Mathematical simulation of radiative–conductive heat transfer in a cylindrical volumeV. K. Shiff

S. I. Vavilov State Optical Institute, St. Petersburg, Russia~Submitted August 27, 1999!Opticheski� Zhurnal67, 56–60~June 2000!

Steady-state radiative–conductive heat transfer in a cylindrical region is considered for variousconditions of heat transfer at the boundaries of the region and over a wide range ofvalues of the absorption coefficient of the medium. A study is made for various functional formsof the density of internal heat release. The transport equation for the radiant energy issolved by the discrete ordinates method. ©2000 The Optical Society of America.@S1070-9762~00!00906-4#

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INTRODUCTION

Simultaneously taking into account the conductive aradiative mechanisms of heat transfer is important frompractical standpoint and is also of interest in respect tocomputational aspect of the problem. The necessity of takradiative–conductive heat transfer~RCHT! into account isencountered in space engineering, various high-temperapower plants, glass production, and other applied probleThe computational aspect of RCHT includes two problemfirst, to develop an efficient algorithm for calculating thtransport equation for the radiant energy, and, second, tointo account jointly the various heat transfer mechaniswhich is a highly nonlinear problem. Analytical methodssolving the RCHT problem run up against statistical difficties that are hard to overcome. Various computational mods for studying radiative heat transfer have been undetensive development in recent years. Foremost among tare the zonal method, the Monte Carlo method, the metof spherical harmonics~the Pn method!, and the discreteordinates~DO! method. The Monte Carlo and zonal methogive acceptable accuracy. These methods are reliable, pcal, and simple, but are not very well adapted for calculatlocal characteristics and are incompatible with finidifference relations for solving the transport equation~theyagree poorly with the equations of hydrodynamics!. Themethod of spherical harmonics~the Pn method! uses an ex-pansion of the radiation intensity and scattering indicatrixLegendre polynomials, which allow one to reduce the ingrodifferential transport equation to a system of differenequations for the moments of the radiation intensity. Indition to certain difficulties in constructing the boundaconditions and the weak convergence, thePn method is lessaccurate than the DO method.1–3 As a rule, the number oharmonics in thePn method is limited toN53, since the useof N.3 makes thePn method too awkward. For the DOmethod one typically discretizes the angular dependencthe radiation intensity and for each radiation direction costructs a finite-difference approximation of the transpequation on a specified spatial mesh.

In this paper we use the DO method to investigate thsteady-state RCHT problems in a ‘‘gray’’ medium distriuted in a cylindrical region bounded by diffusely reflectin‘‘gray’’ boundaries. In the first problem we study heat tran

546 J. Opt. Technol. 67 (6), June 2000 1070-9762/2000/0605

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fer in the absence of sources of heat release at constantues of the temperature on the boundaries of the region. Inthe second and third problems one considers convectivetransfer with the surrounding medium in the presence ofternal heat release. In the second problem the heat releauniform in the volume, while in the third problem it varieradially and represents a one-dimensional model of heatlease in an glass-founding induction furnace. In the secand third problems it is assumed that the thermal conducity of the medium in the cylinder is uniform and equal tol51.5 W/m•K, i.e., close to the thermal conductivity of silicate glasses, and that the heat transfer coefficient for contive heat transfer at the boundary with the surrounding mdium is equal toa5180 W/m•K. This relatively high heattransfer coefficient corresponds to the case of water cooof the ceramic walls of a glass-founding induction furnacThe variation of the intensity of the radiative transport comabout because of variations of the absorption coefficienthe medium.

MATHEMATICAL MODEL. BASIC EQUATIONS

The mathematical model of the RCHT problem includthe radiant energy transport equation~1! and the energy conservation equation~2!, which are written below in dimen-sionless form. As units of measure~normalization param-eters! we chooseT0 for the temperatures and the radiusR ofthe cylindrical region for the characteristic linear dimensiof the investigated region. We introduce the following scafactors: the thermal conductivityl0 and the absorption coefficient kabs0; the scale values of the radiant flux and specpower, 4n2sT0

4 and l0T0 /R2, and also the dimensionlesparameters:N5l0kabs0/(4n2sT0

3), the radial Bouguer num-ber Bu5kabs0R, and the Biot number Bi5aR/l0. Equations~1! and ~2! have the form

cosu]I ~r ,z,u,w!

]z2

]@sinu sinwI ~r ,z,u,w!#

r ]w

1sinu cosw

r

]~rI ~r ,z,u,w!!

]r1kabsBu I ~r ,z,u,w!

5kabsBu I b~T!, ~1!

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Here h(r ,z) is the power of the internal heat release, mesured in units ofl0T0 /R2, l is the thermal conductivity ofthe medium,H andR are the height and radius of the cylinder bounding the region under study,kabs is the absorptioncoefficient, I b(T)5T4/4p is the integrated intensity of themission from an absolute blackbody~the Planck function!,written in the chosen units (4n2sT0

4), n is the refractiveindex of the medium,Qr is the radiant heat flux vectorwhich is the first moment of the radiant intensityI (r ,z,V) atthe point with coordinatesr ,z in the directionV(u,w). Then

Qr5E4p

VI ~r ,z,V!dV. ~3!

An expression for the divergence of the radiant heat flvector in Eq.~2! can be obtained by integrating the transpequation over all directions within a solid angle of 4p:

¹Qr5kabsBu~T42G!, ~4!

whereG is the spatial density of the incident radiation~ze-roth moment of the radiant intensity!,

G5E4p

I ~r ,z,V!dV. ~5!

Boundary conditions for the transport equation

The boundary conditions at the upper boundary spethe radiation entering the region under study. It is the sumthe radiation from external sources and the radiation refleby the surrounding medium:

I ~r ,z,V!5«I b~T!1r

p EnV8.0

~nV8!I ~r ,z,V8!dV8,

nV,0, ~6!

where« and r are, respectively, the emissivity and the rflection coefficient of the boundary surface, and all partsthe external boundary of the region are taken to be nontrparent and diffusely reflecting.

Boundary conditions for the energy equation

The boundary conditions for the energy equation aretermined by the energy balance at the boundary of thegion. In the first problem it is assumed that the values oftemperature are constant on the outer boundary of the reg

T~r ,0!5T0 , 0,r ,1,

T~r ,H !50, 0,r ,1,

T~R,z!50, 0,z,H/R.

On the symmetry axis (r 50) there is no heat flux, i.e.,

]T~0,z!/]r 50, 0,z,H/R. ~7!

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In the second and third problems it is assumed that onparts of the outer boundary of the region there is convecheat transfer with the surrounding medium and thatboundary is nontransparent, i.e.,

2l]T

]r1

Bu

N EnV.0

~nV!I ~r ,z,u,w!dV

5Bu

N EnV,0

~nV!I ~r ,z,u,w!dV1Bi~T2Ta!, ~8!

wheren is the outer normal to the boundary of the region,ais the heat transfer coefficient, andTa is the temperature othe surrounding medium.

METHOD OF SOLUTION

We solve the system of equations~1! and ~2! by an it-eration method. For this we approximate them by consertive finite-difference algebraic equations. For the eneequation we use the integrodifferential method, and fortransport equation, the DO method.

Discrete ordinates method

The construction of a finite-difference approximationthe DO method can be broken into three steps. In thestep an angular discretization of the transport equationcarried out. In the second step a finite-difference approximtion of the transport equation is constructed for each discdirection, specified by the anglesu andw. In the third stepone introduces additional linear equations relating the valof the intensities at the centers of the control volumes wthe corresponding values of the intensities on the faces ofcontrol volumes.

After angular discretization, the transport equation~1!and the corresponding boundary values of the intensity~6!for the angular directionsl ,m become

m l

]I l ,m

]z2

1

r

]~h l ,mI l ,m!

]w1

j l ,m

r

]~rI l ,m!

]r

52BukabsI l ,m1BukabsI b , ~9!

I l ,m5«wI bw1~12«w! (l 8,m8

wl 8,m8j l 8,m8I l 8,m8 ~10!

j l 8,m8.0, j l ,m,0, r 51,

I l ,m5«wI bw1~12«w! (l 8,m8

wl 8,m8um l 8uI l 8,m8 , ~11!

m l 8,0, m l.0, z50,

I l ,m5I l 8,m8 , j l ,m5j l 8,m8 , r 50, ~12!

I l ,m5«wI bw1~12«w! (l 8,m8

wl 8,m8m l 8I l 8,m8 , ~13!

m l 8.0, m l,0, z5H.

In the other 7 octants theEsn quadrature is propagated bsymmetry with respect to the reflections:m→2m, w→2w, p/22w→p/21w. The number of discrete direction

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Nm is determined by the order of approximationn and isgiven by Nm5n(n12). Integrating the transport equatio~9! for each discrete angular directionl ,m over the controlvolumes of the chosen spatial coordinate mesh, we obconservative finite-difference equations. The divergencethe radiant heat flux in the energy equation is expressedthe aid of formula~4!, which explicitly contains the spatiadensity of the incident radiation. Separating out the powlaw temperature dependence in the expression for the dgence of the radiant flux~4!, which appears in the energequation~2! and in the boundary conditions, we can lineariit using the formula

Tn4'Tn21

4 14Tn213 ~Tn2Tn21!. ~14!

Here the subscriptsn andn21 denote the correspondinsuccessive approximations. To separate out the powertemperature dependence in the boundary conditions, wethe corresponding transformation for the boundaryr 5R. Werepresent the radial vector of the radiant heat flux by a cobination of two oppositely directed fluxes:

Qrr~r ,z!5Q1r

r ~r ,z!2Q2rr ~r ,z!. ~15!

The radiant flux emitted by the boundary to the inside ofregion (Q2r) consists of radiation emitted by the boundasurface itself and a reflected flux:

Q2rr 5p«wI b~T!1~12«w!Q1r

r 5«w

T4

41~12«w!Q1r

r .

~16!

After substituting expression~16! into Eq.~8! with a decom-position of the radiant flux vector into two components~15!,we find the boundary condition has acquired a form connient for linearization:

2l]T

]r1«w

Bu

N S Q1rr 2

T4

4 !5Bi~T2Ta! , r 51.

~17!

After linearization according to formula~14!, the boundarycondition becomes

2l]Tn

]r52«w

Bu

NQ1r

r 2Bi Ta23«w

Tn214

4

1Tn~Bi1«wTn213 !. ~18!

The algorithm used to solve the intercoupled equati~1! and ~2! includes independent successive iteration pcesses of solving the transport and energy equations,internal iteration processes done until the specified accuis attained. Of course, the difference form of the eneequation is solved by the linear sweeping method. The mpart of the calculations is done on spatial and angular mewith a number of mesh points equal to 20320 and 80, re-spectively, which in the case of the angular mesh cosponds to S8 — the Carlson quadrature in the DO method

RESULTS

Figure 1a and b shows the values of the spatial denof the incident radiation and the radial projection of the

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diant flux vector for an isothermal emitting and absorbimedium bounded by an isothermal black boundary. Theculations were done for three values of the radial Bougnumber: Bu50.1, 1.0, and 5.0, and were compared with t‘‘exact’’ calculations of Ref. 4, in which the solution of thtransport equation for the case of a nondissipative isothermedium with black boundaries is expressed in definite ingrals.

For comparison of the results of the calculation andperiment, Fig. 2 shows the experimental data for a modelburner cooled with water~the dimensions of the burner arR50.45 m andH55.1 m! and the results of a calculation bthe DO method. As in Refs. 3 and 5, it was assumed thatcylindrical wall of the model burner had an emissivity equto 0.8 and a temperature of 425 K. The end walls weresumed ‘‘black,’’ and their temperature equal to 300 K. Ttemperature distribution in the medium was taken from R5, and the absorption coefficient of the medium in the framwork of the ‘‘graybody’’ approximation is assumed equal0.3 m21. The results obtained by the DO method in Ref.

FIG. 1. Comparison of the results of calculations by the DO method w‘‘exact’’ calculations4 for various values of the Bouguer number:1 — Bu50.1; 2 — Bu51.0; 3 — Bu55.0. R51 m; H52 m. 1,2,3 — The DOmethod, and4 — the data of Ref. 4. The values of the spatial densG(r ,H/2) of the incident radiation~a! and the radial projectionQr

r(r ,H/2)of the radiant heat flux vector~b! in this figure are measured in units ofsT0

4.

FIG. 2. Values of the density of the radial heat fluxQrr(R,z) at the wall of

a model gas burner:1 — experimental values of Ref. 3,2 — calculation bythe DO method.R50.45 m;H55.1 m.

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and in the present study~Fig. 2! agree with each other anare close to the experimental values taken from Ref. 3.

First problem

Calculations were done at constant values of the teperature on the boundaries~9! ~as in Ref. 6! for three valuesof the radiative–conductive parameter:N50.01, 0.1,̀ . Thevalues of the Bouguer number for the radius and heightmained constant and equal to Bu5Rkabs5Hkabs51. The pa-rameterN was varied by varying the thermal conductivitythe medium. The results of the calculations~Fig. 3a and b!reproduce well the data of Ozisik.6

Second and third problems

For these problems the studies were done for a cylincal region (R50.1 m andH50.1 m! and two values of theabsorption coefficient (kabs50.15 m21, kabs5`) at a con-stant value of the thermal conductivity, equal tol51.5W/~m•K!. These values of the absorption coefficient corspond to the following values of the radiative–conductparameter:N50.001 andN5`. The infinite value of theradiative–conductive parameter corresponds to the negleradiative transport. In the second problem, for a uniform d

FIG. 4. Radial profiles of the total heat fluxQt(r ,H/2) ~1,2! and tempera-ture T(r ,H/2) along the linez5H/2 ~3,4! for a uniform density of heatrelease.1,3 — N50.001;2,4 — N→`.

FIG. 3. Radial profile of the temperatureT(r ,H/2) ~a! and of the radiativeheat flux, normalized to the total heat fluxQz

r(r ,0)1Qzc(r ,0) ~b! along the

line z5H/2 at a constant temperature on the outer boundary.1,4 — N50.1; 3 — N5`, 2,5 — N50.01.

549 J. Opt. Technol. 67 (6), June 2000

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tribution of the specific heat release, the total energy releain the volume is equal to 19.2 dimensionless units. Theerage temperature of the medium at this value of the tenergy isT51.75 orT51.5 at the respective values of thradiative–conductive parametersN5` or N50.001. The ra-dial dependence of the temperature and of the heat fluxhibited a monotonic variation~Fig. 4!. The nonuniform heatrelease used in the third problem and was taken from Reand is typical for the median plane of a cylindrical glasfounding induction furnace. For the characteristics of thedial dependence of the heat release we used the funcSer(r ):

Ser~r !52pE0

H

h~r ,z!rdz. ~19!

We see from Fig. 5a that the main part of the heat eneis released in the region near the walls, 0.7,r ,1. This char-acter of the heat release and the intense convectivetransfer at the boundary of the region make for a maximof the temperatures atr /R'0.8 ~Fig. 5b! and the presence otwo oppositely directed heat fluxes. The negatively direcheat flux carries thermal energy into the central reg~toward the axis!, while the positively directed flux carries itoward the boundary of the region. At a value of the toenergy equal to 30.8, the average temperature of the medis T51.39 or T51.38 at the respective values of thradiative–conductive parameterN5` andN50.001.

CONCLUSION

1. We have made a study of the distribution of the teperatures and heat fluxes in radiative–conductive heat tr

FIG. 5. Radial profiles of the internal heat releaseSer(r ) ~a! and of the totalheat flux Qt(r ,H/2) ~1,2! and temperatureT(r ,H/2) ~3,4! along the linez5H/2 ~b!. 1,3 — N 5 0.001;2,4 — N→`.

549V. K. Shiff

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fer for various boundary conditions~constant temperature athe boundary, and convective heat transfer with the extemedium! over a wide range of variation of the radiativeconductive parameter.

2. We have considered radiative–conductive heat trafer for various functional forms of the density of the distbution of internal heat release.

3. We have demonstrated that the discrete ordinamethod can be used in nonlinear problems of radiativconductive heat transfer.

1B. G. Carlson, ‘‘A method of characteristics and other improvementssolution methods for the transport equation,’’ Nucl. Sci. Eng.61, No. 3,408 ~1976!.

2L. P. Bass, A. M. Voloshchenko, and T. A. Germogenova,Discrete

550 J. Opt. Technol. 67 (6), June 2000

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Ordinates Methods in Problems of Radiation Transfer@in Russian#, M. V.Keldysh Institute of Applied Mathematics, Moscow~1986!, 23 pp.

3A. S. Jamaluddin and P. J. Smith, ‘‘Predicting radiative transfer in asymmetric cylindrical enclosures using the discrete ordinates methoCombust. Sci. Tech.62, 173 ~1988!.

4S. S. Dua and P. Chen, ‘‘Multidimensional radiative transfer in noisothermal cylindrical media with nonisothermal bounding walls,’’ Int.Heat Mass Transfer18, 245 ~1975!.

5M. Mengyuch and R. Viskanta, ‘‘Radiation transport in axisymmetric clindrical closed volumes of finite length,’’ Teploperedacha, No. 2,~1986!.

6H. Y. Li and M. N. Ozisik, ‘‘Simultaneous conduction and radiation in2-D participating cylinder,’’ J. Quant. Spectrosc. Radiat. Transfer56, No.5, 393~1991!.

7V. K. Schiff, A. N. Zamyatin, and A. A. Zhilin, ‘‘Numerical simulation ofthermal convection of a glass melt in a cylindrical induction furnaceGlastech. Ber. Glass Sci. Technol.69, No. 12, 1~1996!.

550V. K. Shiff