p¼ IbwðrwÞ �

1� epe

q:nðrwÞ ð2Þ

The non-linear integro-differential eq 1 does not haveclosed form solution, even if in much simplified condi-tions. In this work the philosophy of modified differen-tial approximation [16] has been adopted in order tomodel the radiative transfer equation. The radiationintensity at any point r; sð Þ is considered to be the sum ofthe contributions from surface (Iw) (assuming mediumto be transparent), and medium (Im).

So,

Iðr; sÞ ¼ Iwðr; sÞ þ Imðr; sÞ ð3Þ

As Iw is evaluated assuming medium to be transpar-ent, it satisfies the condition (i.e. Source term=0),Which satisfies

dIwðr; sÞdss

¼ �Iwðr; sÞ ð4Þ

leading to

Iwðr; sÞ ¼JwðrwÞ

pe�ss ð5Þ

The radiosity variation along the enclosure wall canbe determined as the sum of emission plus reflectedirradiation and is represented as

JwðrÞ ¼ epIbwðrÞ þ ð1� eÞZ

s:n\0

Iwðr; sÞ s:nj jdX ð6Þ

For evaluation of surface radiosity, instead of usingthe above Eq 6, discrete ordinate method has been em-ployed. The equation for determination of radiationintensity employing DOM [8] and upon discretizing itwith finite volume method, appears as

Ipi ¼cbVSpi þ nij jAEWIxi þ gij jANSIyi

cbV þ nij jAE þ gij jAWð7Þ

where AEW=(1-c)AE+cAW and ANS=(1-c)AN+cAS

and Ixi and Iyi are the corresponding inlet radiationintensities for x and y directions faces of control vol-umes.

The relation between the cell averaged radiationintensity with cell edge intensities for ith direction in acontrol volume is given as

Ipi ¼ cIxei þ ð1� cÞIxli ¼ cIyei þ ð1� cÞIyli ð7aÞFig. 1 Computational test domain

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In the present work as per symmetry scheme c=0.5has been adopted. The radiation intensity subscriptedwith e,l represents the entering and leaving conditioneither in X or Y direction.

In this Eq 7 making source term (SP) null, the radi-ation intensity determined represents the radiationintensity contributed due to surface radiation only as

Iwi¼ nij jAEWIxi þ gij jANSIyi

cbV þ nij jAE þ gij jAWð7bÞ

Subsequently, Gw(r) and qw(r) representing wall-re-lated incident radiation and heat flux are calculatedusing the following equations as per DOM,

qwðrÞ ¼Z

4p

Iðr; sÞsdX ¼Xn

i¼1wiIwiðrÞsi ð8Þ

GwðrÞ ¼Z

4p

Iðr; sÞdX ¼Xn

i¼1wiIwiðrÞ ð9Þ

where the wi represents the quadrature weights associ-ated with direction i.

The boundary condition while determining IPi fromEq 7, is given as

Iðrw; siÞ ¼JwðrÞ

p

¼ eðrwÞIbðrwÞ þrsðrwÞ4p

Xn:s\0

wjIðrw; sjÞ n:sj

�� �� ð10Þ

Isolating surface radiation, intensity from withinmedium according to P1 approximation, can beexpressed as combination of medium related incidentradiation and heat flux, Modest [22] as

Imðr; sÞ ffi1

4pGmðrÞ þ 3qmðrÞ:s½ � ð11Þ

where

GmðrÞ ¼Z

4p

Imðr; sÞdX and qmðrÞ ¼Z

4p

Imðr; sÞsdX

Substituting Eqs 4 and 11 into Eq into Eq 1, it yields

dImdss¼ srsIm ffi ð1� xÞIb þ

x4p½Gw þ Gm� � Im ð12Þ

After integrating the Eq 12 and taking the divergence, itresembles the Helmhotz equation as

r2sGm ¼ �3½ð1� xÞ4pIb þ xðGw þ GmÞ � Gm� ð13Þ

The necessary boundary condition for the aboveequation is obtained by making an energy balance forthe medium related radiation at a point on the surfacewhich leads to Marshak’s boundary condition [29] for acold surface, in a simplified manner is expressed as

2ð2e� 1Þqm: nþ Gm ¼ 0 ð14Þ

The energy conservation equation for steady statecondition in the absence of convection and heat gener-ation, assuming constant properties in a two-dimen-sional cartesian coordinate system is expressed as

k@2T@x2þ @

2T@y2

� �¼ ð1� xÞb 4pIb � Gð Þ ð15Þ

This equation needs to satisfy the isothermalboundary condition at the vertical walls and insulatedboundary condition at horizontal walls.

The following parameters are used for expressingabove governing equations in dimensionless form.

g ¼ GrT 4

h

; h ¼ TTh; X ¼ x

L; Y ¼ y

L, RC ¼ rT 3

h LK

;

�Ib ¼Ib

rT 4h

; gw ¼Gw

rT 4h

; �S ¼ SrT 4

h

;

�qc ¼qc

kTh=L; �qw ¼

qw

kTh=L; �qm ¼

qmkTh=L

;

�qr ¼ �qm þ �qw; s ¼ bL ð16Þ

The transport equation 13 for medium relatedintensity appears in dimensionless form as

r2sgm � 3ð1� xÞgm ¼ �12ð1� xÞh4 � 3xgw ð17Þ

and the boundary condition as

2

3eð2� eÞrsgmn� gm ¼ 0 ð18Þ

The energy equation 15 for the transport phenomenais represented as

@2h@X 2þ @2h@Y 2¼ RC s 4h4 � gm � gw

� �ð1� xÞ ð19Þ

with boundary conditions for this energy equation isgiven as

h ¼ 1 at X ¼ 0 for 0 � Y � 1

h ¼ Tc=Th at X ¼ 1 for 0 � Y � 1

and �qr þ �qc ¼ 0 at Y ¼ 0; 1 for 0 � X � 1

The volume averaged intensity of Eq 7b in a dimen-sionless form, which has been used to find out theradiation intensity due to surface radiation only, isexpressed as

�IPi¼ nij j�Ixi þ gij j�Iyi

csþ nij j þ gij jð20Þ

and the Eq 7a in dimensionless form appears as

�Ipi ¼ c�Ixei þ ð1� cÞ�Ixli ¼ c�Iyei þ ð1� cÞ�Iyli ð20aÞ

and the boundary condition in dimensionless form ap-pears as

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�Iwi¼ ew�Ib þ

1� ewp

Xnj\0

wj�Ij nj

�� �� ð21Þ

The non-dimensional radiative heat flux and incidentradiation are given as

�qw ¼ RCXn

i¼1wi�Iwi si and gw ¼

Xn

i¼1wi�Iwi ð22Þ

3 Numerical procedure

The computational domain is represented by uniformlyspaced MXN control volumes. Nodes are considered tolie at the centre of control volumes and at the center ofthe faces of the boundary control volumes. The mediumrelated radiation intensity represented by Helmhotzequation upon application of P1 approximation methodhas been discretized using the central difference scheme.In order to obtain the wall related intensity, uponapplication of S4 DOM; the radiative transport equationhas been discretized using the finite volume method. Thesource term in Eq 7 is made null and in order to obtainthe radiation intensity because of the contribution fromthe surface radiation only using Eq 20 and solutionprocedure for DOM is adopted as cited in the work [30].During the iteration, it is observed that the cell averagedintensity predicted from Eq 20 and the relation with celledge intensity from Eq 20a, yield negative intensities,which are meaningless from physical point of view andtherefore they are made zero to obtain the convergence.The nonlinear energy equation is solved employingNewton’s Raphson scheme and under relaxation is donein order to obtain the convergence. It is noted that notonly the governing transport equations are coupled butalso their boundary conditions are also interlinked.Hence, the solution through iteration is inevitable. Theiterative procedure continues till the convergence criteriais met. The flow chart given in the appendix shows thedetailed numerical procedure adopted for obtainingsolution.

4 Results and discussions

The present study has been carried out to investigate theeffects of various influencing parameters, such as radi-

ation–conduction parameter, optical thickness, singlescattering albedo, and surface emissivity. In addition therange of the parameters within which application of thepresent method is acceptable, is outlined.

Before going to the analysis, the grid independencetest has been conducted and presented in Table 1. Thegrid independent study started with a coarse size of mesh(i.e. 21·21) and subsequently refined up to 45·45. It isobserved from the Table 1 that further refinement ofmesh size over 41·41 does not change the values dif-ferent variables considerably, which is thereforeaccepted as the grid size for all computation.

Before synthesis of the results, the energy balance forthe enclosure has been considered and for the purposeFig. 2a and b are presented. Fig. 2a shows the heat fluxdistribution for both the cold wall and hot wall. Thearea between Y-axis and heat flux distribution curveimplying the energy transfer, is same for both the hotwall and cold wall. Neither the radiative heat transfernor the conductive transfer is zero but their sum is zeroat the insulated wall i.e. satisfying the local condition�qr þ �qc ¼ 0; that also leads the sum to be zero for thewhole wall, which is obvious from the Fig. 2b. However,the sum differs from zero implying the numerical error,which is noticed near the isothermal walls because of thechange over from the isothermal boundary condition toinsulated boundary condition. It can be concluded fromboth figures and the results presented in the table thatthe principle of energy conservation holds good. Forrealizing the effect of various parameters on heattransfer Table 2 be referred and further clarification hasbeen given in the following section.

4.1 Radiation–conduction parameter (RC)

The radiation–conduction parameter has been variedkeeping other influencing parameters at constant values.The increase of RC implies that both the scatteringcoefficient and absorption coefficient decreases in aproportional manner, when other parameters get fixed.This is also well reflected in the temperature field asshown in the Fig. 3. The isotherm nearing either ofactive walls become closer and core tends to be inten-sively heated. More or less uniform intense heating ofcore is observed with higher values of RC, from whichdominance of radiation transfer phenomena is con-cluded. At the RC value of zero implying transfer phe-nomena to be conductive in nature, which results the

Table 1 Grid Independent test (hh=1, hc=0.5, RC=10, s=1.0, �=0.5)

Mesh size QCH QRH QRMH � QCC � QRC � QRMC � QCL QRL (-) h(0.5,0.5)

21·21 1.0259 4.3564 1.7832 1.4962 0.3672 �1.4521 0.0995 0.1203 0.800425·25 1.0346 4.3541 1.8101 1.5171 0.3684 1.4522 0.1040 0.1219 0.801233·33 1.0463 4.3512 1.8458 1.5447 0.3705 1.4520 0.1099 0.1238 0.802137·37 1.0504 4.3503 1.8581 1.5542 0.3711 1.4517 0.1121 0.1247 0.802541·41 1.0538 4.3495 1.8682 1.5622 0.3716 1.4515 0.1138 0.1253 0.802745·45 1.0539 4.3496 1.8682 1.5621 0.3716 1.4514 0.1138 0.1253 0.8027

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isotherms to be equally spaced and perpendicular toinsulated walls as shown in Fig. 3a.

For the RC value other than zero, curvature of iso-therms are envisaged implying presence of radiation.This indicates neither the conductive heat transfer norradiative transfer is zero but their sum is zero at theinsulated wall. It is also observed that some portion ofinsulated wall near hot wall receives heat through con-

duction and emits heat through radiation, while nearcold wall reverse phenomena occur. The rise in RC (i.e.,RC=40) is accompanied with substantial increase inradiative transfer leading to intense heating as obviousfrom Fig. 3d. Further clarity with regard to mid-planetemperature distribution has been provided throughFig. 4. It is clearly revealed from the Fig. 4 that thetemperature variation along the mid plane becomesnonlinear when dominance of radiation becomes sig-nificant (i.e., as RC value increases). The heat flux dis-tribution for different radiation–conduction parameteris shown in the Fig. 5. The fig. 5a shows the variation of

Fig. 2 Conduction and radiation heat flux distribution for both hotwall and cold wall when s=1.0, �=0.5, x=0.5, hH=1.0 andhC=0.5

Table 2 Effect of radiation–conduction parameter, single scattering albedo, surface emissivity and optical thickness on heat transfer(hh=1, hc=0.5)

Constant Variables Variablechanged

QCH QRH QRMH � QTH QCC � QRC � QRMC � QTC h0.5,0.5

s=1 x=0.5 �=0.5 RC=0.0 0.5000 0.0 0.0 0.5 0.500 �0.00 �0.00 0.5 0.7500RC=0.1 0.5081 .0433 0.0178 0.5336 0.5169 0.0029 .0125 0.5323 0.7517RC=1.0 .5786 .4337 0.1808 0.8315 0.6591 0.0303 .1386 0.828 0.7640RC=10 1.0538 4.3495 1.8682 3.5351 1.5622 0.4216 1.5415 3.5253 0.80274RC=40 1.7827 17.4134 7.4488 11.5473 3.0158 1.7056 6.7866 11.508 0.8164

s=1 RC=10 x=0.5 �=0.1 1.2827 .9656 0.3927 1.9156 1.4971 0.0299 0.385 1.912 0.79386�=0.5 1.0538 4.3495 1.8682 3.5351 1.5622 0.4216 1.5415 3.5253 0.80274�=1.0 .8007 8.0317 3.3881 5.443 1.4039 1.6746 2.3104 5.3889 0.80303

s=1 RC=10 �=0.5 x=0.0 1.3212 4.353 1.8538 3.8204 1.9142 0.3738 1.5317 3.8197 0.8054x=0.5 1.0538 4.3495 1.8682 3.5351 1.5622 0.4216 1.5415 3.5253 0.80274x=1.0 .5853 4.3425 1.8624 3.0596 0.9589 0.3637 1.6943 3.0169 0.7790

RC=10 �=0.5 x=0.5 s=1 1.0538 4.3495 1.8682 3.5351 1.5622 0.4216 1.5415 3.5253 0.80274s=2 1.1896 4.6134 2.5173 3.2857 1.683 0.3133 1.3069 3.3032 0.80773s=5 1.1946 4.8151 3.3727 2.637 1.4843 0.2745 0.9041 2.6629 0.81562

Fig. 3 Variation of Isotherm pattern with radiation-conductionparameter a RC=0, b RC=1, c RC=10 and d RC=40, whens=1.0, �=0.5, x=0.5, hH=1.0 and hc=0.5

895

the percentage of radiation heat flux contribution out oftotal heat flux contribution with RC, which implies thatthe phenomena becomes non-linear due to dominance ofradiation transfer. The present work is also validatedagainst the earlier works of Mahapatra [30, 31] i.e. withthe results of P1 approximation method and DOM. It isimplied from the figure that result from P1 approxima-tion is an overestimate as compared to DOM andpresent method. For lower values of RC, the result fromthe present work lies in between results obtained fromboth the methods. However, results are more nearertowards the result obtained from DOM.

4.2 Surface emissivity (�)

The surface emissivity plays an important role in thetransport phenomena with regard to the contributionfrom surface radiation and during mathematical mod-eling it appears in the nonlinear boundary condition. Itis observed that with increase in emissivity, both theradiative transfer and core temperature increases asobvious. It is also seen that the conductive transferdecreases near the hot wall. As shown in Fig. 6 thecurvature of isotherm increases with rise in emissivitybecause of increase in surface radiation. Non-uniformlyspaced vertical isotherms in Fig. 6a imply absence ofsurface radiation but ensures the presence of radiationfrom participating medium.

4.3 Single scattering albedo (x)

The rise in the single scattering albedo keeping opticalthickness constant implies that scattering coefficient riseis accompanied by decrease in absorption coefficient.For pure absorption (x=0), it is seen that core getsintensively heated compared to pure scattering situation.In pure scattering situation, divergence of radiativetransfer for the medium vanishes (i.e., energy equationand radiative transport equation gets uncoupled butthey are interlocked through boundary condition).However, the presence of surface radiation causes thebending of isotherms for pure scattering medium to bemore compared to pure absorbing medium as shown inFig. 7.

4.4 Optical thickness(s)

The rise in optical thickness implies rise in extinctioncoefficient, which is contributed by rise in both scatter-ing and absorption coefficient in a proportional manneras ‘x and RC‘ remains constant. The rise in bothabsorption and scattering coefficient gives rise to de-crease in radiative transfer. The decrease in the radiativetransfer, relatively makes conductive transfer to bedominant, which is also reflected from temperature fieldas shown in Fig. 8 i.e., curvature of isotherms get re-duced.

Fig. 4 Variation of Mid-plane temperature (Y=0.5) distributionwith radiation-conduction parameter a RC=0, b RC=1, c RC=10and d RC=40, when s=1.0, �=0.5, x=0.5, hH=1.0 and hc=0.5

Fig. 5 The variation in the heat flux distribution pattern (a) andthe variation of radiation heat flux in percentage (b) with radiation-conduction parameter, when s=1.0, �=0.5, x=0.5, hH=1.0 andhc=0.5

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5 Conclusions

The interaction of radiation and conduction heattransfer within square enclosure with absorbing, emit-ting and isotropic scattering gray medium has beenmodeled using the concepts of modified differentialapproximation by blending discrete ordinate methodand differential approximation and conclusions areoutlined as follows:

– The present problem with nonlinear boundary con-dition imposed by insulated wall is well solvable usingpresent hybrid method. The method obtains thesolution for both scattering and nonscattering med-ium. The dominance of radiation transfer i.e. withhigher value of RC does not put any restriction forobtaining solution. But use of under relaxation factoris necessary for solving energy equation for obtainingconvergence which makes the computation processslower. The hybrid method is found to be accurate formoderately thick medium for present two-dimensionalproblem. The stability and accuracy of presentmethod is found to be more for finer mesh. The results

obtained from the present method seems to be verynear to that of results from DOM compared to P1

approximation method.– The rise in the value of RC and emissivity and

decrease in optical thickness and single scattering al-bedo increases the radiative transfer. The increase inradiative transfer heats the core intensively.

Appendix

Fig. 6 Variation of isothermpattern distribution withsurface emissivity a �=0, b�=0.1 and c �=1.0, whens=1.0, �=0.5, x=0.5, RC=10,hH=1.0 and hc=0.5

Fig. 7 Variation of isotherm pattern distribution with singlescattering albedo a x=0 and b x=1.0, when s=1.0, �=0.5,x=0.5, RC=10, hH=1.0 and hc=0.5

Fig. 8 Variation of isotherm pattern distribution with opticalthickness a s=0.2 and b s=2.0, when �=0.5, x=0.5, RC=10,hH=1.0 and hc=0.5

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