Radiative heat transfer in isothermal spherical media

J. Quant. Specfrosc. Radiat. Tramfer Vol. 43, No. 3, pp. 239-251, 1990 Printed in Great Britain

RADIATIVE

0022-4073/90 $3.00 + 0.00 Pergamon Press plc

HEAT TRANSFER IN ISOTHERMAL SPHERICAL MEDIA

WEIMINGLI and TIMOTHYW. TON@ Departmentof Mechanicaland AerospaceEngineering,ArizonaState University,Tempe, AZ 85287,U.S.A. (Received 5 June 1989)

Abstract-Radiative heat transfer in emitting, absorbing, and scattering spherical media is analyzed. The medium is assumed to be gray, isothermal, and linear-anisotropically scattering.

The medium is confined in the space between two gray concentric spheres, which diffusely emit, and specularly and diffusely reflect radiation. Approximate solutions of the equation of radiative transfer are obtained using the spherical harmonics method. Results presented include the irradiance and the net radiative heat flux. The effects of the different governing parameters and the particular type of boundary reflections are examined. INTRODUCTION There has been strong interest in understanding thermal radiation in participating media because of the high temperatures often encountered in modern technologies. Among previous investigations, most attention has been given to the planar geometry. Publications dealing with radiative heat transfer in spherical media have been relatively few in number and most of them have dealt with nonscattering materials.‘4 Sparrow et al’ analyzed thermal radiation in an absorbing, emitting, heat-generating gray gas confined between two black, concentric, spherical surfaces. Ryhming’ studied a similar problem at about the same time and considered the effects of unequal boundary surface temperatures. Using a differential approximation, Dennar and Sibulkin3 solved the problem of radiative heat transfer between two diffuse, gray concentric spheres enclosing a gray gas. Crosbie and Khali14 studied the case of a gray isothermal nonscattering spherical layer. They obtained closed-form solutions for the local radiative heat flux under various limiting conditions. A comprehensive review of the subject was also presented. In recent years, the scattering effect of participating media has been considered in several publications. Pomraning and Siewert’ derived the integral form of the equation of radiative transfer for an isotropic scattering sphere with internal source and specularly and diffusely reflecting surfaces. Siewert and Thomas6 used integral transformation techniques and the FNmethod to solve radiative transfer problems in spherical geometry for the case of isotropic scattering and uniform internal source. Thynel and Ozisik’ used the Galerkin method to solve a similar problem in a solid sphere with nonuniform internal source. Later, they extended their analysis to include the effects of a point source and various functional forms of space-dependent albedo.8 The spherical harmonics method was used by Tong and Swathi’ to analyze thermal radiation in linear anistropic scattering, heat-generating concentric spherical media. Tsai and Ozisik” studied the interaction of transient, combined conduction and radiation in an isotropically scattering solid sphere. El-Wakil et al” correlated the radiative transfer problem of a diffusely-reflecting sphere containing an inhomogeneous source-generating medium with a source-free radiative transfer problem with isotropic boundary conditions. As far as can be ascertained, no study has addressed isothermal spherical media despite their importance as one of the fundamental cases in radiative heat transfer considerations. It is the objective of this work to establish the basic characteristics of radiative heat transfer in isothermal spherical media. Consideration is given to a medium which absorbs, emits, and linear-anisotropically scatters radiant energy. The medium is confined between two gray concentric spheres which can be both specularly and diffusely reflecting. The spherical harmonics method is used to solve the equation of radiative transfer. A parametric study is conducted to tTo

whom

QSRT4,/3--D

all correspondence

should

be addressed. 239

WEIMINGLI and TIMOTHYW. TONG

240

determine the effects of the various governing parameters and the importance of the particular type of boundary reflection. ANALYSIS The physical problem to be analyzed is shown schematically in Fig. 1. The inner and outer concentric spheres have radii r. and rb, respectively, while the space between these two spheres is filled with an isotropic, homogeneous, absorbing-emitting-scattering medium. The medium is taken to be gray and is characterized by an absorption coefficient rra and a scattering coefficient us. The temperatures of the medium and inner and outer spherical boundary surfaces are constant and equal to T, T,, and T,, respectively. The two boundary surfaces are assumed to be gray, diffusely emitting and specularly and diffusely reflecting. For such a situation, the equation of radiative transfer for the radiant intensity i(r, p) may be written as’* $i(r,

~)>jaz + (1 - ~‘)/z[ai(z, CL)/+] + i(r, p) = (1 - co)oT4/n + w/2

where the optical depth t, the direction cosine of the propagating scattering albedo w are defined as

P(P, ~‘)i(r, p’) dp’, (1) S’-1 radiation p and the single-

z =(o,+g,)r,

(2)

p 3 cos 9,

(3)

0 = o,/(a, + a,).

(4)

The scattering phase function p(p, p’) is assumed to be represented polynomials as P(P9 P’)= i

j=l

(2j +

in a series of Legendre

1)ujpj(P)pj(P')3

(5)

where the values of a, are dependent on the type of scattering being modeled. In the present investigation, linear-anisotropic scattering is considered, and a, have the following values ao= 1;

- l/3 < a, < l/3;

aj = 0

for j = 2,3,. . . ,

where a, = l/3 represents strong forward scattering while a, = - l/3 represents strong backward scattering. In the spherical harmonics method, the intensity of radiation i(t, 1) is expanded in a series of Legendre polynomials in the form i(r, CL)= f

(2m + l)~,(P)$,(r)/47c,

m=O

where I+G~(T)are unknown functions to be determined.

Fig. 1. Schematic of the physical system.

(6)

Radiative heat transfer in spherical media

241

Equations (5) and (6) are substituted into Eq. (1) and the resulting equation is combined with the orthogonality of Legendre polynomials

and the following recurrence formulae

I(II>+ pm-I(~)1/(2m+ I), (1 -~)Wm(~u)ld~l=mPm-I(~)-w~m(~L),

pP&)

(8)

= Km + l)P,+

(9)

to yield m$(m

+ l)dl(/,+,(r)/dr

+(m + l)(m +2)+,+,(2)/r

+m d@,_,(r)/dt

-(m

+(2m + 1)(1 -c%)&(r)

- l)mll/,_,(t)/r

-4(1 -o)aT46,}P,(p)=0.

(10)

The coefficients of P,(p) must vanish identically for Eq. (10) to be valid for any arbitrary p. Thus (m+l)d~,+,(~)/d~+(m+l)(m+2)~,+,(~)/~+(2m+I)(l-~~,)Jl,(~) +m dtj,_,(r)/dr

-(m

- l)m$,_,(t)/r

-4(1 -~)oT4So,=0,

m =O, 1,2,..

..

(11)

For the P, approximation, the terms involving ,jN+, (T) are ignored. Therefore, there are N + 1 simultaneous linear ordinary differential equations (ODES) to be solved for 11/0,$, , . . . , I),,,. As discussed by Davison,,3 only odd-order approximations will be considered. The boundary conditions at the inner and outer spheres are I i(T,,

p) = L,

cT:/n +

,O:i(T,,-p)

+ 2p:

i(T,,

-p’),u’G’,

P > 0

(12)

s0 and

where 6, p’, and pd are the emissivity, specular reflectivity, and diffuse reflectivity, respectively. Subscripts a and b represent the inner and outer boundaries, respectively. It has been assumed that c + p” + pd = 1. The boundary conditions are reformulated by using Marshak’s boundary conditions14 which require the moments of the radiant intensity to be conserved instead of the intensity itself. Therefore, the boundary conditions become

I I ik,, P)PndP = J’ s[ J 1 =STwT:/~ -tpii(q,, CL) +2~: P’h’dP’ P>0 J’i(T,,-P)P”dP J’i(t,, 1N’dp, c,oT:/a

0

+ p:i(z,,-p)

+ 2~:

0

i(L 9-p’)p’

dp’ p”G,

p >0

(14)

0

(15)

0

0

0

where n = (N + 1)/2.

The higher From to the

SOLUTION P, approximation is obtained analytically. Because the derivation becomes very tedious for order approximations, the P3, P5, P,, Pg and P,, approximations are solved numerically. Eqs. (6) and (8) it can be shown that the unknown functions t,bo(z)and @,(T) are related intensity by eo(?)=2n

and $,(T)=27t

’i(T, PU) +u, J-I P) G. J P~(G ’

-1

(16) (17)

242

WEIMINGLl and TIMOTHYW. TONG

By definition, the rhs of Eq. (16) is the irradiance G(z) and that of Eq. (17) is the net radiative heat flux q(r). Since these two quantities are of primary interest in the field of heat transfer, only their solutions will be presented. For the P, approximation, m is set equal to one and the resulting ODES are dq(z)/dz + Zq(r)/r

+ (1 - oa,)G(r)

- 4(1 - o)aT4 = 0,

3(1 - oa,)q(z)

+ dG(r)/dr

= 0.

(18a) (18b)

These two coupled first order ODES can be combined to yield a single, second-order ODE for G(T) as dG2(r)/dt2 + 2 dG(r)/(r

dr) - t2G(t) = -4t2aT4,

(19)

where i; * = 3(1 - ~a,)( 1 - ~a, ) and a,, = 1. The complete solution for Eq. (19) is G(r) = l/&[C,1,,,(4r)

(20)

+ C,&,2(
where C, and CZ are two integration constants. Combining Eqs. (18b) and (20) yields q(r) = - l/13(1 - mQ,)][Cl &:#r)/J

- C, &(i”r)/r3.” -

C25h2(57)/&

-

c2&2(tT)/T3’21.

(21)

The integration constants are determined through the use of Eqs. (6) (14) (15) (20), and (21). Their expressions are given in the Appendix. For the higher-order approximations, Eqs. (1 I), with the two-point boundary conditions represented by Eqs. (14) and (15) are solved numerically using a FORTRAN computer routine named DVCPR in the IMSL library. This routine is designed for solving a system of ODES with boundary conditions at two points by utilizing a variable order, variable step size finite difference with deferred corrections. The convergence criterion in all the calculations was chosen such that a relative tolerance of no greater than 10d4 was met at all mesh points. The computer program was found to work for all values of o, but small values of z, were needed to approximate solid spheres because of singularity in the equations when rrr= 0. RESULTS

AND

DISCUSSION

Since radiant energy in a medium is composed of a component emitted by the medium and a component incident on the medium at the bounding surfaces, one can consider the general problem as a composite of three basic cases: Case 1, T, = T,, = 0 and T # 0; Case 2, T = Tb = 0 and T, # 0; Case 3, T = T, = 0 and T, # 0. Each of the three basic cases represents a fundamental contribution to the radiative transfer problem. Because the problem is linear in terms of the dependent variables, the solution for any general case with To # Tb # T # 0 can be constructed by superposition of the elementary solutions. The results for the irradiance G(T) and the net radiative heat flux q(t) are presented in dimensionless form; G/aT4 and q/4aT4 for Case 1, G/aTt and q/4aTi for Case 2, and G/aT;f and q/4aTi for Case 3. To examine the accuracy of the present analysis, comparisons with results available in the literature are made. Unfortunately, an extensive search of the literature did not reveal any available results for the problem considered here. The closest problem that has been studied by other researchers is for radiative heat transfer in isothermal planar media.” Since the concentric spherical geometry becomes a planar geometry when rb $ 1 and ru/rb A 1, the comparisons are conducted on the basis that T,,is large and T,/T~ is close to one. Results for rb = 100 and ?,/rb = 0.999, 0.99 and 0.9 (i.e., r6 - T, = 0.1, 1 and 10) have been presented in Figs. 2 and 3 along with those given by Dayan and Tien.” Calculations for rb = 50 and 200 and the same values of T~/T,,have also been carried out. The results are practically identical to those shown in Figs. 2 and 3, indicating the conditions used can be regarded as those for planar media. Dayan and Tienls expressed G and q in terms of two coupled integral equations and solved the equations numerically using an iterative method. For discussion purposes, their results will be referred to as exact. It should be noted that for planar geometries, there is no difference between Cases 2 and 3. Thus, the comparisons are carried out for Cases 1 and 2 only.

Radiative heat transfer in spherical media

243

1.0

-.

0.6

0.6

P ca

G B

s 0.4

0.4

__--

“.”

0.1

0.2

0.3

0.4

0.5

0.0

0.1

0.2

(T--bMB - t )

(7-G Y(B -

Fig. 2. (a) Irradiance for Case 1 with TV= 100, a, = l/3, w=o.5, p:,=p;,=p;=p;=o.

013

0:4

0.5

7a )

Fig. 2. (b) Net radiative heat flux for Case 1 with T,, a, = l/3, w = 0.5, pj, = pi = p:’ = p;I = 0.

=

100,

Figures 2 and 3 present results for Case 1 with strong forward scattering (a, = l/3) and for Case 2 with strong backward scattering (a, = - l/3), respectively. It may be seen that the P, approximation gives the worst approximation. There is already considerable improvement when the P3 approximation is used. For example, the difference of G at (z - T,)/(z, - r,) = 0.5 and rb - r. = 1 when compared to the exact solution reduces from 15.7 to 3.3% for the case in Fig. 2, and from 12 to 3.1% for the case in Fig. 3. In general, the higher the order of approximation, the more accurate the approximate solution is. When the medium is optically thick (rb - T, = lo), the approximate solutions converge to the exact solution more quickly than when the medium is either optically thin (rb - r0 = 0.1) or of intermediate optical thickness (56- rrr = 1). Overall, the compari-

0.6

0.5

0.4 0.6 fru

vnl

kl 0.3 2

0.2

0.2

0.1

0.0

0.0

‘r 0.0

0.2

0.4

0.6

0.8

1.0

CT-%M'b-%a)

Fig. 3. (a) lrradiance for Case 2 with TV= 100, a, = - l/3, w = 0.5. p;, = p; = p; = pi = 0.

0.0

I

I

I

I

0.2

0.4

0.6

0.8

@-w@b-

1.0

1

h)

Fig. 3. (b) Net radiative heat flux for Case 2 with ~~= 100, a,= -l/3, w =0.5, p:,=pj,=p;=p;=o.

,

WEIMINGLI and TIMOTHYW. TONG

244

Table 1. Results at both walls for Case 1 with ~~= 10, pi = pi = pi = pi = 0.25. w

0.2

0.5

0.8

+=r,(Inner Wall)

T=T,(Outer Wall)

G/40T4

9/OT4

G/40Ta

q/UT4

0.05 0.50 0.75 0.90

0.7794 0.7296 0.7150 0.6301

-0.5043 -0.4698 -0.4758 -0.4115

0.7217 0.7217 0.7171 0.6509

0.4817 0.4815 0.4745 0.4165

0

0.05 0.50 0.75 0.90

0.7631 0.7350 0.7191 0.6316

-0.5067 -0.4934 -0.4786 -0.4125

0.7270 0.7270 0.7216 0.6531

0.4856 0.4653 0.4778 0.4199

l/3

0.05 0.50 0.75 0.90

0.7666 0.7405 0.7233 0.6334

-0.5092 -0.4972 -0.4814 -0.4136

0.7326 0.7326 0.7266 0.6554

0.4897 0.4894 0.4812 0.4214

-l/3

0.05 0.50 0.75 0.90

0.7582 0.6833 0.6563 0.5421

-0.4911 -0.4590 -0.4370 -0.3540

0.6622 0.6620 0.6517 0.5594

0.4436 0.4431 0.4320 0.3596

0

0.05 0.50 0.75 0.90

0.7664 0.6975 0.6660 0.5452

-0.4979 -0.4686 -0.4435 -0.3560

0.6762 0.6757 0.6629 0.5638

0.4537 0.4530 0.4398 0.3625

l/3

0.05 0.50 0.75 0.90

0.7792 0.7130 0.6760 0.5482

-0.5050 -0.4792 -0.4502 -0.3579

0.6920 0.6912 0.6751 0.5683

0.4652 0.4642 0.4482 0.3655

-l/3

0.05 0.50 0.75 0.90

0.7241 0.5989 0.5133 0.3515

-0.4699 -0.3927 -0.3421 -0.2296

0.5355 0.5327 0.4991 0.3622

0.3618 0.3593 0.3319 0.2329

0

0.05 0.50 0.75 0.90

0.7423 0.6135 0.5236 0.3534

-0.4618 -0.4074 -0.3489 -0.2306

0.5585 0.5540 0.5125 0.3665

0.3762 0.3744 0.3411 0.2350

l/3

0.05 0.50 0.75 0.90

0.7602 0.6371 0.5339 0.3553

-0.4935 -0.4234 -0.3557 -0.2319

0.5870 0.5797 0.5274 0.3690

0.3988 0.3927 0.3514 0.2373

"1

-l/3

za/rb

sons demonstrate that the present analysis of using the spherical harmonics method to solve the equation of radiative transfer is capable of producing reliable results. Next, the basic characteristics of the radiative heat transfer process are considered. Results for different values of w, a,, p”, and pd are obtained using the P,, solution to study their effects. The influence of the optical thickness is determined by varying Z,/Q with zg fixed at 10. Case I: T, = T, = 0 and T # 0

This case is for a medium emitting radiant energy to its boundaries that are non-emitting. The results can be found in Table 1 and Figs. 4(a) and (b). The thicker the medium optically, that is the smaller the ratio T./Q, the higher the G and q at both the inner and outer boundaries. As seen in Fig. 4(a), G(r) has a peak. According to conservation of energy, Q(r) = 4a,aT4 - a,G(r),

(22)

where 4(r) is the volumetric heat generation rate. Hence, G(r) is maximum at the position where the volumetric heat generation rate is minimum. Note that the peak moves toward the center of the medium as T,/T~ + 1. As pointed out before, in the limit t,/rb + 1, the concentric spherical

245

Radiative heat transfer in spherical media IO-I-

0.4

0.9

0.3

-

0.2 0.1 ---..-

p., ..--

.-.-.-

* 0.0 I-

P-5 p.,

____-__

p-0

-

P-11

$-0.1 -0.2 -0.3 -0.4

---_-___. - p-g I

-0.5

II

P-l,

-0.64, 0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

(T-bmb-G)

Fig. 4. (a) Irradiance for Case 1 with Q = 10, a, = l/3, 0 = 0.5, p: = p; = p,”= pi = 0.25.

0.4

0.6

0.8

CT-% Mb - z,) Fig. 4. (b) Net radiative heat flux for Case 1 with th = IO, n, = l/3, w = 0.5, p: = pi = p:’ = p; = 0.25.

geometry becomes a planar system. Hence, symmetry dictates that the maximum value of G(r) must be at the center of the medium. Figure 4(b) also shows that as qJrb + 1, q becomes more symmetrical relative to the center of the medium. Since increasing COwhile keeping rb and 7,/q, fixed has the effect of lowering absorption and hence emission of radiation, G and q decreases at both boundaries (see Table 1). Table 1 also indicates that G and q at both boundaries increase as a, is changed from - l/3 to l/3. This is because radiant energy emitted by the medium is transferred more easily to the boundaries if the medium is forward scattering rather than backward scattering.

0.6

0.1

0.0 0.0

0.2

0.4

0.6

0.8

I0

(.T-wub- T&d

Fig. 5. (a) Irradiance for Case 2 with t,,= 10, a, = l/3, 0 = 0.5, p: = p; = pf = pi = 0.25.

Fig. 5. (b) Net radiative heat flux for Case 2 with T*= 10, 11,= l/3, 0 = 0.5, p: = p; = pi = pf = 0.25.

246

WEMINGLI and TIMOTHY W. TONG Table 2. Effect of interchanging

ps

and

pd

for Case 1 with different r,/r,,

fb = IO,

cl, = l/3,0 = 0.5. Irradiance

‘,/Tb

T=T

a

P6 ,=l.O, PS ,=o.o, 0.05

0.9992

0.60 0.90 0.95 0.99

0.9867 0.6520 0.4459 0.1334

pd,=O.O

pd,=O.O

radiative T=Ta

0.0000 0.0000 0.0000 0.0000 0.0000 =o.o,

0.50 0.90 0.95 0.99

-1.0600 -0.9343 -0.7879 -0.5715 -0.1729

T=Tb

=o.o, ;:;=o.o,

0.4154 0.4154 0.3745 0.3003 0.1211

,=l.O,

;:;:1.0, 0.05

pd,=O.O pd,=O.O

__=o.o,

=1.0, ;:;=o.o, 0.05 0.50 0.90 0.95 0.99

T’T,

P’

Net K./'b

T=Tb

PE

0.5646 0.4743 0.4082 0.3207 0.1260

0.05 0.50 0.90 0.95 0.99

G/4uTQ

0.9992 0.9858 0.6342 0.4136 0.1156 =o.o,

;:;:o.o,

1.0000 0.9970 0.7327 0.5298 0.1559 heat

0.5646 0.4743 0.4122 0.3216 0.1125 flux

=o.o, ;:;=o.o,

pd,=O.O pdb=O.O 0.8703 0.8702 0.7546 0.5621 0.1722

0.0000 0.0000 0.0000 0.0000 0.0000 =o.o, ;I; =o.o,

pd,=O.O pd,=O.O

0.0000 0.0000 0.0000 0.0000 0.0000

pd,=O.O

0.4154 0.4154 0.3762 0.3037 0.1096 pd,=O.O

pd,=l.O 1.0000 0.9963 0.6989 0.4636 0.1247

q/aT4 z:z a

z:+ b

pd,=l.O

-1.0600 -0.9343 -0.7988 -0.5904 -0.1774

T=Tb pd,:l.O #,=O.O O.R703 0.8702 0.7607 0.5713 0.1749 pd,=O.O ,s',=l.O

0.0000 0.0000 0.0000 0.0000 0.0000

-_.

0.2

0.4 0.6 (T-GM%- ‘a 1

0.8

1.0

Fig. 6. (a) Irradiance for Case 3 with TV= 10, a, = l/3, 0 =os, p;,=p;, =p; = p;1= 0.25.

0.0

0.2

0.4

0.6

0.8

1.0

(7-GMB- t, ) Fig. 6. (b) Net radiative heat flux for Case 3 with T*= 10, a, = l/3, w = 0.5, p;, = p; = p: = pi = 0.25.

0.0

0.4 0.0 0.8

0.0

0.4 0.8 0.0

0.0

0.0 0.4 0.0 0.8

0:o 0.4 0.8 0.0

0.4 0.0 0.8

0.0 0.4 0.0 0.8

0.0 0.4 0.8 0.0

0.0

0.0 0.4 0.0 0.8

0.0 0.4 0.8 0.0

0.4 0.8 0.0

Pdb

PSb

0.2448 0.2514 0.2378

0.1073

pg. =0.6,

0.2848 0.2719 0.2780

0.1140

p= a =0.7,

0.1193 0.3348 0.3417 0.3319

ps. =0.8,

0.1017 0.2540 0.2674 0.2412

pd,zO.O

0.1067 0.2917 0.3053 0.2808

pd,=O.O

0.1112 0.3390 0.3520 0.3313

pd,=O.O

0.1162 0.4028 0.4145 0.3947

pd,=O.O

ps,=o.9,

0.1268 0.4021 0.4082 0.3983

0.1211 0.4880 0.5011 0.4798

pd,=O.O

Z=Zb

0.1334 0.4918 0.5002 0.4871

P’a =l.O,

T=Za

Table 3. Effect of interchanging pi and p,” on G for Case a, = l/3, 0 = 0.5. a

=o.o,

=o.o,

0.1027 0.2369 0.2407 0.2332

ps.

0.1077 0.2758 0.2789 0.2728

ps a =o.o.

0.1110 0.3259 0.3280 0.3233

p”.

0.1130 0.3917 0.3956 0.3904

psa =o.o.

0.1156 0.4804 0.4811 0.4799

psa =o.o,

T=Z

1 with 7”/7,,, 7h =

7b =

0.1001 0.2486 0.2577 0.2400

pd,=0.6

0.1025 0.2855 0.2937 0.2778

pd,=0.7

0.1062 0.3332 0.3403 0.3267

~~~-0.8

0.1079 0.3958 0.4034 0.3907

#,=0.9

0.1096 0.4799 0.4837 0.4762

pd,=l.O

+=Tb

0.99,

,,

10,

,, .”

“,

1

0.0

0.4 0.0 0.8

0.0

0.4 0.8 0.0

0.0 0.4 0.0 0.8

0.4 0.0 0.8

0.4 0.8 0.0

0.0 0.4 0.8 0.0

0.0

0.0

0.0

0.4 0.0 0.8

0.0

0.0 0.4 0.0 0.8

0.4 0.8 0.0

0.4 0.8 0.0

0.0

0.1022 0.2443 0.2518 0.2368

~~~-0.6,

0.1071 0.2835 0.2917 0.2760

psb=0.7,

0.3335 0.3417 0.3280

0.1138

psb=0.8,

0.4005 0.4114 0.3943

0.1199

pb=o.9,

0.1213 0.2504 0.2572 0.2445

pd,=O.O

0.1275 0.2913 0.2980 0.2855

pd,=O.O

0.1357 0.3448 0.3520 0.3403

pd,=O.O

0.1460 0.4151 0.4232 0.4058

pd,=O.O

b

pstl

=o.o,

=o.o,

0.0995 0.2390 0.2443 0.2333

P=b=o.o,

0.1025 0.2773 0.2825 0.2736

ps b =o.o,

0.1059 0.3279 0.3319 0.3233

p=b

7,/7,,

=o.o,

0.1125 0.4824 0.4849 0.4798

P”

1 with

0.1092 0.3945 0.4008 0.3889

Table 4. Effects of interchanging p; and pf on G for Case

,“..

=

0.1174 0.2378 0.2418 0.2351

pd,=O.8

0.1225 0.2766 0.2798 0.2753

pJb=o.7

0.1235 0.3290 0.3313 0.3267

pd,=O.R

0.1241 0.3990 0.4060 0.3947

pd,=0.9

0.1257 0.4893 0.4903 0.4882

pd,=l.O

0.99, ‘~b= 1%

“.

WEIMINGLI and TIMOTHY W. TONG

248

Table 2 illustrates the effect due to specular and diffuse reflection by the boundaries. When p’ and pd at either boundary are changed from one extreme to another, that is from one to zero or vice versa, changes for G are < 5% except when r.,/~~ > 0.9, whereas the largest change for q is only around 3% for all ro/rb. Furthermore, Tables 3 and 4 show that the changes in G due to interchanging p’ and pd at a boundary are larger than 5% only when ps or pd at that boundary is 2 0.7. These observations indicate that if an error of no larger than 5% can be tolerated, then: (1) it is always acceptable to simplify the analysis for q by assuming the boundaries are reflecting diffusely only and (2) the same assumption is acceptable for G unless ~,/r, > 0.9 and ps > 0.7. In a study for non-isothermal planar media, Spiga and Spiga16 also concluded that treating a specularly reflecting boundary as a diffusely reflecting surface resulted in negligible differences for temperature distributions and radiative heat fluxes. Hence, the present findings are consistent with the results for planar media except that the aforementioned requirements need to be met if G is the quantity of interest. Case 2: T = T,, = 0 and T, # 0

The results for Case 2 can be found in Table 5 and Figs. 5(a) and (b). Thermal radiation is due to emission by the inner wall only. As r,/r6 decreases, the medium is becoming optically thicker. Table

0.2

0.5

0.8

5. Results at both walls for Case 2 with rb = 10, p:, = pi = pi = pi = 0.25

-l/3

0.05 0.50 0.75 0.90

0.2206 0.2693 0.2705 0.2801

0.5043 0.4908 0.4884 0.4800

0.0000 0.0000 0.0045 0.0638

0.0000 0.0002 0.0070 0.0563

0

0.05 0.50 0.75 0.90

0.2169 0.2638 0.2650 0.2756

0.5068 0.4946 0.4922 0.4831

0.0000 0.0001 O.CO52 0.0661

0.0000 0.0003 0.0076 0.0580

l/3

0.05 0.50 0.75 0.90

0.2131 0.2580 0.2592 0.2709

0.6093 0.4986 0.4962 0.4863

0.0000 0.0001 0.0059 0.0686

0.0000 0.0003 0.0083 0.0598

0.05 0.50 0.75 0.90

0.2417 0.3144 0.3182 0.3320

0.4912 0.4609 0.4571 0.4461

0.0000 0.0003 0.0103 0.0933

0.0000 0.0005 0.0113 0.0755

0

0.05 0.50 0.75 0.90

0.2316 0.2991 0.3033 0.3202

0.4980 0.4714 0.4674 0.4541

0.0000 0.0005 0.0129 0.1002

0.0000 0.0007 0.0134 0.0805

l/3

0.05 0.50 0.75 0.90

0.2204 0.2818 0.2865 0.3076

0.5053 0.4833 0.4791 0.4626

0.0000 0.0006 0.0162 0.1077

0.0000 0.0010 0.0162 0.0859

0.05 0.50 0.75 0.90

0.2743 0.4025 0.4140 0.4322

0.4710 0.4026 0.3943 0.380R

0.0000 0.0028 0.0359 0.1670

0.0000 0.0025 0.0294 0.1232

0

0.05 0.50 0.75 0.90

0.2544 0.3736 0.3869 0.4128

0.4640 0.4224 0.4129 0.3940

0.0000 0.0044 0.0448 0.1809

0.0000 0.0038 0.0361 0.1330

l/3

0.05 0.50 0.75 0.90

0.2322 0.3373 0.3539 0.3911

0.4984 0.4473 0.4357 0.4088

0.0000 0.007% 0.0568 0.1966

0.0000 0.0060 0.0450 0.1441

-l/3

-l/3

Radiative heat transfer in spherical media Table 6.

0.2

0.8

Resultsat

249

both walls for Case 3 with r,, = 10, p:, = pi = pt = p; = 0.25.

0.05 0.50 0.75 0.90

0.0000 0.0010 0.0146 0.0898

-0.0000 -0.0010 -0.0126 -0.0685

0.2783 0.2783 0.2784 0.2853

-0.4817 -0.4817 -0.4815 -0.4748

0

0.05 0.50 0.75 0.90

0.0000 0.0012 0.0160 0.0927

-0.0000 -0.0012 -0.0136 -0.0706

0.2730 0.2730 0.2731 0.2808

-0.4856 -0.4866 -0.4854 -0.4780

l/3

0.05 0.50 0.75 0.90

0.0001 0.0015 0.0175 0.0958

-0.0000 -0.0014 -0.0148 -0.0727

0.2674 0.2674 0.2675 0.2761

-0.4897 -0.4897 -0.4894 -0.4812

l/3

0.05 0.50 0.75 0.90

0.0004 0.0051 0.0375 0.1442

-0.0002 -0.0041 -0.0289 -0.1048

0.3080 0.3080 0.3087 0.3240

-0.4652 -0.4651 -0.4644 -0.4524

0.05 0.50 0.75 0.90

0.0016 0.0137 0.0727 0.2162

-0.0011 -0.0100 -0.0523 -0.1512

0.4645 0.4645 0.4650 0.4708

-0.3618 -0.3618 -0.3613 -0.3560

0

0.05 0.50 0.75 0.90

0.0034 0.0210 0.0895 0.2338

-0.0022 -0.0151 -0.0641 -0.1632

0.4415 0.4416 0.4427 0.4536

-0.3782 -0.3782 -0.3772 -0.3680

l/3

0.05 0.50 0.75 0.90

0.0076 0.0336 0.1123 0.2537

-0.0050 -0.0240 -0.0800 -0.1768

0.4130 0.4130 0.4158 0.4344

-0.3988 -0.3988 -0.3964 -0.3814

-l/3

-l/3

This has the effect of making it more difficult for radiant energy to be transferred away from the inner surface. Therefore, the results in the table as well as those in the figures show that both G and q decrease as z,/q, decreases. Note that a larger o results in higher G at both walls and higher q at the outer wall, but lower q at the inner wall. Strong forward scattering of the medium makes it easier for radiant energy to travel from the inner wall to the outer wall. Thus, G at the outer wall and q at both walls increase as a, changes from - l/3 to l/3. Results such as those in Tables 2-4 for Case 1 were also obtained to examine the effect of p” and pd for this case. These results are not presented in order to save space. It suffices to point out that the results indicated negligible impact on G and q for all situations when p” was approximated as pd. Thus, assuming diffuse reflection only is always acceptable. Case 3: T, = T = 0 and Tb # 0

This case is very similar to Case 2 except the wall emitting radiant energy is on the outside instead of on the inside. Thus, the trends exhibited in Table 6 and Figs. 6(a) and (b) are similar to those in Table 5 and Figs. 5(a) and (b) for Case 2. What was true for G and q at the inner and outer walls for Case 2 is now true at the outer and inner walls, respectively. Also, the diffuse-reflectiononly assumption was found to be always acceptable for both G and q.

250

WEIMINGLI and TIMOTHYW. TONG

CONCLUSIONS

Radiative heat transfer in emitting, absorbing, linear-anisotropically scattering media has been studied in this work. The medium is isothermal and confined in the space between two concentric spherical surfaces. The spherical harmonics method has been used to obtain different orders of approximate solutions of the equation of radiative transfer. Analytical solution has been obtained for the P, approximation while finite-difference numerical results have been obtained for the P3, P,, P, , Ps and P,, approximations. Based on the results presented, the following conclusions have been drawn: q increase at the boundaries for Case 1. The opposite is true for Case 2 and Case 3. (2) Increasing w results in higher G at the boundaries for all three cases. As far as q is concerned, it increases with increasing o at both boundaries for Case 1, but it increases only at the emitting boundary for Case 2 and Case 3. (3) As a, is changed from -l/3 to l/3, G and q increase at both boundaries for Case 1, but increase only at the non-emitting wall for Case 2 and Case 3. (4) If a 5% error can be tolerated, then whether specular reflection is analyzed as specular or diffuse reflection has negligible influence on q for all three cases. The same is true for G except for Case 1 and when r,/rb > 0.9 and p” > 0.7. These findings imply that other than conditions falling within these specified ranges of t,/rb and p” for G in Case 1, the analysis can always be simplified by treating the boundaries as diffusely reflecting only.

(1) As r,/tb decreases, G and

REFERENCES 1. E. M. Sparrow, C. M. Usiskin, and H. A. Hubbard, J. Heat Transfer 83, 199 (1961). 2. I. L. Ryhming, Int. J. Heat Mass Transfer 9, 315 (1966). 3. E. A. Dennar and M. Sibulkin, J. Heat Transfer 91, 73 (1969). 4. A. L. Crosbie and H. K. Khalil, JQSRT 12, 1465 (1972). 5. G. C. Pomraning and C. E. Siewert, JQSRT 28, 503 (1982). 6. C. E. Siewert and J. R. Thomas, JQSRT 34, 59 (1985). 7. S. T. Thynell and M. N. Ozisik, JQSRT 33, 319 (1985). 8. S. T. Thynell and M. N. Ozisik, JQSRT 35, 349 (1986). 9. T. W. Tong and P. S. Swathi, J. Thermophys. Heat Transfer 1, 162 (1987). 10. J. R. Tsai and M. N. Ozisik, JQSRT 38, 243 (1987). 11. S. A. El-Wakil, M. H. Haggag, M. T. Attia, and E. A. Saad, JQSRT 40, 71 (1988). 12. M. N. Ozisik, Radiutiue Transfer, Chap. 8, Wiley, New York, NY (1973). 13. B. Davison, Neutron Transport Theory, Chap. 10, Oxford, London (1957). 14. R. E. Marshak, Phys. Rev. 71, 443 (1947). 15. A. Dayan and C. L. Tien, JQSRT 16, 113 (1976). 16. G. Spiga and M. Spiga, Int. J. Heat Fluid Flow 4, 235 (1985).

APPENDIX Integration C, = (A,A,

Constants for

- A,A,)I(A,A,

C, = (A, A, - A,AMA,

the P, Approximation

- A,A,), A, - AzA&

where A, = [E,/T;‘*-

wA1~,,2

A, = [E,/r;2’*-

WJr,l~,,,&J

- vs 5/Jz,1~-1,2b3,)~ + [E, U,/d~-1,2W~

A 3 = c(ca T”, + 4F, T4), A4 = -[E2/2~‘*+Fz/~lZ,,2(55*)

+ [E25/Jtbl~-1,2(5%),

A, = -[E2/~~‘2+F2/~1~,,2(~~~)

- [E2CI&l~-,,2(5d>

Radiative heat transfer in spherical media

A, = O(Q,T; + 4F2 T4), 5 = [3(1 - o)(l

- oa,)]“2,

E, = (1 + P: + d)l[6(1

- ma,

11,

E2

-

)I,

=

(1

+

P;

+

d)l[W

F,=(P:+d-

1)/4,

F2=(p:+pf--

1)/4.

oal

251