Radiative transfer in a spherical inhomogeneous medium with anisotropic scattering

RADIATIVE TRANSFER IN A SPHERICAL INHOMOGENEOUS MEDIUM WITH ANISOTROPIC SCATTERING S.

A.

EL-WAKIL.

IRerewed

M.

T.

24 Julr

and E. M.

ATTIA.

Fasulr~ oi Scxnse.

Phbslcs Dcpanmenl.

Mansoura

ABLILWAFA

limrcrsn~.

hlanioura.

ior publrcarron IT Januarr

1990. recmed

I991

Egypt

I

Abstract-The radialIke heal flu\: a~ the boundary oi a sphere comalnmp an mternal energ! source and subject to general boundary condnlons IS obtained m lerms ol’ the albedo of the corresponding source-free problem wh lsolropic boundaq condIllon. The relallons obiamed apply IO the general case of amsotroplc scatiennp m an inhomogeneous medium. The adkanlage of these relalions IS the result of the l-act thal there is no need LO obtain a particular solutlon l-or specilicd mlernal sources Therefore. calculations can be done easily ior a non-uniform source dislrlbulion The phase funcrlon 1s approalmaled b> usmg a linear anisotropIc relallon. The linear coefficient IS laken IO Ix the sum of Ihe coefficients of the Legendre expansion of the phase funcuon. The resullmg relalions are used IO calculate the partial heal flux and emiswlt> for a gl\en mlernal enerp) source and mhomogeneous medium

I

The

radiative

scattering

and

anisotropic boundary

transfer

problem

INTRODUCTION

for an internal

inhomogeneous

sphere

energy source in an absorbing.

is important

scattering effects leads to mathematical surface emits and

reflects radiation

in numerous

complexit). diffusely

source. I--I In recent pap ers.’ ’ the problem of radiation

anisotroplcally

applications.

Inclusion

of

especially for the case in which the

and

the medium

contains

transfer in spherically-symmetric

an energy media with

an arbitrary distrrbution of internal energy sources and diffusely-reflecting boundary conditions has been connected to the source-free problem with lsotroplc boundary conditions. The resulting equation

for the partial heat flux

problem and the arbltrar)

IS

expressed in terms of the albedo of the corresponding

simple

energy source. An expression for the emisslvity is also obtained in terms

of the albedo and the energy source. Here, we consider calculations problem in inhomogeneous

for the anisotropic

scattering

media. We use a linear anisotropic phase function. The linear coefficient

is forced to be equal to the sum of the expansion coefficients of the Legendre series of the phase function.’

Numerical

anisotropic

results are obtained

scattering in an inhomogeneous 2

BASIC

for the partial

heat flux and emissivity

for linear

medium with an internal energy source. EQUATIONS

We consider the radiation transfer equation for an internal energy source Q(r) anisotropicall}-scattering and inhomogeneous spherical medium in the form

in an absorbing.

+I

-I
-p)

and

O
(1)

condition

= cl./‘(~)(~)

+ Clfk(C)

I(R.p’)p’dp’.

+ 2Bz

/3:=

I -cl.

p >O.

(2)

1’(I Here, /(r. p) IS the radiation Intensity. r the radial coordinate and p the direction cosine of the intensity. I, is the blackbody radiation of the boundary which is at the temperature 7.. The !I

S .4

32

EL-WAKIL

CI d

emissrvttres are L, and 6: from the outer and Inner surfaces respectiveI>. and ,/‘I ~1) is the externally incident radtation.

This problem can be solLed in terms of the source-free

As proven b) El-Wakil of the first problem
pa(r)>

IS

et al.” the forward radiation

problem

heat flux at the boundary of the sphere y +(R)

given in terms of the albedo A of the second problem

and the functional

bj

q+(R)=

‘I/J(R.p)dp .I”

=(I

-p,.~,-‘[(~,f+c,f,)(A,?‘I+(r’Qt,r).p,(r)),’R’].

where ,f‘( p I is taken to be the constant

The hemispherical

emiswit!

(5)

F and pr,(r) is defined b>

is

where the albedo A of the second prot Aem

is’

-1 .4 =2

$f’tRR.~td/l

=

.I ” The notation

\: Itrt.

y,,(r):

is used for .R

\

The scattering phase function polynomials.

For unpolarized

I‘crJ.p,,vJ, = PI p. 11’ I ma)

radiation,

J II

I‘(r w,,(r) dr.

19)

be represented by a series cxpanrton

it ma> be written

of Legendre

as

where the a, are expansion

coeffictents of the Legendre

difficult and time-consuming

to Hark ntth the complete scattering phase functton. we represent the

linear anisotropic

pol)nomtals

R,,,t P J. Smce it IS usually

scattering as Ptp.p

r=I+a/.y.

(II)

and express a b! a, of Eq. (IO) or in terms of the coeffictents a, as’ i -2

Tables for a, literature.’ ’

as functions

In Sec. 3. we use Galerkin’s in order to obtain the partial

i [(-l)"u~,+,I2nlJ!]~[2?"+'m!lm + II!]. m _4

of the refractive

index ‘I and particle

(12)

size x are available

method to calculate the albedo and the functional heat flux 9 ‘(RI and the emissiwty cu.

!r’Qlr).

in the p~rj)

RT III a sphericalinhomogeneousmedium

3. CALCULATIONS

AND

33

RESULTS

The integral equations corresponding to Eqs. (3) and (4) have been obtained by &Sit and Thynell.’ For linear anisotropic scattering, they may be reduced to two coupled integral equations, viz.)

R

rp,(rj = r Y,(r) + i

I lJ

~(~j[P~(x)~(r,~j+ap,(.rjK,o(r.x)ldx.

(13)

R rp,(rj = ‘Y,(T) +;

~(xjcP,(x)&,(~.

xj + a~,(-v)K,,(r.

s0

~11d.r.

(14)

where rY,(r)

= (-

I)“RK”,(R.

r)

(15)

and

s

1, + 1)

KAr. xl =

lr - II

- x2 + r’)/2rr]p,[(rz

[exp(-r)/rlp,[V

- x’ - r’)/2xr] dt.

(16)

Here, the p,(z) are Legendre polynomials. The kernels K,(r. x) for n. m = 0, I are given in the Appendix. In order to calculate the radiation intensity p,(r) and the net radiant heat flux p,(r), we expand these in finite power series in the forms’ p,(r) = i

C,r’.

(17)

I’ll

where the C, and 0, are unknown expansion coefficients. The single-scattering albedo w(r) for the inhomogeneous medium is taken in a space-dependent form as w(r) = 2 w,(r/R)j.

(19)

1-O

where the o, are known coefficients. Substituting Eqs. (17x19) into Eqs. (I 3) and (14). multiplying the two resultant equations by rk and r’, respectively, and integrating over r~(0, R). we find ‘p+k+z’/(i + k + 2)]

-f

2 (o,/R’)K&‘+

‘.k(R)

- (u/2) i

r-0

X

D,

1-I

2” (q/R’)K;;‘+

‘.k(R) = RKi,(R.

R j.

k = 0. I. 2,. . ., I,

(20)

r-0

~,~o~,~o(w,,~7~;;~+~.~(R~ +(ni2),f; _

\;I+ l.‘(R) - [R”*‘+z’/(l + j + 2)] ~J,IR~)K

D,{?

_

1-O =

RK:,(R.

R),

I = I. 2,3,. . ..J,

(21)

where R x’

Kz(R)= I

0

R r’K,,,,,(r, x) dx dr,

(22)

R x’KJr,

(23)

s0

and Kb(r,

R) =

x) dx.

50 Expressions for the integrals (22) and (23) for n, m = 0 and I are given in the Appendix.

34

S -4 EL-WAKIL

el al

The I + J + I se1 of Eqs. (20) and (II 1 can be obtained simultaneouslv to find the coefficients C, and D,. The two fluxes pO(rj and p,(r) are. respectively. determined from

-1111 ,=(I

‘, t-p,(r) = RK,,,(R. t-1 + $ 1

((IJ.R’r

i

c,K;.:‘+’

(r. Rj+a

I=, i

D,h’:,;‘+‘(r.

RI

1

24)

and

w,(r)= Substituting

-M,,(R.r)+

-,=8.9 $i

,=I

(r. R) + a i

(LfJ, R’)

R)

1

Eq. (19) into Eq. f8~ and ustng Eq. 124~. the albedo IS given by

.4 = I - IZ,R’I

jr-‘.p,(r),b

‘, x

-

energy source Q(r)

(CJ,

r26)

R’I!r’+‘.p,,(r)\,

=II

[ If the internal

D,h’l,+‘+‘(r.

15 represented

Qtrt=[l

I

by

--t,,r,]i

(Qj

R*w”.

(27)

j I 8.4 where the Q, are known coet?ictents. the parttal q+(R)=(l

-BAA,-“(A:Z)[r,F.+r,/,J+(I I

heat flus becomes

R-‘I f (Q/R”, ,, I ,i

=II \

X

;.‘r”+‘.p,,(ri/

-

x

((IJ,

R’lc:r’+‘+‘.p,(r))

Consequently. we may calculate the hemispherical emtsstvity from Eq. 17). In this work. the calculations have been carried out by usmg the radiation

I 1.!

(28)

intensity PO(r) from

Eq. (24) in order to obtain the most accurate values The coefficients C, are gtven by Eqs. (20) and (21). Two

different

phase functions

Table hrward

n

are used for anisotropic

scattering.

I Legendre evpanwn soefficlenls for IV = I 2. r = 2) and hackuard 19 = r: . t = I I scallermg Forward Scattering

Backward Scattering

0

1.0

1

1.98398

2

1.50823

0.29783

3

0.70075

0.08571

4

0.23489

0.01003

3

0.05133

0.00063

6

0.00760

0.00000

7

0.00048

8

0.00000

ii

1.81517

1.0 -0.56524

-0.58659

One of these refers to

0.56298 0.55514 0.54807 0.54483 0.54181 0.53645 0.53208 0.57249 0.55606 0.54899 0.53739

2z43

0.2+2
0.4+2c;fI5

0.5

0.6-2
O.S-2
l.O-2iy3

4iyi5+<=n

o.I-lzrl5+~=n

0.6-8Z:/i5+c2d

l.O-l6~/l5+~"/2

al

0.52774

0.54226 0.53941 0.53440 0.53038 0.56871 0.55292 0.54619 0.53534

0.54278 0.53997 0.53502 0.53103 0.56882 0.55321 0.54618

0.54225 0.53941 0.53441 ,0.53038 0.56871 ~0.55291 '0.54618

0.54539 0.54240 0.53707 0.53273 0.57271 0.55641 0.54939

0.54493 0.54190 0.53653 0.53214 0.57263 0.55617 0.54909 0.53741

0.53793

0.53582 0.53533

0.53590

0.54532

0.54581

0.54531

0.54860

0.54133

0.55469

0.52427

0.52696

0.53062

0.53280

0.53519

0.54064

0.54827

0.55526

0.55205

0.56341

0.56311

0.54692

81

1

0.52834

0.52840

0.54210

0.!35568

0.52495

0.52766

0.53136

0.53356

0.53599

0.54151

0.54788

Rel CC)

0.53660

T

L ng

0.53680

0.54235

0.55593

0.52480

0.52761

0.53139

0.53362

0.53608

0.54164

0.54803

b

Scatter

Present.

0.55247

Ref (8) f

0.55204

Ref CC)

0.55563

Present.

Forward

0.55957

(4)

Isotropic Scattering

0.55990

a

Ref

T

Scatlering

0.55956

Present

Backward

Table2 The albednA for an mhomogeneoussphcncal medium with Q(r) = I). F = 0. I, = I. and 0 = I for anrsolroplc scallering with radius R = I.

forward

scattering

while

the

other

refers

coefficients of these 1~0 phase functions may be used m either the firsr Legendre The hemispherical Q(f)

carried and

reflectivity

= 0. and an isotroprc OUI for diAPren[

backward

anisotropy

phase

factors

to

backirard

scattering.

.4 for an inhomogeneous

boundary

condlrion

space-dependem.

functions

and

The

Legendre

are given tn Table I.’ The hnear expansion coefficient P, or In the sum In Eq I 121. spherical

medium

has been calculated.

single-scarrcring

isolroplc

\rith

with

(r)(r b. using both

The calculated

\alucs

those of Refs

4 and

u

R = I.

radius

These calculations

albedos

scallering.

u, and U have been compared

expansron coefficient

ha\e

been

the iorward

of .-I using

8 In Table

the

2. The

medium wirh radu R = I and 5 and differem hemlspherical emissil iries t,., for a homogeneous scattering albedos try,:, are giLen in Table 3. The calculalcd values are compared with [hose of Thynell.! with

We find thar both the hemrspherical

the exact

Tables

4 and 5 contam

an internal

rhe forward

radlation

The values of q +( RI

c,., at the boundar]. with

reHecri\i[!

.-I and ernlsslvll>

tH are m good agreement

values.

energy

source

heat flu\

y ‘I R b and the hem\ipherlcal

and cH \tere
emlssi\il)

for an inhomogeneous

medium.

giLen b)

Q(rl = [I - OJli-I)[1 - I-’ R’]. at different

radii

R = I and 5.

The space-dependent. here. The amsotropic

single-scattering scattering

albedo

tr~lr

I

has both the linear and quadratic

factors a = a, and (I for both rhe backward

iorrns

and forward

used

scalrering

phase functions

are the same as those gIlen in Table 2. by usmg Tables 2. 4 and 5. we note rhat the values ol’ .-l. y ‘I R I and tH. calculated a = a, and a = cf for each of the iormj oi cr~(r~. are the same 10 ar leas1 three sigmficanl From

figures.

-I Radiarlon comaining

transfer

through

an mlernal

forward

radiation

medium

have been calculated

Table

a spherical.

energ)

heal

flux

? The hemlspherlcal

CONCLL~SIONS inhomogeneous

source and general 9 ‘(RI

and

the hemIspherIcal

In terms of the internal

emlswq

medium

boundary

t,, Ibr d homogeneous and I, = I for anlsolroplc

wl[h anlsolropic

condilions ernlssliir>

scarlerlng

and

has been considered.

The

cH al the boundar)

source Qlr j. rhe albedo

spherIcal xaltermg

medium

ulth

QI~I

= 0, f = 11. t: =

R-1 -2.7

a

0.0 (3)

of lhls

.4. and the radlatton

+2.7

-0

Present.

Ref

0.1

0.658364

0.658364

0.663137

0.663137

0.668026

0.668026

0.5

0.445797

0.445797

0.457751

0.457752

0.470768

0.470768

0.9

0.120273

0.120273

0.12199s

0.121995

0.123790

0.123790

0.1

0.934295

0.934295

0.932679

0.952679

0.972641

0.972641

0.5

0.723546

0.723546

0.794826

0.794629

0.903043

0.903838

0.9

0.327929

0.327932

0.376519

0.376523

0.460709

0.460711

Present.

Ref <3>

Present.

Ref (3)

I

RT

III a spherical inhomogeneous medwm

37

Table 4 Tbe forward radialIon heal flux at the boundary q ‘(RJ for an inhomogeneous spherical medwn and Tar amsolroplc scattering w10.1 Q(r) as in Eq. (28). t, = 0.5. F = I 0. C: = 0.0. and I, = 1.0.

isotropic

B.W. Scatt. a'

a1

0.0

F.V. Scatt. a1

b

ae-

0.32619

0.32621

0.32570

0.32391

0.32406

0*2+2z/3

0.31724

0.31726

0.31600

0.31520

0.31534

0.4+2cri5

0.30047

0.30849

0.30808

0.30665

0.30678

0.5

0.30415

0.30416

0.30377

0.30242

0.30234

0.6-2
0.29905

0.29906

0.29949

0.29022

0.29833

0. a-2
0.29133

0.29134

0.29101

0.20907

0.28997

I. 0-2r/3

0.20206

0.20207

0.28250

0.28156

0.28165

4zr15+q2n

0.33671

0.33673

0.33620

0.33300

0.33407

0.31050

0.31806

0.31623

0.31630

0.4-4
0.31048

0.6-0~ri5+~'~

0.30967

0.30960

0.30926

0.30765

0.30770

0.29264

0.29322

0.29200

0.29070

0.29000

1.0-16
R-0 2w3

0.19317

0.19325

0.19103

0.10334

0.10400

0.2+2
0.18021

0.18028

0.17842

0.17233

0.17282

O.C+2
0.16054

0.16859

0.16706

0.16233

0.16269

0.5

0.16311

0.16316

0.16177

0.15764

0.15790

0.6-2c/15

0.15793

0.15797

0.15672

0.15315

0.15339

0.0-2x/5

0.14010

0.14022

0.14722

0.14456

0.14473

l.O-2c/3

0.13911

0.13914

0.13036

0.13633

0.13645

4r/i5+z2n

0.21201

0.21291

0.21003

0.19941

0.20037

o.4-4~/15+~2R2

0.10439

o.la446

0.18239

0.17540

0.17590 0.16554 0.14702

0.6-0~r15+~2&

0.17233

0.17240

0.17067

0.16510

1.0~16fri5+c2/2

0.15136

0.15140

0.35022

0.14670

intensityp,(r) of this medium. without an internal energy source and with an isotropic boundary condition. The single-scaltermg albedo Q(T) is taken to have both a linear and quadratic space dependence. The scattering phase function is considered to be a linear, anisotropic scattering phase function. The anisotropic coefficient is represented by the first Legendre expansion coefficient or as given by Eq. (12). The radiation intensity p,,(r) and the net radiant heat flux p,(r) of the simple problem are expressed in terms of finite power series. The Galerkin approach is used to find a set of linear equations which are solLed simultaneously to give the expansion coefficients of p,(,r) and p,(r). The internal energy source is used in the form given by Eq. (29). Comparisons of A and cH (see Tables 2 and 3) with the values given in Refs. 3. 4 and 8 show good agreement and our results are nearly exact. The calculations of q+(R) and cH Tar the general problem with the given internal energy source and a constant external incident radiation at the boundary and outer emissivitb tl = 0.5 and zero inner emissivity. are given in Tables 4 and 5, respectively. These calculated values of the anisotropic coefficients a = a, and d = d are the same as the exact values to the third decimal digit.

3&

EL-H’AKIL

S 4

CI al

Table 5 The hemlsphcncal emlssltq tn 01 an Inhomogeneous mechum ioranwolroplc scaltenng \clth Q(r)asIII Eq 131. t, = 0 5. f= IO.t:=OO. and 1.= IO

R-1

T:sotrop1c I

8.V. ScatA..

a

al

0.0

r

F.W. Scatt.

1

al

2z/3

0.17381

0.17379

0.17430

0.17609

0.17594

0.2+2iy5

0.18276

0.18274

0.18320

0.18480

0.18466

0.4+2
0.19153

0.19151

0.19192

0.19335

0.19322

0.5

0.19585

0.19584

0.19623

0.19758

0.19746

0.6-2ty15

0.20015

0.20014

0.20051

0.20178

0.20167

0.8-2cfl

0.20867

0.20866

0.20899

0.21013

0.21003

l.O-2Jy3

0.21714

0.21713

0.21742

0.21844

0.21835

4
0.16329

0.16327

0.16380

0.16612

0.16593

0.4-4~ris+&yn

0.18152

0.18150

Q.i8i94

0.18377

0.18362

0.6-8~ris+<2/2

0.19033

0.19032

0.19074

0.19235

0.19222

l.0-i6tylS+~2~

0.20736

0.20678

0.20800

0.20922

0.20912

R-5 2w3 0.2+2
0.30683

0.30675

0.30897

0.31666

0.31600

0.31979

0.31972

0.32158

0.32767

0.32718

0.4+2
0.33146

0.33141

0.33294

0.33767

0.33731

0.5

0.33689

0.33684

0.33823

0.34236

0.34202

0.6-2c/15

0.34207

0.34203

0.34328

0.34685

0.34661

0.8-Z
0.35182

0.35178

0.35278

0.35544

0.35527

l.O-zyn

0.36089

0.36086

0.36164

0.36367

0.36355

4?/i5+<2n

0.28719

0.28709

0.28997

0.30059

0.29963

0.4-4
0.31561

0.31554

0.31761

Q-32460

0.32402

0.6-8~rl5+~'/2

0.32767

0.32760

0.32933

0.33490

0.33446

l.O-l6~riJ+C:'/Z? 0.34864

0.34860

0.34978

0.35322

0.35298

L

REFERENCES I. S. T. Thynell and M N &I$I~. 1M.4 J .ippl ,Jfdr 2. A. K. Kolesot and V. j’u Perot. .tvrohrrka 26. 239 3. S. T. Thynell. JQSRT 41. 383 I 19891.

34. 32.7I I9851 I 19E’1.

4. J R. Tsai. M N. t)zi$ik. and F Samarelh. JQSRr 42, Iti7 I 13891 5. S. A. E -Wakil and E. A Saad. .dsrrph~ r SF)UY Sc-r 129. X7 t 19871. 6. S. A. E.-Wakll. M H. Haggag. M. T .AIII~. and E. 4 Gad. JQ.SRr 40. 71 11988) 7. M. P. hlenguc and R. Wskanla. JQSRr 29. 382 I 13831 8 S. T. Thynell and hl. N &i)ik. J. Hear Tranz/Pr 107. 732 I 19851

.APPENDIS Kerrwls

These have the following

K’,, (r. .x I .lor

II. m = 0. I

forms. &,lr..vJ=

E,rIr

Kl,(r..r)=si$nrr-.r)El(lr-rlI+E,lr+,l-t[E,llr-rI)-E,(r+r~j.

-

.sll-

E,Ir

+.vj.

(Al) (AZ)

RT In a spherrcnl mhomogeneous medium

39

I ~h(f,x)=sign(f-.r)E,(lf-.rI)-E,(f+x)+-[E,(If-xI)-E,(f+~)],

(’+e~f~4fx)&(f +x1-[sip(f

K,,h,x)=

(A3)

r l-J&(lf -xl)

-x)(z)-

where E,(:) is the exponential integral function. The integrals

s R

.r’K&.

R) =

KL(f.

x) d.r

(A%

0

for n, m = 0, I are given by

Kbo(~,R) = B;(f. R) - .4',(f. R). ‘(r, R) + A’,- ‘(r, R),

Klo(f. RI = C#. RI + ASV.

R) - B;-

K&V, R) = CW. R) - A;(r.

R) + ; [B;(r.

W.R)=f

{

(A61

R) - A;@,

R)],

A;-‘(r.Rj+:A;(f,R)

[

II, r
+~A;-‘(f.R,-C:-‘(r.R,+5c;(r.Rj-~8:-I(r.R)

(A9)

where

s R

YE&

(A@

A:‘%.Rj+;A;(r.Rj+4A’Jr.R)-C;(r,Rj

+~C;(r.Rj+JB;(r.R)+3

A;@. R) =

(A7)

+ .r)d.r = i!

0

R

B’,(r. Rj =

.u’E,,(lr

-xl)d.r

= i!

1

-RI’-“‘E,+,+,(R-r)

-(-I)‘&+,,,(r)

and

s

1121 andr
(All)

R

C:(f,

R)

=

sign(f

-

x)

-.rI)d.rc

.r’E,(lf

= i!{$o&[((-ll”-I)

0

fII -In, x+ R”-“‘Em,,,, (n +nt1

(R

--r)

1

,

-(-IYE,,,+,

n 2 I and r < R.

(AIZ)

The integrals K’,‘,(R)

=

r’K;,(r.

R) dr

(A13)

for n, m = 0, I may be represented as follows: Kg(R)= K;d(R)=C;‘(R)+

B’i’(R)-

A;‘(R),

A;‘(R)-B;-‘,‘(R)+

Kh’(R)=C’?‘(R)-A;!(R)+&!-l(R)-Al;’-’(R),

(Al4) A;-‘.‘(R).

(A15) (Al6)

40

S A EL-WAKIL et al

K’,j’(R)

= A;‘(R

)+sli’(R)+![A;-“‘(R)+.4:i’-‘(R)-C;-”(R)

+ c;* -‘(R)]+;i[A:-“rR)+.4;‘-‘(R)+.4:-”-l(R) - c;-

‘,‘fR)+C;‘-l(R)-B;-“-‘(R)].

(A17)

where

A;‘(R)

I” R

=

J I:,

r’A;(r.

B;‘(R) =

r’B:(r,

R)dr=r!

R) dr = i!

-(-I)‘B:,,,+,(O.R)

(A191

R C:‘(R)

=

s rJ +I-

r’C;(r.

R)dr

I,‘Ci,+,+,cO.

= I! ~~:.,~[“,‘l,,“(~+~~~+ 1 RI 1 . !

n 2 I.

,,+C..+,tR.RI]

(A20)