a step-function approximation for the experimental

,I. Quant. Spectrosc. Radiat. Transfer Vol. 43, No. 3, pp. 253-265, 1990 Printed in Great Britain

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

A STEP-FUNCTION APPROXIMATION F O R THE EXPERIMENTAL D E T E R M I N A T I O N OF THE EFFECTIVE SCATTERING PHASE F U N C T I O N OF PARTICLES M. P. MENGOC and S. SUBRAMANIAM Department of Mechanical Engineering, University of Kentucky, Lexington, KY 40506-0046, U.S.A. (Received 17 April 1989; receivedfor publication 16 October 1989) Abstract--A step-function approximation is introduced to determine the coefficients of the

Legendre polynomial expansion of the highly-forward-scattering phase-functions from experiments. It is shown that by using this approximation with a limited number of angular intensity data, as many as l0 coefficients can be recovered within a 10% error bound. The errors in the first two coefficients determined by employing a delta-phase-function approximation are at least an order of magnitude greater than that for the step-function approximation. The effect of the recovered phase-function on the accuracy of radiative transfer calculations is also discussed. The results show that the phase-function coefficients may be determined from experiments using the new step-function. For radiative transfer calculations, a delta-function approximation with coefficients based on those derived from the step-function approximation should be employed.

1. I N T R O D U C T I O N For practical systems operating at elevated temperatures such as utility boilers and furnaces, radiation transfer is the major heat-transfer mechanism. The thermal efficiency of these systems can be improved only if details of radiation heat transfer and its effect on the combustion of fossil fuels are fully understood. Thus, accurate and reliable radiative transfer models are required. The accuracies of these models are, however, limited by the accuracy of the available radiative properties of combustion products. Although radiative properties of combustion gases are well known, particle properties (especially of coal, char, fly-ash, and soot) are not widely available. For spherical or cylindrical particles, the Lorenz-Mie theory may be used to determine radiative properties such as absorption, extinction and scattering coefficients and the scattering phasefunction if the characteristic dimensions and complex index of refraction of the particles are available. ~ For irregularly-shaped particles, the extended boundary condition method or the discrete-dipole method may be employed to evaluate the properties ~'2 if the exact sizes and shapes of the particles are known. Usually, however, properties of these particles are obtained from the Lorenz-Mie theory after introducing several, sometimes unrealistic assumptions, such as spherical shape and an effective, spectrally-independent, complex index of refraction. With these oversimplifications, detailed physical characteristics of the real system are lost. In order to obtain a good grasp of complicated local radiation-transfer-combustion interactions in a coal-fired system, it is desirable to conduct controlled experiments and determine in situ radiative properties. The need for more experimental data of radiative properties of particulates has been discussed by Buckius. 3 Also, a detailed review of radiative transfer in combustion systems and the related literature have recently been given by Viskanta and Mengiic. 4 Radiative properties may be obtained in situ by performing experiments using optical diagnostic techniques and then reducing the data by employing an inverse solution for the radiative transfer equation. Our objective in this paper is to introduce a new scattering phase-function approximation, which can be used in inverse solution algorithms such as an inverse Monte-Carlo method? In Sec. 2, we discuss the new step phase-function approximation. Next, we evaluate its accuracy in two ways by comparing it against the exact phase-function for spherical particles and by comparing the radiative transfer equation solutions based on the exact, delta, and step phase-functions. 253

254

M.P. MENGUCand S. SUBRAMANIAM 2. S C A T T E R I N G PHASE F U N C T I O N

A plane electromagnetic wave incident on an obstacle (particle) in a dispersed medium is absorbed, diffracted, refracted, and reflected. The extent of each of these physical phenomena is a function of the wavelength of the incident beam and the complex index of refraction, shape, and orientation of the particle. The ratio of the redistributed wave intensity to the incident intensity is known as the scattering phase-function. The integral of this function over the entire solid angle 4g is equal to the scattering cross section. The single-scattering albedo ~ is defined as the ratio of the scattering cross section to the extinction cross section. If we normalize the scattering phase-function with respect to the scattering cross section, we obtain the normalized scattering phase-function • (n', n) d~,

(1)

which is interpreted as the probability of light incident from the direction fl' scattered into the direction ~ within a solid angle dfl. By definition, ( 1 / 4 n ) f qb(f2', f~) dff2= 1. d4tr

(2)

For radiative transfer calculations, it is convenient to write the phase-function in terms of Legendre polynomials, viz. N

¢(cos 0) = ~ akPk(COS 0),

(3)

k=0

where 0 is the scattering angle (the angle between the fl and f2' directions), Pk is the Legendre polynomial of degree k, and the ak are coefficients to be determined. The number of terms (N) required in the series to simulate the phase-function correctly depends on the size parameter (x = reD~2) and the complex index of refraction (~ = n - ik). For spherical particles with diameters much larger than the wavelength of the radiation, N may be of the order of several hundred. Although it is possible to consider all or several terms of the Legendre polynomial expansion in the solution of the radiative transfer equation (RTE) for simple, homogeneous, plane-parallel media, it is not possible to do so for practical systems, which are usually inhomogeneous and multidimensional. Thus, simple but accurate approximations of the phase-function are required. For highly-forward scattering particles, the correct behavior of the scattering phase-function cannot be retained if its Legendre polynomial expansion is truncated after a few terms. A better approximation may be introduced by accounting for the forward-peak separately. One of the most frequently used and convenient phase-function approximations is the Dirac-delta (or delta-M) phase-function 6'7 with the sharp forward-peak of the actual phase-function expressed in terms of the delta-function, viz. q~6v(cos 0) = 2f 6(1 - cos 0) + (1 - f ) c b ' ( c o s 0),

(4)

where f is the fraction of scattered intensity in the forward (cos 0 = 1) direction. The ¢ ' is the reduced phase-function expressed as in Eq. (3), viz. K

4~'(cos 0) = ~ a~, Pk(COS 0),

(5)

k = 0

where K < N. The number of terms required for the q~'-polynomial is much smaller than that for the original phase-function q~ [Eq. (3)] to obtain the same accuracy in the solution of the RTE. For example, for large water droplets in air (x >~ 100), up to 240 terms may be required for the full phase-function. However, using Eq. (4) with only 10 terms for the ~'-function, the main features of the full phase-function may be approximated very accurately. 7 If the q~'-function is considered to be a linearly anisotropic scattering phase function, then Eq. (4) is reduced to the convenient delta-Eddington phase-function s 4)6E(COS 0) = 2f6(1 -- COS0) + (1 - - f ) ( l + g ' COS0),

(6)

Effective scattering phase function of particles

255

where

g=al/3,

f=a2/5,

g'=a~/3=(g-f)/(1-f).

(7)

We use parameters with single-prime (') for the Dirac-delta phase-function and parameters without any prime for the full phase-function. The delta-Eddington phase-function was discussed in detail by Joseph et al, 8 McKellar and Box, 9 and Crosbie and Davidson. 7 The significance of this approximation may be understood best by considering that if Eq. (6) is used, the RTE for a highly forward-scattering medium can be scaled down to that for a linearly-anisotropically scattering medium. Such approximations are crucial for modeling the RTE in multidimensional enclosures, ~°'1~ especially for application to practical systems. The scattering phase-function of a cloud of pulverized-coal particles may be a function of several different parameters, including coal type, shape, size distribution, as well as combustion characteristics. For these systems, the phase-function cannot be determined from the theory accurately. However, an effective phase function can be obtained after conducting a series of experiments to measure scattered intensity at several angular orientations including 0 ° and by fitting a least-squareerror polynomial to the data. The 0 ° reading is required to evaluate the forward peak of the phase function. However, it is extremely difficult to separate the forward scattered intensity from the transmitted intensity. Thus, a different procedure is required to obtain the effective phase-function from experiments. Here, we introduce the step-function approximation ~sF (COS0) = 2 f " H ( c o s 0 -- cos A01 ) q- [1 - f ' ( 1 - cos A01 )]~"(cos 0),

(8)

where sF stands for the step-function, H is the step of Heaviside function, and A01 is the small angle around the forward peak over which we define the step-function. In Eq. (8), ~ " is the reduced phase-function given as in Eq. (3), viz. K

q~"(cos 0) = ~ a~ Pk(COS0),

(9)

k=0

where K < N. If Eq. (9) is written as a linearly anisotropic scattering phase-function, we obtain the step-Eddington (sE) approximation ~sE(COS 0) = 2f"H(cos 0 -- COS01 ) + [1 --f"(1 -- COSA01)](1 + g" cos 0),

(10)

where the double prime C) is used to indicate the parameters of the step-function approximation. An azimuthally-symmetric phase-function is obtained by integrating Eq. (8) over the azimuthal angle ~b from 0 to 2zr. The integral involving the Heaviside function is evaluated as H(cos0 -cosA0t)d~b =

2roll(cos0 -cosA01)6(~b - q~')dq5 = 2rcH(/~ - / h ) ,

(lla)

where/~ = cos 0 and ~ = cos A01. The azimuthally-symmetric 4~" function [see Eq. (9)] is p

(*2tt

K

~"(/~, U ) = (1/27r) | q~"(cos 0) d~b = ~ a~Pk(l~)Pk(I.t'), k=O d~ = 0

(1 lb)

where /~' is the direction cosine for the incident radiation. The normalization relations of the aximuthally-symmetric phase function are given f o r / ~ ' = 1 as 1

I ' q~"(/~) d/~ 2 d- i -

2

1

QSRT 43/3--E

j

1,

~ " ( u ) P , ( ~ ) d u = a173 = g",

('[J q~"(/~)Pk(#)dp = a;/(2k + l).

(12a)

(12b)

(12c)

256

M.P. MENGUCand S. SUBRAMANIAM

Using these relations, we relate the a~ of Eq. (9) to the a, coefficients of Eq. (3) by

[

a2= ak--(2k+l)f" where

f

Pk(~)d# os A01

1/

[1-f"(l-~,)],

k=l,2

. . . . K,

(13a)

/

f"=a~+l /[(2K + 3) fc]ao PK+l(lOd#l.

(13b)

For the step-Eddington approximation, we need only g" a n d f f , which are obtained from Eqs. (13) for K = 1 as

g"=[g--f'---2(l- ~ U 2 ) ] / [ l - - , " ( l -

#,)3,

(14a)

f " = 2a2/[5/~, (1 -- #~)].

(14b)

For the delta-Eddington phase-function, the g and f parameters are given by Eq. (7). When the size of the step is decreased, the step-function approximation approaches the Dirac-delta function approximation, as expected, because [see Eqs. (4) and (8)] lim (1 - / ~ ) --- lim (1 - cos AG ) = 0,

/~l~l

l i m f f ( l - U,) = lim #,~I

(15a)

A01~0

#I'I

lim g" = lim .,., .,~,

2a2 5#i(I +/~,)

a2- f ,

(15b)

5

#~)f"/2 = g- - - f

g - (1 l - f " ( 1 - #,)

lim 2 f " ( c o s 0 - cos A0~) = 2 f 6 ( 1 - cos 0).

0140

1-f

= g',

(15c)

(15d)

These relations will allow us to obtain any of the phase-functions if the coefficients of one of the others are available. If the radiation intensity at the forward scattering angle ( 0 - - 0 ) is not available, we may assume that the energy scattered in the forward direction is uniformly distributed over the A0~ range. Based on this approximation, we obtain the parameters of the step phase-function, which may be used to retrieve full and delta phase-function coefficients. Although it is possible to obtain the phase-function from the experiments using a delta phase-function approximation, as will be shown later, this procedure yields very poor results compared to those from the step-function approximation. 3. N U M E R I C A L

PROCEDURE

In order to obtain the phase-function coefficients for a dynamic system, such as a pulverized-coal flame, the radiation intensity scattered from a control volume should be measured at several angular locations simultaneously. It is impractical to make measurements at a very large number of angles. Therefore, the true behavior of the phase-function should be determined from a limited number of measurements. In this section, we discuss a numerical procedure to obtain the desired coefficients from a small number of angular intensity measurements. We require an accurate interpolation scheme. For this reason, we use Chebyshev points, i.e,, the roots of Chebyshev polynomials, to determine the observation angles for the measurements. There are basically three advantages in using Chebyshev points: (i) they are easy to locate for measurements; (ii) if the number of observation angles is increased in multiples of three, then the earlier observation angles are included in the new set of Chebyshev points to allow better and higher order interpolation; (iii) if the number of measurements is < 10, the error in the interpolating polynomial may only be about 2.5 times worse than the error in the best possible interpolating polynomial of the same orderY '~3 Once the data at M observation angles (corresponding to the Chebyshev and other random


257

points) are obtained, we use a basis-spline (B-spline) routine to fit a piecewise continuous polynomial of order J (degree J - 1) to the set of M points. The B-spline fit of order 4 is considerably better than the conventional cubic spline fit, which is also order of 4, owing to better placement of knots ~2'~3and is therefore preferable. After the coefficients of the B-spline curve are obtained, the coefficients of the Legendre polynomial expansion of the fitted curve are determined from

bk=If~,Pk(#)~iBi(I.t)dlt]/If I , = ~

, Pk (/t)Pk (/t) d# 1,

(16a)

where bk = a~[1 --f"(1 --/t,)]

(16b)

for k = 0, 1. . . . K. Here, the ~i are the B-spline coefficients and M is the number of angular observations. Preliminary runs indicated that a higher order K is required for a closer match between the Legendre polynomial expansion and the B-spline fit. For example, the values of the integrals of these two functions differed in the second digit for K = 10 and in the third digit for K = 20. The order K has a small effect on the accuarcy of the ak obtained using the a~ coefficients of the step-function approximation. The ak coefficients calculated using K = 5 were different only in the third significant digit. In this study, we used K = 50, which is acceptable for all of the phase functions we considered. Recently, Battistelli et a114described a methodology to obtain the forward scattered peak from the experiments. They recorded transmission of laser light at two small solid angles in the forward direction. The differences between these readings are considered to be deviation from true forward-scattering values. They calculated the forward peak of the phase-function by interpolating the data. In some of the numerical predictions given in the next section, we included the forward peak, assuming that is it obtained from the experiments. By doing so, we evaluated the advantage of knowing the forward-scattering value in determining the phase-function. It should be noted that the forward-scattering intensity is difficult to obtain and, even if it is available, it may not be possible to include it in the B-spline fit because of its large value, which is a couple of orders of magnitude larger than the other phase-function readings. If the size parameter is small, the phase-function has a smooth variation in the forward direction, with magnitude comparable to the values of the phase-function in other directions. In this case, the coefficients obtained from Eq. (16) will be those for the full phase-function, i.e., of Eq. (3). For smooth phase-functions, i.e., for functions with relatively small forward peak, the experimental technique proposed by Battistelli et a114may be used effectively to determine q~(cos 0), which may also be incorporated into the B-spline fit. However, even for these cases, it is possible to obtain a B-spline fit to points away from the forward direction, in which case the step-function approximation can be used. If the particle size parameter is large, the phase function will be highly-forward peaked. In determining the effective phase function from the experiments, the forward peak may be approximated by a step-function. The B-spline least-square-fit technique can be employed to determine the bk coefficients of Eq. (17) without using ff~(cos0). These are related to the ak coefficients of the ~"-polynomial as

bk- ak

a; =bk/bo,

(17a)

b0 = 1 - f ' ( 1 - #1).

(17b)

where

The f" is related to the forward-scattering value by the following relation: K

• ( c o s 0 ) = 2 f " + ~ a;,

(17c)

k~0

which yieldsf' and, consequently,/~l from Eq. (l 7b) if ~(cos 0) is available. For the delta-function, b0 is expressed as in Eq. (17b), viz. bo = 1 - f (18a)

258

M.P. MENG!JCand S. SUBRAMANIAM

Once the f value is determined, the a~ functions are obtained from a'~ = bk/bo,

(18b)

and the ak coefficients from the inversion of a~, = [ak -- (2k + l)f]/(1 - f ) ,

(18c)

which is similar to Eq. (13a). The details of the numerical procedure are outlined in the Appendix. For most of the calculations, we used six Chebyshev points corresponding to equal division of the hemisphere and a few other observations at random angles. Thus, we employed a B-spline function of order four or five. The number of points used was usually more than that required for the spline fit we considered; therefore, it was possible to place some knots away from the Chebyshev points and enforce the continuity condition. In the calculations, IMSL Library 15 subroutines are employed. All the results presented here were generated using the IBM 3090 supercomputer at the University of Kentucky. 4. C O M P A R I S O N S

OF STEP, D E L T A , AND L O R E N Z - M I E PHASE FUNCTIONS

In this section, we compare the step and delta phase-function approximations obtained from imaginary experiments against the full phase-function calculated from the Lorenz-Mie theory. We assume that experiments are conducted to measure the intensity of radiation scattered by a homogeneous, optically-thin, absorbing, and single-scattering medium. Also, we assume that there is an incident collimated light source towards the center of the medium and that detectors placed at the periphery receive energy from the small control volume at the center. Angular locations of the detectors are determined from the Chebyshev points. For a fixed arrangement of the detector solid angle, the physical dimension of the control volume from which the radiation is scattered may change, being smallest for 0 = 90 °, and largest at 0 and 180 °. In real applications, this variation, as well as variation of the phase-function over the solid angle of interest, must be considered. ~6 Here, it is assumed that these effects are accounted for in the experiments. In evaluating the new step phase-function, we first calculate the exact phase function using a modified form of the Lorenz-Mie theory algorithm given by Dave, ~7and assume that the angular distribution of the radiative intensity at the detectors will be identical to this distribution. Using the procedure described in Sec. 3, we determine the coefficients of the full Lorenz-Mie, step and delta phase-functions. The scattered intensity data at nine observation points employed in the calculations include six Chebyshev points corresponding to scattering angles of 15, 45, 75, 105, 135, and 165", as well as three additional points (for 25, 155 and 180°). An arbitrary +_ 10% error is introduced into all true readings at all nine angles. This random error set with values of +8.28, + 1.70, +5.32, --8.98, - 9 . 7 7 , - 9 . 2 8 , +8.95, +6.68, and - 8 . 0 7 % for increasing values of the angles is kept constant for all calculations and provides a standard for comparison. The exact phase-function is obtained for a monodisperse coal cloud with the size parameter x = 10 and corresponds to ), = 10.6 p m and Dco,~ = 33.7/~m. It is also obtained for narrow size distributions with size-parameter ranges of 3 . 0 < x < 5.9 (Dco,~= 10-20/~m) and of 5.9 < x < 10.4 (D~o~l = 20-35 #m). In Fig. 1, comparisons are depicted between the exact (input) phase function (solid line) and the recovered full (dashed line) and step phase-functions (points). It was assumed that the forward value 4~(cos 0) was known and that the complex index of refraction was ti = 1.85 - 0.04i. A + 25% error was introduced into the forward value and a fifth-order B-spline curve was fitted to all of the data points. For the step-function approximation, a forward step was used to enforce the integral to be equal to unity and to satisfy Eq. (17c). Using these curve-fitted expressions, a new set of coefficients of the Legendre-polynomial expansion of the phase-function was obtained from Eqs. (13) and (16)-(18). As seen from Fig. 1, both the full and step-function approximations follow the general trend of the exact (input) Lorenz-Mie phase-function. Introduction of a - 25% error into the forward value did not change the appearance of the recovered phase functions significantly. In Table 1, we compare the ak coefficients determined from the angular intensity data using the step, delta or exact phase-functions. The results were obtained by assuming that the


259

3'

2'

o o

r.. t= n.

e.Q

-I

-2

. . . . -1.0

i

. . . .

1

-0.5

. . . .

0.0

i

0.5

1.0

Cosine of Scattering Angle,/z

Fig. I. Comparison of the exact input and recovered phase functions. The exact Lorenz-Mie phase function ( ) is for pulverized-coal particles; ~ = 1.85 - 0 . 0 4 i ; x = 10. Recovered phase functions are based on the step-function approximation (C)) and full phase function expression ( - - ~ . + 2 5 % error is imposed to the forward direction readings; + 10% error in the values at other directions. Based on a total of 10 observation points.

forward-scattering value at 0 ° was known, and had either a + 25% or a - 2 5 % error. As seen from this table, the first seven or eight ak coefficients recovered by using the step-function approximation are not more than 10% different than the exact Lorenz-Mie coefficients. When a B-spline fit was obtained using the full phase-function (i.e., without considering a step approximation), we could obtain only the first three coefficients with < 10% error, if there was a + 2 5 % error in the forward peak. I f the error in the forward peak was - 2 5 % , then up to l0 coefficients were recovered within 10% error. Here, the full phase-function coefficients were obtained by dividing all the bk coefficients by b 0 for normalization. It should be understood, however, that the measurement by the scattering intensity at 0 ° is extremely difficult, because it cannot be separated from the transmitted intensity. Without having the forward peak O(cos 0), we cannot use the full phase-function expression to determine the coefficients. The step-function, on the other hand, can be used even if the O(cos 0) is not known. It is also instructive to see if similar conclusions can be reached for different complex index of refraction of particles. We repeated the calculations for several possible refractive indices of combustion products, i.e., ~ = 1.85 - 1.00i (Table 2), ~ = 1.85 - 0.22i (Table 3), r~ = 1.85 - 0.04i

Table 1. Comparison of the phase function coefficients obtained using step and delta-function approximations with those of the exact 29-term Lorenz-Mie phase function; x = 10; ~ = 1 . 8 5 - i0.04. The forward value is assumed to be known, with either + 2 5 % or - 2 5 % error; + 1 0 % error imposed on all other experimental observations; a fifth-order B-spline curve fit is used. No A0~ value is required. Lorenz-Mie phase function coefficients k

1 2 3 4 5 6 7 8 9 I0

ak

2.50495 3.82209 4.98929 6.09389 6.88569 7.52022 7.86640 7.98464 7.93731 7.68703

Percentage error in the step, full and delta-function coefficients Step +25% error -25% error 2.25 4.89 6.16 8.89 9.45 8.47 9.40 10.79 11.19 12.54

1.88 3.70 3.66 4.64 2.80 1.17 4.05 7.38 12.66 18.26

Full +25% error -25% error 4,20 7.66 9,61 12.41 13.59 13.12 14.29 15,58 15,43 15.50

-0.61 0.77 0.88 3.13 2.16 -0.69 -1.94 -3.23 -6.20 -8.97

Delta -13.31 -10.62 -9.00 -6.20 -4.81 -4.-17 -2.18 0.91 3.78 8.21

260

M . P . MENGrJC a n d S. SUBRAMANIAM

Table 2. Comparison of the phase function coefficients obtained using step and delta-function approximations with those of the exact 29-term Lorenz-Mie phase function; x = 10; r7 = 1.85 - i 1.00. _ 10% error imposed on all of the experimental observations; a fifth order B-spline curve fit is used. Lorenz-Mie phase function coefficients

Percentage error in the step and delta-function coefficients

k

ak

A0t=l °

~01~--..5 °

1 2 3 4 5 6 7 8 9 10

2.47667 3.87094 5.04218 6.00675 6.77090 7.34027 7.72001 7.91776 7.94189 7.79909

0.04 -0.4,t -1.23 -1.52 -2.44 -2.25 1.10 5.58 10.48 18.42

0.50 0.00 -1.17 -1,60 -3.16 -3.75 -1.03 2.51 6.21 10.37

Step A01~10 ° 0.30 -0.65 -2.55 -4.05 -7.09 -9.63 -9.43 -9.13 -9.54 -10.59

I

Delta

~01~15 ° -0.04 -1.71 -4.78 -7.94 -13.15 -18.,tl -21.52 -25.15 -30.12 -36.,14

-18.58 -18.13 -17.93 -17.01 -16.61 -15.27 -11.18 -6.03 -0.12 6.96

(Table 4), ti = 1 . 7 0 - 0.04i (Table 5), and r~ = 1 . 5 0 - 0.04i (Table 6). In these calculations, we assumed that forward-scattering value was unknown, and we chose an arbitrary step width within the range of 1° < A0~ < 15 °. Thenf" was obtained from Eq. (17b), the a~ coefficients from Eq. (17a), and the a k coefficients from inversion of Eq. (13). The results presented in these tables depict the sensitivity of the recovered ak coefficients on the choice of the width of the forward-step. Also, cross-comparisons of these tables indicate the sensitivity of the results on the real and the imaginary parts of the complex index of refraction. As seen from Tables 2, 3, 4, and 6, up to 10 coefficients of Legendre polynomial expansion can be obtained with + 10% deviation from the true values if the step-function approximation is used. On the other hand, if the delta-function approximation is used, the errors in the first two coefficients are always more than 10%. It is important to note that usually we need only the first two terms of the phase-function to scale the radiative transfer equation to simplify its solution. Also, with Table 3. A s in Table 2 for ~q = 1.85 - i0.22. Percentage error in the step and de[ta-functlon coefficients

Lorenz-Mie phase

function coefficients k

ak

~0t~l °

A01z5 °

I 2 3 4 5 6 7 8 9 10

2.66393 4.17698 5.45688 6.51477 7.35285 7.97619 8.39028 8.60286 8.62734 8.46568

0.04 -0.44 -1.23 -1.52 -2.44 -2.25 1.10 5.58 10.48 16.42

-0.02 -0.64 -1.67 -2.29 -3.69 -4.16 -1.67 1.66 5.05 8.99

Step ~01~10 ° -0.22 -1.27 -3.00 -4.65 -7.46 -9.80 -9.75 -9.52 -10.08 -11.17

Delta ~01~15 ° -0.55 -2.30 -5.15 -8.39 -13.28 -18.24 -21.36 -24.90 -29.86 -36.03

-20.11 -19.76 -19.49 -18.73 -18.76 -16.84 -12.99 -8.06 -2.48 4.31

Table 4. As in Table 2 for r~ = 1.85 - i 0 . 0 4 . Lorenz-Mie phase function coefficients


k

ak

A01~I°

A01m5 °

1 2 3 4 5 6 7 8 9 10

2.50495 3.82209 4.98929 6.09389 6.88569 7.52022 7.86640 7.98464 7.93731 7.68703

2.32 5.13 6.68 9.80 10.92 10.68 12.63 15.38 17.58 21.36

2.28 4.99 6.37 9,27 10.06 9.37 10.72 12.66 13.77 16.08

Step A01~10 ° 2.14 4.55 5.43 7.64 7.46 5.51 5.16 4.88 3.16 1.76

Delta A01ml5 ° 1.92 3.82 3.91 5.06 3.45 -0.26 -7.83 -5.81 -10.71 -15.91

-13.31 -10.62 -9.00 -6.20 -4.81 -4.47 -2.18 0.94 3.78 8.21


261

Table 5. As in Table 2 for ~ = 1 . 7 0 - i0.04. Lorenz-Mie phase function coefficients


k

ak

~0t~l °

AOl=,,5 °

1 2 3 4 5 6 7 8 9 10

2.43104 3.44584 4.16032 4.79639 5.30776 5.79490 6.13670 6.39770 6.50742 6.49360

-1.23 -7.35 -17.29 -25.96 -31.00 -28.48 -27.18 -14.51 -5.75 4.93

-1.30 -7.62 -17.90 -27.10 -32.88 -31.31 -26.28 -20.21 13.53 -5.53

Step A0t--10 ° -1.53 -8.44 -19.79 -30.56 -38.52 -39.71 -38.20 -36.45 -35.20 -33.92

Delta A0t--15 ° -1.92 -9.79 -22.84 -36.05 -47.23 -52.25 -55.35 -58.80 -63.52 -68.93

-20.44 -22.10 -26.14 -29.75 -30.90 -27.79 -20.82 -13.30 -4.47 6.22

Table 6. A s in Table 2 for ~ = 1 . 5 0 - i0.04. Lorenz-Mie phase function coefficients


k

ak

A0I~I °

A 0t =,,5°

1 2 3 4 5 6 7 8 9 10

2.62478 4.02828 5.16914 6.28965 7.13437 7.79307 8.13278 8.27548 8.17833 7.83149

1.85 4.32 7.22 11.04 12.03 12.33 13.11 14.86 16.05 19.54

1.82 4.21 6.98 10.61 11.35 11.29 11.59 12.70 13.01 15.28

Step A01~10 ° 1.71 3.86 6.23 9.32 9.28 8.23 7.18 6.53 4.55 3.73

Delta A0t~15 ° 1.53 3.30 5.02 7.26 6.10 3.65 0.83 -1.94 -6.51 -10.51

-11.76 -9.43 -6.70 -3.29 -2.15 -1.55 -0..11 1.64 3.39 7.31

increasing material and shape inhomogeneity of the scattering particles, the higher order terms of the phase function cannot be determined accurately. In practical systems, particle clouds are usually polydispersed; therefore, a size distribution of particles is to be considered. We repeated the calculations for two narrow size distributions: one for 10-20/zm dia range (3.0 < x < 5.9) and the other for 20-35/zm dia range (5.9 < x < 10.4) pulverized-coal particles with ~ = 1.85 -0.22i. These ranges were divided into five equal-number density intervals, and the phase-function of the cloud was determined after obtaining the phase-function for each interval from the Lorenz-Mie theory. Then, the coefficients of the composite phase-function were recovered from experimental angular intensity data using either a step or a delta phase-function approximation. Comparisons of results shown in Tables 7 and 8 indicate that the errors in the first two coefficients of the recovered phase-function are about an

Table 7. C o m p a r i s o n o f the phase function coefficients obtained from step and delta-function approximations with those of the exact 19-term L o r e n z - M i e phase function; equal-weight pulverized-coal size distribution; x range is 3.0-5.9; = 1 . 8 5 - i0.22; _+ 10% random error is imposed at eight observation ( f o u r Chebyshev and four other) points; a fourth order B-spline fit is used. Lorenz-Mie phase function coefficients

Percentage error in the step and delta-functlon coefficients

k

a~

A0tffil °

1 2 3 4 5 6 7 8 9 I0

2.40939 3.43253 3.99119 4.15230 3.89657 3.32739 2.57797 1.82369 1.13669 0.65341

Step A0t,~5 °

Delta

A0tffil0 °

A0t--15 °

0.46 0.65 -0.58 -1.66 -1.04 0.65

0.43 0.57 -0.77 -2.05 oi.81 0.84

0.36 0.32 -1.36 -3.26 -4.13

0.24 -0.09 -2.32 -5.16 -7.70

-5.24

- 11.81

-8.37 -7.99 -8.78 -9.33 -8.10 -5.47

6.46

3.53

28.13 81.52 202.47

22.11 68.12 171.18

-6.02 4.96 30.78 86.24

-17.31 -18.64 -18.03 -18.48

24.89 80.67 205.72

1.75

M. P. MENGI~C a n d S. SUBRAMANIAM

262

Table 8. As in Table 7, but for nine observation (six Chebyshev and three other) points; x range is 5.9-10.4; 30 term Lorenz-Mie phase function. Lorenz-Mie pha~e function coefficients


k

a,

AOI=I °

,5 0w-=,5°

1 2 3 4 5 6 7 8 9 10

2.64055 4.07339 5.20711 6.06136 6.64803 6.97977 7.07069 6.93352 6.59099 6.06803

1.34 4.21 8.18 14.57 24.18 36.97 54.16 76.64 106.45 146,94

1.19 3.73 7.12 12.64 20.95 31.91 46.51 65.34 89.93 122.88

Step A01=10 ° 0.73 2.23 3.88 6.74 11.26 16.92 24.26 33.11 43.93 57.54

Delta

A0t=15 ° -0.04 -0.22 -1.36 -2.59 -3.69 -5.46 -7.75 -11.23 -16.20 -23.01

-12.89 -7.37 -0.23 8.82 20.29 34.62 52,8,1 76.21 106.90 148.19

order of magnitude smaller if the step-function approximation is used in inversion rather than the delta-function. Also, the width of the forward step has little effect on the accuracy of the recovered coefficients. In general, we conclude that the effective phase-function can be determined relatively accurately if a step-function approximation is employed. The delta-function approximation is never as effective as the step-function in determining the original phase-function coefficients. 5. EFFECT OF PHASE F U N C T I O N A P P R O X I M A T I O N S RADIATIVE TRANSFER CALCULATIONS

ON

Although accurate knowledge of scattering phase-function is important, for practical applications the accuracy of the solution of the radiative transfer equation is more critical. Thus, the effect of the approximate phase-functions on the accuracy of radiative flux calculations should also be evaluated. In this section, we compare solutions of the radiative transfer equation (RTE) obtained from the exact Fs-method using the true and recovered phase functions. The FN-method was first introduced by Siewer0 8 and known to converge to exact results rapidly with increasing order. Earlier studies show that if the order N t> 7 radiative flux values are accurate for the first five significant digits. 19 The physical system being considered here is a one-dimensional, plane-parallel cold medium with one surface being transparent and is subjected to diffuse incident radiation, whereas the other being opaque and cold. We do not consider emission from the medium to not overshadow the effect of scattering phase-function on the predictions. In order to account for the forward-peak of the delta-function or the forward-step of the step-function approximation in the solution, we need to scale the RTE. For the delta-function, scaling yields the following expressions: 6 r * = (1 - e ) f ) r ,

m*=oo(1-f)/(l-~f),

(19)

where * is used to indicate the scaled optical thickness or the single scattering albedo. For the step-function, we define the scaled quantities as z * = [1 - m f (1 - cos A01 )]z,

co* = w [1 - - f (1 -- cos A01 )] 1 - o~f(1 - cos A01) '

(20)

These relations are obtained by using the following approximation: H(cos 0 - cos A01) - (1 - cos A01)2~5(# -- #')6(th -- ¢ ' ) ,

(21)

which is valid only for small A01 values. Because of this assumption, some error in the solution of the RTE using the scaled step-function approximation with a finite A0~ value is expected. Radiative heat transfer calculations were performed for a polydispersed pulverized-coal cloud using the phase functions given in Tables 7 and 8. The radiative heat fluxes at the boundaries were calculated using the F9-method (see Tables 9 and 10). Here, Q(1) corresponds to slab transmittance and [1 - Q ( 0 ) ] can be considered as slab reflectance. In calculating the phase-function coefficients using the imaginary experimental data, an arbitrary


263

Table 9. The effect of the phase function approximation on radiative transfer calculations. Fg-method results are given. For pulverized-coal particles, ~ = 1.85-0.22i; x range is 3.0-5.9; _ 10% error is imposed at all of the observation angles. The exact Lorenz-Mie phase function has 19 terms; for the step-function AOi = 10°.

Full Lorenz-M]e Phase Function Results Step Phase Function (a) 10 terms, with error (b) 5 terms, with error (c) 2 terms, with error Delta Phase Function (based on coefficients of the step-function) (d) 10 terms, with error (e) 5 terms, with error (f) 2 terms, with error Delta Phase-Function (based on the deltafunction approximation with B-spline fit) (g) 10 terms, with error (h) 5 terms, with error (i) 2 terms, with error

Q(O) 0.96818

Q(1) 0.40608

0.96875 0.96987 0.97176

0.40668 0.40817 0.41069

0.96858 0.96861 0.96940

0.40647

0.95801 0.95803 0.95869

0.39596 0.39599 0.39676

0.40650 0.40747

A0~ value was used for the step-function approximation and the value at the forward 0 ° was assumed unknown. A random _+ 10% error was introduced to input intensity values at all observation angles. Two sets of ak coefficients were recovered from the B-spline fit using either a delta-function [Eq. (18a)] or a step-function [Eq. (13)]. These ak coefficients were used to obtain the coefficients of the 2, 5, or 10 term delta or step-function approximations. Also, the reduced delta-function coefficients (a~) were obtained from the a[ coefficients of the step-function. In Table 9, the results obtained from the true or recovered phase-functions for pulverized-coal size distribution of 10-20/~m dia range are compared. The results obtained using the delta phase-function with coefficients based on the step-function approximation are most accurate. This result is not surprising because the delta function approximation scales the RTE accurately; however, it is not as efficient as the step-function approximation in recovering the phase function coefficients. The step-function approximation, on the other hand, is very convenient to recover the phase function coefficients; however, it cannot be used successfully in scaling the RTE. In Table 10, a similar comparison is given for a pulverized-coal size distribution of 20-35/~m dia range. The exact phase-function was obtained after determining the monosize phase-functions Table 10. The effect of the phase function approximation on radiative transfer calculations. Fg-method results are given. For pulverized-coal particles, ~ = 1 . 8 5 - 0.22i; x range is 5.9-10.4; _+ I0% error is imposed at all of the observation angles. The exact Lorenz-Mie phase function has 30 terms; for the step-function A0, = 5 °.

Full Lorenz-Mie Phase Function Results I Step Phase Function (a) 20 terms, no error (b) I0 terms, with error (c) 5 terms, with error (d) 2 terms, with error Delta Phase Function (based on coefficients of the step-function) (e) 20 terms, no error (f) 10 terms, with error (g) 5 terms, with error (h) 2 terms, with error Delta Phase Function (based on the deltafunction approximation with B-spllne fit) (i) 10 terms, with error (j) 5 terms, with error (k) 2 terms, with error

Q(o) 0.97949

Q(1) 0.41980

0.98021 0.98100 0.98143 0.98212

0.41978 0.42073 0.42139 0.42243

0.98021 0.98070 0.08071 0.98118

0.41978 0.42031 0.42036 0.42100

0.98g17 0.96018 0.96958

0.40848 0.40852 0.40910

264

M.P. MENG~Cand S. SUBRAMANIAM

at 15 equal intervals from the Lorenz-Mie theory. The approximate phase-functions used in the F9 method had either 10, 5 or 2 terms. The coefficients of the step-function were obtained after assuming that there were random ___10% experimental error at all angles and a step-width of A0 = 5 ° was used. Again, the results show that the delta-function approximation based on the coefficients derived from those of the step-function yields more accurate radiative flux predictions than the others. Even the two-term delta-function, which is nothing but the delta-Eddington approximation, yields very good predictions. 6. C O N C L U S I O N S We introduced a methodology to determine the scattering phase-function of mono- and polydispersions using a limited number of experimental angular intensity data. The motivation for this approach stemmed from the need of determining the scattering phase-functions of highlyforward scattering polydispersions in situ from experiments and the extreme difficulty in making accurate measurements in the forward-direction. A step-function approximation was introduced and shown to be useful in retrieving several coefficients of the full phase-function as well as those of the delta-function approximation. Comparisons of the angular distribution of the approximate phase-functions yielded acceptable agreement with that of the true phase-function. The effect of these approximate phase-functions on the radiative transfer calculations was also evaluated using the exact Fg-method, and it was shown that the recovered coefficients can be used to calculate the radiative fluxes accurately. The results presented here show that the coefficients of the phasefunction should be determined from the experiments using a step-function approximation, with a narrow forward-step. Then, a delta-function approximation based on the coefficients obtained from the step-function should be used to scale the radiative transfer equation. The most subjective aspect of the step phase-function approximation is the value of #~ = cos A0~ to be used in the analysis. Our preliminary studies indicated that any A0~ value within 5-15 ° range yielded almost similar and acceptably good results. The possible smallest value for A0t is always limited by the experimental set up. The largest A0~ value should be smaller than the angle at which the first observation is made in the experiments and included in the B-spline fit. Some improvements of the model, such as consideration of detector solid-angle range and variation of the effective scattering control-volume with angular orientation of the detector, are to be introduced to the model before it can be used for practical applications. Recently, this approach was used to determine the phase-function coefficients of latex particle polydispersions from experiments. ~6The agreement achieved between the theoretical and experimental results were very good, especially for the first two coefficients of highly-forward scattering phase-functions. Acknowledgements--Partial support for this research was obtained from NSF Grant No. CBT-8708679and DOE Grant No. DE-FG22-87PC79916.Support for S. Subramaniam in the 1988-1989academicyear from the Universityof Kentucky Center for Computational Sciencesis gratefully acknowledged.

REFERENCES 1. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley, New York, NY (1983). 2. W. J. Wiscombe and A. Mugnai, "Single Scattering from Nonspherical Chebyshev Particles: a Compendium of Calculations," NASA Reference Publication 1157, Washington, DC (1986). 3. R. O. Buckius, "Radiative Heat Transfer in Scattering Media: Real Property Contributions," in Heat Transfer 1986, Proc. Eighth Int. Heat Transfer. Conf., Vol. 1, pp. 141-150, C. L. Tien, V. P. Carey, and J. K. Ferrell eds., Hemisphere, New York, NY (1986). 4. R. Viskanta and M. P. Mengiic, Prog, Energy. Combust. Sci. 13, 97 (1987). 5. S. Subramaniam and M. P. Mengfic, Int. J. Heat Mass Transfer, in press (1990). 6. W. J. Wiscombe, J. Atmos. Sci. 34, 1408 (1977). 7. A. L. Crosbie and G. W. Davidson, JQSRT 33, 391 (1985). 8. J. Joseph, W. J. Wiscombe and J. A. Weinman, J. Atmos. Sci. 33, 2452 (1976). 9. B. H. J. McKellar and M. A. Box, J. Atmos. Sci. 38, 1063 (1981). 10. M. P. Mengiic and R. Viskanta, JQSRT 33, 533 (1985). 11. M. P. Mengiic and R. Viskanta, J. Heat Transfer 108, 271 (1986).


265

12. C. DeBoor, .4 Practical Guide to Splines, Applied Mathematics Series, Vol. 27, Springer, New York, NY (1978). 13. T. J. Rivlin, An Introduction to Approximation of Functions, Blaisdell, MA (1969). 14. E. Battistelli, P. Bruscaglioni, A. Ismaelli, L. LoPorto, and G. Zaccanti, Appl. Opt. 25, 420 (1986). 15. IMSL Library, Edition 1.0, NBC Building, 7500 Ballaire Boulevard, Houston, TX (1984). 16. B. M. Agarwal and M. P. Mengiic, Int. J. Heat Mass Transfer, submitted for publication (1990). 17. J. V. Dave, IBM J. Res. 13, 302 (1969). 18. C. E. Siewert, Astrophys. Space Sci. 58, 131 (1978). 19. M. P. Mengiic and R. Viskanta, JQSRT 29, 381 (1983). APPENDIX

A Procedure to Determine the Coej~cients

The following is a step-by-step procedure for finding the coefficients of the phase-function approximations: (i) we select the observation points; observations at the four, five or six Chebyshev points are recommended. It is not necessary to include any 0 ° reading to the B-spline fit. (ii) We fit a B-spline curve of order J( --- number of Chebyshev points - 1) to the angular data obtained at M points. The knots of the B-spline curve fit should be at points other than the Chebyshev points and mth observation point should be between knots m and (m + j ) t h . (iii) Using a Gaussian quadrature scheme [see Eq. (16)], we find the coefficients of the best Legendre polynomial fit to the B-spline curve from I1--f(1--/.~l)](~)tt(]A) :

K E bkek(~)"

(AI)

k=O

The order K should be equal to the number of coefficients required. (iv) If b0 > 1, we first try a lower order B-spline fit with the same points or by using a different set of points. Once it is verified that b0 < 1, we evaluate the a)~ coefficients from (A2)

a'~ = bk/bo

and proceed to the next step. (v) If ¢ (cos 0) is not known, we first estimate a/~z value based on the experiments or a possible phase-function and then determine the f"-coefficient from Eq. (17b), i.e., cos A0~ =/~l = 1 - (1 - bo)/f"

(A3)

and proceed to step (vii). (vi) If ¢ (cos 0) is known, we determine the value of the forward peak from K

• (cos 0) = 2 f " + ~ a~

(A4)

k=0

and find Pt from Eq. (A3). (vii) We evaluate the coefficients of the Legendre polynomial fit for the entire phase-function ak=a'~[1 - - f " ( 1 - ~ ) ] + f " ( 2 k

+ 1)

Pk(~) d~.

(A5)

I

(viii) We use the coefficients determined in step (vii) to obtain a lower order (2 or 3 term) step or delta phase-function approximation to be used for the solution of radiative transfer equation. In determining the lower-order step functions, we first obtained a higher-order polynomial (K = 50) and then retained the forward peak value [4~(cos 0)]; we finally adjusted the width of the step, i.e., the/~ value. In this procedure, the key factor is finding a B-spline fit to experimental data which yields b0 < 1. Once this condition was satisfied, it was found that the first three ak coefficients determined by using a step-function were very close (within 5%) to those of the exact Lorenz-Mie phase-function coefficients for all of the cases considered here.