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Radiative transfer in random media with Rayleigh scattering
Journal of Quantitative Spectroscopy & Radiative Transfer 76 (2003) 359 – 372
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Radiative transfer in random media with Rayleigh scattering A.R. Deghiedy, M. Sallah ∗ , M.A. Abdou Theoretical Physics Research Group, Physics Department, Faculty of Science, Mansoura University, Mansoura, Egypt Received 17 January 2002; accepted 22 April 2002
Abstract This paper considers radiation transfer through a 0nite plane-parallel random medium consisting of two immiscible mixed materials. The mixing statistics of the two components of the medium is assumed to be described by the two-state homogeneous Markovian statistics. The problem treats a medium contains an internal source and has di4usely- and specularly-re5ecting boundaries obeying Rayleigh scattering law. The problem with this generalized boundary conditions is solved in terms of the solution of the corresponding free source problem with simple boundary conditions. Pomraning–Eddington approach is used to obtain an explicit solution of the simple problem in the deterministic case. A formalism, developed to treat radiative transfer in statistical mixtures, is used to obtain the ensemble-averaged solution. Numerical results for the ensemble-average partial heat 5uxes of the problem under consideration are obtained using three di4erent weight functions for the sake of comparison. ? 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction A subject of recent interest has been the problem of formulating and solving of linear transport problem in statistical medium composed of two or more immiscible mixed materials. [1– 4]. The applications of this theory can be found in astrophysics, vision, atmospheric and oceanic sensing and medical optics during Laser therapy of cancer tumors. Also the theory of nuclear reactors and elementary particle physics are connected with this theory. A complete and rigorous description has been given for time-independent problems in the absence of the scattering interaction in the underlying physics of the transport process, both for Markov statistics [1,2] and non-Markov statistics [3,4]. With scattering present, a general phenomenological model has been suggested [4] for Markov statistics, using the master equation approach [5]. Recently, much of the literatures has been directed towards the development of a mathematical formalism to estimate the ensemble-averaged ∗
Corresponding author. E-mail address: [email protected] (M. Sallah).
0022-4073/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 0 2 ) 0 0 0 6 3 - 8
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intensity of radiation [1–11]. Levermore et al. [1] and Pomraning [11] developed a formalism, for a binary Markovian mixtures, to treat a certain class of time-independent transport problems in a planar geometry. In this paper, we consider the problem of stochastic radiative transfer in a 0nite plane-parallel medium of binary Markovian mixture with Rayleigh scattering phase function. In the next sections we solve this problem with di4usely- and specular-re5ecting boundaries and internal energy source in terms of the solution of the source-free problem with a simple boundary conditions in the deterministic case. We use the formalism obtained by Levermore et al. [1] and Pomraning [11] to average the solution given by Pomraning–Eddington method [12,13] for the deterministic case. The obtained average solution is used to give explicit analytical forms for the average re5ectivity R and transmissivity T , which are used to calculate the average partial heat 5uxes J ∓ at the boundaries of the medium. Section 4 tabulates numerical results for the average partial heat 5uxes for media with di4erent thicknesses and single scattering albedo. For the sake of comparison we use three di4erent weight functions. 2. Problem formulation The starting point of the analysis is the time-independent transport equation in a planar medium, which is given by [6]
@(z; E; ) + (z; E)(z; E; ) @z ∞ 1 = S(z; E; ) + dE d s (z; E → E; → )(z; E ; ); 0
−1
0 6 z 6 t:
(1)
Here (z; E; ) is the radiation intensity, with z; E; and representing the spatial, energy, and angular variables, respectively. The quantity (z; E) is the total cross section, s (z; E → E; → ) is the scattering kernel and S(z; E; ) represents an internal energy source. The quantities ; s and S in Eq. (1) are treated as discrete random variables, which obey the same statistics. The key to the analysis is the introduction of the optical depth variable, de0ned by [6] z x(z; E) = d z :(z ; E): (2) 0
In terms of x Eq. (1) becomes @I (x; E; ) + I (x; E; ) = Q(x; E; ) + @x
0
∞
dE
1
−1
0 6 x 6 a;
d !(x; E → E; → )I (z; E ; ); (3)
where I (x; E; ) = (z; E; );
(4)
!(x; E → E; → ) = s (z; E → E; → )= (z; E)
(5)
which is the single-scattering albedo of the medium,
A.R. Deghiedy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 76 (2003) 359 – 372
361
and Q(x; E; ) = S(z; E; )= (z; E):
(6)
The optical thickness a of the medium is given by t a(t; E) = d z (z ; E): 0
(7)
In this paper we consider the monoenergetic radiative transfer equation for a 0nite slab medium, viz. @ ! 1 P(; )I (x; ) d + Q(x); −1 6 6 1; 0 6 x 6 a (8) + 1 I (x; ) = @x 2 −1 subjected to boundary conditions I (0; ) = F1 + s1 I (0; −) + d1 J − ;
¿ 0;
(9a)
I (a; −) = F2 + s2 I (a; ) + d2 J + ;
¿ 0;
(9b)
where F1 and F2 are the externally-incident 5uxes on the left and right boundaries, respectively, si and di are the specular and di4use re5ectivities of the boundaries, respectively. We now assume that ; s and S obey the same statistics in the sense that ! = s = and Q = S=
are non-stochastic [6]. This means that ! and Q take the same values inside the two immiscible 5uids of the medium. Hence the transport problem described by Eqs. (8) and (9) is only stochastic through the optical depth variable x and the optical size of the system a. The statistics of the problem are entirely described by the joint probability density P(x; a; z; t) , de0ned such that P d x da is the probability that for a given geometric position z and a given geometric system thickness t, the position z corresponds to an optical depth lying between x and x + d x and the system thickness t corresponds to an optical depth lying between a and a + da. The ensemble-averaged intensity is thus given by [1,6] ∞ ∞ I (x; E; ) = da d x P(x; a; z; t)I (x; a; E; ): (10) 0
0
All of the statistical informations are embodied in the joint probability density function P(x; a; z; t). From the transport point of view, this function is assumed to be known. In the special case of binary homogeneous Markov statistics, one has the near separable form [5] denoting the two materials making up the binary mixture by subscripts 0 and 1 P(x; a; z; t) =
1
pi fi (x; z)fi (a − x; t − z):
(11)
i=0
Here pi represents the probability of 0nding material i at any point in the system. In terms of mean slab thickness i of the alternating slabs of the two materials making up the planar system, one has [1] pi =
i : 0 + 1
(12)
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For homogeneous Markov statistics, these slab thicknesses are exponentially distributed with mean (average) i for material i [6]. The function fi (x; z) in Eq. (11) is the probability density function de0ned such that fi d x is the probability that the planar system has an optical depth lying between x and x + d x, given that the geometric depth is z and given that the point z lies in material i. For the case of homogeneous binary Markov statistics being discussed, the function fi (x; z) is known in closed analytic form [6,10]. It is given as the derivative of the cumulative distribution function Fi (x; z) de0ned as fi (x; z) =
@Fi (x; z) : @x
(13)
By ordering the cross sections of the two materials making up the binary mixture such that
1 ¿ 0 , the function Fi (x; z) is given by 0; x ¡ 0 z; (14) Fi (x; z) = 1; x ¿ 1 z and F0 (x; z) = exp(− ) + 2g( ; u);
0 z ¡ x ¡ 1 z;
F1 (x; z) = 1 − exp(−u) − 2g(u; );
0 z ¡ x ¡ 1 z;
where g(u; ) = exp(−u)
√
u
0
x2 d x I1 (2x) exp − u
(15a) (15b)
(16)
with I1 (x) denoting the modi0ed Bessel function. The quantities u and are given by u=
x − 0 z ; 1 ( 1 − 0 )
=
1 z − x : 0 ( 1 − 0 )
(17)
In view of Eq. (11), Eq. (10) becomes ∞ 1 I (x; E; ) = pi d x fi (x; z) i=0
0
0
∞
da fi (a − x; t − z)I (x; a; E; ):
(18)
Such an average is easily computed [6] for homogeneous binary Markovian statistics. We consider a pure exponential in optical depth space exp(−kx) with k being a constant, and ask for the ensemble average of this exponential. If we denote this ensemble average by Ei (k; z) we have ∞ Ei (k; z) = d x fi (x; z) exp(−kx): (19) 0
It was found by Pomraning [6] that Ei (k; z) = !i exp(−–+ z) + (1 − !i ) exp(−–− z)
(20)
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with Ei (Nk; 0) = 1 k j + (1=c ) − –+ ; !i = (–− − –+ ) and 1 ∓ 2–± = ( 1 + 0 )k + c
i = 0; 1; j = 0; 1; i = j ( 1 − 0 )2 k 2 −
2k 1 (p1 − p0 )( 1 − 0 ) + 2 c c
363
(21)
(22)
where 1 1 1 = + : c 0 1
(23)
The above analysis is used in the next sections to evaluate the ensemble-average solution of the problem under consideration. 3. The solution of the problem in terms of source-free problem Eq. (8) with the generalized boundary conditions (9) can be solved by connecting it with the solution of the corresponding source-free problem ! 1 @ + 1 %(x; ) = P(; )%(x; ) d ; −1 6 6 1; 0 6 x 6 a (24) @x 2 −1 with simple boundary conditions as %(0; ) = 1 + s1 %(0; −);
(25a)
%(a; −) = s2 %(a; ):
(25b)
The connection between the two problems may be demonstrated as follows. Eq. (24) has two adjoint equations, viz ! 1 @ − + 1 %(x; −) = P(; )%(x; − ) d ; (26a) @x 2 −1 ! 1 @ P(; )%(a − x; ) d : (26b) − + 1 %(a − x; ) = @x 2 −1 Multiplying Eq. (26a) by I (x; ) and Eq. (8) by %(x; −), subtracting the resultant equations, and then integrating over ∈ [ − 1; 1] and x ∈ [0; a] we get A11 J + + A12 J − = H1 ;
(27)
where A11 = −d2 T;
(28a)
A12 = 1 − d1 R;
(28b)
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H1 = Q1 + F2 T + F1 R; Q1 =
1
0
T=
Q(x)E(x) d x;
(28d)
%(0; −) d;
(29a)
%(a; ) d:
(29b)
0
R=
a
(28c)
1
0
Further, multiplying Eq. (8) by %(a−x; ) and Eq. (26b) by I (x; ), subtracting the resultant equations and then integrating over ∈ [ − 1; 1] and x ∈ [0; a] one 0nds A21 J + + A22 J − = H2 ;
(30)
where A21 = 1 − d2 R + 2(s1 − s2 )R;
(31a)
A22 = −d1 T + 2(s1 − s2 )T;
(31b)
H2 = Q2 + F1 T + F2 R;
(31c)
Q2 =
a
Q(a − x)E(x) d x;
0
1
(31d)
%(x; ) d:
(32)
The partial heat 5uxes J ∓ are de0ned by 1 J− = I (0; −) d;
(33a)
E(x) =
+
−1
J =
0
1
0
I (a; ) d:
(33b)
Eqs. (27) and (30) give the solutions of J − and J + in terms of R; T; Q1 ; and Q2 . The ensembleaverage of the partial heat 5uxes J ∓ can be obtained from the calculation of the corresponding stochastic values R; T ; Q1 ; and Q2 , where 1 R = %(0; −) d; (34) 0
T =
0
1
%(a; ) d;
where %(x; ) is the ensemble-averaged solution of the source-free problem.
(35)
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365
3.1. Solution of the source-free problem for Rayleigh scattering In this work we consider the Rayleigh scattering phase function P(; ) which has the form P(; ) = 38 [(3 − 2 ) + (32 − 1)2 ]: Then Eq. (24) becomes 1 @ 3! 2 2 2 (3 − )E(x) + (3 − 1) %(x; ) d : + 1 %(x; ) = @x 16 −1
(36)
(37)
We use Pomraning–Eddington approximation [12,13] to solve Eq. (37) %(x; ) = ((x; )E(x) + o(x; )F(x);
(38)
where E(x) and F(x) are the radiant energy and the net 5ux, respectively and ((x; ) and o(x; ) are even and odd functions in which are assumed to be slowly varying in x and normalized as 1 1 d ((x; ) = d o(x; ) = 1: (39) −1
−1
Substituting Eq. (38) in Eq. (37), and integrating over ∈ [ − 1; 1]; one gets dF(x) + *E(x) = 0: dx
(40)
Multiplying Eq. (37) by and integrating over ∈ [ − 1; 1] gives d [D(x)E(x)] + F(x) = 0; dx
(41)
*=1−!
(42)
where
and
D(x) =
1
−1
d 2 ((x; ):
(43)
The even and odd functions are, respectively, given by ((x; ) =
3! (3 − 2 ) + (*=k 2 )(32 − 1) ; 16 1 − k 2 2
(44)
o(x; ) =
3!k 2 (3 − 2 ) + (*=k 2 )(32 − 1) ; 16* 1 − k 2 2
(45)
where k 2 = *=D:
(46)
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A.R. Deghiedy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 76 (2003) 359 – 372
Using Eq. (44) in Eq. (43) for |k| ¡ 1 leads to the transcendental equation 1+k 2ky = ln ! 1−k with
8 y= 9
k 4 − 3!k 2 =8 + 9*!=8 k 4 − (1 + *)k 2 =3 + *
(47)
(48)
since D(x) is a slowly varying function in x, Eqs. (40) and (41) lead to the di4usion equation d 2 E(x) − k 2 E(x) = 0 d x2
(49)
which has the solution E(x) = Aekx + Be−kx
(50)
and F(x) will be * F(x) = − [Aekx − Be−kx ]; k
(51)
where A and B are constants to be determined. Using Eqs. (44) – (51) in Eq. (38), we get the solution of the free-source problem %(x; ) = Ah+ ()ekx + Bh− ()e−kx ;
(52)
where h± () =
3! (3 − 2 ) + (*=k 2 )(32 − 1) : 16 1 ± k
(53)
To calculate the constants A and B we use a weight function W () which forces the boundary conditions to be ful0lled, 1 d W ()[%(0; ) − 1 − s1 %(0; −)] = 0; ¿ 0; (54a) 0
1
0
d W ()[%(a; −) − s2 %(a; )] = 0;
¿ 0:
(54b)
Using Eq. (52) we obtain (1 − Cs1 )A + (C − s1 )B = I0 =I+ ;
(55a)
(C − s2 )A + (1 − Cs2 )e−2ka B = 0;
(55b)
where
I0 =
0
1
d W ();
I± =
0
1
d W ()h± ()
and
C = I− =I+ :
(56)
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367
Then the intensity %(x; ) takes the form %(x; ) = −A0 {A4 h+ ()
∞
An5 exp[ − k(2a(n + 1) − x)]
n=0
− A3 h− ()
∞
An5 exp[ − k(2na + x)]};
(57)
n=0
where A1 = 1 − Cs1 ;
A2 = C − s1 ;
(58a)
A3 = C − s2 ;
A4 = 1 − Cs2 ;
(58b)
A5 =
A1 A4 ; A2 A3
I0 : I+ A 2 A 3
A0 =
(58c)
Therefore, the ensemble-averaged solution can be obtained by using Eq. (57) in Eq. (10) ∞ ∞ 1 n A5 pi d x fi (x; z)exp[ − k(2n + 1)x] %(z; t; ) = −A0 A4 h+ () ×
n=0
∞
0
d2 fi (2; t − z) exp[ − 2k(n + 1)2]
+ A0 A3 h− () ×
∞
An5
n=0
∞
0
0
i=0
1
pi
i=0
0
∞
d x fi (x; z) exp[ − k(2n + 1)x]
d2fi (2; t − z) exp[ − 2kn2];
(59)
where 2 = a − x. Using the de0nition of ensemble-average function, Eqs. (10) – (23), %(z; t; ) could be written in the form %(z; t; ) = A0
∞ n=0
An5
1
pi Ei [(2n + 1)k; z]{−A4 h+ ()Ei [2(n + 1)k; t − z]
i=0
+ A3 h− ()Ei [2nk; t − z]}:
(60)
3.2. The average partial heat 3uxes The average of re5ectivity R and transmissivity T are R = A0
∞ n=0
An5
1 i=0
pi [A3 I1 Ei (2nk; a) − A4 I2 Ei [2(n + 1)k; a]]
(61)
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A.R. Deghiedy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 76 (2003) 359 – 372
and T = A0
∞
An5
n=0
1
pi Ei [(2n + 1)k; a](A3 I2 − A4 I1 )
(62)
i=0
with I1 =
1
d h+ ()
0
and
I2 =
0
1
d h− ():
(63)
Therefore, the ensemble-average partial heat 5uxes J ∓ for the general problem could be calculated from J − =
HK 2 AK 11 − HK 1 AK 21 AK 11 AK 22 − AK 12 AK 21
(64)
J + =
HK 1 AK 22 − HK 2 AK 12 ; AK 11 AK 22 − AK 12 AK 21
(65)
and
where the bar over each symbol represents the corresponding average quantities of the deterministic values of these quantities which are given by Eqs. (28) and (31).
4. Numerical results The ensemble-average partial heat 5uxes J ∓ at the boundaries of the considered medium are computed for di4erent values of i ; i and thickness t, for Rayleigh scattering and with internal energy source. The values of i and i are taken from the Refs. [5,7,8]. For the sake of comparison we use three weight functions with the special forms [12] W1 () = ; W2 () =
(66a)
H () ; 1 − k
(66b)
√
3 3 1+ : W3 () = 2 2
(66c)
The Chandrasekhar H -function H () is given in an approximate form by [12] H () = 1 −
!(1 − k) : 1 + k
(67)
A.R. Deghiedy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 76 (2003) 359 – 372
369
Table 1 Ensemble-averaged partial heat 5uxes J ∓ with ! = 0:9; si = di = Q(x) = 0 and F1 = 2; F2 = 0
t
W1 ()
W2 ()
W3 ()
J
J
J
J
J −
J +
Case 1 0.1 1.0 2.0 5.0 10 20
0.041859 0.235607 0.347606 0.451197 0.468660 0.469507
0.940487 0.621029 0.402189 0.118780 0.017168 0.000378
0.040239 0.235207 0.347851 0.451876 0.469516 0.470367
0.942946 0.622335 0.402878 0.118919 0.017185 0.000379
0.064207 0.239813 0.342093 0.437519 0.453755 0.454547
0.900815 0.599173 0.390193 0.116136 0.016833 0.000371
Case 2 0.1 1.0 2.0 5.0 10 20
0.037743 0.126883 0.195300 0.326545 0.419382 0.462914
0.944949 0.783935 0.665879 0.415776 0.197280 0.047613
0.036088 0.125807 0.194665 0.326688 0.420012 0.463747
0.947441 0.785867 0.667387 0.416533 0.197557 0.047664
0.060586 0.141157 0.203046 0.322613 0.408010 0.448396
0.904798 0.752460 0.640942 0.402718 0.192228 0.046612
Case 3 0.1 1.0 2.0 5.0 10 20
0.054201 0.231260 0.286884 0.372032 0.434646 0.464991
0.927269 0.617741 0.474677 0.265741 0.108751 0.018643
0.052689 0.230783 0.286715 0.372357 0.435322 0.465829
0.929623 0.619126 0.475749 0.266306 0.108954 0.018671
0.075006 0.236502 0.287465 0.365189 0.422488 0.450374
0.889095 0.594850 0.456981 0.256315 0.105274 0.018139
−
−
+
+
with !=
k=(1 + k) + ln(1 + k) : ln(1 + k) − yk=!
(68)
Table 1 gives the results of the ensemble-average partial heat 5uxes J ∓ for the transparent free-source medium with F1 = 2; F2 = 0 and for the single scattering albedo ! = 0:9. The results are done for di4erent cases of i and i . Case
0
1
0
1
1 2 3
99/100 99/10 101/20
11/100 11/10 101/20
10/99 10/99 2/101
100/11 100/11 200/101
Table 2 shows the same results as in Table 1, but for only di4used-re5ecting boundary medium, di = 0:5, and internal source Q(x) = (1 − !)e−x , for F1 = 1; F2 = 0 and di4erent values of ! and of thicknesses t. In Table 3 we calculate the ensemble-average partial heat 5uxes J ∓ for both specular and di4use re5ecting boundary medium, si = 0:5 and di = 0:25.
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Table 2 Ensemble-averaged partial heat 5uxes J ∓ with si = 0; di = 0:5; Q(x) = (1 − !)e−x , and F1 = 1; F2 = 0 !
W1 () −
t = 0:1 Case 1 0.95 0.90 0.80 Case 2 0.95 0.90 0.80 Case 3 0.95 0.90 0.80 t = 1:0 Case 1 0.95 0.90 0.80 Case 2 0.95 0.90 0.80 Case 3 0.95 0.90 0.80 t = 5:0 Case 1 0.95 0.90 0.80 Case 2 0.95 0.90 0.80 Case 3 0.95 0.90 0.80 t = 10:0 Case 1 0.95 0.90 0.80 Case 2 0.95 0.90 0.80 Case 3 0.95 0.90 0.80
W2 () +
−
W3 () +
J
J
J
J
J −
J +
0.150795 0.151114 0.151609
0.516949 0.517616 0.518851
0.149335 0.150816 0.149083
0.520148 0.518675 0.500863
0.156474 0.153785 0.148187
0.503534 0.499361 0.490612
0.149033 0.149086 0.149140
0.518008 0.518311 0.518914
0.147539 0.148777 0.146733
0.521238 0.519378 0.500891
0.154845 0.151904 0.145908
0.504470 0.499949 0.490617
0.156072 0.157172 0.159011
0.513691 0.515438 0.518605
0.154720 0.156904 0.156121
0.516790 0.516472 0.500740
0.161322 0.159380 0.155006
0.500676 0.497536 0.490571
0.241631 0.235608 0.228474
0.405260 0.400226 0.395558
0.241474 0.235701 0.222390
0.407467 0.401006 0.381674
0.241938 0.232462 0.219187
0.395929 0.386552 0.373781
0.184772 0.180308 0.176722
0.456451 0.453145 0.451434
0.183813 0.180153 0.173165
0.459280 0.454086 0.435571
0.188384 0.180909 0.171556
0.444574 0.436930 0.426530
0.240301 0.232519 0.159011
0.404162 0.396507 0.518605
0.240035 0.232584 0.216399
0.406498 0.397319 0.373143
0.241018 0.229770 0.213405
0.394337 0.382415 0.365229
0.388404 0.355146 0.322638
0.126232 0.104297 0.086453
0.390397 0.355843 0.311368
0.126642 0.104464 0.083486
0.379826 0.342938 0.304877
0.124417 0.101196 0.081805
0.289163 0.274675 0.261579
0.292056 0.285300 0.281596
0.289796 0.274983 0.253818
0.293580 0.285843 0.271752
0.286237 0.268401 0.249519
0.285567 0.275688 0.266166
0.323683 0.296772 0.271768
0.201526 0.185032 0.172223
0.324663 0.297175 0.263518
0.202804 0.185454 0.165686
0.319248 0.289075 0.258860
0.196168 0.177830 0.161953
0.410713 0.368432 0.330293
0.028595 0.016814 0.010069
0.412988 0.369196 0.318536
0.028669 0.016840 0.009726
0.400946 0.355217 0.311745
0.028260 0.016335 0.009531
0.356985 0.330859 0.306148
0.163642 0.154016 0.146823
0.358587 0.331443 0.295881
0.164337 0.154282 0.141761
0.350047 0.320405 0.290010
0.160633 0.149172 0.138896
0.375482 0.340086 0.307879
0.092844 0.081626 0.073731
0.377254 0.340709 0.297505
0.093383 0.081805 0.070942
0.367815 0.329043 0.291548
0.090565 0.078532 0.069353
A.R. Deghiedy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 76 (2003) 359 – 372 Table 3 Ensemble-averaged partial heat 5uxes J ∓ with si = 0:5 di = 0:25; Q(x) = (1 − !)e−x , and F1 = 1; F2 = 0 !
W1 () −
t = 0:1 Case 1 0.95 0.90 0.80 Case 2 0.95 0.90 0.80 Case 3 0.95 0.90 0.80 t = 1:0 Case 1 0.95 0.90 0.80 Case 2 0.95 0.90 0.80 Case 3 0.95 0.90 0.80 t = 5:0 Case 1 0.95 0.90 0.80 Case 2 0.95 0.90 0.80 Case 3 0.95 0.90 0.80 t = 10:0 Case 1 0.95 0.90 0.80 Case 2 0.95 0.90 0.80 Case 3 0.95 0.90 0.80
W2 () +
−
W3 () +
J
J
J
J
J −
J +
0.518293 0.512866 0.504850
0.801466 0.796328 0.788923
0.518025 0.513045 0.491533
0.805358 0.797716 0.761966
0.518697 0.506280 0.484627
0.784991 0.771281 0.746882
0.517388 0.511847 0.504038
0.801409 0.796102 0.788802
0.517101 0.512018 0.490842
0.805305 0.797489 0.761913
0.517872 0.505394 0.484016
0.784920 0.771097 0.746877
0.520933 0.516026 0.507764
0.801768 0.797296 0.789924
0.520719 0.516230 0.494082
0.805646 0.798692 0.762752
0.521117 0.509067 0.486935
0.785338 0.772133 0.747519
0.535561 0.495005 0.449599
0.645920 0.607457 0.566016
0.537072 0.495683 0.435001
0.649268 0.608668 0.545435
0.528714 0.481837 0.426848
0.631667 0.586229 0.533700
0.516148 0.494778 0.472570
0.710900 0.684976 0.668088
0.516638 0.495145 0.459283
0.705486 0.686236 0.644900
0.513472 0.485866 0.452155
0.686734 0.662560 0.631837
0.533923 0.488935 0.440150
0.636294 0.592337 0.546052
0.535506 0.489619 0.426010
0.639637 0.593529 0.526320
0.526777 0.475750 0.418125
0.622075 0.571508 0.515087
0.554252 0.464415 0.386667
0.218822 0.158812 0.115570
0.558690 0.465737 0.370587
0.220045 0.159186 0.110983
0.535552 0.442784 0.361139
0.213550 0.152446 0.108315
0.530829 0.475824 0.421574
0.463372 0.429332 0.396351
0.533314 0.476713 0.406698
0.465916 0.430229 0.381633
0.520052 0.460040 0.398180
0.452560 0.413774 0.373185
0.531347 0.457381 0.393378
0.303634 0.260906 0.229970
0.534712 0.458424 0.378876
0.305441 0.261477 0.221533
0.517000 0.439673 0.370482
0.295992 0.251121 0.216705
0.557593 0.461333 0.382144
0.051072 0.025651 0.013188
0.562523 0.462729 0.365905
0.051358 0.025713 0.012653
0.533905 0.438726 0.356342
0.049834 0.024594 0.012342
0.543388 0.466250 0.395772
0.269748 0.233225 0.201093
0.547198 0.467440 0.380047
0.271275 0.233749 0.193306
0.527243 0.446443 0.370857
0.263223 0.224243 0.188797
0.544765 0.458089 0.385117
0.142504 0.115226 0.097074
0.549065 0.459346 0.369589
0.143373 0.115490 0.093422
0.526631 0.437439 0.360491
0.138818 0.110748 0.091321
371
372
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It can be shown that the results given by the three weight functions are comparable with each other, especially at large !: 5. Conclusion The average partial heat 5uxes at the boundaries of a plane-parallel Markovian statistical medium containing an internal source with general boundary conditions obeying Rayleigh scattering law are calculated in terms of the average solution of the corresponding source-free problem. The Pomraning– Eddington approximation is used to solve the source-free problem in the deterministic case. The statistical formalism of Levermore et al. [1] is used to formulate the average solution of the source-free problem. Calculations are done for statistical (random) medium with specular and di4use re5ecting boundaries assuming an internal energy source of the form Q(x) = (1 − !)e−x for di4erent values of the single scattering albedo !. Acknowledgements The authors would like to thank Prof. S. A. El-Wakil for his encouragement and reviews of this work. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
Levermore CD, Pomraning GC, Sanzo DL, Wong J. J Math Phys 1986;27:2526. Vanderhaegen D. JQSRT 1986;36:557. Vanderhaegen D. JQSRT 1988;39:333. Levermore CD, Pomraning GC, Wong J. J Math Phys 1988;29:995. Pomraning GC. JQSRT 1988;40:479. Pomraning GC. Linear kinetic theory and particle transport in stochastic mixtures. Singapore: World Scienti0c, 1991. Adams ML, Larsen EW, Pomraning GC. JQSRT 1989;42:253. Vandehaegen D, Deutch C. J Stat Phys 1989;54:331. Pomraning GC. Transp Theory Stat Phys 1988;17:595. Pomraning GC. JQSRT 1989;41:103. Pomraning GC. Transp Theory Stat Phys 1986;15:773. El-Wakil SA, Aboulwafa EM, Degheidy AR, Radwan NK. Waves Random Media 1994;4:127. El-Wakil SA, Aboulwafa EM, Degheidy AR, El-Shahat A. Phys Scr 1994;50:135.
www.elsevier.com/locate/jqsrt
Radiative transfer in random media with Rayleigh scattering A.R. Deghiedy, M. Sallah ∗ , M.A. Abdou Theoretical Physics Research Group, Physics Department, Faculty of Science, Mansoura University, Mansoura, Egypt Received 17 January 2002; accepted 22 April 2002
Abstract This paper considers radiation transfer through a 0nite plane-parallel random medium consisting of two immiscible mixed materials. The mixing statistics of the two components of the medium is assumed to be described by the two-state homogeneous Markovian statistics. The problem treats a medium contains an internal source and has di4usely- and specularly-re5ecting boundaries obeying Rayleigh scattering law. The problem with this generalized boundary conditions is solved in terms of the solution of the corresponding free source problem with simple boundary conditions. Pomraning–Eddington approach is used to obtain an explicit solution of the simple problem in the deterministic case. A formalism, developed to treat radiative transfer in statistical mixtures, is used to obtain the ensemble-averaged solution. Numerical results for the ensemble-average partial heat 5uxes of the problem under consideration are obtained using three di4erent weight functions for the sake of comparison. ? 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction A subject of recent interest has been the problem of formulating and solving of linear transport problem in statistical medium composed of two or more immiscible mixed materials. [1– 4]. The applications of this theory can be found in astrophysics, vision, atmospheric and oceanic sensing and medical optics during Laser therapy of cancer tumors. Also the theory of nuclear reactors and elementary particle physics are connected with this theory. A complete and rigorous description has been given for time-independent problems in the absence of the scattering interaction in the underlying physics of the transport process, both for Markov statistics [1,2] and non-Markov statistics [3,4]. With scattering present, a general phenomenological model has been suggested [4] for Markov statistics, using the master equation approach [5]. Recently, much of the literatures has been directed towards the development of a mathematical formalism to estimate the ensemble-averaged ∗
Corresponding author. E-mail address: [email protected] (M. Sallah).
0022-4073/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 0 2 ) 0 0 0 6 3 - 8
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A.R. Deghiedy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 76 (2003) 359 – 372
intensity of radiation [1–11]. Levermore et al. [1] and Pomraning [11] developed a formalism, for a binary Markovian mixtures, to treat a certain class of time-independent transport problems in a planar geometry. In this paper, we consider the problem of stochastic radiative transfer in a 0nite plane-parallel medium of binary Markovian mixture with Rayleigh scattering phase function. In the next sections we solve this problem with di4usely- and specular-re5ecting boundaries and internal energy source in terms of the solution of the source-free problem with a simple boundary conditions in the deterministic case. We use the formalism obtained by Levermore et al. [1] and Pomraning [11] to average the solution given by Pomraning–Eddington method [12,13] for the deterministic case. The obtained average solution is used to give explicit analytical forms for the average re5ectivity R and transmissivity T , which are used to calculate the average partial heat 5uxes J ∓ at the boundaries of the medium. Section 4 tabulates numerical results for the average partial heat 5uxes for media with di4erent thicknesses and single scattering albedo. For the sake of comparison we use three di4erent weight functions. 2. Problem formulation The starting point of the analysis is the time-independent transport equation in a planar medium, which is given by [6]
@(z; E; ) + (z; E)(z; E; ) @z ∞ 1 = S(z; E; ) + dE d s (z; E → E; → )(z; E ; ); 0
−1
0 6 z 6 t:
(1)
Here (z; E; ) is the radiation intensity, with z; E; and representing the spatial, energy, and angular variables, respectively. The quantity (z; E) is the total cross section, s (z; E → E; → ) is the scattering kernel and S(z; E; ) represents an internal energy source. The quantities ; s and S in Eq. (1) are treated as discrete random variables, which obey the same statistics. The key to the analysis is the introduction of the optical depth variable, de0ned by [6] z x(z; E) = d z :(z ; E): (2) 0
In terms of x Eq. (1) becomes @I (x; E; ) + I (x; E; ) = Q(x; E; ) + @x
0
∞
dE
1
−1
0 6 x 6 a;
d !(x; E → E; → )I (z; E ; ); (3)
where I (x; E; ) = (z; E; );
(4)
!(x; E → E; → ) = s (z; E → E; → )= (z; E)
(5)
which is the single-scattering albedo of the medium,
A.R. Deghiedy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 76 (2003) 359 – 372
361
and Q(x; E; ) = S(z; E; )= (z; E):
(6)
The optical thickness a of the medium is given by t a(t; E) = d z (z ; E): 0
(7)
In this paper we consider the monoenergetic radiative transfer equation for a 0nite slab medium, viz. @ ! 1 P(; )I (x; ) d + Q(x); −1 6 6 1; 0 6 x 6 a (8) + 1 I (x; ) = @x 2 −1 subjected to boundary conditions I (0; ) = F1 + s1 I (0; −) + d1 J − ;
¿ 0;
(9a)
I (a; −) = F2 + s2 I (a; ) + d2 J + ;
¿ 0;
(9b)
where F1 and F2 are the externally-incident 5uxes on the left and right boundaries, respectively, si and di are the specular and di4use re5ectivities of the boundaries, respectively. We now assume that ; s and S obey the same statistics in the sense that ! = s = and Q = S=
are non-stochastic [6]. This means that ! and Q take the same values inside the two immiscible 5uids of the medium. Hence the transport problem described by Eqs. (8) and (9) is only stochastic through the optical depth variable x and the optical size of the system a. The statistics of the problem are entirely described by the joint probability density P(x; a; z; t) , de0ned such that P d x da is the probability that for a given geometric position z and a given geometric system thickness t, the position z corresponds to an optical depth lying between x and x + d x and the system thickness t corresponds to an optical depth lying between a and a + da. The ensemble-averaged intensity is thus given by [1,6] ∞ ∞ I (x; E; ) = da d x P(x; a; z; t)I (x; a; E; ): (10) 0
0
All of the statistical informations are embodied in the joint probability density function P(x; a; z; t). From the transport point of view, this function is assumed to be known. In the special case of binary homogeneous Markov statistics, one has the near separable form [5] denoting the two materials making up the binary mixture by subscripts 0 and 1 P(x; a; z; t) =
1
pi fi (x; z)fi (a − x; t − z):
(11)
i=0
Here pi represents the probability of 0nding material i at any point in the system. In terms of mean slab thickness i of the alternating slabs of the two materials making up the planar system, one has [1] pi =
i : 0 + 1
(12)
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A.R. Deghiedy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 76 (2003) 359 – 372
For homogeneous Markov statistics, these slab thicknesses are exponentially distributed with mean (average) i for material i [6]. The function fi (x; z) in Eq. (11) is the probability density function de0ned such that fi d x is the probability that the planar system has an optical depth lying between x and x + d x, given that the geometric depth is z and given that the point z lies in material i. For the case of homogeneous binary Markov statistics being discussed, the function fi (x; z) is known in closed analytic form [6,10]. It is given as the derivative of the cumulative distribution function Fi (x; z) de0ned as fi (x; z) =
@Fi (x; z) : @x
(13)
By ordering the cross sections of the two materials making up the binary mixture such that
1 ¿ 0 , the function Fi (x; z) is given by 0; x ¡ 0 z; (14) Fi (x; z) = 1; x ¿ 1 z and F0 (x; z) = exp(− ) + 2g( ; u);
0 z ¡ x ¡ 1 z;
F1 (x; z) = 1 − exp(−u) − 2g(u; );
0 z ¡ x ¡ 1 z;
where g(u; ) = exp(−u)
√
u
0
x2 d x I1 (2x) exp − u
(15a) (15b)
(16)
with I1 (x) denoting the modi0ed Bessel function. The quantities u and are given by u=
x − 0 z ; 1 ( 1 − 0 )
=
1 z − x : 0 ( 1 − 0 )
(17)
In view of Eq. (11), Eq. (10) becomes ∞ 1 I (x; E; ) = pi d x fi (x; z) i=0
0
0
∞
da fi (a − x; t − z)I (x; a; E; ):
(18)
Such an average is easily computed [6] for homogeneous binary Markovian statistics. We consider a pure exponential in optical depth space exp(−kx) with k being a constant, and ask for the ensemble average of this exponential. If we denote this ensemble average by Ei (k; z) we have ∞ Ei (k; z) = d x fi (x; z) exp(−kx): (19) 0
It was found by Pomraning [6] that Ei (k; z) = !i exp(−–+ z) + (1 − !i ) exp(−–− z)
(20)
A.R. Deghiedy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 76 (2003) 359 – 372
with Ei (Nk; 0) = 1 k j + (1=c ) − –+ ; !i = (–− − –+ ) and 1 ∓ 2–± = ( 1 + 0 )k + c
i = 0; 1; j = 0; 1; i = j ( 1 − 0 )2 k 2 −
2k 1 (p1 − p0 )( 1 − 0 ) + 2 c c
363
(21)
(22)
where 1 1 1 = + : c 0 1
(23)
The above analysis is used in the next sections to evaluate the ensemble-average solution of the problem under consideration. 3. The solution of the problem in terms of source-free problem Eq. (8) with the generalized boundary conditions (9) can be solved by connecting it with the solution of the corresponding source-free problem ! 1 @ + 1 %(x; ) = P(; )%(x; ) d ; −1 6 6 1; 0 6 x 6 a (24) @x 2 −1 with simple boundary conditions as %(0; ) = 1 + s1 %(0; −);
(25a)
%(a; −) = s2 %(a; ):
(25b)
The connection between the two problems may be demonstrated as follows. Eq. (24) has two adjoint equations, viz ! 1 @ − + 1 %(x; −) = P(; )%(x; − ) d ; (26a) @x 2 −1 ! 1 @ P(; )%(a − x; ) d : (26b) − + 1 %(a − x; ) = @x 2 −1 Multiplying Eq. (26a) by I (x; ) and Eq. (8) by %(x; −), subtracting the resultant equations, and then integrating over ∈ [ − 1; 1] and x ∈ [0; a] we get A11 J + + A12 J − = H1 ;
(27)
where A11 = −d2 T;
(28a)
A12 = 1 − d1 R;
(28b)
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A.R. Deghiedy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 76 (2003) 359 – 372
H1 = Q1 + F2 T + F1 R; Q1 =
1
0
T=
Q(x)E(x) d x;
(28d)
%(0; −) d;
(29a)
%(a; ) d:
(29b)
0
R=
a
(28c)
1
0
Further, multiplying Eq. (8) by %(a−x; ) and Eq. (26b) by I (x; ), subtracting the resultant equations and then integrating over ∈ [ − 1; 1] and x ∈ [0; a] one 0nds A21 J + + A22 J − = H2 ;
(30)
where A21 = 1 − d2 R + 2(s1 − s2 )R;
(31a)
A22 = −d1 T + 2(s1 − s2 )T;
(31b)
H2 = Q2 + F1 T + F2 R;
(31c)
Q2 =
a
Q(a − x)E(x) d x;
0
1
(31d)
%(x; ) d:
(32)
The partial heat 5uxes J ∓ are de0ned by 1 J− = I (0; −) d;
(33a)
E(x) =
+
−1
J =
0
1
0
I (a; ) d:
(33b)
Eqs. (27) and (30) give the solutions of J − and J + in terms of R; T; Q1 ; and Q2 . The ensembleaverage of the partial heat 5uxes J ∓ can be obtained from the calculation of the corresponding stochastic values R; T ; Q1 ; and Q2 , where 1 R = %(0; −) d; (34) 0
T =
0
1
%(a; ) d;
where %(x; ) is the ensemble-averaged solution of the source-free problem.
(35)
A.R. Deghiedy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 76 (2003) 359 – 372
365
3.1. Solution of the source-free problem for Rayleigh scattering In this work we consider the Rayleigh scattering phase function P(; ) which has the form P(; ) = 38 [(3 − 2 ) + (32 − 1)2 ]: Then Eq. (24) becomes 1 @ 3! 2 2 2 (3 − )E(x) + (3 − 1) %(x; ) d : + 1 %(x; ) = @x 16 −1
(36)
(37)
We use Pomraning–Eddington approximation [12,13] to solve Eq. (37) %(x; ) = ((x; )E(x) + o(x; )F(x);
(38)
where E(x) and F(x) are the radiant energy and the net 5ux, respectively and ((x; ) and o(x; ) are even and odd functions in which are assumed to be slowly varying in x and normalized as 1 1 d ((x; ) = d o(x; ) = 1: (39) −1
−1
Substituting Eq. (38) in Eq. (37), and integrating over ∈ [ − 1; 1]; one gets dF(x) + *E(x) = 0: dx
(40)
Multiplying Eq. (37) by and integrating over ∈ [ − 1; 1] gives d [D(x)E(x)] + F(x) = 0; dx
(41)
*=1−!
(42)
where
and
D(x) =
1
−1
d 2 ((x; ):
(43)
The even and odd functions are, respectively, given by ((x; ) =
3! (3 − 2 ) + (*=k 2 )(32 − 1) ; 16 1 − k 2 2
(44)
o(x; ) =
3!k 2 (3 − 2 ) + (*=k 2 )(32 − 1) ; 16* 1 − k 2 2
(45)
where k 2 = *=D:
(46)
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A.R. Deghiedy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 76 (2003) 359 – 372
Using Eq. (44) in Eq. (43) for |k| ¡ 1 leads to the transcendental equation 1+k 2ky = ln ! 1−k with
8 y= 9
k 4 − 3!k 2 =8 + 9*!=8 k 4 − (1 + *)k 2 =3 + *
(47)
(48)
since D(x) is a slowly varying function in x, Eqs. (40) and (41) lead to the di4usion equation d 2 E(x) − k 2 E(x) = 0 d x2
(49)
which has the solution E(x) = Aekx + Be−kx
(50)
and F(x) will be * F(x) = − [Aekx − Be−kx ]; k
(51)
where A and B are constants to be determined. Using Eqs. (44) – (51) in Eq. (38), we get the solution of the free-source problem %(x; ) = Ah+ ()ekx + Bh− ()e−kx ;
(52)
where h± () =
3! (3 − 2 ) + (*=k 2 )(32 − 1) : 16 1 ± k
(53)
To calculate the constants A and B we use a weight function W () which forces the boundary conditions to be ful0lled, 1 d W ()[%(0; ) − 1 − s1 %(0; −)] = 0; ¿ 0; (54a) 0
1
0
d W ()[%(a; −) − s2 %(a; )] = 0;
¿ 0:
(54b)
Using Eq. (52) we obtain (1 − Cs1 )A + (C − s1 )B = I0 =I+ ;
(55a)
(C − s2 )A + (1 − Cs2 )e−2ka B = 0;
(55b)
where
I0 =
0
1
d W ();
I± =
0
1
d W ()h± ()
and
C = I− =I+ :
(56)
A.R. Deghiedy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 76 (2003) 359 – 372
367
Then the intensity %(x; ) takes the form %(x; ) = −A0 {A4 h+ ()
∞
An5 exp[ − k(2a(n + 1) − x)]
n=0
− A3 h− ()
∞
An5 exp[ − k(2na + x)]};
(57)
n=0
where A1 = 1 − Cs1 ;
A2 = C − s1 ;
(58a)
A3 = C − s2 ;
A4 = 1 − Cs2 ;
(58b)
A5 =
A1 A4 ; A2 A3
I0 : I+ A 2 A 3
A0 =
(58c)
Therefore, the ensemble-averaged solution can be obtained by using Eq. (57) in Eq. (10) ∞ ∞ 1 n A5 pi d x fi (x; z)exp[ − k(2n + 1)x] %(z; t; ) = −A0 A4 h+ () ×
n=0
∞
0
d2 fi (2; t − z) exp[ − 2k(n + 1)2]
+ A0 A3 h− () ×
∞
An5
n=0
∞
0
0
i=0
1
pi
i=0
0
∞
d x fi (x; z) exp[ − k(2n + 1)x]
d2fi (2; t − z) exp[ − 2kn2];
(59)
where 2 = a − x. Using the de0nition of ensemble-average function, Eqs. (10) – (23), %(z; t; ) could be written in the form %(z; t; ) = A0
∞ n=0
An5
1
pi Ei [(2n + 1)k; z]{−A4 h+ ()Ei [2(n + 1)k; t − z]
i=0
+ A3 h− ()Ei [2nk; t − z]}:
(60)
3.2. The average partial heat 3uxes The average of re5ectivity R and transmissivity T are R = A0
∞ n=0
An5
1 i=0
pi [A3 I1 Ei (2nk; a) − A4 I2 Ei [2(n + 1)k; a]]
(61)
368
A.R. Deghiedy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 76 (2003) 359 – 372
and T = A0
∞
An5
n=0
1
pi Ei [(2n + 1)k; a](A3 I2 − A4 I1 )
(62)
i=0
with I1 =
1
d h+ ()
0
and
I2 =
0
1
d h− ():
(63)
Therefore, the ensemble-average partial heat 5uxes J ∓ for the general problem could be calculated from J − =
HK 2 AK 11 − HK 1 AK 21 AK 11 AK 22 − AK 12 AK 21
(64)
J + =
HK 1 AK 22 − HK 2 AK 12 ; AK 11 AK 22 − AK 12 AK 21
(65)
and
where the bar over each symbol represents the corresponding average quantities of the deterministic values of these quantities which are given by Eqs. (28) and (31).
4. Numerical results The ensemble-average partial heat 5uxes J ∓ at the boundaries of the considered medium are computed for di4erent values of i ; i and thickness t, for Rayleigh scattering and with internal energy source. The values of i and i are taken from the Refs. [5,7,8]. For the sake of comparison we use three weight functions with the special forms [12] W1 () = ; W2 () =
(66a)
H () ; 1 − k
(66b)
√
3 3 1+ : W3 () = 2 2
(66c)
The Chandrasekhar H -function H () is given in an approximate form by [12] H () = 1 −
!(1 − k) : 1 + k
(67)
A.R. Deghiedy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 76 (2003) 359 – 372
369
Table 1 Ensemble-averaged partial heat 5uxes J ∓ with ! = 0:9; si = di = Q(x) = 0 and F1 = 2; F2 = 0
t
W1 ()
W2 ()
W3 ()
J
J
J
J
J −
J +
Case 1 0.1 1.0 2.0 5.0 10 20
0.041859 0.235607 0.347606 0.451197 0.468660 0.469507
0.940487 0.621029 0.402189 0.118780 0.017168 0.000378
0.040239 0.235207 0.347851 0.451876 0.469516 0.470367
0.942946 0.622335 0.402878 0.118919 0.017185 0.000379
0.064207 0.239813 0.342093 0.437519 0.453755 0.454547
0.900815 0.599173 0.390193 0.116136 0.016833 0.000371
Case 2 0.1 1.0 2.0 5.0 10 20
0.037743 0.126883 0.195300 0.326545 0.419382 0.462914
0.944949 0.783935 0.665879 0.415776 0.197280 0.047613
0.036088 0.125807 0.194665 0.326688 0.420012 0.463747
0.947441 0.785867 0.667387 0.416533 0.197557 0.047664
0.060586 0.141157 0.203046 0.322613 0.408010 0.448396
0.904798 0.752460 0.640942 0.402718 0.192228 0.046612
Case 3 0.1 1.0 2.0 5.0 10 20
0.054201 0.231260 0.286884 0.372032 0.434646 0.464991
0.927269 0.617741 0.474677 0.265741 0.108751 0.018643
0.052689 0.230783 0.286715 0.372357 0.435322 0.465829
0.929623 0.619126 0.475749 0.266306 0.108954 0.018671
0.075006 0.236502 0.287465 0.365189 0.422488 0.450374
0.889095 0.594850 0.456981 0.256315 0.105274 0.018139
−
−
+
+
with !=
k=(1 + k) + ln(1 + k) : ln(1 + k) − yk=!
(68)
Table 1 gives the results of the ensemble-average partial heat 5uxes J ∓ for the transparent free-source medium with F1 = 2; F2 = 0 and for the single scattering albedo ! = 0:9. The results are done for di4erent cases of i and i . Case
0
1
0
1
1 2 3
99/100 99/10 101/20
11/100 11/10 101/20
10/99 10/99 2/101
100/11 100/11 200/101
Table 2 shows the same results as in Table 1, but for only di4used-re5ecting boundary medium, di = 0:5, and internal source Q(x) = (1 − !)e−x , for F1 = 1; F2 = 0 and di4erent values of ! and of thicknesses t. In Table 3 we calculate the ensemble-average partial heat 5uxes J ∓ for both specular and di4use re5ecting boundary medium, si = 0:5 and di = 0:25.
370
A.R. Deghiedy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 76 (2003) 359 – 372
Table 2 Ensemble-averaged partial heat 5uxes J ∓ with si = 0; di = 0:5; Q(x) = (1 − !)e−x , and F1 = 1; F2 = 0 !
W1 () −
t = 0:1 Case 1 0.95 0.90 0.80 Case 2 0.95 0.90 0.80 Case 3 0.95 0.90 0.80 t = 1:0 Case 1 0.95 0.90 0.80 Case 2 0.95 0.90 0.80 Case 3 0.95 0.90 0.80 t = 5:0 Case 1 0.95 0.90 0.80 Case 2 0.95 0.90 0.80 Case 3 0.95 0.90 0.80 t = 10:0 Case 1 0.95 0.90 0.80 Case 2 0.95 0.90 0.80 Case 3 0.95 0.90 0.80
W2 () +
−
W3 () +
J
J
J
J
J −
J +
0.150795 0.151114 0.151609
0.516949 0.517616 0.518851
0.149335 0.150816 0.149083
0.520148 0.518675 0.500863
0.156474 0.153785 0.148187
0.503534 0.499361 0.490612
0.149033 0.149086 0.149140
0.518008 0.518311 0.518914
0.147539 0.148777 0.146733
0.521238 0.519378 0.500891
0.154845 0.151904 0.145908
0.504470 0.499949 0.490617
0.156072 0.157172 0.159011
0.513691 0.515438 0.518605
0.154720 0.156904 0.156121
0.516790 0.516472 0.500740
0.161322 0.159380 0.155006
0.500676 0.497536 0.490571
0.241631 0.235608 0.228474
0.405260 0.400226 0.395558
0.241474 0.235701 0.222390
0.407467 0.401006 0.381674
0.241938 0.232462 0.219187
0.395929 0.386552 0.373781
0.184772 0.180308 0.176722
0.456451 0.453145 0.451434
0.183813 0.180153 0.173165
0.459280 0.454086 0.435571
0.188384 0.180909 0.171556
0.444574 0.436930 0.426530
0.240301 0.232519 0.159011
0.404162 0.396507 0.518605
0.240035 0.232584 0.216399
0.406498 0.397319 0.373143
0.241018 0.229770 0.213405
0.394337 0.382415 0.365229
0.388404 0.355146 0.322638
0.126232 0.104297 0.086453
0.390397 0.355843 0.311368
0.126642 0.104464 0.083486
0.379826 0.342938 0.304877
0.124417 0.101196 0.081805
0.289163 0.274675 0.261579
0.292056 0.285300 0.281596
0.289796 0.274983 0.253818
0.293580 0.285843 0.271752
0.286237 0.268401 0.249519
0.285567 0.275688 0.266166
0.323683 0.296772 0.271768
0.201526 0.185032 0.172223
0.324663 0.297175 0.263518
0.202804 0.185454 0.165686
0.319248 0.289075 0.258860
0.196168 0.177830 0.161953
0.410713 0.368432 0.330293
0.028595 0.016814 0.010069
0.412988 0.369196 0.318536
0.028669 0.016840 0.009726
0.400946 0.355217 0.311745
0.028260 0.016335 0.009531
0.356985 0.330859 0.306148
0.163642 0.154016 0.146823
0.358587 0.331443 0.295881
0.164337 0.154282 0.141761
0.350047 0.320405 0.290010
0.160633 0.149172 0.138896
0.375482 0.340086 0.307879
0.092844 0.081626 0.073731
0.377254 0.340709 0.297505
0.093383 0.081805 0.070942
0.367815 0.329043 0.291548
0.090565 0.078532 0.069353
A.R. Deghiedy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 76 (2003) 359 – 372 Table 3 Ensemble-averaged partial heat 5uxes J ∓ with si = 0:5 di = 0:25; Q(x) = (1 − !)e−x , and F1 = 1; F2 = 0 !
W1 () −
t = 0:1 Case 1 0.95 0.90 0.80 Case 2 0.95 0.90 0.80 Case 3 0.95 0.90 0.80 t = 1:0 Case 1 0.95 0.90 0.80 Case 2 0.95 0.90 0.80 Case 3 0.95 0.90 0.80 t = 5:0 Case 1 0.95 0.90 0.80 Case 2 0.95 0.90 0.80 Case 3 0.95 0.90 0.80 t = 10:0 Case 1 0.95 0.90 0.80 Case 2 0.95 0.90 0.80 Case 3 0.95 0.90 0.80
W2 () +
−
W3 () +
J
J
J
J
J −
J +
0.518293 0.512866 0.504850
0.801466 0.796328 0.788923
0.518025 0.513045 0.491533
0.805358 0.797716 0.761966
0.518697 0.506280 0.484627
0.784991 0.771281 0.746882
0.517388 0.511847 0.504038
0.801409 0.796102 0.788802
0.517101 0.512018 0.490842
0.805305 0.797489 0.761913
0.517872 0.505394 0.484016
0.784920 0.771097 0.746877
0.520933 0.516026 0.507764
0.801768 0.797296 0.789924
0.520719 0.516230 0.494082
0.805646 0.798692 0.762752
0.521117 0.509067 0.486935
0.785338 0.772133 0.747519
0.535561 0.495005 0.449599
0.645920 0.607457 0.566016
0.537072 0.495683 0.435001
0.649268 0.608668 0.545435
0.528714 0.481837 0.426848
0.631667 0.586229 0.533700
0.516148 0.494778 0.472570
0.710900 0.684976 0.668088
0.516638 0.495145 0.459283
0.705486 0.686236 0.644900
0.513472 0.485866 0.452155
0.686734 0.662560 0.631837
0.533923 0.488935 0.440150
0.636294 0.592337 0.546052
0.535506 0.489619 0.426010
0.639637 0.593529 0.526320
0.526777 0.475750 0.418125
0.622075 0.571508 0.515087
0.554252 0.464415 0.386667
0.218822 0.158812 0.115570
0.558690 0.465737 0.370587
0.220045 0.159186 0.110983
0.535552 0.442784 0.361139
0.213550 0.152446 0.108315
0.530829 0.475824 0.421574
0.463372 0.429332 0.396351
0.533314 0.476713 0.406698
0.465916 0.430229 0.381633
0.520052 0.460040 0.398180
0.452560 0.413774 0.373185
0.531347 0.457381 0.393378
0.303634 0.260906 0.229970
0.534712 0.458424 0.378876
0.305441 0.261477 0.221533
0.517000 0.439673 0.370482
0.295992 0.251121 0.216705
0.557593 0.461333 0.382144
0.051072 0.025651 0.013188
0.562523 0.462729 0.365905
0.051358 0.025713 0.012653
0.533905 0.438726 0.356342
0.049834 0.024594 0.012342
0.543388 0.466250 0.395772
0.269748 0.233225 0.201093
0.547198 0.467440 0.380047
0.271275 0.233749 0.193306
0.527243 0.446443 0.370857
0.263223 0.224243 0.188797
0.544765 0.458089 0.385117
0.142504 0.115226 0.097074
0.549065 0.459346 0.369589
0.143373 0.115490 0.093422
0.526631 0.437439 0.360491
0.138818 0.110748 0.091321
371
372
A.R. Deghiedy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 76 (2003) 359 – 372
It can be shown that the results given by the three weight functions are comparable with each other, especially at large !: 5. Conclusion The average partial heat 5uxes at the boundaries of a plane-parallel Markovian statistical medium containing an internal source with general boundary conditions obeying Rayleigh scattering law are calculated in terms of the average solution of the corresponding source-free problem. The Pomraning– Eddington approximation is used to solve the source-free problem in the deterministic case. The statistical formalism of Levermore et al. [1] is used to formulate the average solution of the source-free problem. Calculations are done for statistical (random) medium with specular and di4use re5ecting boundaries assuming an internal energy source of the form Q(x) = (1 − !)e−x for di4erent values of the single scattering albedo !. Acknowledgements The authors would like to thank Prof. S. A. El-Wakil for his encouragement and reviews of this work. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
Levermore CD, Pomraning GC, Sanzo DL, Wong J. J Math Phys 1986;27:2526. Vanderhaegen D. JQSRT 1986;36:557. Vanderhaegen D. JQSRT 1988;39:333. Levermore CD, Pomraning GC, Wong J. J Math Phys 1988;29:995. Pomraning GC. JQSRT 1988;40:479. Pomraning GC. Linear kinetic theory and particle transport in stochastic mixtures. Singapore: World Scienti0c, 1991. Adams ML, Larsen EW, Pomraning GC. JQSRT 1989;42:253. Vandehaegen D, Deutch C. J Stat Phys 1989;54:331. Pomraning GC. Transp Theory Stat Phys 1988;17:595. Pomraning GC. JQSRT 1989;41:103. Pomraning GC. Transp Theory Stat Phys 1986;15:773. El-Wakil SA, Aboulwafa EM, Degheidy AR, Radwan NK. Waves Random Media 1994;4:127. El-Wakil SA, Aboulwafa EM, Degheidy AR, El-Shahat A. Phys Scr 1994;50:135.