Radiative Transfer Problem in two Concentric Spheres …7) 34-2011...Arab Journal of Nuclear Science...

Arab Journal of Nuclear Science and Applications, 46(3), (153-166) 2013

153

Radiative Transfer Problem in two Concentric Spheres with Reflective

Boundary Conditions and internal Source

S. El-Konsol, A.S. Sabbah*, A. Elghazaly and M. Hosni

Reactor &Neutron Physics Department, Nuclear Research Center, Atomic Energy Authority, Cairo, Egypt.

* Mathematical Department, Faculty of Science, Zagazig University, Zagazig, Egypt

Received: 20/6/2011 Accepted: 29/8/2011

ABSTRACT

Radiation transfer problem for anisotropic scattering in a spherical

homogeneous, turbid medium with angular dependent (specular) and diffuse

reflecting boundaries is considered. The solution of the problem containing an

energy source in a medium of specular and diffuse reflecting boundaries is given in

terms of the solution of the source-free problem. The angular dependent reflectivity

of the boundary is considered as Fresnel’s reflection probability function. The

source-free problem is solved by using the Pomraning-Eddington approximation.

Two different weight functions are used to find the solution constants. The partial

heat fluxes at the boundaries of solid sphere and spherical shell of transparent and

reflecting boundaries are calculated. The calculations are carried out for various

values of refractive index and different radii. The results are compared with those

of the Galerkin technique.

Key Words: Radiative Transfer / homogeneous spherical shell Medium / Heat Flux/ Pomraning-

Eddington /Sspecular Reflecting/ anisotropic scattering.

INTRODUCTION

The radiative transfer through a spherical, participating medium has numerous applications in

engineering and space science. In most radiative transfer problem, the angular variation of the

reflectivities for spherical medium is neglected and the interfaces are usually assumed to have constant

reflectivities. Few works have taken into consideration the direction–dependent reflectivities. (1-7) Also,

the radiative transfer through a spherical shell medium can be calculated with the use of different

methods.( 8-11) The case in which the spherical medium has boundaries that emit and reflect radiation

and contains an energy source has mathematical complexity.

In this work, we find the solution of radiative transfer through a spherical medium with

specular and diffuse reflecting boundaries and energy source in terms of the solution of the source-free

problem. The Pomraning-Eddington approximation(12-14) is used to solve the source-free problem. The

constants of the solution are calculated by using weight functions technique. The partial heat fluxes at

the boundaries of solid sphere and spherical shells of transparent and reflecting boundaries are

calculated. The angular dependent reflectivity of the boundary is taken as Fresnel’s reflection

probability function. The calculations are compared with those given by the Galerkin technique(10).

THEORETICAL TECHNIQUE

Consider the radiative transfer equation through an absorbing, emitting, anisotropically scattering,

and homogeneous, medium. This medium occupies the region between two concentric spheres of inner

radius R1 and outer radius R2. The radiation problem with internal source is formulated mathematically

by the equation (15)


154

1

1

0

2

),(,,2

,11

rQrIpdrI

rr (1)

Where R1 r R2 and -1 μ 1,

subject to diffuse and specular boundary conditions

),-(R )n,( )(Rq2)(TI ),( 111111b11 IRIsd

(2a)

),(R )n,( )(Rq2)(TI ),( 222222b22 IRIsd

(2b)

Here ),( rI is the radiance at optical distance r, while is the cosine of the angle between the

propagation direction and the positive direction of the r-axis. The single scattering albedo is taken as

0 while P (,’) is the scattering phase function which can be expanded in terms of the Legendre

polynomial Pn() as (16)

0

)()(),(n

nnn ppap 10 a (3a)

The expansion coefficients an can be calculated by Mie scattering theory(12) using the refractive index

η and the size parameter of the particles x of the medium. (10) For a linear anisotropic approximation

this phase function can be stated as

ap 1),( (3b)

Where a is given by (12)

])!1(!2[])!2()1[(2

12

0

mmmaam

m

m

m

),( rQ is an internal energy source which may be considered as )]([)1( 0 rTIb , where )(rT is

the temperature distribution of the medium. ,iiand where 1i and 2,are the emissivities and

diffuse reflectivities of the inner and outer surfaces, respectively, and are related by ii 1 .

)( ib TI is the black-body radiation where 21andTT are the temperatures of the inner and outer surfaces,

respectively. in addition ),( i

sn are the specular reflectivities at the boundaries which are

dependent upon the refractive indices, and 1n and 2n are the relative refractive indices of the inner

and outer surfaces, respectively.

n

n - S

S

- S

2

1 ),(

2

2

22

Sn (5)


155

where

)1(n 22 S (6)

These functions ),( n are equal to zero for refractive index n equal to unity and equal to unity for

all angles larger than the critical angle c cos c , which is given by 22

/)1( nnc .

The partial heat fluxes )(

Rq

is defined by

),( )(

1

0

RIdRq (7)

This general problem can be treated in terms of the solution of the source-free problem with

specular boundary conditions formulated mathematically as

1

1

0

2

,,2

,11

rpdr

rr (8)

1121 andRrR

with boundary conditions

),(),(, 111 RnRS

(9a)

),(),()(, 22222 RnTIRS

b (9b)

The solution of general problem (1-2) can be given in terms of the solution of the source-free problem

(8-9) demonstrated as follows:

Changing to in equation (3-3), multiplying the resultant equation by ),(2 rIr and

equation (1) by ),(2 rr subtracting the obtained equations and then integrating the resultant

equation over ),()1,1( 21 RRroverand and using the boundary conditions of equations (2)

and (9) leads to

13112211 )()( ARqARqA

(10)

Where

2

22211 )(21 RRAd

(11a)

2

11112 )(2 RRAd

(11b)

)()( 22

2

211

2

1113 RFRRFRQA

(11c)

1

0

),()( RdR (12)

)( ibii TIF (13a)


156

2

1

1

1

2

1 ),(),(

R

R

rrQdrdrQ (13b)

Put and rr in equation (3-3) and multiplying equation(3-1) and (3-3) by

),,(2 rIr respectively, subtracting the resultant equation, integrating over )1,1( and over

),( 21 RRr and using the boundary conditions of equations (2) and (9) gives

23122221 )()( ARqARqA

(14)

Where

2

21221 )(2 RRAd

(15a)

2

121

2

222 )(2 RRRAd

(15b)

)()( 12

2

221

2

1223 RFRRFRQA

(15c)

And

2

1

1

1

2

2 ),(),(

R

R

rrQdrdrQ (16a)

21 RR (16b)

The simultaneous solution of both equations (3-5) and (3-10) gives the partial heat fluxes at the

boundaries, )()( 12 RqandRq

of the source problem in terms of the partial heat fluxes of the

source free problem and the integration of the energy sources ).,( rQ then

21122211

23122213

2 )(AAAA

AAAARq

(17)

21122211

21132311

2 )(AAAA

AAAARq

(18)

SOLUATION of THE SOURCE FREE PROBLEM

The source free problem of radiative transfer described by equations (8) and (9) can be solved

by using the Pomraning Eddington approximation given in the form.(12)

)O(r, q(r) )(r, G(r) ),( r (19)

Where (r,) and O(r,) are even and odd functions of , respectively, which are spatially slowly

varying and are defined by normalization relations.

1 )(r, O )(r,

1

1

1

1

dd (20)

The space irradiant, G(r) is defined as

)(r, )(

1

1

drG (21a)


157

While the net heat flux q(r), is given as

1

1

),()( rdrq (21b)

Integrating the source free problem equation, (8) over (-1,1), using Eqs. (19-21) one get the first–

order differential equations for G(r) and q(r) as17

0 G(r) q(r) (2/r) /)( drrdq (22a)

where

1 0 (22b)

Multiplying equation (1) by and integrating over (-1,1), using Eqs. (19-21) lead to

0 q(r) )/( G(r) /r)( /)( DdrrdG (23a)

where

/3)a( - 1 0 (23b)

1

1

2),( rEdD (23c)

(1/D) - 3 (23d)

The function ),( r and ),( rO can be defined by substituting equation (19) into equation (8),

separating the even and odd terms of and using eqs. (22a) and (23a) to obtain

) -1/()1)(2/( ),( 22

0

2

0 ar (24a)

and

) - )/(1 a ( )]2/([ ),(22

0

2

00 rO (24b)

Where

D / 2

0 (25)

which can be calculated using the transcendental equation

) - 1(

) (1ln Y ) / 2(

0

0

00

(26a)

Where

)a ( / )a ( 2

00

2

0 Y (26b)

This equation can be solved numerically to find the value of 0 for each value of the scattering albedo

0 .


158

The two-coupled first order differential equations (22a) and (23a) lead to the second - order

differential equation

0 G(r) ] - r

[ dr

dG(r) 2)/r] [(

)( 2

022

2

dr

rGd (27)

This equation is in the form of Bessel equation which has the solution

)()()()( 2/)1(22/)1(1

2/)1(vrKCvrICvrrG

(28)

Where )()( xKandxI are modified Bessel functions of the first and second kind.

Substituting (28) into (23a) leads to

)()()()(- )( 2/)3(22/)3(1

2/)1(

0

vrKCvrICvrrq

(29)

The constants C1 and C2 appearing in (28) and (29) can be defined using the boundary conditions

of our problem given by (9). This boundary conditions can be verified using a weight function.

Weight function method

In this method the boundary conditions given by (9a) and (9b) are forced to be satisfied using a

weight function )(W as follows(18-20)

1

0

1111b11 0 ] ),-(R ),( - )(TI - ),(R[ )( nWds

(30a)

1

0

2222b22 0 ] ),(R ),( - )(TI - ),-(R[ )( nWds

(30b)

Here, we shall use two different weight functions to verify the boundary conditions and determine the

constants C1 and C2. The first weight function WA, is taken in terms of the adjoint of the radiation

intensity as(18)

sphere surfaceouter at ),(R ),-(R

sphere surfaceinner at ),-(R ),(R )(

22

11

AW (31)

This weight function is the Lagrange multiplier of the boundary condition. The second weight function

WH, is taken in terms of Chandrasekher’s H-function as(21)

3 / )H( )(1/2 HW (32)

In the calculations Chandrasekher’s H-function will be taken by its approximate form (18)

1 ]) 1(

) - 1( - [1 C )( 0

0

0

0

H (33a)

1 ) 2 / 3 1 )( 3(2/ )H( 0 (33b)


159

Where

]1) 1( /[2)(Z exp 2/12

000 00

C (34a)

) - 1ln(

)(1 /)ln(1 2 -

2

0

000

(34b)

and Z0 is the extrapolated distance of the Milne problem.

These weight function methods lead to a set of two linear equations for the constants. The

simultaneous solution of these set of equations gives the values of C1 and C2 which used to find

irradiant, G(r) and the net heat flux q(r). The values of G(r), q(r), and

at the boundaries of the

source-free problem will be used in Eqs. (10) to (18) to calculate the partial fluxes at the boundaries q-

and q+ of the general problem described by Eqs. (1) and (2).

RESULTS AND DISCUSSION

Here, we examine the Pomraning Eddington method in solving the source-free medium in the

determination of the partial heat flux at the boundaries )(

Rq

and )( -

Rq of the spherical shell of

different radii (0.5, 1) and (1,2). In Table 1 The calculation for source-free medium with transparent

inner surface and diffuse reflecting outer surface of emissivity2 =0.5. The calculations are carried out

for transparent boundaries and the same media of Table 1 but with black-body radiation given by

)1/(1)( 0TI b and are given in Tables 2 and 3, respectively.

All the calculations are normalized to the outer surface black-body radiation, compared with

Ref.( 11) and show good agreement.

Also, we examine the variation involved in the determination of the partial heat flux at the

boundaries )(

Rq

and )( -

Rq , when the angular dependent Fresnel’s reflectivity at the boundary

used instead of constant reflectivity. The calculations are carried out for of the spherical shell of

different radii (0.5, 1) and (1,2) mfp using the weight function method with two weight functions WH

and WA for homogeneous media of isotropic and anisotropic scattering with single scattering albedo

0 between .5 and 1.0. The same calculations are carried out for the same media of Tables (1-3) but

with diffuse reflecting surface and specular reflecting outer surface with refractive index n2=1.33 and

n2=1.5 and are given in Tables (4-6).

The caculations are carried out for homogeneous media of forward anisotropic, isotropic and

backward anisotropic scattering. The phase functions of forward and backward anisotropic scattering

describe media of refractive index and size parameter (η, x)given by (1.2, 2) and (∞, 1), respectively.

The Legendre polynomials expansion coefficients of these phase functions are taken from Ref. (7) and

give linear anisotropic parameters a =1.81517 and -0.58659, respectively.

The comparison shows that the results of the Pomraning Eddington approximation become more

accurate for media of high scattering albedo.


160

Table 1: The partial heat fluxes q+(R2) and q-(R1) at the surfaces of a homogeneous spherical shell of diffuse reflecting boundaries , of

emissivities ε1=1.0 and ε2=0.5, black-body radiation Ib (T1)=0, Ib(T2)=1.0, Ib (T)=0 and for different radii R1, R2

ω0

R1=0.5 and R2=1.0 R1=1.0 and R2=2.0

Q+(R2) q-(R1) q+(R2) q-(R1)

WA WH Ref[11] WA WH Ref[11 ] WA WH Ref[ 11] WA WH Ref[11 ]

Forward Scattering

0.3 0.0557 0.0815 0.1083 0.0997 0.1431 0.1937 0.0293 0.0412 0.0354 0.0619 0.0869 0.1147

0.5 0.0845 0.1147 0.1405 0.1464 0.1847 0.2270 0.0573 0.0703 0.0834 0.1023 0.1256 0.1502

0.7 0.1720 0.1825 0.1858 0.2378 0.2525 0.2726 0.1191 0.1259 0.1325 0.1820 0.1922 0.2069

0.9 0.2501 0.2482 0.2544 0.3334 0.3310 0.3393 0.2221 0.2206 0.2259 0.3021 0.3001 0.3073

0.99 0.2981 0.2969 0.2982 0.3804 0.3793 0.3807 0.2997 0.2989 0.2999 0.3835 0.3824 0.3836

0.999 0.3032 03029 0.3032 0.3854 0.3850 0.3854 0.3093 0.3090 0.3093 0.3931 0.3928 0.3931

Isotropic Scattering

0.3 0.0572 0.0840 0.1124 0.0959 0.1409 0.1896 0.0327 0.0471 0.0620 0.0551 0.0794 0.1048

0.5 0.0961 0.1228 0.1475 0.1438 0.1807 0.2193 0.0573 0.0810 0.0966 0.0779 0.1102 0.1319

0.7 0.1668 0.1864 0.1954 0.2190 0.2460 0.2603 0.1266 0.1405 0.1513 0.1479 0.1642 0.1771

0.9 0.2600 0.2592 0.2652 0.3148 0.3137 0.3210 0.2476 0.2469 0.2525 0.2608 0.2600 0.2659

0.99 0.3081 0.3072 0.3083 0.3587 0.3578 0.3590 0.3339 0.3326 0.3340 0.3396 0.3385 0.3398

.999 0.3131 0.3127 0.3132 0.3633 0.3628 0.3633 0.3445 0.3440 0.3444 0.3494 .3491 0.3494

Backward Scattering

0.3 0.0571 0.0851 0.1136 0.0947 0.1411 0.1885 0.0336 0.0491 0.0644 0.0532 0.0776 0.1019

0.5 0.0991 0.1227 0.1496 0.1347 0.1794 0.2173 0.0574 0.0828 0.1005 0.0725 0.1047 0.1265

0.7 0.1625 0.1873 0.1982 0.2111 0.2434 0.2571 0.1267 0.1448 0.1565 0.1371 0.1567 0.1690

0.9 0.2630 0.2625 0.2684 0.3096 0.3090 0.3159 0.2538 0.2534 0.2590 0.2501 0.2497 0.2552

0.99 0.3114 0.3102 0.3115 0.3523 0.3513 0.3525 0.3416 0.3405 0.3418 0.3289 0.3278 0.3291

0.999 0.3263 0.3159 0.3163 0.3566 0.3562 0.3566 0.3525 0.3521 0.3527 0.3388 0.3384 0.3388


161

Table 2: The partial heat fluxes q+(R2) and q-(R1) at the surfaces of a homogeneous spherical shell of transparent boundaries , black-body

radiation Ib (T1)=0, Ib(T2)=1.0, Ib (T) = 1/(1- ω0) and for different radii R1, R2

ω0

R1=0.5 and R2=1.0 R1=1.0 and R2=2.0

Q+(R2) q-(R1) q+(R2) q-(R1)


Forward Scattering

0.3 0.4477 0.4861 0.5240 0.4305 0.4979 0.5689 0.5557 0.5766 0.5986 0.5321 0.5757 0.6231

0.5 0.5210 0.5629 0.6015 0.5274 0.5774 0.6245 0.6866 0.7071 0.7291 0.6604 0.6970 0.7349

0.7 0.6639 0.6754 0.6958 0.6381 0.6541 0.6901 0.8894 0.8981 0.9113 0.8499 0.8631 0.8833

0.9 0.8076 0.8059 0.8118 0.7632 0.7610 0.7682 1.1742 1.1515 1.1779 1.0829 1.0808 1.0878

0.99 0.8727 0.8720 0.8729 0.8081 0.8074 0.8083 1.3408 1.3400 1.3409 1.2070 1.2062 1.2071

0.999 0.8794 0.8790 0.8794 0.8125 0.8120 0.8125 1.3592 1.3587 1.3592 1.2202 1.2196 1.2202


0.3 0.4429 04841 0.5248 0.4278 0.4969 0.5651 0.5516 0.5763 0.6006 0.5263 0.5679 0.6090

0.5 0.5318 0.5678 0.6018 0.5208 0.5698 0.6196 0.6694 0.7060 0.7289 0.6329 0.6827 0.7139

0.7 0.6561 0.6776 0.6949 0.6355 0.6653 0.6865 0.8743 0.8917 0.9057 0.8265 0.8468 0.8631

0.9 0.8052 0.8045 0.8098 0.7648 0.7640 0.7703 1.1629 1.1622 1.1672 1.0868 1.0861 1.0913

0.99 0.8710 0.8702 0.8710 0.8152 0.8145 0.8153 1.3315 1.3307 1.3216 1.2380 1.2372 1.2342

.999 0.8775 0.8773 0.8775 0.8201 0.8198 0.8201 1.3503 1.3499 1.3503 1.2549 1.2543 1.2550

Backward Scattering

0.3 0.4418 0.4848 0.5251 0.4259 0.4971 0.5639 0.5499 0.5763 0.6011 0.5239 0.5657 0.6048

0.5 0.5347 0.5677 0.6019 05188 0.5683 0.6181 0.6639 0.7030 0.7286 0.6268 0.6764 0.7076

0.7 0.6500 0.6772 0.6946 0.6273 0.6628 0.6853 0.8671 0.8895 0.9039 0.8172 0.8414 0.8570

0.9 0.8044 0.8040 0.8092 0.7653 0.7648 0.7709 1.1595 1.1591 1.1640 1.0869 1.0865 1.0914

0.99 0.8703 0.8696 0.8704 0.8173 0.8165 0.8174 1.3291 1.3284 1.3292 1.2457 1.2449 1.2458

0.999 0.8814 0.8811 0.8769 0.8223 0.8219 0.8223 1.3481 1.3476 1.3481 1.2636 1.2634 1.2637


162

Table 3: The partial heat fluxes q+(R2) and q-(R1) at the surfaces of a homogeneous spherical shell of diffuse reflecting boundaries , of

emissivities ε1=1.0 and ε2=0.5 , black-body radiation Ib (T1)=0, Ib(T2)=1.0, Ib (T) = 1/(1- ω0) and for different radii R1, R2

ω0

R1=0.5 and R2=1.0 R1=1.0 and R2=2.0

Q+(R2) q-(R1) q+(R2) q-(R1)


Forward Scattering

0.3 0.4403 0.4816 0.5292 0.4192 0.4929 0.5782 0.5592 0.5830 0.6093 0.5396 0.5900 0.6457

0.5 0.5456 0.5857 0.6305 0.5334 0.5982 0.6706 0.7088 0.7367 0.7670 0.6987 0.7491 0.8038

0.7 0.7236 0.7417 0.7701 0.7285 0.7549 0.7966 0.9786 0.9942 1.0186 0.9934 1.0147 1.0529

0.9 0.9653 0.9618 0.9739 0.9671 0.9623 0.9786 1.4681 1.4639 1.4674 1.4880 1.4821 1.5020

0.99 1.0989 1.0966 1.0991 1.0894 1.0872 1.0898 1.8353 1.8327 1.8357 1.8472 1.8441 1.8476

0.999 1.1133 1.1128 1.1133 1.1023 1.1017 1.1023 1.8804 1.8796 1.8804 1.8908 1.8899 1.8908


0.3 0.4370 0.4818 0.5304 0.4183 0.4932 0.5746 0.57511 0.6044 0.6301 0.4944 0.5507 0.6301

0.5 0.5214 0.5867 0.6320 0.5001 0.5971 0.6644 0.6909 0.7232 0.7729 0.6591 0.7247 0.7738

0.7 0.7080 0.7425 0.7715 0.7040 0.7498 0.7885 0.9675 0.9999 1.0271 0.9347 0.9727 1.0045

0.9 0.9657 0.9642 0.9756 0.9587 0.9567 0.9706 1.4833 1.4815 1.4961 1.4208 1.4188 1.4343

0.99 1.1017 1.0998 1.1019 1.0832 1.0813 1.0836 1.8677 1.8642 1.8681 1.7767 1.7734 1.7771

.999 1.1161 1.1153 1.1161 1.0964 1.0954 1.0964 1.9152 1.9140 1.9152 1.8206 1.8196 1.8206

Backward Scattering

0.3 0.4356 0.4823 0.5308 0.4153 0.4931 0.5734 0.5533 0.5839 0.6141 0.5291 0. 5774 0.6253

0.5 0.5160 0.5844 0.6324 0.4934 0.5928 0.6625 0.6833 0.7374 0.7744 0.6503 0.7186 0.7650

0.7 0.6988 0.7423 0.7720 0.6910 0.7402 0.7859 0.9579 1.0001 1.0290 0.9147 0.9600 0.9907

0.9 0.9657 0.9648 0.9761 0.9560 0.9549 0.9682 1.4867 1.4855 1.5003 1.4024 1.4012 1.4157

0.99 1.1024 1.1001 1.1027 1.0813 1.0793 1.0817 1.8761 1.8729 1.8765 1.7580 1.7551 1.7584

0.999 1.1170 1.1165 1.1170 1.0946 1.0942 1.0946 1.9243 1.9231 1.9243 1.8021 1.8009 1.8021


163

Table 4: The partial heat flux q+(R2) and q-(R1) at the surfaces of a homogeneous spherical shell

of diffuse reflecting and specular reflecting outer boundaries with reflective index

n2=1.33 , and n2=1.5 respectively, of emissivities ε1=1.0 and ε2=0.5 , black-body

radiation Ib (T1)=0, Ib(T2)=1.0, Ib (T) = 0.0 and for different radii R1, R2,

Ω0

R1=0.5 and R2=1.0 R1=1.0 and R2=2.0

q+(R2) q-(R1) q+(R2) q-(R1)

WA WH WA WH WA WH WA WH

Forward Scattering

0.3 0.0558 0.0823 0.0999 0.1444 0.0293 0.0416 0.0620 0.0877

0.5 0.0858 0.1171 0.1487 0.1886 0.0582 0.0718 0.1039 0.1283

0.7 0.1770 0.1885 0.2448 0.2608 0.1226 0.1301 0.1874 0.1985

0.9 0.2673 0.2669 0.3566 0.3560 0.2399 0.2366 0.3264 0.3217

0.99 0.3322 0.3251 0.4240 0.4132 0.3342 0.3272 0.4275 0.4187

0.999 0.3382 0.3315 0.4298 0.4213 0.3384 0.3408 0.4400 0.4301


0.3 0.0574 0.0848 0.0962 0.1422 0.0328 0.0475 0.0553 0.0802

0.5 0.0974 0.1254 0.1457 0.1845 0.0581 0.0827 0.0789 0.1125

0.7 0.1717 0.1930 0.2254 0.2547 0.1303 0.1455 0.1522 0.1700

0.9 0.2819 0.2796 0.3413 0.3385 0.2681 0.2660 0.2824 0.2801

0.99 0.3439 0.3363 0.4004 0.3916 0.3741 0.3652 0.3805 0.3716

0.999 0.3507 0.3423 0.4068 0.3971 0.3875 0.3779 0.3931 0.3835

Backward Scattering

0.3 0.0576 0.0859 0.0948 0.1424 0.0340 0.0496 0.0537 0.0783

0.5 0.1003 0.1253 0.1363 0.1832 0.0581 0.0846 0.0734 0.1069

0.7 0.1671 0.1938 0.2171 0.2518 0.1303 0.1498 0.1410 0.1621

0.9 0.2852 0.2834 0.3357 0.3336 0.2750 0.2733 0.2710 0.2693

0.99 0.3478 0.3399 0.3934 0.3849 0.3831 0.3744 0.3689 0.3605

0.999 0.3545 0.3459 0.3996 0.3901 0.3972 0.3871 0.3817 0.3720

n2=1.5

Forward Scattering

0.3 0.0561 0.0828 0.1004 0.1453 0.0295 0.0418 0.0623 0.0883

0.5 0.0867 0.1185 0.1502 0.1907 0.0588 0.0726 0.1049 0.1297

0.7 0.1798 0.1916 0.2486 0.2651 0.1245 0.1322 0.1903 0.2018

0.9 0.2808 0.2763 0.3746 0.3685 0.2484 0.2445 0.3379 0.3325

0.99 0.3480 0.3394 0.4441 0.4335 0.3501 0.3417 0.4478 0.4372

0.999 0.3556 0.3501 0.4520 0.4449 0.3633 0.3576 0.4616 0.4544


0.3 0.0576 0.0852 0.0965 0.1429 0.0329 0.0478 0.0556 0.0805

0.5 0.0983 0.1268 0.1471 0.1866 0.0586 0.0837 0.0797 0.1138

0.7 0.1744 0.1960 0.2290 0.2587 0.1324 0.1477 0.1546 0.1727

0.9 0.2924 0.2876 0.3541 0.3482 02779 0.2735 0.2972 0.2880

0.99 0.3604 0.3516 0.4195 0.4094 0.3926 0.3824 0.3994 0.3891

0.999 0.3681 0.3584 0.4270 0.4158 0.4076 0.3965 0.4135 0.4023

Backward Scattering

0.3 0.0574 0.0863 0.0952 0.1430 0.0338 0.0498 0.0535 0.0787

0.5 0.1012 0.1267 0.1375 0.1853 0.0586 0.0855 0.0740 0.1081

0.7 0.1697 0.1970 0.2204 0.2560 0.1323 0.1523 0.1431 0.1648

0.9 0.2960 0.2940 0.3483 0.3460 0.2852 0.2833 0.2791 0.2792

0.99 0.3645 0.3555 0.4124 0.4025 0.4024 0.3922 0.3874 0.3777

0.999 0.3721 0.3622 0.4194 0.4087 0.4179 0.4063 0.4016 0.3905


164


of transparent inner boundary and specular reflecting outer boundary with reflective

index n2=1.33 and 1.5 respectively, black-body radiation Ib (T1)=0, Ib(T2)=1.0, Ib (T) =

1/(1- ω0) and for different radii R1, R2,

ω0

R1=0.5 and R2=1.0 R1=1.0 and R2=2.0

q+(R2) q-(R1) q+(R2) q-(R1)


Forward Scattering

0.3 0.4484 0.4907 0.4312 0.5026 0.5565 0.5820 0.5329 0.5811

0.5 0.5290 0.5548 0.5355 0.5896 0.6972 0.7221 0.6706 0.7118

0.7 0.6834 0.6977 0.6568 0.6757 0.9155 0.9277 0.8748 0.8735

0.9 0.8514 0.8456 0.8046 0.7984 1.2378 1.2082 1.1416 1.1340

0.99 0.9327 0.9221 0.8637 0.8537 1.4331 1.4169 1.2901 1.2754

0.999 0.9414 0.9288 0.8697 0.8580 1.4550 1.4356 1.3062 1.2886


0.3 0.4444 0.4887 0.4292 0.5016 0.5534 0.5818 0.5280 0.5733

0.5 0.5389 0.5799 0.5277 0.5819 0.6783 0.7210 0.6413 0.6972

0.7 0.6752 0.7017 0.6540 0.6889 0.8997 0.9234 0.8506 0.8796

0.9 0.8486 0.8451 0.8060 0.8026 1.2256 1.2209 1.1454 1.1409

0.99 0.9309 0.9195 0.8713 0.8606 1.4231 1.4060 1.3232 1.3072

0.999 0.9393 0.9265 0.8778 0.8658 1.4454 1.4256 1.3432 1.3247

Backward Scattering

0.3 0.4423 0.4894 0.4263 0.5018 0.5505 0.5817 0.5244 0.5710

0.5 0.5410 0.5797 0.5250 0.5804 0.6718 0.7179 0.6342 0.6907

0.7 0.6683 0.7006 0.6450 0.6857 0.8916 0.9202 0.8403 0.8704

0.9 0.8477 0.8448 0.8065 0.8037 1.2219 1.2180 1.1454 1.1417

0.99 0.9302 0.9191 0.8735 0.8630 1.4205 1.4040 1.3314 1.3157

0.999 0.9436 0.9304 0.8804 0.8679 1.4433 1.4231 1.3528 1.3342

n2=1.5

Forward Scattering

0.3 0.4505 0.4936 0.4332 0.5056 0.5592 0.5855 0.5354 0.5846

0.5 0.5345 0.5813 0.5411 0.5963 0.7044 0.7302 0.6775 0.7198

0.7 0.6942 0.7091 0.6672 0.6867 0.9299 0.9427 0.8887 0.9062

0.9 0.8714 0.8647 0.8235 0.8165 1.2670 1.2354 1.1685 1.1596

0.99 0.9588 0.9462 0.8879 0.8761 1.4731 1.4540 1.3261 1.3089

0.999 0.9682 0.9594 0.8946 0.8863 1.4965 1.4830 1.3434 1.3312


0.3 0.4457 0.4910 0.4305 0.5040 0.5550 0.5845 0.5296 0.5760

0.5 0.5440 0.5865 0.5327 0.5886 0.6847 0.7292 0.6474 0.7052

0.7 0.6860 0.7126 0.6645 0.6996 0.9142 0.9377 0.8642 0.8905

0.9 0.8686 0.8605 0.8250 0.8172 1.2544 1.2431 1.1723 1.1617

0.99 0.9568 0.9440 0.8955 0.8836 1.4627 1.4435 1.3599 1.3421

0.999 0.9661 0.9519 0.9029 0.8895 1.4867 1.4646 1.3816 1.3609

Backward Scattering

0.3 0.4436 0.4914 0.4276 0.5039 0.5521 0.5842 0.5260 0.5735

0.5 0.5458 0.5863 0.5295 0.5869 0.6776 0.7260 0.6398 0.6985

0.7 0.6786 0.7122 0.6549 0.6971 0.9053 0.9355 0.8532 0.8849

0.9 0.8677 0.8646 0.8255 0.8225 1.2508 1.2465 1.1724 1.1684

0.99 0.9561 0.9437 0.8979 0.8861 1.4602 1.4416 1.3685 1.3510

0.999 0.9704 0.9560 0.9054 0.8918 1.4843 1.4622 1.3912 1.3599


165


of diffuse and specular reflecting outer boundaries with reflective index n2=1.33 and 1.5

respectively, of emissivities ε1=1.0 and ε2=0.5, black-body radiation Ib (T1)=0,

Ib(T2)=1.0, Ib (T) = 1/(1- ω0) and for different radii R1, R2,

ω0

R1=0.5 and R2=1.0 R1=1.0 and R2=2.0

q+(R2) q-(R1) q+(R2) q-(R1)


Forward Scattering

0.3 0.4411 0.4816 0.4201 0.4975 0.5601 0.5885 0.5404 0.5956

0.5 0.5540 0.5981 0.5416 0.6109 0.7197 0.7523 0.7095 0.7650

0.7 0.7448 0.7662 0.7498 0.7798 1.0073 1.0270 1.0225 1.0482

0.9 1.0176 1.0157 1.0195 1.0096 1.5477 1.5359 1.5687 1.5550

0.99 1.1745 1.1595 1.1644 1.1496 1.9616 1.9379 1.9743 1.9500

0.999 1.1917 1.1758 1.1799 1.1641 2.0129 1.9860 2.0240 1.9969


0.3 0.4383 0.4864 0.4196 0.4979 0.5768 0.6101 0.4959 0.5559

0.5 0.5294 0.5991 0.5078 0.6098 0. 7001 0.7386 0.6679 0.7401

0.7 0.7286 0.7689 0.7245 0.7764 0.9957 1.0354 0.9619 1.0072

0.9 1.0178 1.0129 1.0104 1.0050 1.5633 1.5563 1.4974 1.4905

0.99 1.1775 1.1621 1.1577 1.1425 1.9962 1.9697 1.8989 1.8741

0.999 1.1947 1.1779 1.1736 1.1569 2. 0501 2.0214 1.9488 1.9217

Backward Scattering

0.3 0.4397 0.4868 0.4195 0.4977 0.5589 0.5894 0.5344 0.5828

0.5 0.5221 0.5968 0.4993 0.6054 0.6914 0.7530 0.6580 0.7338

0.7 0.7185 0.7679 0.7105 0.7657 0.9849 1.0346 0.9405 0.9931

0.9 1.0177 1.0138 1.0074 1.0034 1.5667 1.5610 1.4779 1.4724

0.99 1.1783 1.1627 1.1557 1.1407 2.0052 1.9795 1.8789 1.8550

0.999 1.1959 1.1790 1.1719 1.1555 2.0602 2.0308 1.9293 1.9957

n2=1.5

Forward Scattering

0.3 0.4430 0.4891 0.4218 0.5005 0.5627 0.5920 0.5430 0.5991

0.5 0.5597 0.6049 0.5472 0.6178 0.7272 0.7608 0.7168 0.7736

0.7 0.7566 0.7787 0.7617 0.7926 1.0232 1.0438 1.0387 1.0653

0.9 1.0416 1.0319 1.0435 1.0325 1.5841 1.5706 1.6056 1.5902

0.99 1.2074 1.1899 1.1969 1.1797 2.0164 1.9887 2.0295 2.2025

0.999 1.2257 1.2146 1.2136 1.2025 2.0703 2.0516 2.0818 2.0628


0.3 0.4397 0.4886 0.4209 0.5002 0.5787 0.6130 0.4975 0.5585

0.5 0.5333 0.6060 0.5116 0.6167 0.7067 0.7470 0.6742 0.7485

0.7 0.7403 0.7808 0.7361 0.7885 1.0116 1.0515 0.9773 1.0229

0.9 1.0417 1.0310 1.0342 1.0233 1.6001 1.5842 1.5326 1.5176

0.99 1.2102 1.1931 1.1899 1.1750 2.0517 2.0223 1.9517 1.9238

0.999 1.2288 1.2101 1.2071 1.1885 2.1086 2.0767 2.0045 1.9743

Backward Scattering

0.3 0.4383 0.4889 0.4170 0.4999 0.5555 0.5919 0.5312 0.5853

0.5 0.5267 0.6035 0.5036 0.6122 0.6974 0.7615 0.6638 0.7421

0.7 0.7296 0.7807 0.7214 0.7785 1.0001 1.0518 .9550 1.0096

0.9 1.0417 1.0376 1.0312 1.0269 1.6037 1.5975 1.5128 1.5069

0.99 1.2111 1.1938 1.1879 1.1713 2.0611 2.0325 1.9313 1.9046

0.999 1.2298 1.2114 1.2052 1.1872 2.1187 2.0866 1.9841 1.9540


166

CONCLUSION

The Pomraning Eddington method is used to solve the source-free radiative transfer problem

with transparent boundaries and specular reflectivity of the surface satisfied Fresnel’s law. The

solution of this problem is used to find the partial heat flux at the spherical shell boundaries of a

source radiative transfer problem with diffuse and specular reflectivity boundaries. The caculations are

carried out for spherical shell media of different radii, different single scattering albedo and different

refractive index. The media are taken with or without internal black-body radiation. The caculations

are given for homogeneous media of forward anisotropic, isotropic and backward anisotropic

scattering. The results of Pomraning Eddington method give agreement with those of the Galerkin

technique.

ACKNOWLEDGEMENTS

The authors would like to acknowledge Prof. S. A. El-Wakil for his continuous guidance,

encouragement, reviews and the useful discussions regarding this manuscript.

RERERENCES

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