Cdm Using Dom

8/3/2019 Cdm Using Dom

1/12


2/12

In earlier works on the CDM [1421], the ray tracingapproach similar to the DTM [6,9,10] was used. The raytracing approach although converges fast, has an inher-ent drawback that it requires prior knowledge of thepoints of origin of the radiative intensities on the enclo-sure boundaries, and in this the source term evaluationprocedure is more complicated. Because of this, themethod not only becomes difficult to implement in com-plicated geometries, but it also becomes computationallyinefficient [9,19].

Combined mode problems often require non-uniformor unstructured grids. Such problems, therefore, necessi-tate a radiative transfer method to be compatible to non-uniform or unstructured grids used for solving themomentum and energy equations. In complex or evenin simple geometries, with non-uniform or unstructuredgrids, the ray tracing and the source term evaluation be-

come more difficult. The DTM and the existing ap-proach of the CDM suffer from this drawback.

Since the DOM [35] begins with the differential formof the radiative transfer equation (RTE) and in a givencontrol volume it makes use of the FVM approach tobalance the radiative energy in a given direction, unlikethe DTM, it is more flexible to the type of the grids [23].As the DOM does not require ray tracing like the DTMand the source term evaluation in this method is simpler,it is easier to implement and is thus computationally fas-ter than the DTM [24].

To make the CDM computationally more efficientand also to extend its applications to various types ofgeometries and grids, in the present work the DOM likeapproach is proposed. Since in the CDM, the effectiveintensities lie in the 2-D solution plane and the conceptof solid angle does not arise, unlike the DOM, in this

Nomenclature

A cell-face area

a anisotropy factorc weight factor, Eq. (24)

f weight factor, Eq. (20)I effective intensityIb blackbody effective intensity,

rT4b2

i intensityib blackbody intensity,

rT4bp

M number of effective intensitiesq heat fluxr position, r(x, z)S source term in the RTE dealing with inten-

sity i

eS source term in the RTE dealing with effec-

tive intensity Is direction of intensity i~s direction of the effective intensity IT temperatureV volume of the cellX length of the enclosure in x-directionZ length of the enclosure in z-directionx, z coordinate axis directions

Greek symbols

a planar angleb extinction coefficientc linear interpolation factore emissivity

g collapsing coefficientn direction cosine with respect to z-axish polar angleja absorption coefficientl direction cosine with respect to x-axis

r StefanBoltzmann constant (= 5.670

10

8 W/m2 K4)rs scattering coefficients optical thickness/optical distance~s optical thickness defined in the 2-D solution

planeU scattering phase function while dealing with

intensity iH scattering phase function while dealing with

the effective intensity I/ azimuthal anglew non-dimensional heat fluxx scattering albedoX solid angle, sin h dh d/DX elemental solid angle

Subscripts

b blackbodyB boundarye,w, n,s east, west, north and southin entering the control volume

j upstream point dealing with intensity ij+ 1 downstream point dealing with intensity i~j upstream point dealing with effective inten-

sity I~j 1 downstream point dealing with effective

intensity Iout leaving the control volume

P value at cell centerx, z x- and z-reference faces

Superscript

m index for direction

S.C. Mishra et al. / International Journal of Heat and Mass Transfer 49 (2006) 3041 31


3/12

method the selection of discrete directions is simple andevaluation of their corresponding weights is much easierthan the DOM.

To validate and also to compare the performance of

the proposed formulation of the CDM, hereafter calledthe modified collapsed dimension method (MCDM),with the earlier approach of the CDM [1321] and theDOM [35], three test problems have been taken up.For various parameters like the extinction coefficient,scattering albedo, etc., heat flux and temperature resultshave been compared with the benchmark results. CPUtimes of the MCDM, CDM and the DOM have beenalso compared.

In previous works [17,19], performance of the CDMhas been compared with that of the DTM. Although inearlier applications [2,1422], the CDM has been foundto provide accurate results, as far as comparison of thecomputational efficiencies of the CDM and the DOMis concerned, so far no study has been reported. Sincethe proposed formulation of the CDM (MCDM) usesa DOM like formulation, the present work is, therefore,also aimed at comparing the computational efficienciesof the two methods.

2. Formulation

At any point P of an area element dA as shown inFig. 1(a), the RTE for direction s(X) is given by [25]

di X

dsX jaib bi X

rs

4p

Z4pX0

i X U X0;X dX 1

where ja is the absorption coefficient, b is the extinctioncoefficient, rs is the scattering coefficient, U is the scat-tering phase function and ib

rT4b

pis the Planck black-

body intensity. Eq. (1) can be written as

di X

dsX bi X bS X 2

In optical coordinate, Eq. (2) is written as

di X

dsX i X S X 3

where s is the optical-distance bs and the source term Sis given by

S 1 x ib x

4p

Z4pX0

i X UX0;X dX 4

dA

d

d

P

d

Solution plane:xz plane

y

x

z

dA

Effectiveintensity

P

d

(a) (b)

x

z Solution plane:xz plane

y

Fig. 1. (a) 3-D radiative information i(h, u) at a point Pof an area element dA in actual situation and (b) collapsing of 3-D radiativeinformation to the 2-D solution plane in the CDM.

32 S.C. Mishra et al. / International Journal of Heat and Mass Transfer 49 (2006) 3041


4/12


5/12

the CDM computationally more efficient, in the follow-ing pages, a new formulation is presented.

In the CDM, the differential form of the RTE (Eq.(8)) for the discrete direction m in the solution plane is

written as

lmoIm

ox nm

oIm

oz bgIm bgeSm 13

where lm cos am and nm sin am are the direction co-sines (Fig. 2) of the effective intensity Im having planarangle am in the solution plane. Integrating Eq. (13) overthe volume element V= DxDz (with unit depth in

y-direction) and applying Gauss divergence theoremand Greens theorem, we get

lm Ime Ae ImwAw

nm ImnAn I

ms As

bgI

m

P bgeSmP 14where Ime ; I

mw; I

ms and I

mn are the average effective intensi-

ties in the discrete direction m through the east, west,south and north faces of the control volume, respectively(Fig. 3). In Eq. (14), ImP and

eSmP are the volume-averagedeffective intensity and the source term for direction m atthe geometric center P of the control volume.

To reduce the number of unknowns from Eq. (14),for a given direction m, a relation between the cell-faceand volume-averaged intensities is used. For a givencoordinate direction, if a linear relationship among thetwo cell-face intensities and ImP is assumed, then

ImP cxIme 1 cxImw czImn 1 czImn 15

where cx and cz are constants126 cx; cz6 1. From Eqs.

(14) and (15), unknown effective intensities Ime and Imn

can be eliminated and the following relation is obtained.

ImP bgVeSm

P lm

AewImw

cx n

m

AnsIms

cz

bgV lm Aewcx

nm Anscz

16

where

Aew 1 cxAe cxAw

Ans 1 czAn czAs17

are average face areas. Eqs. (16) and (17) are valid whenboth the direction cosines lm and nm are positive. In thissituation, when marching from the bottom-left corner(Fig. 3), as given in Eq. (16), ImP is calculated fromknown values ofImw and I

ms and then using Eq. (15), un-

known cell-face intensities Ime and Imn are calculated.

However, when one or both of the direction cosinesare negative, marching will have to be started from theother three corners (Fig. 3) and for these cases, eliminat-ing the unknown effective intensities, we get a generalform of the Eq. (16).

ImP bgVeSmP lmj jAx Imx;incx nmj jAzImz;incz

bgV lmj j Ax;outcx

nmj j Az;outcz

18

where Imx;in and Imz;in are the effective intensities entering

a control volume through x- and z-faces, respectively.Ax,out and Az,out are x- and z-cell-face areas, respectivelythrough which effective intensities leave.

Ax 1 cxAx;out cxAx;in

Az 1 czAz;out czAz;in 19

Fig. 3. In a 2-D rectangular enclosure, marching scheme in the MCDM for four sample equally spaced directions with one in everyquadrant.



6/12

The expression of ImP given in Eq. (18) is valid for any2-D rectangular control volume. In case of 2-D irregulargeometries, the control volumes can be of irregularshapes (Fig. 4) and normal vectors to the cell-facesmay not be aligned with the coordinate directions. Also,unlike rectangular control volumes, in case of irregularshape control volumes, the concepts of quadrants andthe direction cosines get modified. In this case, theymay be acute or obtuse as per the geometry of the con-trol volume. Therefore, rather than taking the direction

cosines as cos a and sin a, in this case, we find out thedirection cosines by taking the scalar product of theray direction sm and the outward normal vector nk tothe kth surface. Thus for a general control volume(Fig. 4), ImP is calculated from the following:

ImP bgVeSmP Pk;inImk sm nkj jAk=ck

bgVP

k;out sm nkj jAk=ck

20

In Eq. (20), ck stands for cx or cz depending uponwhether the cell surface is the east (or west) or the north(or south) surface. Similarly Imk stands for intensitythrough the kth surface of the control volume. Thein and out on the summation signs denote summa-

tion over the cell faces with incoming ^sm ^nk > 0 oroutgoing sm nk < 0 intensities, respectively.

Evaluation of ImP in Eq. (20) requires prior informa-tion about the source term eSmP. For this, Eq. (10) is writ-ten as

eSmP 1 xIb;P x2pXMm01

Im0

P Ham0 ; amfm

0

21

where in Eq. (21) fm0

is the weight and M is the totalnumber of effective intensities Im considered over the0 6 am6 2p. If scattering is approximated by a linear

anisotropic phase function H am0; am

1 a sin am

0

sin am [14], then Eq. (21) is written as

eSm

P

1 xIb;P x

2p XM

m01

Im0

P

Dam0

xa sin am

2p

XMm01

2sin am0

sinDam

0

2

Im

0

P 22

where in Eq. (22), a is the anisotropy factor,1 6 a 6 +1. For forward scattering, a > 0, for back-ward scattering a < 0 and for isotropic scatteringa = 0. Eq. (22) can be written in a simplified form as

eSmP 1 xIb;P x2p G xa sin am

2pq 23

where effective incident radiation G and heat flux qexpressions in the CDM are given by

GZ2p

a0

I a da %XMm1

ImPDam 24

q

Z2pa0

I a sin ada %XMm1

2sinam sinDam

2

ImP

XMm1

cmImP

25

where the weight factor cm is given by

cm 2sin am sinDam

2

26

In above equations, Dam is the angular span over whichthe effective intensity Im having planar angle am is as-

sumed to be isotropic.D

a

m

need not be equal. However,in the present work, Dam has been assumed equal for allvalues of m.

To calculate the unknown intensities, before march-ing from any corner, knowledge about the knowncell-face intensities is required. If the boundary is dif-fuse-gray having temperature TB and emissivity eB, inthe CDM, the boundary intensity IB is given by

IB eBrT

4B

2

1 eB2

Zpa0

I a sin a da

%eBrT

4B

2XM=2m1

cmImP 27

In Eq. (27), the first and the second terms on the right-hand-side represent emitted and reflected components ofthe boundary effective intensity, respectively.

If a heat transfer problem involves thermal radiation,then in the CDM, the divergence of radiative heat fluxr ~q is calculated from the following:

r ~q jag 2pIb G jag 2prT4b

2 G

28

In case of radiative equilibrium, r ~q 0 and the un-known temperature is calculated from

Fig. 4. An irregular shape 2-D geometry with control volumes

along with outward normals to the cell surfaces of a controlvolume.



7/12

T G

pr

14

29

In the CDM, one can either follow the approach used in

earlier works [1421] or the one proposed above. In boththe approaches, values of the collapsing coefficient g arerequired. g is a function of the properties of the partici-pating medium and its evaluation procedure has beenlaid out in [14]. Expressions ofg for absorbingemittingcases of radiative equilibrium and non-equilibrium situa-tions are given in Appendix A. For easy access to others,expressions of g for different situations can be down-loaded from the website address given in Appendix A.

Expressions of g are general ones and have beenfound to be valid for 1-D and 2-D geometries. Withproblems involving radiation along with conductionand/or convection, g expressions for non-radiative equi-librium are used.

3. Results and discussion

In the following pages, a new formulation of theCDM, called the MCDM, presented above is validatedby solving three test problems dealing with an absorb-ing, emitting and scattering homogeneous gray medium.Both radiative equilibrium and non-radiative equilib-rium cases are considered. To compare the performanceof the MCDM, the same problems are also solved usingthe CDM (having ray tracing approach) and the DOM.Heat flux and temperature results of the three methods

are compared against benchmark results. CPU timesof the three methods are also compared for some repre-sentative cases.

In validation studies, 20 20 control volumes and 24discrete directions were considered in the three methods.In the DOM, the widely used SN quadrature scheme[35,23] was used. For the 2-D problems, these 24 dis-crete directions were equivalent to S-6 approximationof the DOM. In the CDM and the MCDM, equallyspaced 24 directions were considered. In both the meth-ods, weight factors cm were calculated using Eq. (25) andin the MCDM the direction cosines were computedusing lm cos am and nm sin am. Both in the MCDMand the DOM, cx = cz = 0.5 was considered.

3.1. 2-D heat transfer in an enclosure with one hot wall

In this case, the south boundary of the 2-D rectangu-lar enclosure (Fig. 3) is at a known temperature Ts.Other three boundaries are cold. All four boundariesare considered black. The enclosed gray and homoge-nous medium is absorbing, emitting and isotropicallyscattering. This is a representative of a radiative equilib-rium problem. In this case, the medium temperature isunknown.

In Fig. 5(a) and (b), for aspect ratio X/Z= 1, non-dimensional heat flux W q

rT4sresults along the hot

(south) wall obtained from the CDM, the MCDM andthe DOM have been compared with the exact results[26]. In Fig. 5(a), heat flux results have been comparedfor extinction coefficient b = 0.25, 0.5 and 1.0, whereasin Fig. 5(b), the same have been compared for b = 2.0,4.0 and 10.0. It can be seen from the figures that the re-sults of the three methods match very well with eachother and they are also in good agreement with the exactresults [26].

Since the MCDM and the DOM compute heat flux atthe same points (cell-surface centers), in Fig. 5, thevalues of the two methods are seen at the same x/X

Distance, x/X

Heatflux

0 0.1 0.2 0.3 0.4 0.50.7

0.75

0.8

0.85

0.9

0.95

1

Exact

CDM

DOM

MCDM

0.25

0.5

=

1.0

(a)

Distance, x/X

Heatflux

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Exact

CDMDOM

MCDM

2.0

4.0

= 10.0

(b)

Fig. 5. Variation of non-dimensional heat flux W along the hotwall; comparison of the CDM, the MCDM and the DOMresults with the exact results [26] for (a) extinction coefficientb = 0.25, 0.5 and 1.0, and (b) b = 2.0, 4.0 and 10.0.

http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


8/12

locations. In the CDM, although heat flux can be calcu-lated at cell-surface centers also, in the present case, it hasbeen evaluated at the cell corners, and thus like the exactresults [26], the CDM results can be seenalso tocover the

corner and the middle points along the hot wall.In Fig. 6, for b = 0.5, heat flux results along the hot

wall have been compared with the exact results [26] foraspect ratio X/Z= 0.2, 2.0, 4.0 and 10.0. In studyingthe performance of the three methods for different val-ues of the aspect ratio, the value of X was kept unity.For different values of X/Z, results of the CDM, theDOM, and the MCDM compare very well with eachother and they also match very well with the exact re-sults [26]. In Fig. 6, it can be seen that at x/X= 0.5for the aspect ratio X/Z= 10.0, results of the four meth-ods closely approximate its values (0.7040) for the 1-Dplanar medium [27].

For X/Z= 1.0 and extinction coefficient b = 1.0, 4.0and 10.0, non-dimensional centerline (x/X= 0.5) emis-

sive power U rT4=p

rT4s=p

has been compared with exact

results [26] in Fig. 7(a)(c), respectively. For b = 1.0,the DOM results are more scattered than the MCDMresults. CDM results closely match with the exact results[26]. Since the MCDM uses DOM like formulation inwhich for absorbing, emitting and isotropically scatter-ing medium, in a given control volume, source term(Eq. (22)) is the same for all the directions and in a givendirection, an intensity is also homogeneous over a cell-surface, for less number of rays and control volumes,for lower values of b some fluctuations in temperature

results are observed. The CDM and MCDM use equally

spaced directions. Therefore, in these methods, fluctua-tions are lower then the DOM in which the directionsare not equally spaced. Further, in the CDM, unlikethe MCDM and the DOM, even for absorbingemitting

case, source term is different for different directions andover a cell surface, in a given direction, intensity is notconsidered homogeneous, results are relatively moreaccurate.

3.2. 2-D black enclosure with absorbingemitting

medium

In this case, the four boundaries of the 2-D rectangu-lar enclosure (Fig. 3) are cold. All four boundaries areconsidered black. The enclosed gray and homogenousmedium is absorbing, emitting and isotropically scatter-ing and is at uniform temperature Tg. This is a represen-tative of a non-radiative equilibrium problem.

In Fig. 8, for X/Z= 1.0 non-dimensional heat fluxW q=qT4g along the south wall computed using theCDM, the MCDM and the DOM have been comparedwith the DTM results. For 20 20 control volumes, theDTM results with 12 24 (h /) directions, in the pres-ent case, have been considered as benchmark as theDTM results were benchmarked against the MonteCarlo method results.

For absorbingemitting medium (x = 0.0), heat fluxresults have been compared for extinction coefficientb = 0.25, 0.5 and 1.0 in Fig. 8(a), while in Fig. 8(b), com-parisons have been made for b = 2.0, 4.0 and 10.0.

It can be seen from Fig. 8(a) and (b) that results of

the three methods match very well with each other andare also in close agreement with the DTM results. Minorfluctuations in the DOM is attributed to the reason citedbefore.

In Fig. 8(c), for b = 1.0, comparisons of south wallheat flux have been made for absorbing, emitting andisotropically scattering situation with scattering albedox = 0.0, 0.5 and 0.9. Like previous cases, here too the re-sults of the three methods are found to compare verywell with the benchmark results.

3.3. A quadrilateral enclosure with constant gas

temperature

To demonstrate the applicability of the MCDM toirregular geometries, we next consider radiative heattransfer in an irregular quadrilateral enclosure (withdimensions in meters) (Fig. 9(a)). The four walls of theenclosure are cold and the absorbing, emitting mediumis maintained at a constant temperature Tg. The compu-tational grids used in the calculation are also shown inFig. 9(a).

For 20 20 and 24 ray directions, in Fig. 9(b), non-dimensional heat flux W q=rT4g along the south wallcomputed using the CDM and the MCDM have been

Distance, x/X

Heatflux

0 0.1 0.2 0.3 0.4 0.50.6

0.7

0.8

0.9

1

Exact

CDM

DOM

MCDM

X/Z = 10.0

4.0

2.0

0.2

= 0.5

Fig. 6. Variation of non-dimensional heat flux W along the hotwall; comparison of the CDM, the MCDM and the DOMresults with the exact results [26] for aspect ratio X/Z= 0.2, 2.0,4.0 and 10.0.



9/12

plotted against the benchmark DOM results (taken asexact here) available in the literature [28]. For absorb-ingemitting medium (x = 0.0), heat flux results havebeen compared for extinction coefficient b = 0.1, 1.0and 10.0. It can be seen that the results of the MCDMresults are in very good agreements with that CDMand the DOM results [28].

3.4. Comparison of CPU times and iterations

Especially while solving conjugate mode problems,for steady state solutions, number of iterations andCPU times are one of the vital points that form the basisfor selection of a radiative transfer method, because for

a large number of control volumes and directions, allmethods will provide accurate results. Keeping abovepoint in mind and having MCDM found able to provideaccurate results, to further justify its usage, next it isimportant to know how its CPU times and number ofiterations compare with the CDM and the DOM.

In Table 1, CPU times (in second) and number ofiterations required to get converged solutions in threemethods have been compared for results presented inFig. 4(a) and (b). In all three methods, 20 20 controlvolumes and 24 directions have been used, and solutionswere found converged when between two consecutiveiteration levels, the change in source term at all pointswas within 1.0 106.

Distance, z/Z

Emissivepower

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Exact

CDM

DOM

MCDM

= 1.0

Distance, z/Z

Emissivepower

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Exact

CDM

DOM

MCDM

= 4.0

(a) (b)

Distance, z/Z

Emissivepower

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Exact

CDM

DOM

MCDM

= 10.0

(c)

Fig. 7. Variation of non-dimensional centerline emissive power U; comparison of the CDM, the MCDM and the DOM results with theexact results [26] for (a) extinction coefficient b = 1.0, (b) 4.0 and (c) 10.0.



10/12

It can be seen from Table 1 that the CDM that isbased on ray tracing approach [2,1421] and evaluatessource term for every intensity using bilinear interpola-tion, takes less number of iterations for the convergedsolutions. However, per iteration it spends more timethan the MCDM and the DOM. Further, because ofthe more number of calculations involved in the evalua-tion of ray tracing and source term evaluation, totalCPU time in this method is much more than the MCDMand the DOM. Since assigning the source term to everyeffective intensity is more accurate in the CDM, thismethod converges fast.

The number of iterations of the MCDM and theDOM are found to be comparable. However, theMCDM is found faster than the DOM. In the MCDMand the DOM, the number of operations in all steps isexactly the same, except in evaluation of the incidentradiation and heat flux. In calculation of heat flux, inthe DOM, in addition to the weight factor, every inten-sity term is multiplied by corresponding directioncosines. Whereas in the MCDM, no such directioncosine term appears in the evaluation of heat flux (seeEq. (24)). Also, in the DOM, in calculation of incidentradiation, every intensity is multiplied by a direction

Distance, x/X

Heatflux

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

DTM

CDM

DOM

MCDM

= 0.25

0.5

1.0

= 0.0

Distance, x/X

Heatflux

0 0.1 0.2 0.3 0.4 0.50.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

DTM

CDM

DOM

MCDM

= 2.0

4.0

10.0

= 0.0

(a) (b)

Distance, x/X

Heatflux

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

DTM

CDM

DOM

MCDM

0.0

0.5

= 0.9

= 1.0

(c)

Fig. 8. Variation of non-dimensional heat flux W along the bottom wall; comparison of the CDM, the MCDM and the DOM resultswith the DTM results for (a) absorbingemitting medium (x = 0.0) with b = 0.25, 0.5 and 1.0, (b) absorbingemitting medium(x = 0.0) with b = 2.0, 4.0 and 10.0 and (c) with b = 1.0 and absorbingemittingisotropically scattering medium with x = 0.0, 0.5 and0.9.



11/12

specific weight factor. However, in the MCDM, thisweight factor is simply Da which is constant for alldirections.

Since in the present test cases, anisotropy effect wasnot considered and the boundaries of the enclosure werealso assumed black, heat flux term in all three methodswas calculated at the end of the iteration. However, ifeither the medium has anisotropic scattering (Eq. (22))and/or boundaries are reflecting (Eq. (26)), then heatflux term has to be within the iteration loop and in thatcase, the computational efficiency of the MCDM willfurther improve.

4. Conclusions

A new formulation of the CDM, called the MCDM,

was proposed. After collapsing the radiative informationto the 2-D solution plane, formulation procedure of theMCDMwas found similar to the DOM. MCDM was val-idated by solving three test problems in 2-D rectangularand irregular quadrilateral enclosures containing absorb-ing, emitting and isotropically scattering media. To checkhow the MCDM compares against the earlier formulationof the CDM and the DOM, the same problems were alsosolved using three methods. In all cases, the MCDM wasfound to provide accurate results. Since in the MCDM,number of calculations is less than the CDM and theDOM, it was also found faster than both the methods.

Appendix A

Expression of the collapsing coefficient g for the caseof absorbingemitting medium at radiative equilibrium[14], r.~q 0.

g

1:27276 1:22505b 23:6226b2 136:935b3

22:2416b4 for 104 6 b 6 0:1

1:25684 0:108915b 0:075431b2 0:028079b3

0:005164b4 0:000363b5 for 0:1 < b 6 5

1:18129; for bP 5

8>>>>>>>>>:

A:1

Expression of the collapsing coefficient g for the case of

absorbingemitting medium (x = 0) at non-radiativeequilibrium [14], r.~q 6 0.

g

1:27338 1:34631b 26:78364b2 159:953b3

for 104 6 b 6 0:1

1:26002 0:145619b 0:125726b2 0:067566b3

0:006472b4 0:006152b5 for 0:1 < b 6 1

1:2376 0:068275b 0:018395b2 0:002886b3

0:000185b4 for 1 < b 6 5

1:110975 for bP 5

8>>>>>>>>>>>>>>>>>>>:

A:2

Fig. 9. (a) The geometry and coordinates of the quadrangularenclosure under consideration along with the non-uniform gridsused and (b) variation of non-dimensional heat flux W along thesouth wall; comparison of the CDM and the MCDM resultswith the exact results [28] for extinction coefficient b = 0.0, 1.0and 10.0.

Table 1Comparison of CPU times (second) and number of iterations ofthe CDM, the MCDM and the DOM for radiative equilibriumproblem with 20 20 control volumes and 24 directions

b CDM MCDM DOM

CPUtime

Iterations CPUtime

Iterations CPUtime

Iterations

0.25 10.413 8 0.256 12 0.522 120.5 14.734 11 0.328 16 0.625 161 21.384 16 0.483 25 0.890 252 36.949 28 0.826 44 1.542 444 76.617 59 1.707 93 2.958 9310 259.061 196 6.269 348 10.259 337



12/12

Along with the above two expressions, for scattering sit-uations, for both radiative and non-radiative equilib-rium cases, expressions of g can be obtained from:http://www.iitg.ac.in/scm. From the same location, the

CDM, MCDM and the DOM codes used for generatingresults for the present work can also be downloaded.

References

[1] I. Malico, J.C.F. Pereira, Numerical study on the influenceof radiative properties in porous media combustion,J. Heat Transfer 123 (2001) 951957.

[2] P. Talukdar, S.C. Mishra, D. Trimis, F. Durst, Heattransfer characteristics of a porous radiant burner underthe influence of a two-dimensional radiation field, J.Quant. Spectrosc. Radiat. Transfer 84 (2004) 527537.

[3] W.A. Fiveland, Discrete ordinates solutions of the radia-

tive transport equations for rectangular enclosures, J. HeatTransfer 106 (1984) 699706.[4] W.A. Fiveland, Discrete ordinate methods for radiative

heat transfer in isotropically and anisotropiocally scatter-ing media, J. Heat Transfer 109 (1987) 809812.

[5] W.A. Fiveland, Three-dimensional radiative heat transfersolutions by the discrete-ordinate methods, J. Thermophys.Heat Transfer 2 (1988) 309316.

[6] N.G. Shah, New method of computation of radiation heattransfer in combustion chambers, Ph.D. Thesis, ImperialCollege, University of London, UK, 1979.

[7] G.D. Raithby, E.H. Chui, A finite-volume method forpredicting a radiant heat transfer in enclosures withparticipating media, J. Heat Transfer 112 (1990) 415423.

[8] J.C. Chai, H.S. Lee, S.V. Patankar, Finite volume method

for radiation heat transfer, J. Thermophys. Heat Transfer 8(1994) 419425.

[9] P.S. Cumber, Improvements to the discrete transfermethod of calculating radiative heat transfer, Int. J. HeatMass Transfer 38 (1995) 22512258.

[10] P.J. Coelho, M.G. Carvalho, A conservative formulationof the discrete transfer method, J. Heat Transfer 119 (1997)118128.

[11] M.A. Ramankutty, A.L Crosbie, Modified discrete ordi-nates solution of radiative transfer in two-dimensionalradiative rectangular enclosures, J. Quant. Spectrosc.Radiat. Transfer 57 (1997) 107140.

[12] B.-W. Li, Q. Yao, X.-Y. Cuo, K.-F. Cen, A new discreteordinates quadrature scheme for three-dimensional radia-tive heat transfer, J. Heat Transfer 120 (1998) 514

518.[13] S.H. Kim, K.Y. Huh, A new angular discretization scheme

of the finite volume method for 3-D radiative heat transferin absorbing, emitting and anisotropically scatteringmedia, Int J. Heat Mass Transfer 43 (2000) 12331242.

[14] S.C. Mishra, A novel computational approach for thesolution of radiative transport problems in participatingmedia, Ph.D. Thesis, IIT Kanpur, India, 1997.

[15] P. Talukdar, S.C. Mishra, Transient conduction and

radiation heat transfer with heat generation in a partici-pating medium using the collapsed dimension method,Numer. Heat Transfer, Part A 39 (2001) 79100.

[16] S.C. Mishra, M. Prasad, Radiative heat transfer inabsorbingemittingscattering gray media inside 1-D grayCartesian enclosure using the collapsed dimension method,Int. J. Heat Mass Transfer 45 (2002) 697700.

[17] P. Talukdar, S.C. Mishra, Analysis of conductionradia-tion problem in absorbing, emitting and anisotropicallyscattering media using the collapsed dimension method,Int. J. Heat Mass Transfer 45 (2002) 21592168.

[18] P. Mahanta, S.C. Mishra, Collapsed dimension methodapplied to radiative transfer problems in complex enclo-sures with participating medium, Numer. Heat Transfer,Part A 42 (2002) 367388.

[19] S.C. Mishra, P. Talukdar, D. Trimis, F. Durst, Compu-tational efficiency improvements of the radiative transferproblems with or without conductiona comparison ofthe collapsed dimension method and the discrete transfermethod, Int. J. Heat Mass Transfer 46 (2003) 30833095.

[20] S.C. Mishra, P. Talukdar, D. Trimis, F. Durst, Two-dimensional transient conduction and radiation heattransfer with variable thermal conductivity, Int. Commun.Heat Mass Transfer 32 (2005) 305314.

[21] S.C. Mishra, A. Lankadasu, K. Beronov, Application ofthe lattice Boltzmann method for solving the energyequation of a 2-D transient conductionradiation problem,Int. J. Heat Mass Transfer 48 (2005) 36483659.

[22] S.C. Mishra, A. Shukla, V. Yadav, View factor calculationin the 2-Dgeometries using the collapsed dimensionmethod,

Int. J. Thermal Sciences, submitted for publication.[23] M. Sakami, A. Charette, Application of a modified discrete

ordinates method to two-dimensional enclosures of irreg-ular geometries, J. Quant. Spectrosc. Radiat. Transfer 64(2000) 275298.

[24] H. Xue, J.C. Ho, Y.M. Cheng, Comparison of differentcombustion models in enclosure fire simulation, Fire SafetyJ. 36 (2001) 3754.

[25] M.F. Modest, Radiative Heat Transfer, second ed., Aca-demic Press, New York, 2003.

[26] A.L. Crosbie, R.G. Schrenker, Radiative transfer in atwo-dimensional rectangular medium exposed to diffuseradiation, J. Quant. Spectrosc. Radiat. Transfer 31 (1984)339372.

[27] M.A. Heaslet, R.F. Warming, Radiative transport and

wall temperature slip in a absorbing planar medium, Int. J.Heat Mass Transfer 8 (1965) 979994.

[28] K.-B. Cheong, T.-H. Song, An alternative discrete ordi-nates method with interpolation and source differencing fortwo-dimensional radiative transfer problems, Numer. HeatTransfer, Part B 32 (1997) 107125.

http://www.iitg.ac.in/scmhttp://www.iitg.ac.in/scm

Cdm Using Dom

Documents

Transcript of Cdm Using Dom