J. Quant. Spectrosc. Radiat. Transfer Vol. 29. No. 5, pp. 381 394, 1983 0022 4073/83/050381 14503.00/0 Printed in Great Britain. © 1983 Pergamon Press Ltd.

C O M P A R I S O N O F R A D I A T I V E T R A N S F E R

A P P R O X I M A T I O N S F O R A H I G H L Y F O R W A R D

S C A T T E R I N G P L A N A R M E D I U M

M. P. MENGU(~ and R. VISKANTA Heat Transfer Laboratory, School of Mechanical Engineering, Purdue University, West Lafayette,

IN 47907, U.S.A.

(Received 12 July 1982)

Abstract--We examine critically the accuracy of the two-flux, spherical harmonics and discrete ordinates methods for predicting radiative transfer in a planar, highly-forward scattering and absorbing medium. Numerical results for the radiative fluxes show that the two-flux and P3-approximations yield accurate results compared to solutions based on the Fu-method. Indeed, these approximate methods are relatively simple and have potential for generalization to predict radiative transfer in multidimensional systems, as long as an appropriate simplification of the phase function is utilized.

INTRODUCTION

Radiative transfer in coal-fired combustion and conversion systems, in lightweight fibrous insulations, in the atmospheres and hydrospheres, and in other systems containing small, highly scattering particles has been a subject of fundamental importance due to its wide applications. Some of the systems of physical interest can be accurately treated as one- dimensional planar, and others are definitely multidimensional. Multidimensional radiative transfer in media containing particulate matter which absorb, emit, and scatter radiation in a highly forward manner is much more complex and has received relatively little research attention. This paper is motivated by the need to develop realistic models for predicting radiation transfer in coal-fired combustion products containing pulverized coal, char, fly-ash and slag particles.

Some approximate, numerical and rigorous solutions of the RTE (radiative transfer equation) for planar, radiation participating media with nonisotropic scattering have been reported.i J3 The literature relevant to geophysical problems is particularly large, and no attempt will be made here to review it. For example, approximate analytical, ~ , iterative, 5 two-flUX, 6'7 spherical harmonics, 5.s discrete ordinates, 9 II numerical,2.3.9.12 and rigorous 13 approaches have been used. Exact solutions of the radiative transfer equation for non- isotropic scattering are difficult and time consuming, but the Fu-method ~3,~4 is faster and easier than all the other exact methods for planar problems with arbitrary boundary conditions. There are other rigorous solution methods of the RTE in the literature. 1~19 However, most of these methods, as well as the Fu-method cannot be readily extended to multidimensional geometries. Those that can be extended are usually not practical for highly nonisotropically scattering media. On the other hand, other techniques such as the Hottel zonal 2° and Monte Carlo 2~ methods are too time consuming and usually not suitable for many problems. Therefore, some reliable approximate methods of solving of the RTE together with models of the scattering phase function are needed for use in engineering design calculations. An up-to-date review of radiative transfer has been prepared 22 and there is no need to repeat the survey.

The purpose of this paper is to compare critically the radiative transfer results for a planar, participating and highly-forward scattering medium based on several different methods of solution and to establish their accuracy and validity. The scope of the paper is limited to the radiative transfer approximations, which appear to have the potential for extension to multidimensional geometries and would be compatible with finite-difference numerical methods for solving fluid dynamics equations. Specifically, the results based on the two-flux, spherical harmonics and discrete ordinates methods are compared with those obtained using the low- and high-order FN-method of solution. The Fu-method has been

381 QSRT Vol. 29~ No 5 A

382 M. P. M E N G U ( , ' and R. VISKANTA

,~ ' ! X o

I : z , ~ '

• ' r ~2;

t ' i l J~

Fig I. Physical model and coordinate sys tem

developed for the planar geometry and can not be applied to multidimensional problems, however, since it yields essentially exact results with less computational effort than other rigorous methods, it was chosen as the benchmark for comparison.

ANALYSIS

Physical model and basic equations The problem considered in the paper is that of radiative transfer in a plane-parallel

medium consisting of highly forward scattering particles which are suspended in a gas. Since the emphasis here is primarily on scattering, the gas is assumed to be nonparticipating and unbounded by opaque, reflecting walls. These assumptions were made in order to eliminate the radiation characteristics of the bounding surfaces and emission from the walls as parameters of the problem and focus the attention on scattering. The medium is irradiated by an isotropic radiation field of intensity It~ (see Fig. 1 for coordinates). For a plane layer of participating (absorbing, emitting, and scattering) medium having azimuthal symmetry, the radiative transfer equation becomes

(?"V uJ (" - - - - + q ~ 0 1 , / ~ ) = ( 1 - e ~ ) f l 0 / ) + 7 j p(l~o)W(~l,l~)d#', (11

where W is normalized (with respect to 10) intensity, q is the normalized distance, It is the direction cosine, z0 is the optical depth of the layer, ~o is the albedo for single scattering, and fl is dimensionless emission source (B/lo). The boundary conditions are assumed to be

ud(q./~)=l a t r / = 0 (/~>0) 2a)

and

q'(q, - / ~ ) = 0 at ~/ = I (I~ >0). (2b~

Depending on the complex index of refraction of the particles and the size parameter, the scattering phase function P(#0) may be quite complicated, and its determination from the Mie theory may require the calculations to be carried out for a large number of scattering angles. However, the phase function may be represented by a series expansion of Legendre polynomials, and for unpolarized radiation can be written as 2~

P(Po) = I + £ amPm(m), I3) m = l

where a,, are the expansion coefficients and Pm are the Legendre polynomials of the first kind. The cosine of the scattering angle #0, which is the angle between the incoming ray and the scattered ray, and is given in terms of/~ and/x ' such that

/xo =/~/x' + (I -/12)1'2(1 -- IZ ,2)1,2 COS (0 -- 0 '). (41

C o m p a r i s o n of radia t ive t ransfer a p p r o x i m a t i o n s 383

For scattering which is independent of the azimuthal angle, #0 reduces to ~/~'. It is sometimes convenient to approximate the phase function only by one coefficient

such that

P(#0) = ~ (2n + 1)(g,)"/zP,(m). (5) n=0

The coefficient ga is the asymmetry factor and is calculated by the least square approach. This phase function approximation is also known in the literature as the Henyey-Greenstein approximation. 18

Since it is usually difficult and time consuming to work with the complete scattering phase function, one can define 24 the backward (b) and forward Or) scattering fractions and express them in terms of the expansion coefficients a,, defined in Eq. (3), viz.,

j~'O I ~ m I . . . . . ~ ( - 1 ) a2,,+l(2m ). / = (1/2) p(#0) d/~' = 1/2 + t,/z) 2~_ 0 ~5;~-g ~m .--m~m ~ ~.v

and

(6a)

b = 1 - f (6b)

The factors b and f a r e especially useful when obtaining solutions of the radiative transfer equation using flux methods.

For linearly anisotropic scattering, the phase function reduces to

P(/~0) = 1 + 5##'. (7)

There are two different ways of determining the linearly anisotropic scattering coefficient 5. One can either take it as the first expansion coefficient of the phase function, Eq. (3), or evaluate it after substituting Eq. (7) into Eq. (6a), which yields

m y V ( ~ l)ma2"+l(2m)! 5 = 2 zT_0 22,,+lm!(m + 1)!" (8)

In some cases, the coefficients obtained can become greater than unity and result in negative phase functions at certain angles.

Rigorous solution of Eq. (1) for planar media have been obtained, but the solutions are quite complex and are not suitable for engineering design calculations. In the following subsections, the approximate methods which have a potential extension to multi- dimensional problems, such as the two-flux, the spherical harmonics and the discrete ordinate methods, and the exact Fu-method are described for the solution of radiative transfer equation in a highly-forward scattering medium.

Two-flux methods In spite of their approximate nature, the two-flux methods yield reasonably accurate

radiative flux predictions at least for isotropically and linear-anisotropically scattering media. Their detailed development can be found elsewhere. 15,~7,25

For a one-dimensional planar medium, the two-flux approximations of the radiative transfer equation can be written as

d4, +

d~ _ _ _ A C b + ( q ) + B ~ - ( q ) + C , (9a)

d~- - A ~ - ( q ) + B ~ + ( q ) + C , (9b)

dr/

384 M. P. MENGUq and R. VISKANTA

Table 1. Coefficients for different flux methods.

Two-Flux Methods A B C

Isotropic Scattering

Schuster-Hamaker [25]

Schuster-Schwarzchild [25]

Linearly Anisotropic Scattering

Schuster Schwarzchild [6]

Modified Two-Flux [7]

1 ( 1 -~,):/

(2-'-,~) ,, 2(I-,)T~

2 ( l - ( l - b ) , ) 2,b 2 (I -:.,)~( ~

where the forward ((1)+) and the backward ( ~ ) radiative fluxes are defined as

f ° dO+(q) = 2 ud(q, p)p dp and • (r/) = 2 ud(q, p)p dp. (10) - I

The coefficients A, B and C corresponding to the different flux methods are listed in Table 1. In this table, b represents the backward scattering fraction factor and/3 = B[T(rl)]/I o.

The boundary conditions for Eqs. (9a) and (9b) can be obtained from Eqs. (2a) and (2b), i.e.,

(1)+O1)] ,=o = 1. (l) - (r/ ) l , = t = 0 . (11)

The local radiative flux can be written as

Q(r/) = @+(r/) - (I) (r/). (12)

Spherical harmonics method The method of spherical harmonics can be used to obtain the solution for the radiative

transfer equation. It yields quite accurate results at the expense of some additional work. The detailed discussion of this method can be found in neutron transport 26 and radiative transfer 17 literature. Some applications have also been recently reported. 8'27 3o

The main idea of the spherical harmonics method is to expand the radiation intensity in terms of Legendre polynomials such that

.5. / 2n + 1 \ ~(r/, P) = ~ ° L ~ ) P~(P)q~Ol). 03)

In practice, a finite number of functions q~, are considered by neglecting W~,+L, and a simultaneous solution of N + 1 linear differential equations yields the desired functions. This is called the Pu-approximation. The solution of the system of. equations can be written as a linear sum of the solution to the homogeneous part and the particular solution. For the homogeneous part, we seek a solution in the form of

~ u ( q ) = g , ek,, n = 0, 1,2 . . . . . N. (14)

The coefficients g, are evaluated using the auxiliary function given by Davison 26, i.e.,

g , ( k ) - ( - 1)" P, - ~ - Q0 x P , - Q , ~ , (15)

w ere ege. re olynomia,s secon

Comparison of radiative transfer approximations 385

kind with order n, respectively. The coefficients g. are calculated for each permissible value of k,Y '26 Once k~ and g.(k,) are determined, one can express the radiation intensity as

~(r/,/~) = 2n + 1 P,,(~) Ag,,(k,) e ~'" + ~.e(tl) , n = (I 4 ~ i

(16)

where the unknown coefficients A, are determined by constraining the complete solution to meet the boundary conditions of the problem. For a constant temperature medium, the particular solution ud.P(r/) is equal to 4zrfl(r/).

There are two different types of boundary conditions, namely, Mark's and Marshak'sY '26 Although it is believed that for higher order approximations, the Mark boundary conditions are more appropriate, in most of the problems the Marshak boundary conditions yield accurate results for all Px-approximat ionsY .26

For any arbitrary boundary conditions at the faces of the layer, the Marshak condition can be written as

f' W(0, p)/~ -'/ ' d p = H,(,u)p2i-~d/,, /~>0, (17a) I I

and

j*O f 0 tP(l,#)p?/ ~ d # = H2(#)g2J-~dl~, # > 0 , (17b) I I

where j = 1, _~, 3 . . . . , ~(N + 1). Here, H~ and H,. represents the boundary conditions at r /= 0 and I, respectively. For the boundary conditions specified by Eqs. (2a) and (2b) and a constant temperature medium, Eqs. (17a) and (17b) become

,'~ ~ 2 n + l x 1 }" A~ L~ - - - g . ( k , ) R , , j + f l ~ (2n + 1)R,, i=-- (lSa)

,=% ,,=0 4rE " .=0 ' 2j

and

' L L A,e k' g . ( k ) ( - l ) " R , , j + f l (2n + 1)(-I)"R.,j t =0 n =0 n=O

=0, (18b)

where the coefficients R,,.j for ,j = 1,2 . . . . . (N + 1)/2 are given by

' ~ ~i2r(2j) R . . ,= , P,,(It)IL 2/ ~ d l ~ = g r ( j ~ + l / 2 + n / 2 ) F ( j + l + n / 2 ) (19)

with E being the gamma function) ~ For any arbitrary boundary conditions at the faces of the layer, the Mark boundary

conditions can be written as

and

• (0, ~/) = H~(&) (20a)

ud(1, - p/) = H2( - #/), (20b)

where & are the positive roots of Px+ ~(#j) = 0. For the specific boundary conditions given by Eqs. (2a) and (2b) and a constant temperature medium, the Mark boundary conditions become, respectively,

~ A , 'v 2n + 1 N i_ Z 4~ g.(k,)P.(#j) + fl 2 (2n + l lP.(#j)= 1 (21a)

i = 0 =0 n = 0

386

and

M. P. MEN(i{}(" and R V1SKANI'~

~ 2n + I A~e'a ~, -- ~,,(k,)(--ll"P,,(lzO+fl ~ (2n + 1 ) ( - I)~P,,(Iz,)=O, (21bt

where j = 1.2 . . . . . (N + 1),,2. The coefficients .4, are evaluated either from Eqs. (18) or Eqs. (21) by matrix inversion technique. Once the coefficients have been determined, the radiative heat flux can easily be written as

and

(I) ( q ) = R,~ ~ A,g,,(k,)e~,"+4~l ~ n 0 1 i

( ..~,i )

q~ (fl = + 2n + 1

a.., 2rr t,, ~ p

(1)"R .... tl ~ A.~,,,(k,)e~.'.'+4xfl]. (22bl

Then the net local radiative flux can be evaluated from Eq. (12).

Di.screle ordinates method One of the methods o f solving the radiative transfer equat ion is to divide the radiative

field at any optical depth r into 2N streams in various I*, directions, where i = -'c I, -+ 2 . . . . . + N. By doing so we eliminate the difficulty of representing the radiation intensity at a surface by a cont inuous function and replace the radiative transfer equat ion by a system of 2N differential equations. The details o f the application o f the method arc available2 m>'~: Equat ion (1) can be written in terms of the discrete ordinates.

C,7 + + _. , v 'e(,t. .,,Y,, {23~

where 1 ~< i 4 K, and wa are the weights of Loba l to integration. The reason for using Loba t to integration here is to have quadra ture points at I* = 1 and I. Without going into the details of algebraic manipulat ion, we can write the solution of Eq. (23) as. ~

k

~ ( t l , #,t = ~ ('~ I",, exp (;'~fl) + .~., 24)

where the ~'a and Y,.a are the eigenvalues and eigenvectors of the homogeneous solution, respectively. The g, are the constants for an isothermal medium and are to be evaluated from the particular solution. The coefficients Ca are the integration constants, and are to be determined from the boundary conditions. Once Ca. )'.~. ;'~. and ,< are evaluated, the local radiative flux can be obtained from

~j l , i

(...!

To avoid ill-conditioned behavior in the numerical solution, the number o f ordinates K should not be greater than 25. ~ However, the accuracy of method decreases if the number o f terms used for the phase function exceeds twice the number of ordinates chosen. ~4

F~-rnethod The iW.-method (F for facile) was first developed by Siewert t4 for isotropic and

nonisotropic scattering media and has been used to solve various radiative transfer problems. ~~'3~ This method utilizes the properties o f the exact solution and leads to final equat ions those are particularly concise and easy to apply. By following Case ~6 and a

C o m p a r i s o n o f radiative transfer a p p r o x i m a t i o n s 3 8 7

relatively recent paper by McCormick and Kuscer, 37 we can write the solution of Eq. (l) a s

k - I

q?(rl, #) = ~ [A(v:)4>(v,, Ix) e -"1'' + A ( - v:)4J(-v~, Ix) e ~/''] ~1=0

f, + A(v)@(v,!a)e-"/~dv - 1

(26)

where A's are the expansion coefficients ~b(v,,/1) and q~(v,/~) are the eigenfunctions. The v's are the continuum eigenvalues, and v='s are the discrete eigenvalues. Since the details of analysis is available in the literature, 13'1a'16'37 there is no need to repeat it here. In the FN-method, the exit distributions are defined as ~4

N

• ( 0 , - # ) ~ ak# k, # > 0 , (27a) k = 0

N

W(I,#)~ ~ bkt xk, /x>0, (27b) k = 0

T a b l e 2. The coefficients of the phase functions (a) phase function I (ti = 2 .20 - i 1 . 1 2 , x = 1.0), (b ) phase function I I (ri = 1 . 8 5 - i 0 .22 , x = 4 .0) , (c) phase function I I I (~ = 1 . 5 0 - 0 . 1 0 , x = 8.0) .

Phase Function I Phase Function I I Phase Function I l l

~za I 0.643833 2.319461 Z.6@~844

T,Eq. (8) 0.618017 1.701301 1.843041

ga 0,06 0.57 0.75

b 0.345 0.075 0.D39

f 0.655 0.925 0.961

a o 1.0 1.0 1.0 a I 0.643833 2.319461 2.60Z844

a 2 0.554231 3.194909 3.904987

a 3 0.103545 3.676597 4.961962

a 4 0.010498 3.689530 5.827835

a 5 0.000563 3.501532 6.449251

a 6 0.000019 2.773759 6.857353

a 7 0.000000 1.971312 6.986990

a 8 1.057293 6,876321

a 9 0.329689 6.564448

alO 0.I02908 6.083206

all 0.017849 5.466910

al2 0.002439 4.874364

al3 0.000268 4.202631

al4 0.000025 3.417068

al5 0.000002 2.441223

al6 O.0OODO0 1.375775

al7 0.580278

al8 0.240728

a19 0.058807

a20 0.013133

a21 0.002492

a22 0.000410

a23 0.000060

a24 0.000008 a25 0.00000~ 426 0.000000

388 M . P . MENGi)t" and R, VlsKayta

where a k and hk are exit distribution coefficients. It is clear that with increasing ;\, the F.~.-method solution converges to the exact solution. The ultin'late value of N ma? be determined by comparing the solutions t\~r the two successive integers \ ' . For engineering purposes, the radiative flux is the quantity of primary interest, which in dimensionless form is given bx

QOI) ::: q'(T ~'0 i ,'r/,~ : 2 , vp(~l,/~ )ll d ,

By using this definition of Q(~I), the boundary conditions and exit distributtons, Q~I)~ au,i Q( I ) can be evaluated as. ~'

Q ( 0 ) = I 2 ~,,/, , ~. ~. ~-' ¢

RES[!I. IS AND I)IS(I S<,I('~N

Scatlerin~ di.x'trihution /iozcti(m.s There are a large number of independent parameters x~hleh ini]ucnce radiatixc transi',.'r

through a participating medium. However. since the emphasis ,~1" the pape~ i, on scattering. only the parameters such as the scattering distribution f'unction, ,,c.,tttet-mg albedo. ;tnd optical thickness of the layer are varied. The numerical results reported in the paper arc for the special case of a "'cold" (non-emitting) medium As an illustration, three differen~ phase functions, ranging from nearly isotropic to highly forward scattering are considered The functions used in the calculations were predicted from ihe Mie theory b',' specil-vio.~: the complex index of refraction of the unaterial and the size parameter ot lhe particlc, The computat ions were carried out on the gray basis.

In order to express the phase functions in terms of Legendre polynomials ~he expanslo~ coefficients were determined by the least square app roach ) * These coeflicients are givcr~ in Table 2. The backward and forward scattering fractions are calculated from Eqs !¢,~ and the linearly anisotropic scattering coefficient is taken as the tirst term of the phast function, which are also given in Table 2.

In this work, the full phase functions were used in the f ' , and discrete ordimilc,, methods, and the first 9 terms of the expansion were employed in the P,-method "Ihc approximate phase function was used in the discrete ordinates method, and suilabte linearly anisotropic scattering phase functions in the spherical harmonics and the two-flu:,. methods.

Comparison o[" radiative fluxes In radiative heat transfer calculations it is the flux and the flux divergence which are the

quantities of primary interest and not the intensity (radiance). For this reason on!v the fluxes and flux divergence at the boundaries are presented in the pape r The reflectance and the transmittance of the slab can be readily evaluated from the values ,:q" Q!O~ and Q(I) such that " ref lec tance"= 1 - Q(0) and " ' transmittance" - Q t l ~

The effect of the scattering distribution function on the radiative fluxes at the ' ~ boundaries can be obtained from the examination of the resulls given in Table 3 and Fig. 2. If the F~ results are taken as the benchmark it is noted that the error in the two flux methods are slightly higher than those of the spherical harmonics me thod Note that in obtaining the p~,-approximation the Marshak boundary condition was used.

In Table 3 the results for Pg-approximation are given for linearly anisotropic scattering as well as for the 9 term phase function. The error in the P~-apwoximation for phase function I. which consists of only seven terms, is only 0.1"o; however, for the phase functions described by more than 9 terms, the error increases. In this table the results based on the discrete ordinates method tbr two different phase functions are also compared. Here, the approximate phase function refers to the one obtained by using the asymmetry factor g,, Eq. (5) (Henyey-Greenstein phase function approximation). This approximation may become particularly useful when trying to fit experimetal phase function data since

Comparison of radiative transfer approximations 389

Table 3. Effect of phase function on the dimensionless radiative fluxes at the boundaries, % = 1.0, e) = 0.8.

Methods

Two-Flux Schuster- Schwarzchild Mod.Two-Flux

Spherical Harmonics Pl

P3 P5 P9

TM

Discrete Ordinates

Approx. P,F. Full Mie P.F.

FN-Method

F O

F 1

F 2

F 3

F 5

F 9

Phase Functions

I I I I I I

Q(O) Q(1) QIO) Q(1) Q(O) Q(1)

.74265 .42256

.75740 .47124

.78743 .46513

76590 .45636

.76317 .45631

.76192 .45604

.92518 .59765

.93134 .64003

,94485

,91225

,90851

.90701

.95925 .63069

.96279 .67078

76143

.76759 76097

.45637

.46204 ,45610

.78757 .49377

76026 .45456

.76057 .45591

76061 .a5596

.76058 .45588

.76957 ,45588

.61671 .98025 .65110

.59739 .94477 .62899

.59644 .94077 ..62779

.59596 .93920 .62726

.9081 0 .60296

.90092 .59543

.90752 .60274

.94297 .63910

.94069 .63688

.94217 .63904

.96662 .70723

.95219 .65387

94593 .64336

.94330 .63857

.94177 .53816

,94178 .63874

.93679 .86205

.91707 .61568

.91028 .60408

.90729 .60152

,9Q709 .60257

.90706 .60251

Q (0) Qfl)

Fig. 2. Comparison of radiative

o

02--Lr' ............. 01,,

O[ I0 I00

To fluxes at the boundaries for phase functions I and Ill, eJ = 0 . 8 .

there in only a single correlating parameter. It is clear from Table 3 that this approximate phase function yields accurate results, and the relative error remains less than 1.0%, while the error for the discrete ordinates method, which utilizes the full phase function, is always less than 0.1%. A graphical representation of the results is given in Fig. 2. For the sake of clarity only the results for phase functions I and II] are included, The results for the Schuster-Schwarzchild and discrete ordinate methods are not drawn in the figure since they fall between the results displayed and cannot be clearly shown.

The effects of phase function and optical thickness on the dimensionless heat flux Q (0) are shown in Figs. 3 and 4, respectively. For phase function I, the general trend for all approximate methods are the same. It is clear from Fig. 3 that the two-flux approximations

390 M. P. ME>;(~(iC and R. VISKANTA

q (0) .,\\

0 9 2 , "\

';; 9L,

' : " > ~ . . . . E_

Fig. 3. C o m p a r i s o n of radia t ive fluxes Q(0) as a funct ion of opt ica l th ickness q, for phase i 'unctkm I and ~.,~ = 0 5

IO01 . . . . r----~--~ . . . . . . , ' - . . . . . . . r . . . .

I . . . . . . . . . t"" "-, ,",~\ "~.. ~ - " . I - - -

Q ( O ) il " ' ~ , . , ~ _ . - : ~ M2F t

. . . . . . . . . l ,c- 9 " ' . . . . . . . . . . . . . . . i

ogri J I i l i t _ _ i k . _ J _ _ l ~ i I I ~ - - l t I I l i l [ . . . .

O I I0 I0.0

To Fig. 4. C o m p a r i s o n of radia t ive fluxes Q(0) as a funct ion of opt ica l th ickness :,, l o t phase function

[II and ,,~ = ).5.

underpredict Q(0) lor z. >~ 0.5 and give more accurate results than the &-approximatmn. For highly scattering medium, the spherical harmonics method with linearly anisotropic scattering approximation does not yield reliable results, especially with increasing optical thickness; however, as seen from Fig. 4, the general trends for the two-flux approximations are as expected. Indeed, similarly poor behavior of &-approximation (Milne-Eddington) has also been observed for semi-infinite media. ~9 For the problem considered here it can be readily concluded that when a 1 >-(1/~O), the &-approximation yields negative reflectances for the slab. This occurs for phase function III and ~,J ~> 0.4. As expected, with increasing optical thickness, this poor prediction becomes more pronounced. Similar limiting situations for the higher P~.-approximations are also expected. Therefore, it is clear that the first term of the phase function, which is chosen as linearly anisotropic scattering coefficient, does not always represent the general characteristics of the phase function. Although, the linearly anisotropic scattering coefficient can be evaluated also from Eq. (8). this usually gives poorer results.

The effect of the optical thickness r,, on the dimensionless radiative fluxes at the boundaries are presented in Table 4. Examination of the results given in this table shows that the &-approximation yields less accurate results than two-flux approximation particularly for large optical thicknesses.

A comparison of the dimensionless radiative fluxes at the boundaries, Q(0) and Q(I). obtained by the modified two flux approximation, discrete ordinates method and P3-approximation with the F9 results are given in Figs. 5 and 6, respectively. Note that these results are for phase function III, which represents a highly forward scattering

C o m p a r i s o n o f r a d i a t i v e t r a n s f e r a p p r o x i m a t i o n s 391

Table 4. Effect of o p t i c a l t h i c k n e s s o n t h e d i m e n s i o n l e s s r a d i a t i v e flux at t h e b o u n d a r i e s f o r p h a s e

Methods

Two-Flux Schuster- Schwarzchild Mod.Two-Flux

Spherical Harmonics

P1

P3

P5 P9,9 term

P.F,

Discrete Ordinates Approx.P.F. Full P.F.

FN-Method

F 1

F 9

f u n c t i o n III, o9 --0.8.

~o

0. I 2.0 I0 .0

Q(O) Q(1) Q(O) Q(1) Q(O) Q(1)

.99404 .95483

.99481 .96076

.99718 .95797

98761 .94865

.98440 .94557

.98885 .95020

.98713 .94860 ,98797 .94946

.99293 .95838

.98692 .94856

.94302 .39844

.94603 .45057

.97187 .42409

.93699 ,41826

.93345 .41779

.92660 .43072

.92482 .42836

.92574 ,43048

.93336 .43903

.92523 .43029

.93222 .01019

.93224 .01883

.96570 .01377

.93336 .01747

.92971 .01740

.91464 ,02456

.91356 .02370

.91380 .02455

.92011 .02216

.91332 .02454

[.(~ i 1 ,11 i i , i J i l l I i , i F i i i i I

0.98 ~ ~ " , X ,

096

0 . 9 4 .8

~ - ~ Two-Flux Modified

. . . . P3

0.92 - - - Oi~re. Ord~,= - - %

. . . . . . Schuster - Schworzchild

0.90 ' " .I 1.0 10.0 To

Fig. 5. C o m p a r i s o n o f r a d i a t i v e fluxes Q(0) as a function o f optical thickness r 0 for p h a s e f u n c t i o n III and d i f f e r e n t s i n g l e s c a t t e r i n g a l b e d o e s .

medium. Based on the results on the P]-approximation, since the linearly anisotropic scattering coefficient 6 for this phase function is greater than 1/o9 for both (o = 0.5 and 0.8, the P3-approximation yields poor predictions of Q(0) for both of these values of the single scattering albedo which is not unexpected. From Fig. 5 it is seen that all of the approximate methods overpredict Q(0), but the maximum overprediction is less than 3%. The general trends of Q(I) predicted by the approximate methods are consistent, except

392

IO,

08 l

0 . G -

Q(I) 04-

l

0.2/ F

o t . . . . , © 1

M. P. MENGi]I(,' and R. VISKANTA

I l l I i i r * , ' ' ' 1 t J , i , r w r f - - - - ]

co = 0 5 - / X \ _ _ \ \ \

. . . . . . <"k M2F '\ i

1 _ 1 _ ~ I I l t J

I0 I00 To

Fig. 6. ( ' ompar i son of radia t ive f luxes Q( I ) as a function of optical thickness r, for phase l'unct]~m lit and different s ingle scat ter ing a l b e d o e s

T a b l e 5. Effect o f opt ica l t h i c k n e s s o n the d i v e r g e n c e o f rad ia t ive heal flux at the boundaries for phase func t ion I l l and ~,~ = 0.8

o

. . . . . . . 1 . . . . . . . . T _

0. I '~ 2.0 It! :

~=0 '=~ i ::=0 ,=1 .:n ,-~ :

Schuster Schwarzchi ld -0.40238 -0.3819 -0.42279 -0.15937 -C).,12711 -0.00408 !

M o d i f i e d Two Flux -0.3482l -0.33282 I-0.36511 -0.15608 i-0.36988 -0.00652 i

. . . . . . . . . . . . ÷ . . . . . . . . . . . . . . . . . . . . . . i

Spherical Harmonics: i

P]

P5

P9

P9, 9 term Phase Fun.

Discrete Ordinates

H e n y e y - G r e e n s t e i n Approx . Phase Fun.

Full Mie Phase Fun,

-0.40113 -0.38319 l -0.41i25 -0.16964 j-o 437g -,~.~o551 i

-0.40907 -0.37068 I-0.44357 -0.1422] -0 .44495 --) 0059

t i -0.41356 -0.36392 I-0.45057 -0.14002 i-0.45192 0.00580

J i i -0.41857 -0.35679 1-0,45521 -0.13863 i-0.45654 ~J.!10574

i

i -0.40890 -0.36498 I-0.44270 -0.I3456 i -O.ac%6e - ?] /~ .

- - 4 ~ 1

-0.41817 -0.35491 -r~.45251 - ~.13%37 ]-0.45~-24 -) 006~9

-0.4]71( -~ .35579 i-0.45043 -i).~3359, -0.454~6 ,, r~070_;

at large optical thicknesses (Fig. 6). In heat transfer calculations the divergence of the radiative flux is also a quantity ol

interest. Because of this, a comparison of the flux divergences at the boundaries is useful In Table 5, this comparison is given for phase function III with a single scattering albedo of 0.8. The reason for choosing this phase function and large ~o is to examine the effect of scattering more critically. The discrete ordinates results for the full phase function are used as the benchmark of comparison.

As seen from Table 5, with increasing optical thickness of the medium, the absolute value of (dQ/dq) at r /= 1 is decreasing, as expected. Between the simplest three approximate methods, namely, the Schuster-Schwarzchild, the modified-two-flux and the

Comparison of radiative transfer approximations 393

P~-approximations, the Schuster-Schwarzchild model yields relatively good agreement with the benchmark solution, in calculating the divergence of the radiative flux, especially at q = 0. The P3-approximation is superior to all these three methods for all optical thicknesses and at both boundaries. With increasing order of approximation the spherical harmonics method yields more accurate results. Instead of using one term scattering parameter, if the first 9 coefficients of the phase functions are employed for Pg-approximation the predictions of the radiative flux divergence at r /= 1 are improved significantly especially for large optical thicknesses. However, the same improvement does not result for the predictions at r /= 0. Here, it should be noted that phase function III consists of 25 terms. On the other hand, for discrete ordinates method, the Henyey-Greenstein phase function approximation is quite satisfactory and yields accurate results also for the prediction of the radiative flux divergence.

CONCLUSIONS

A review of different methods for solving the RTE in a planar, highly forward scattering medium has been presented, and accuracy of the methods has been examined by comparing the results obtained with those based on the Fg-method which is considered to be exact.

The two-flux approximations yielded surprisingly accurate results for all optical thick- nesses and scattering functions considered. This is in agreement with the most recent experimental findings. 4° The P~-approximation (Milne-Eddington) yielded particularly poor results for Q(1) in comparison to the other methods. On the other hand, the results based on the P3-approximation agreed much better with those of the Fg-method. However, the higher spherical harmonics approximation with full phase function (P9 with seven term phase function) does not agree very well with the "exact" solutions. This is believed to be due to the Marshak boundary condition for which the higher order approximations are not accurate. For a highly scattering medium, the spherical harmonics method with linearly anisotropic scattering assumption did not yield accurate predictions of radiative transfer at large optical thicknesses. The discrete ordinates method with approximate phase functions did not give as accurate solutions as did the full phase functions. Nevertheless, for the highly forward scattering media these type of phase function approximations are very useful.

In a way of summary, it can be concluded that the two flux approximations are very promising, simple, methods which have a potential of being extended to multidimensional radiative transfer problems. With increasing optical thickness and increasing peakness in the phase function, the accuracy of both flux methods decreases; however, for these cases, the modified two-flux approximation yields much better agreement with the exact results than does the Schuster-Schwarzchild approximation. Although the P~-approximation does not yield as accurate results as the two-flux methods, the P3- and even Ps-approximations are also very promising for phase functions with moderate peaks. For phase functions with extremely high peaks, a delta-Eddington approximation may be more useful. 41 This can be done by separating the phase function into an isotropically scattering and a delta function components. Since there will be one more coefficient to express the phase function, the approximation may yield better radiative transfer predictions using the spherical harmonics method. The discrete ordinates method also has potential of being applied for multi- dimensional problems, but at the expense of complexity and computational effort.

Acknowledgements--This work was supported in part by CONOCO, Inc. through a grant to the Coal Research Center of Purdue University. This account is an abbreviated version of an ASME Paper No. 82-HT-17 which was presented at the AIAA/ASME 3rd Joint Thermophysics, Fluids, Plasma and Heat Transfer Conference, 7-11 June 1982, St. Louis, Missouri.

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