Radiative transfer in an isotropically scattering two-region slab with reflecting boundaries

0022-4073/81/077ooo1~$02.W/0 Pergamon Press Ltd.

1. Quont. Sprctrosc. Radial. Transjer Vol. 26, PP. l-9. 1981 Printed in Great Britain.

RADIATIVE TRANSFER IN AN ISOTROPICALLY SCATTERING TWO-REGION SLAB WITH REFLECTING BOUNDARIES S. M. SHOUMAN and M. N. ~ZISIK Departments of Nuclear and Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27650,U.S.A. (Received 24 Nouember 1980)

Abstract-The problem of radiative heat transfer in an absorbing, emitting, isotropically scattering two-layer slab with diffusely and specularly reflecting boundaries is solved by the FN method and results are presented for the transmissivity and reflectivity of the slab.

NOTATION

a,b optical thickness defined by Eq. (2) a,, b,, a,, ho coefficients in the angular expansion, Eq. (13) A,(Y), WY) functions defined byEqs. (lj), (16) or (i7) C(v), D(w) expansion coefficients. Eas. (7) and (8) fi &face source, i = 1to 4’ 11(7,CL),I?(79II) radiation intensity for regions I and 2, respectively IlP(7, P), IZ,(T,CL) particular solutions PI, Pz ~&Io~~~I) and {voIU(O,I), respectively ; P

extinction coefficient reflectivity optical variable defined by Eqs. (2) transmissivity of the interface direction cosine single scattering albedo for regions 1 and 2 eigenfunctions for regions 1 and 2 eigenvalues for regions 1and 2 INTRODUCTION

Studies on the solution of radiative heat transfer in a composite medium, including the effects of scattering are very limited and they deal with very specific situations.‘-3 Here we apply the recently developed FN methodb7 to solve a quite general radiative heat transfer problem for a two-layer slab which absorbs, emits, and scatters radiation and has boundaries which reflect radiation diffusely and specularly and contain diffuse surface sources. The layers 1 and 2 are considered to be confined to the regions 0 5 x i II and II 5 x 5 l2 in the physical space variable x. If PI and p2 are the extinction coefficients for the regions I and 2, respectively, it is convenient to define an optical variable 7 as forOlx5I,, ’ = 1(PI - /%)r, +/32x for I, 5 x 5 /*. PIX

(I)

The domain of the regions 1 and 2 becomes, in the optical variable T, region 1 in 0 5 T 5 p,l, = a,

(2a)

region 2 in a I 7 5 [(p, - pz)ll + p&] = b,

(2b)

and the radiative heat transfer problem considered in this paper is written in the optical variable

2

S. M. SHOUMANand M. N.

~ZISIK

7 as*

CL

& I,(T,CL)+ I,(?, CL)= S,(T)+ y I_‘, I,(T,CL’)b’, (3)

p $4(7,

cl) + 12(7, CL) = S?(T) + y

I,

/

12(7, CL’) b’,

(4) where the source terms

s,(T)

and

S2(7)

are defined as

s,(T)= (1- WI)Lc.p, s2(T)

=

(1 -

02)

qd.

(5)

Here, I(T, CL)is the radiation intensity, T(T) is the temperature of the medium, I_Lis the cosine of the angle between the radiation beam and the positive T axis, o is the single scattering albedo, and the subscripts 1 and 2 refer to the regions 1 and 2, respectively. The boundary conditions that allow for specular and diffuse reflection, interface transmission and diffuse surface sources are taken as I

Ido, CL)= fl + /hs~l(o,-/.‘L)+ 2pld

I P’I,(O,-CL’)dtL’, >O, I I I p’Mb, dp’, p > 0, CL

0

(64

I

I,(&

-II)

= f2 + 72,12(4

I2(4

CL) = f3 + y,2~l(a,

@2d

--CL) + PZs~,(ar CL) +

0

P'~,(G

F') dp',

CL >O,

(6b)

P > 0,

(6~)

I

CL) + p3sl2(&

-IL)

+

@3d

0

p’12(a,

--CL’) dp’,

I

Mb, - PL)= f4+ ddb, CL) + 2P4d

0

CL’)

(64

where f represents a diffusely emitting surface source, pd and ps refer respectively to diffuse and specular reflectivity and y to the interface transmissivity. Figure 1 illustrates the locations of these various quantities.

Fig. 1. Illustration of the definitions of various quantities.

3

Radiative transfer in an isotropically scattering two-region slab

In the following section, we first develop the solution of this radiation problem and then present results for the transmissivity and reflectivity of the composite region over a wide range of parameters. ANALYSIS

The general solution of the specified radiation problem for the intensities Z,(T,CL)and MT, EL) in the regions 1 and 2 respectively, may be expressed in terms of the known eigenfunctions 8.9 and particular solutions of the equation of radiative transfer as

(7) and 12(7,/A)

=

D(qo)42(so,

+

I

p)

e-""O+D(-770)42(-770,

II) e""O

I

-I

CL)e-T’9 dq + hd7, CL),

D(~hh(q,

where ~i(+ v, CL),i = 1 or 2. v = lo, q. or E (-- 1,1) are known eigenfunctions; to and no are discrete eigenvalues; Ilp(7, CL) and 12p(~, p) are particular solutions of the equations of radiative transfer and are considered to be known; C(?v), D(+v), with v = to, nor 5, 7, are unknown expansion coefficients to be determined by application of boundary conditions. We note that the solutions (7) and (8) satisfy the equations of radiative transfer exactly. As is discussed in Ref. 4 the full-range orthogonality relations of the eigenfunctions +(v, IL) can be used to transform Eqs. (7) and (8) into a system of singular integral equations; the FN method may then be applied to reduce the system to a set of coupled algebraic equations for the coefficients associated with the determination of the exit distribution of the radiation intensity. The procedure will now be described. Equation (7) is evaluated at T = 0 and T = a and the resulting equations are then operated on first by the operator

I

I

_I CL~I(-V, II) dcL,v E PI = {WU(o,

I),

and then by the operator

I

I

-I

P$I(V,EL)dp,v E PI,

and the full-range orthogonality relations is utilized. After some manipulation, we obtain

IoI o1 CI

/41(-v,

-e

-e/v

PM-V,

CL)II(O>cL)dp -

PMI(~,cL)dp -

Io’

ddv,

CL)II(O,-cL) dk

Io1

/.+(v, P)ZI(~,

1

-

-cL)dp = y

K,(v),

Pa)

where

Kdv) =

$ [1’ &JI(- v, -I

CLUI~(O,

cL)dp -

II

ema’”_I 1-41(-v,

PVI~(~,

p)

dp]

(9b)

4

S. hf.

and M. N.

SHOUMAN

&I$IK

with v E P,; also, I

I

CL+I(V',/-+I@,

0

I*) dF

- [I I.LMV, I

e4v

FL)ZI@,cL)dp -

0

-I,’/.41(-v, P)ZI(& -cL) dcL-

Io’/41(-v,

P)ZI(~> -cL)

$1=y

G(v),

(lOa)

where 2 &(V)

=

I

’

G

I.L$I(V, [I

pUl,(O,

~1

dp

-e””

-I

I -I

P~I(V,

FUIJG

IL)

dp

1

(lob)

with v E PI. Equation (8) is now manipulated in a similar manner. That is, it is evaluated at r = a and 7 = b, the resulting equations are operated on by the operator I

f

_]

P#Q(-v’,

CL) b,

v’

E p2 =

bJo)U(O,

11,

and then by the operator

I’ -I

/.42(v’,

CL) dp,

v’ E P2,

and the full range orthogonality relation is utilized. We obtain e

I

[I

-n/v’

0

_ e-b/v'

/42(-v',

[I,' P$z(-v',

where

~)Zz(a,

cL)Zz@,

cL)dcL

cL) dp

o/v’ f

j-0'

-I,'

/.&W,

/42(v',

~U2(4

~U2(b,

I

2 K3Wyj-J

e-

F

-1

142(-v',

p)I2Ja,

I.L)~F

-embi"'

-CL)

-p)dr]

I f

_] /.42(-v',

dp]-

=y

K,(v'h

~U2p(b,

P)~CL

(lla)

1(lib)

with v’ E P2; also, e

4”’

[I 0

_ eb/v'

[I'

f

I

I

/-42(v',

/.42(v'9

~)Zz(a,

p)Z2@,

PI&J

cL) dp

-I,'

-

o

I-+-V',

/42(-v',

P)MG

~V2@,

-CL)~P

-0p]

I

=y

-

WV'),

(124

where

f-I I

_

e b/u'

P+z(v',

pUzp(b,

CL)~CL

1

(12b)

with v’ E P2, We note that Eqs. (9-12) contain the boundary surface intensities Z,(O,CL), Zr(a, -CL), 4(a, CL),and Zz(b, -p) which are obtainable, respectively, from the boundary conditions given

Radiative

transfer

in an isotropically

scattering

two-region

slab

5

by Eqs. (6a-d). When the boundary conditions (6) are introduced into Eqs. (9-12), the resulting system yields four singular integral equations for the four unknown exit distributions I,(O, -p), Ir(a, p) of region 1 and &(a, --CL),Iz(b, p) of region 2. Up to this point, the analysis has been exact. We do not try to solve such a system of integral equations but apply the basic concepts of the FN method and represent the unknown exit distributions by the polynomials in the form

(13b)

When the boundary conditions (6) and the polynomial representations (13) are introduced into Eqs. (9-12) we obtain, respectively, p,,A,(v) + ~PMAO(V) $j

+ 2P2dB

-e

6”(V)

--&

-

A,(V)]

pzsB‘b’(V)+

-

+ ~21 C”“g,Bb”(u))

= K,(v)

-.fAo(v)

-

-“” . f2Bf'(v),

+

W)

+mvL(~) -& - Bh”(v)]

I

2m40(~)

-fzAo(v), XI [

= e-a’YI&(v) -f&?(v)1

(14b)

N

a=0 +

g, pl,A,(v’)+2p,dAo(v’)~-~~)(~‘)]+e-(b-”””’b,[pl,Bh2’(v’)+

2p4J3f’( 4

--& -A,(d)]+

y12h,A,(v')}

=e”‘“‘K3(v’)-

- f3AO(v’) - e-(b-a”“’ . f4B s’(u’),

+ be

=e

~4sAu(~‘)

+ &J,dAo(d

--&

(14c)

-

1

B:‘(w’) t_Y,2huB~)(V’)

-b’“‘K4(~‘)- e-(b-a)‘“‘f~B&2)(y’) - f4AO(v’),

e-(b-a)lv’

I

=

(144

where v E PI = {&)U(O, 1) and V’E PZ = {~o}U(O, 1). The functions K,(Y), Kz(v), K3(v’), and K4(v’) are defined by Eqs. (9b), (lob), (1 lb), and (12b), respectively, and the functions A,(y) and BE)(y) are defined as

A&)=&~P”“4i(-y, CL)dp,

UW

B:‘(y)= $1,’PLl+‘4+(y, CL)dp,

(15b)

6

S.M. SHOUMANand M. N.6ZISIK

where i = 1 or 2 and y = v or VI. By utilizing the definitions of the eigenfunctions +i(?y, p), it can be shown that the functions A,(y) and BE’(y) are determined for (Y2 1 from the following recursive relations:

A,(Y) = 2 YA-I(Y) +

A,

Y =

50 or

~0,

(164

with Ao(r)= 1-ylog

(

l++

)

and B”‘(y) = YB:‘,(~)a

-

’

1’

a +

Y =

50, ~0 or E (0,11,

(174

with y=50or

B$)CY)=AO(~)+$--~,

70,

(17b)

I

y E (0, l), i = 1 or 2,

(17c)

and to and 770refer to the discrete eigenvalues for the regions 1 and 2, respectively. Equations (14) provide a system of algebraic equations for the determination of the constants a,, h,, g, and b, required in Eqs. (13), since the functions A,(y) and B:‘(r) can be computed for Eqs. (16) and (17). The system involves 4(N + 1) equations for LY= 0, 1,2,. . . , N. In selecting the (N + 1) discrete values of y for each region, we have used the scheme discussed in Ref. 7. For the lowest order F. solution, we have chosen the discrete eigenvalues y. = to for region 1 and y. = q. for region 2. For the F, solution, we have chosen y. = to, yl = 4, for region 1 and yo = 770,yI = t, for region 2. In general, for the FN solution, the first value of y is taken to be ‘y. = to for region 1 and ‘y. = q. for region 2; then, the remaining values are chosen for each region according to the following relation: yj = (2j - 1)/2N, where j = 1,2,3,. . . , N with N being the order of approximation. RESULTS

In engineering applications, the hemispherical reflectivity and the transmissivity of the slab for externally incident isotropic radiation are of practical interest. To illustrate the application of the preceding analysis, we now present some numerical results for reflectivity and transmissivity of the slab irradiated externally by isotropic radiation at the boundary surface T = 0. We assume that all other sources in the medium are negligibly small compared with the external radiation source. Then, this radiation problem becomes a special case of the radiation problem with S,(T) = S*(T)= f2 = f3 = f4 = 0 and the surface source term f, remaining finite in order to take into account the external radiation source. For a transparent boundary at T = 0, the source term f, may be interpreted to be the intensity of the externally incident isotropic radiation and the externally incident radiative heat flux, qincdat the boundary surface T = 0 becomes qincd =

2a

fl/J Ioi

dF =

rf~.

(18)

The hemispherical reflectivity of the slab is defined as Reflectivity of the slab

(19)

Radiative transfer in an isotropically scattering two-region slab Table 1. Effects of wall thickness and o on the transmissivity and reflectivity of the slab (transparent boundaries, P,~= pid= 0, i = I, 2,3,4).

Optical thickness ?

a -

0.1

0.2

0.2

0.2

0.2

0.2

0.2

r

slab

0.5

0.5

FO

0.5

F5

F6

VSlW

FO

‘C b.24625

0.24625

0.24627

0.03194

F4

of

r

slab

F6

F5

-

E. V.

I.04393 I.05696

(I.(

0.37225

).27381

‘( I.27376

0.27378

0.27378

0.05300

1.043930.04393 3.05687 0.05695

j.31333

‘( j.31336

0.31338

0.31338

0.08840

3.07511

0.07532

I.07535

(I.(

(I. 13243 (I. 16133 (I.21374

0.13245

0.13245

0.03510

3.04561

0.04561

I.04561

(1.c

0.16136

0.16135

0.06424

0.06329

0.06336

3.06337

1I.(

0.21376

0.21375

0.11785

0.09280

0.09298

5.09300

(I.(

I1.04052 II.05778 13.10279

0.04052

0.04052

0.03669

0.04620

0.04620

3.04621

II.(

0.05778

0.05778

0.07018

0.06620

0.06628

3.06629

I3.c

0.10280

0.10279

0.13849

0.10506

0.10525

0.10528

,3.1

0.13245

0.13245

0.03510

0.04561

0.04561

0.04561

,0.C

0.04884

0.04884

,0.C

0.05341

0.05341

0.C

0.36748

0.5

0.5

Reflectivity

-

Exact

(b-s

0.2

0.1

Trsnsmissivity of

w2

j.24633

0.8

0.5

0.5

0.40190

0.2

0.5

1.0

0.22207

I.13240

0.5

0.5

1.0

0.22987

I.16130

0.8

0.5

1.0

0.27446

I.21366

0.2

0.5

2.0

0.08199

1.04051

0.5

0.5

2.0

0.08808

3.05777

0.8

0.5

2.0

0.13237

3.10276

0.2

1.0

0.5

0.22287

3.13240

().(

0.8

1.0

0.5

0.2

1.0

1.0

0.13518

0.07271

I3.13243 13.14629 (0.16624 1 j 0.07274

0.07271

0.07274

0.03626

0.04607

0.04607

0.04607

0.C

0.5

1.0

1.0

0.13952

0.08763

0.08763

0.08761

0.08764

0.04699

0.05059

0.05058

0.05058

0.C

0.8

1.0

1.0

0.16679

0.11460

0.11459

0.1145!

0.11460

0.06676

0.05818

0.05816

0.05816

0.C

0.2

1.0

2.0

0.04973

0.02281

0.02281

0.02281

0.02281

0.03685

0.04625

0.04624

0.04625

0.C

0.5

1.0

2.0

0.05347

0.03196

0.03195

0.03191

0.03195

0.04918

0.05143

0.05142

0.05142

0.1

0.8

1.0

2.0

0.08048

0.05565

0.05565

0.05561

0.05565

0.07439

0.06164

0.06162

0.06162

0.c

0.2

2.0

0.5

0.08199

0.04051

0.04052

0.0405;

0.04052

0.03669

0.04620

0.04620

0.04621

0.c

0.5

2.0

0.5

0.08312

0.04446

0.04447

0.0444;

0.04447

0.03774

0.04646

0.04646

0.04646

0.0

0.8

2.0

0.5

0.08985

0.05013

0.05014

0.0501~

0.05014

0.03951

0.04683

0.04683

0.04683

0.0

0.5

1.0

0.22587

0.5

0.24407

0.14632 0.16624

0.14631

0.14631

0.04285

0.04884

0.16625

0.16626

0.05589

0.05342

0.2

2.0

1.0

0.04973

0.02281

0.02281

0.02281

0.02281

0.03685

0.04625

0.04624

0.04625

0.0

0.5

2.0

1.0

0.05135

0.02715

0.02715

0.02711

0.02715

0.0383C

0.04661

0.04661

0.04661

0.0

0.8

2.0

1.0

0.06142

0.03497

I 0.03498

0.0349t

0.03498

0.04098

0.04724

0.04724

0.04724

0.0

0.2

2.0

2.0

0.01830

0.00739

0.00731

0.00739

0.03693

0.04626

0.04626

0.04626

0.0

0.2

0.5

2.0

2.0

0.01968

0.01014

O.OlOlL

0.01014

0.03860

0.04669

0.04669

0.04669

0.0’

-

0.8 -

2.0

2.0

0.02965

0.01721

0.01721

0.01721

0.04201

0.04755

0.04755

0.04755

-

0.0, -

Table 2. Effects of interchange of 01 and wz on transmissivity and reflectivity of the slab (transparent boundaries. pls = pid= 0, i = I, 2,3,4). Optical thickness a1

w2

a

b-s

0.2

0.8

0.5

0.5

0.8

0.2

0.5

0.5

0.2

0.8

1.0

0.5

0.8

0.2

1.0

0.5

0.2

0.8

1.0

2.0

0.8

0.2

1.0

2.0

l-

T Trsnsmissivity

! FO

F4

F5

0.40190

0.31333

0.31336

0.40190

0.31333

Reflectivity Exact VSIUS

5

F6

-

F.

-.

EXS<

F5

F6

VSll

0.31338

0.31338

0.08840

0.07511

0.07532

0.07535

0.0;

0.31336

0.31337

0.31338

0.18658

0.21854

0.21844

0.21843

0.21

0.05341

0.05

0.24407

0.16624

0.16624

0.16625

0.16626

0.05589

0.05342

0.27446

0.21363

0.21373

0.21375

0.21375

0.26050

0.28569

0.28569

0.28569

0.28

0.08048

0.05565

0.05565

0.05565

0.05565

0.07439

0.06164

0.06162

0.06162

0.06

0.03498

0.03498

0.03498

0.03498

0.26315

0.28741

0.28742

1

0.06142

L

0.28742 L

0.28

-

0

0.4 0.4

0

0.4

0.4

0.6

0.8 -

0.:

0.f

a.5 0.6

0.8

O.@

0.5

0.2

0.2

0.2

O.L

0.2

LI.L 0.8

0.2

0.8

0

0

0.2 0.5

I.6

1.8

3.6

3.8

0

0

0

0.2 0.8

-

p3

0

-

112

1

3. Effects

0.2 0.5

Ol

Table

0.115Y2 0.08195 0.08193

0.11592 3.088Y2 O.Ot3894 0.08894 0.08894

0

0

I)

0

0.14625 0.10490 0.10493 0.10493 0.10493

Cl.11816 0.08983 0.08985 U.cl8986 0.08986

-1

0.14625 0.09571 (3.09568

0.11816 0.08286 G 3.08284

0.14416 Cl.09487 (3.09484

0.10092 0.07460 0.07458 0.07458 0.07458

0.10092 0.07929 0.07930 0.07930 0.07930

0

0.14416 0.10406 0.10409 0.10409 0.10409

0.09246 0.07126 0.07125 0.07125 0.07125

0.09246 0.07594 0.07596 0.07596 0.07596

0

0.9

0.5

0.09954 0.07411 0.07409 0.07409 0.07410

F5

0.09954 0.07525 0.07523 0.07523 0.07523

F4

-7

0.9

FO

0.06654 0.05847 ) 0.05847 0.05847 0.05847

Vallle -__

with diffusely reflecting boundary

0.06654 0.06011 0.06010 0.06009 0.06010

F6

T

0.08277 0.06584, 0.06583

F5

t

a = I,

0.08277 0.06661 0.06660 0.06660 0.06660

F4

Exzic

with specular reflectinK boundary

thickness

0.05732 0.05471 0.05471

I

FO

on reflectivity for the optical of 7 = 0, i.e. PI = 0.

i of slab Reflectivit)

of specular and diffuse reflection c1 = I and a transparent boundary

0.05732 0.05566 0.05566 0.05566 0.05566

0.5

I

Ic

___~

of interchange b-

Radiative transfer in an isotropically scattering two-region slab and

the hemispherical transmissivity of the slab as ;&ys;sasbivity)

. 2T

Id

12f;,p)p

=

(1 -

dp

= ;

p4s

-

P4d)

2

=

(1 -

p4s -

P4d)

’

(1 - p4S- ,&,) I,’ I,(b, j~)p dp.

(20)

In Eqs. (19) and (20), the quantities Z,(O,-II) and Z2(b,p) are the exit distributions which are given by Eqs. (13a) and (13b), respectively. Then Eqs. (19) and (20) are written, respectively, as reflectivity of the slab

2 N a, =&Y +2’

transmissivity = ; (1 of the slab >

p4s -

P4d)

f.

(21)

5.

(22)

Clearly, the hemispherical reflectivity and transmissivity of the slab are obtainable from Eqs. (21) and (22), respectively, once the coefficients a, and b, are determined from the present analysis. We present in Table 1 the effects of wall optical thickness and of the single scattering albedo on the transmissivity and the reflectivity of the slab for the case when all boundaries are transparent (i.e. pis = Pid= 0 for i = 1.2,3,). In Table 1, F,-,,F4, Fs, and Fe solutions are listed together with the exact results, which are obtained from the convergent solutions of FN as N varied from 10 to 15. It appears the Fs and F6 solutions are accurate to about three and four significant figures, respectively. Increasing the single scattering albedo, 02, for the slab 2, increases the reflectivity of the slab. The transmissivity of the slab also increases with increasing w2. Table 2 shows the effects of the interchange of the values of wl and w2on the transmissivity and reflectivity of the slab. For various combinations of the optical thicknesses considered in Table 2, the reflectivity is higher with wl higher than w2; the transmissivity is influenced not only by the relative magnitudes of wl and w2 but also by the relative optical thicknesses of the plates. Table 3 shows the effects of interchange of specular and diffuse reflection at the surfaces on the reflectivity of the slab. The reflectivity of the slab is slightly higher with the specularly reflecting boundary than with the diffusely reflecting boundary at T = b. Acknowledgements-The authors would like to express their gratitude to Dr. C. E. Siewert for many helpful and stimulating discussions. This work was supported by the Scientific Affairs Division of NATO through NATO Research Grant No. 1606.

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