- Email: [email protected]
Nongray radiative transfer in a planar medium exposed to a collimated flux
J. Quant. Spectrosc. Radiat. Transfer. Vol. 10, pp. 465~.85. Pergamon Press 1970. Printed in Great Britain
NONGRAY RADIATIVE TRANSFER IN A PLANAR MEDIUM EXPOSED TO A COLLIMATED FLUX A. L.
CROSBIE
Department of Mechanical and Aerospace Engineering, University of Missouri at Rolla, Rolla, Missouri 65401
and R . VISKANTA School of Mechanical Engineering, Purdue University, Lafayette, Indiana 47907 (Received 2 September 1969) Abstract -The problem of energy transfer through a plane layer of an absorbing and emitting medium exposed on one side to a monochromatic collimated radiation flux is analyzed to determine the effect of line or band shape on radiative transfer. The analysis is restricted to a medium in local thermodynamic equilibrium and an absorption coefficient xv of the Milne-Eddington type, i.e. rcv(T) = ct(v)fl(T). For the case when ~¢~consists of an array of equal intensity, nonoverlapping bands or lines, the solution for the source function and the radiative flux are expressed in terms of universal functions. Numerical solutions for the Chandrasekhar X and Y-functions are presented in graphical and tabular form for the rectangular, triangular, exponential, Lorentz and Doppler profiles.
INTRODUCTION
BECAUSEof the mathematical complexity of the nongray analysis, the gray approximation is usually employed in most non-isothermal radiative transfer studies. While the use of a mean absorption coefficient appears to be a logical compromise to the nongray problem, the validity of this approach has yet to be established. Even when the medium is isothermal, the analysis is complicated by the lack of detail information on the spectral absorption coefficient. In this paper we investigate the effect of line or band shape on the radiative transfer. Because of its simplicity and its relation to other problems, nongray radiative transfer in a plane layer exposed to a collimated flux is studied. The medium can absorb and emit but is unable to scatter thermal radiation. The other modes of heat transfer, i.e., conduction and convection have been neglected to simplify the analysis. A review of the analytical investigations of radiative transfer in a nongray absorbing and emitting media is found in Ref. 1. Since the integral equations describing noncoherent scattering are similar to those of radiative transfer in a nongray absorbing and emitting medium under the conditions of radiative equilibrium, a discussion and review of noncoherent scattering is appropriate. In the theory of spectral line formulation by noncoherent scattering the simplest model is that of complete redistribution. In coherent scattering, the absorbed photon is assumed 465
466
A . L . CROSBIEand R. VISKANTA
to be reemitted with no change in frequency, while with complete redistribution the frequencies of the absorbed and reemitted photons are completely unrelated. The work of JEEFERIES(2) provides a comprehensive review of the physical phenomena involved. The formulation of the complete redistribution model often results in a linear integral equation for the source function similar to that derived for the coherent scattering case. Implicit in such a formulation is the assumption that the source function is independent of frequency. The line shape is accounted for in the kernel and the nonhomogeneous term. BIBERMAN (3) and HOLSTEIN (4 5) were the first to obtain solutions for the source function. While Biberman has replaced the integral equation for a planar medium by a set of linear equations, Holstein has employed a variational approach for planar and cylindrical geometries. More recent work includes that of HEARN,"~ 7) HUMMER et al. (s ~ and ('UPERMAN et al., (ts ~6) ENGELMANN(17) and GIUFFRE and ENGELMANN. (18} SOBOLEV,(l'~i AVRETT and KALKOFEN(2°) and FINN (2~) have studied line radiation in multilevel atoms. IVANOV(22 2,*) IVANOV and SHEHERBAKOV, (25 26) IVANOV and NAGIRNER (27 29) and SOBOLEV(3°) have used the analytical methods of SOBOLEV(3~) for resonance radiation in finite and semi-infinite atmospheres, lvanov and associates have developed generalized 1t, X and Y functions for a single line. These results are summarized and extended by HEASLET and WARMING. (32) A couple of additional references worth noting are those of YAKOVKIN and ZELDINA (33) and KOSTIC. (34) ANALYSIS
Physical model and energy equation The physical system considered consists of a plane layer of absorbing and emitting medium of thickness L. One side of the layer is exposed to a collimated monochromatic flux Fo of frequency vo and direction po(COS0o), while on the other side there is no incident radiation. The absorption coefficient ~ is of the Milne Eddington type, i.e. ~ = ~(v)fl(T). The dimensionless function ~(v) can range from zero to unity. The index of refraction of the medium is considered to be unity, and the medium is assumed to be in local thermodynamic equilibrium. For the physical model and form of spectral absorption coefficient considered, the local radiative flux becomes ,~-(z) = Fol~oexp[-~(Vo)Z/t~o]+2iiE~v(t)sign(z-t)~(v)E2[ct(v),z-tl]dtdv
(1)
0 0
where Ebb(T) = 7tlb,(T) is the Planck function, E,(t) is the exponential function of order n, and r = J'~ fl(T)dx and ro = f~6 fl(T)dx are respectively the optical depth and optical thickness. The conservation of radiant energy equation under the conditions of radiative equilibrium becomes
f ~(v)Ehv(T) dv = ~Fo~(Vo) 1 e x p [ - ~(Vo)ro//Lo]+ 12if Eb~(t)~2(v)El[~(v)lz- t]] dt dr.
0
(2)
o 0
Five different models for ~(v) have been selected for investigation to determine the effect of ~(v) on the radiative transfer. The rectangular, triangular (3s) and exponential (36 371
N o n g r a y r a d i a t i v e transfer in a p l a n a r m e d i u m exposed to a c o l l i m a t e d flux
467
shapes are usually associated with bands, while the D O P P L E R (38) and L O R E N T Z ( 3 8 - 3 9 ) profiles characterize spectral lines. These models have been selected because they approximate the absorption characteristics of a wide range-of substances. The mathematical definition of the models is given in Table 1. Each model is the superposition of m symmetric lines or bands. In Table 1, c~i is the amplitude, v~ is the center, di is the damping factor and Av~ is the width of the i-th band or line. The unit step function is defined as u(t) = 0 for t < 0 and u(t) = 1 for t > 0. In all models frequency vi+ 1 is larger than frequency v~. T A B L E 1. M O D E L S FOR THE ABSORPTION COEFFICIENT ct(v)
Model
M a t h e m a t i c a l definition ra
Rectangular
~Zu(v - v, + ½av,) - u(v - vl - ½Av,)]
ct(v) = i=1
Triangular
ct(v) = ~, i=1
Exponential
ct(v) =
t
t
~ 1- - - I v - v~l Eu(v-vi+½Av~)-u(v- vi- ½A~'i)] Av i
cti exp[ - dl[ v - vii ] i=l
Doppler
cti e x p [ - d,(v
~(v) =
-
vi)2]
i=1
Lorentz
l
ct(v) = i=1
o~] + d~v- vl) 2
Narrow band approximation The narrow band approximation assumes that Planck's function does not vary greatly across the band or line. Mathematically, this approximation can be stated as follows
f Eb~(t)f(v, t) dv ~- Ebi(t) f f ( v , t) dv R
(3)
R
where R is the region of the spectrum over which the integration is performed and Ebi is the function Ebv evaluated at the center of the band or line. The narrow band approximation is obviously more applicable to lines than to bands. Since many bands are too broad for this approximation to apply, an alternate procedure must be worked out. A broad band can be divided into smaller bands which can be considered narrow. The narrow band approximation may be applied to the individual lines which compose the band if there are not too many important lines. Dividing the portion of the spectrum which is capable of absorbing radiation [ct(v) ~ 0] into m regions and applying the narrow band approximation to each region, the local radiative flux becomes
mf ~o
~(z) = Fop 0 e x p [ - ct(Vo)Z/po] + 2 ,~=1=Vi
0
Eb,(t) sign(z-- t)K/2([z-- tl) dt.
(4)
468
A . L . CROSB1Eand R. VISKANTA
The kernels K~ and quantity 7~ are defined as follows K~(r) = ~ ~(V)Ez[0~(vjz ] dv/"7;
(5)
Ri
~(v) dv
Yi
(6)
,d
Ri
where R i is the i-th region of the spectrum. Note that K~(0) ~ 1. Similarly, the energy equation (2) becomes 7~Eb~[T(z)] =
Fo,:~(vo) exp[ -:~(Vo)r/po] + 2
i=1
,,~Eb~[r(t)lK~(lr - t]) dt 0
(7)
i=
where
K](z) = [ 0~2(V)El[~(v)r] dv/Ti. Ri
If the absorption coefficient in each region is identical in shape [Av~ = Av 2 . . . . . Av., = Av or dl = d2 . . . . . d,. = d] and intensity [e 1 = c~2 . . . . . ~., = 1] and the profiles are nonoverlapping, the quantities K](t), K~2(t) and ),~ are independent of i [K ~(t) = Ki~(t), K d t ) = K~(t) and 7 = 7i where i = 1, 2 . . . . , m]. Thus, equation t8) reduces to the linear integral equation
J('r,Z, Zo) = e ~":+2
J(t'z'z°)Kx(lr-t[)dt
(9)
0
where z = po/~(Vo) and m
J(z, z, z o) = 47 ~ Ehi[T(r)]/'~(vo)Fo.
t l0)
i=1
Equation (9) represents the dimensionless energy equation. The function J(r, z, to) is a universal function of the optical depth, the shape of the absorption coefficient and z. The case ofz > 1 for the rectangular profile has no physical meaning, since Po < 1 and c~(vo) - I. However, for the other profiles the values o f z greater than unity imply that ,:~(v0) < 1. Once J(z, z, %) is known, the t e m p e r a t u r e distribution in the m e d i u m can be obtained by solving the transcendental equation (10). The assumption that the a b s o r p t i o n coefficient in each spectral region is identical in shape and intensity reduces equation (4) for the radiative flux to ,~(z) = Fo/~oQc(r) wh'ere ro o
ro) s i g n ( r - t)K2([z- t[) dt.
(1 1)
o
It can readily be shown from the conservation of energy that Q~(z) is a constant, i.e. independent of r.
Nongray radiative transfer in a planar medium exposed to a collimated flux
469
Representation of Kernels K.(t) Substitution of the definition of El(t) into expression for Kl(t) with the assumption that the absorption coefficient is symmetric about v = vc yields a
1/0tIy)
K,(t) = 2_f f ~Z(y)exp(-t/x) dx dy 7
x
0
(12)
0
where 7 = 2.fo c~(y)dy with y = (v-vc)/D. The constants a and D for the various profiles are defined in Table 1. Letting x = la/c~(y),the function Kl(t) can be expressed as follows a
1
Kl(t) = 2 f f ct2(y)exp[_ct(y)t/it]d#dy. 0
(13)
0
Changing the order of integration requires that the integral be broken into two parts
KI(t)--i[i ~2(y)dy]exp(-t/x)d~Xx+ i [ 0
0
1
f =2(y)dy] exp(-t/x)d;.
(14)
f(x)
Rewriting equation (14) in a more compact form yields
t/x)G(x)dXx
Kl(t) = i exp( -
(15)
0
where a
G(x) = G1 = 7/f=2(y) dy
0 < x < 1
(16a)
0
G(x) =
2i
G2 = ~
0~2(y)dy
x > 1.
(16b)
f(x)
The functions G(x) and f(x) for the various profiles are summarized in Table 2. Similarly, the function K2(t) can be expressed as
Kz(t) =
j exp(-
t/x)G(x) dx.
(17)
0
X and Y functions Following closely the procedure outlined by SOBOLEV,~31) the integrodifferential equations for J(z, z, To) are obtained. Replacing z by Zo - z , equation (9) becomes ro
J(ro-r. z. to) = e
f J ( r o - t, z, zo)Kl([z-t])dt. 0
118)
A. L. CROSBIE and R. VISKANTA
470
TABLE 2. FUNCTIONS G(x) AND f(x) FOR THE VARIOUS PROFILES Profile
:¢(y)(0 _< y _< a)
a
D
7
Gl
G2
Rectangular
1
Av
Av
1
Triangular
1 -.i,
1
Av f
Av 2
2 3
2 1 3 x3
~
Exponential
e x p ( - y)
~;
d
2 ~i
1 2
I1 2 ,¢2
In
Doppler
exp( - >,2)
~
,/d
d
Lorentz
1/(1 ky 2
cf~
,/d
~[)d
0
I
erli'[x/(2 In ~:j]l'x/2
\/2
/r
1
.'~
/(x)
I
I
-'~-/z tan
I ' "v/x-I
1 V'(x-]) n x
ll,,x
\,,lln ~-}
x
1
Differentiating equations (9) and (18) with respect to to and superimposing solutions results in the integrodifferential equations
~J (¢, z, To) ~ro (~J (T O - "C, Z, tO)
= J(zo, z, To)~(Zo-T, To)
J ( t 0 - t, z, To)
=
Z
~-J(0, z, ro)@(Zo-r, to)
(19a) (19b)
where ~(T, To) is defined by O(z, to) = ~ K l ( r ) + ~
~(t, r o ) K d l z - t l ) dr.
{201
0
Multiplying equation (9) by ½G(z) dz/z and integrating from 0 to vo yields
J(z,Z, zo)G(z)dz/z.
~(r, to) = ~
(21)
0
The initial conditions for equations (19) are determined by equation (9) and are J(t, z, 0) = 1 f o r z > 0. The X and Y functions correspond to the values of J(r, z, To) at z = 0 and r = r o, respectively, X(z, To) = J(0, z, to) and Y(z, To) = J(zo, z, to). The X and Y functions for the rectangular profile are identical to Chandrasekhar's X and Y functions. ~4°1 Setting z = 0, the integrodifferential equations (19) b e c o m e
OX (z,
1 Y(z, z o) [ Y(z', ro)G(z') dz'/z' 2
"CO)
OTo Y (z, zo) ~"C0
(22a)
0
-
g(z, Zo) 1 Z
(
~-~X(z, to) j Y(z', zo)G(z') dz'/z' 0
{22b)
Nongray radiative transfer in a planar medium exposed to a collimated flux
471
with the initial conditions, X(z, 0) = 1 and Y(z, 0) = 1 for z > 0. Using the definition of qb(z, "Co)and Y(z, "Co),one can show that the function ~("Co,"Co)has the form
q~("Co,"Co)= ~
Y(z', "Co)G(z')dz'/z'.
(23)
o Equations (22) were derived in a somewhat different manner by HEASLETand WARMING.(a2)
Calculation of X and Y functions In summary of the mathematical development, the values of J("C, z, "Co)at the boundaries can be obtained by solving integrodifferential equations (22). The integrodifferential equations are nonlinear, and the possibility of an analytical solution is remote. Since modern digital computers possess the ability to solve large numbers of ordinary differential equations, equations (22) are reduced to a system of ordinary differential equations in a manner similar to that of BELLMANet al. (41) Breaking the integral term ~("Co,Zo) into two integrals and applying the transform z I = 1/z to the second integral, equation (23) becomes 1
[Gl Y(zl, "Co)+ Y(1/zl, "Co)G2(1/zO] dzl/z 1.
O("Co, "Co)= ~
(24)
0
The integral (24) is evaluated by using Gaussian Quadrature 1
"
qb("Co,"Co)= ~ k=~l ~OR[G1Y(Zk, "CO)+Y(1/ZR, "Co)Gz(I/Zk)]/ZR
(25)
where ~ok and z k are Gaussian weights and abscissas, respectively. The evaluation of the integrodifferential equations (22) at discrete values of z yields dXi d"Co
-- ~ Yi ~ O)k[GiYk q- Yk+nG2(l/Zk)]/Zk k 1
dY~ dzo
(26a)
=
Y~ 1
--
.L
"~i 'l'~-'Jt-~Xi kL 1= 09k[Gl Yk-t- Yk+nG2(1/Zk)]/Z k
(265)
where
Xl = X(zl, to)
Xn+ i = X(1/zi, "Co) i = 1, 2 . . . . . n
(27)
= r(zi, "Co)
Y,+i = Y(1/zl, Zo). The system of ordinary differential equations was solved numerically by a fourth-order Runge-Kutta method. The step size used depends on n and Zo. In general a step size of Azo = 0.005 was used for n < 20, while Az o = 0.002 for n = 40. When the argument "Covanishes the function *("Co, %) is undefined. The derivatives of X and Y at z o = 0 are therefore infinite. While this result would discourage the use of
472
A.L.
CROSB1E and R. VISKANTA
equations (26), actually the influence of the singularity decreases quickly. The results for a given fixed step size may be in error for small values of %, but for larger values of ~o the accuracy improves quickly. In actual computations, a smaller step size was used for small values of Zo. The X and Y functions at equally spaced values of z are determined by interpolation and extrapolation. RESULTS AND DISCUSSION X and Y junctions Since a large number of results have been obtained only some representative ones will be presented in this section, and general trends nointed out. The X and Y functions for the various profiles are presented graphically in Figs. 1 10. The abscissa in the figures consists of two different scales to cover the complete range of z from 0 to ~ . For z _< 1 the scale is linear, and for z > 1 the scale is the inverse of z. The curves corresponding to r0 -" ~v in Figs. 1 and 4 are taken from Refs. 42 and 27, respectively. Numerical values of the X and Y functions at z = 0.5, 1.0 and 2.0 are tabulated in Tables 3 8. These results are obtained by solving equations (26). The computations for n = 40 were terminated at ro = 10 due to the excessive computer time required. The values in tables correspond to n=40forro< 10 and n = 20 for r0 > 10. It is seen from the graphs that the X and Y functions for the various profiles all exhibit similar trends. For a given optical thickness X and Y are monotonically increasing functions of z. However, for a fixed value of z, X and Y functions behave quite differently. The X function is monotonically increasing with ro for all z, while the Y function is monotonically decreasing with ro for only z < 1. In the region z > 1 no general trends are noted in the t ^
40
"
3.0
5 N X
X
2.0 I
I-~z
vz-~
FIG. l. X(z, "Co)for the r e c t a n g u l a r profile.
)
IlO
12
--z
14
16
,8
IlO .8
16
.4
12
,/z~-~
FIG. 2. S{z, r o) for the t r i a n g u l a r profile.
Nongray radiative transfer in a planar medium exposed to a collimated flux
473
4.0
4oI .................. 30
3C
% =loo
N X
v
N
20 20
2.(
3 2 I
5 I.C . . . . . . . . . . .
'
.2 .4 6
.8 1,0 .8 6
'
'
.4 .2 0
Z
I/Z
I.C
F'lc~.3. X(z, %) for the exponential profile.
4.0
I
'
I
I
i
,
i
,
l
,
i
,
i
,
i
,
i
,
j
"~"-~
i
FIG. 4. X(z, r0) for the Doppler profile.
•
3.0 @ i ~
t
5
I 20
To=IO0 ~/ N
/
X
>- .05
50
iO
2.0
5 3 2
,
I
,
I
1.0~ .2 .4
l--z
,
I
.
i
.
.
.
i
,
i
o21 .
.6 .8 II0 .8 .6 .4 .2 0
,,,-I
FIG. 5. X(z, z0) for the Lorentz profile.
• 001
o
.2
F'-z
.4
.6
.8
to
.8
.6
.4
.z
o
,~z--I
FIG. 6. Y(z, %) for the rectangular profile.
474
A. L. CROSBIE and R. VISKANTA
-% N v N ),-
.I - ' 0
~--z
.2
.4
.6
.8
1.0
8
.6
4
.2
0
vzM
i:I(;. 7. Y(Z, ~o) for the triangular profile.
FI(;. 8. Y(z, ~0) for the exponential profile.
I0 5
N v >--
>-
.05
.or
O(
005
00[0 .2 ~,--Z
4
6
8
I0
.8
6
.4
2 0 ,lZ-~
FIG. 9. Y(z, %) for the Doppler profile.
"Ot"O
.2
4
.6
.8
LO .8
6
.4
.2
0
,,z-4
l-'lo. I0. Y(z, r0) for the Lorentz profile.
Nongray radiative transfer in a planar medium exposed to a collimated flux
475
TABLE 3. VALUESOF X(0.5, "t-0)FOR THE VARIOUSPROFILES
X(0.5, r0) to
Rectangular
.01 .02 .04 .06 .08 .10 .20 .30 .40 .50 .60 .70 .80 .90 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 60.0 70.0 80.0 90.0 100.0 oc
1.02564 1.04485 1.07711 1.10484 1,12966 1.15232 1,24479 1.31548 1.37252 1.41989 1.46000 1.49446 1.52442 1.55073 1.57403 1.65982 1.71535 1.75480 1.78459 1.80803 1.82702 1.84276 1.85602 1.87716 1.89327 1.90595 1.91620 1.92466 1.95149 1.96580 1.97469 1.98075 1.98515 1.98848 1.99110 1.99321 1.99639 1.99869 2.00042 2.00177 2.00286 2.01278
Triangular 1.01807 1.03171 1.05462 1.07426 1.09175 1.10766 1.17160 1.21920 1.25660 1.28687 1.31186 1.33283 1.35063 1.36592 1.37919 1.42561 1.45367 1.47281 1.48693 1.49791 1.50677 1.51409 1.52026 1.53013 1.53770 1.54370 1.54859 1.55266 1.56582 1.57307 1.57769 1.58089 1.58326 1.58508 1.58652 1.58769 1.58949 1.59080 1.59180 1.59259 1.59323
Exponential
Doppler
Lorentz
1.01391 1.02443 1.04206 1.05712 1.07047 1.08256 1.13049 1.16533 1.19207 1.21320 1.23024 1.24420 1.25578 1.26550 1.27374 1.30092 1.31587 1.32535 1.33199 1.33695 1.34083 1.34396 1.34653 1.35055 1.35354 1.35585 1.35770 1.35920 1.36390 1.36636 1.36788 1.36892 1.36966 1.37023 1.37067 1.37103 1.37157 1.37196 1.37225 1.37248 1.37266
1.01889 1.03309 1.05690 1.07726 1.09537 1.11182 1.17773 1.22658 1.26481 1,29561 1,32093 1,34207 1.35994 1.3752l 1.38839 1.43375 1.46028 1.47777 1.49026 1.49968 •.50706 1.51301 1.51790 1.52549 1.53110 1.53540 1.53880 •.54156 1.5500 1.5543 1.5569 1.5586 1.5599 1.5608 1.5615 1.5621 1.5629 1.5635 1.5640 1.5643 1.5646 1.5671
1.01362 1.02385 1.04089 1.05534 1.06809 1.07957 1.12455 1.15658 1.18069 1.19939 1.21419 1.22607 1.23574 1.24370 1.25030 1.27091 1.28109 1.28695 1.29070 1.29331 1.29522 1.29668 1.29784 !.29953 1.30072 1.30159 1.30226 1.30279 1.30434 1.30509 1.30553 1.30582 1.30603 1.30618 1.30630 1.30640 1.30654 1.30664 1.30672 1.30678 1,30683
behavior of the Y function. At one limit z = 1 the Y function is monotonically increasing with to- In general for a fixed ro and z, the X function for the rectangular profile has the largest numerical value, followed by the triangular, Doppler, exponential and Lorentz profiles in that order. The same behavior is true for the Y function in the region 0 < z < 1. Comparison of the X and Y functions for the rectangular profile obtained in the present study with those reported by CARLSTEDT and MULLIKIN, (43) BELLMAN e t a/. (41) and SOBOUTI (44~ s h o w s excellent agreement. (45) The X and Y functions for the Doppler profile are compared with those of FULLER (46) in Ref. 45 and are found in excellent agreement except for large and small to. The calculation procedure of FULLER and HYETT (46 47)
476
A. L. CROSBIE a n d R. VISKANTA
TABLE 4. VAI.UI~SOF X( 1.0, r{}) FOR tHE VARIOUS PROFILFS
X( I.t/, to} ro .01 .02 .04 .06 .08 .10 .20 .30 .40 .50 .60 .70 .80 .90 1.0 1.5 2.0 2.5 3.1} 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9,0 I 0.0 15.{1 20.(/ 25.0 30.0 35.0 40.0 45.(I 50.0 60.0 70.(I 80.0 90.0 100.0
Rectangular 1.02576 1.04525 1.07846 I. 10754 1.13406 1.15872 1.26452 1.35233 1.42876 1.49682 1.55830 1.61435 1.66582 1.71333 1.75737 1.93756 2.07065 2.17286 2.25358 2.31878 2.37242 . 2.41725 2.45525 2.51606 2.56253 2.59916 2.62878 2,65320 2,73074 2.77207 2.79776 2.81527 2.82798 2.83761 2.84517 2.85127 2.86047 2.86710 2.87210 2.87601 2.87914 2.90781
Triangular
Exponential
Doppler
Lorenlz
1.01816 1.03199 1.05558 1.07619 1.09490 1.11224 1.18561 1.24515 1.29584 1.34007 1.37923 1.41427 1.44586 1.47452 1.50(t66 1.60305 1.67387 1.72547 1.76456 1.79513 1.81968 1.83984 1.85671 1.88339 1.90363 1.91957 1.93248 1.94318 1.97762 1.99647 2.00846 2.01678 2.02291 2.[)2761 2.03135 2.03438 2.03903 2.04242 2.04501 2.04704 2.04869
1.01397 1.02465 1.04281 1.05861 1.07290 1.08608 1.14119 1.18497 i.22151 1.25277 1.27993 1.30379 1.32492 1.34375 1.36063 1.4237(/ 1.46404 1.49141 1.51087 I. 52527 1.53628 1.54496 1.55196 l. 56257 1.57026 1.57611 1.58071 1.58444 1.59594 1.60189 1.60554 1.60800 1.6(1979 1.61114 !.61219 1.61304 1.61432 1.61524 1.61594 1.61648 1.61692
1.01898 1.03339 1.05790 1.07926 1.09864 1.11655 1.19217 1.25324 1.30501 1.34998 1.38963 1.42494 1.45664 1.48526 1.51124 1.61156 1.679 [ I 1.72695 1.76215 1.78890 1.80977 1.82645 1.84(/05 I. 86083 1.87593 1.88739 1,89638 1.90362 1.9256 1.9366 1.9432 1.9476 1.9508 1.9531 1.9549 1.9564 1.9585 1.9601 1.9612 1.9621 1.9628 1.9691
I.{) 1368 1.02406 1.04161 1.05678 1.07043 1.08295 1.13466 I. 17495 1.21)797 1.23573 1.25944 1.27992 1.29775 1.31339 1.32717 1.37{,33 1.4(}528 1,42338 1.43527 1.44343 1.44925 1.45355 1.45683 1.46145 1.46453 1.46673 1.46839 1.46967 1.47337 1.47513 1.47615 1.47683 1.47730 1.47766 1.47793 1.47815 1.47848 1.4787 I 1.47888 1.479(t2 1.47013
differs from the one used in the present analysis in several ways. A special quadrature formula was developed by Fuller to evaluate the integral term in equations (22), and the X and Y functions at regularly spaced values of z were obtained by solving the differential equations for these values along with those required to evaluate the integral term. The differential equations were solved with a variable step size Adams Moulton method, and values of the X and Y functions at regularly spaced values of To were obtained by cubic interpolation.
Nongray radiative transfer in a planar medium exposed to a collimated flux
477
TABLE 5. VALUES OF X(2.0, l"o) FOR THE VARIOUS PROFILES
X(2.0, ~o) ro
Rectangular
Triangular
Exponential
Doppler
Lorentz
.01 .02 .04 .06 .08 .[0 .20 .30 .40 .50 .60 .70 .80 .90 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 60.0 70.0 80.0 90.0 100.0
1.02582 1.04545 1.07914 1.10893 1.13635 1.16207 1.27531 1.37335 1.46213 1.54424 1.62113 1.69370 1.76259 1.82823 1.89099 2.16956 2.40257 2.60109 2.77211 2.92064 3,05047 3,16459 3.26541 3.43467 3.57029 3.68068 3.77183 3,84810 4.09418 4.22653 4.30887 4.36500 4.40572 4.43661 4.46085 4.48037 4.50988 4.53112 4.54715 4.55967 4.56972
1.01820 1.03214 1.05607 1.07718 1.09654 1.11463 1.19328 1.25996 1.31916 1.37292 1.42239 1.46832 1.51124 1.55155 1.58953 1.75194 1.88035 1.98471 2.07108 2.14356 2.20507 2.25777 2.30332 2.37779 2.43579 2.48206 2.51972 2.55093 2,65070 2,70467 2.73875 2.76232 2.77965 2.79294 2.80347 2.81203 2.82512 2.83466 2.84193 2.84767 2.85231
1.01401 1.02476 1.04319 1.05938 1.07416 1.08793 1.14704 1.19618 1.23901 1.27722 1.31179 1.34337 1.37242 1.39929 1.42424 1.52668 1.60258 1.66073 1.70632 1.74270 1.77217 1.79636 1.81645 1.84758 1.87033 1.88750 1.90084 1.91147 1.94298 1.95862 1.96803 1.97434 1.97887 1.98229 1.98495 1.98709 1,99031 1,99262 1.99436 1.99571 1.99680
1.01902 1.03354 1.05841 1.08029 1.10033 1.I1903 1.20007 1.26845 1.32889 1.38353 1.43360 1.47987 1.52292 1.56317 1.60092 1.76018 1.88309 1.98055 2.05923 2.12364 2.17697 2.22159 2.25925 2.31881 2.36325 2.39731 2.42404 2.44547 2.5092 2.5403 2.5585 2.5707 2.5793 2.5857 2.5906 2.5945 2.6004 2.6045 2.6076 2.6100 2.6119 2.6288
1.01372 1.02417 1.04197 1.05752 1.07164 1.08472 1.14020 1.18544 1.22417 1.25815 1.28840 1.31559 1.34022 1.36265 1.38316 1.46392 1.51964 1.55956 1.58891 1.61096 1.62783 1.64095 1.65129 1.66624 1.67625 1.68324 1.68832 1.69213 [.70229 1.70675 1.70927 1.71091 1.71205 1.71290 1.71355 1.71407 1.71484 1.71539 1.71580 1.71612 1.71637
Dimensionless radiative flux Qc(z, %) The dimensionless
r a d i a t i v e f l u x c a n b e e x p r e s s e d a s (32)
9.c(z, ~o) = ½flo[X(z, ~o)+ Y(z, ~o)] where the zeroth moment
(28}
o f t h e Y f u n c t i o n is d e f i n e d a s
Bo(~o) = r Y(z, ~o)G(z) dz. o
(29)
A. L. CROSBIE and R. VISKANTA
478
TABLE 6. VALUES OF Y(0.5, r o) FOR THE VARIOUS PROFILES Y(0.5, to) ~o .01 .02 .04 .06 .08 • 10 .20 .30 .40 •50 .60 .70 .80 .911 1,0 1.5 2.(t 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0
9.11 I0.0 15.0 20.0 25.0 311.0 35,0 411.0 45.0 50.11 60.1/ 70.0 80.0 90.0 100.0
Rectangular 1.00579 1.00544 .99946 .99006 .97884 .96651 .89858 .83039 .76676 .70895 .65704 .61070 .56944 .53273 .50005 .38176 .31056 .26393 .23091 .20601 . 18635 • 17031 . 15690 • 13566 . 11952
Triangular
Exponential
Doppler
Loremz
.99824 .99237 .97723 .96005 .94193 .92338 .83098 .74564 .66972 .6/1310 .5451/3 .49455 .45071 .41262 .37948 .26675 .20582 • 16953 • 14569 . 12867 . 11573 • 10543 ./196970 .083795 .073930
.99408 .98512 .96480 .94319 ,92114 .89903 .79266 .69753 .61457 .54294 .48135 .42849 .38315 .34422 .31078 .20081 . 14517 • 11417 .(194999 .081956 .072397 .065002 .059062 .050032 .043449
.99905 .99374 .97947 .96298 .94543 .92735 •83647 .75173 .67585 .60893 .55033 .49917 .45456 .41565 .38168 .26504 .20109 . 16268 • 13739 • 11939 . 10579 .095079 .086372 .073012 .063212
.99380 .98454 .96362 .9414 I .91876 .89606 .78676 .68891 .60349 .52966 .46612 .41156 36474 .32455 .29002 . 17677 • 12013 .089272 .070785 .1/58672 .050132 .043765 .038822 .031630 .02665 I
. 10683
.066233
.038422
.055709
.023006
•096575 .088119 .061287 .046982 .038091 ,032029 .027632 .024297 .024925 .019572 .016385 ,014091 .012360 .011008 .009923
.060045 .054952 .038792 .030108 .024656 .020904 .018159 .(tl 6061 .014404 .013061 .011017 .009532 .008404 .007517 .006801
.034450 .031232 .021330 .016219 .013092 .010980 .009457 X~)8305 .(107405 .I106681 .005589 .004804 .004213 .003752 .003381
.049780 .(144978 .030265 ,022747 .01818 .01513 .01294 ,01130 .010113 ,00901 .00748 ,00639 •00557 .(10494 .00443
.020227 .018041 .0l 1696 .008643 .006852 .005675 .004843 .004224 ,003746 .003366 .002798 .002396 ,1/112096 ,001863 .001678
The values of flo are tabulated in Refs. 45 and 48. The dimensionless radiative flux for tho various profiles is compared graphically in Fig. 11 for z = 1.0. Numerical results are given in Tables 9, 10 and 11. All the profiles show similar trends. The radiative flux Qc for the triangular, exponential and Doppler profiles is bounded by the results for the rectangular and Lorentz profiles. The variation of Qc with the profiles increases with z0 and decreases with z. The results for the various profiles collapse to the same curve as Zo --" 0. The radiative flux vanishes as Zo --~ x:~. From Tables 9 and 11 the
479
Nongray radiative transfer in a planar medium exposed to a collimated flux TABLE 7. VALUES OF Y(1.0, z o) FOR THE VARIOUS PROFILES Y(1.O, ~o) •0 .01 .02 .04 .06 •08 .10 •20 .30 .40 .50 .60 .70 .80 .90 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 60.0 70.0 80.0 90.0 100.0
Rectangular
Triangular
Exponential
Doppler
1.01578 1.02535 1.03886 1.04843 1.05564 1.06118 1.07415 1.07360 1.06588 1.05379 1.03886 1.02208 1.00407 .98529 .96606 .86966 .78075 .70281 .63571 .57828 .52911 .48687 .45039 .39099 .34501 .30855 .27900 .25459 .17707 .13574 .11005 .092540 .079836 .070199 .062638 .056547 .047341 .040712 .035712 .031806 .028670
1.00819 1.01213 1.016!I 1.01737 1.01700 1.01549 .99832 .97307 .94434 .91412 .88347 .85301 .82312 .79403 .76589 .64127 .54265 .46563 .40538 .35784 .31988 .28915 .26391 .22514 .19683 .17524 .15818 .14431 .10110 .078234 .063965 .054177 .047030 .041576 .037274 .033790 .028489 .024642 .021721 .019426 .017574
1.00401 1.00480 1.00340 .99994 .99525 .98973 .95543 .91621 .87566 .83532 .79593 .75792 .72150 .68679 .65382 .51416 .41059 .33426 .27770 .23534 .20313 .17824 .15866 .13017 .11064 .096440 .085623 .077077 .051716 .039028 .031370 .026237 .022553 .019780 .017615 .015879 .013267 .011393 .009985 .008886 .008006
1.00901 1.01352 1.01841 1.02041 1.02067 1.01971 1.00452 .98041 .95226 .92220 .89139 .86053 .83003 .80018 .77115 .64109 .53666 .45439 .38974 .33870 .29804 .26529 .23859 .19812 .16921 .14764 .13095 .11766 .07800 .05823 .04636 .03848 .03287 .02867 .02541 .02280 .01891 .01614 .01407 .01246 .01118
Lorentz 1.00372 1.00421 1,00220 .99811 .99278 .98661 .94893 .90627 .86231 .81861 .77599 .73487 .69550 .65800 .62241 .47219 .36189 .28181 .22368 .18120 .14983 .12635 .10850 .083849 .068087 .057325 .049550 .043670 .027521 .020111 .015845 .013073 .011127 .009687 .008577 .007697 .006388 .005462 .004774 .004241 .003818
maximum difference between the flux Qc for the rectangular and Lorentz profiles is less than 6 per cent for Zo < l and z = 0.5 and 0.04 per cent for Zo < 1 and z = 2. At an optical thickness o f z o = 100 the flux Qc for the Lorentz profile is about a factor of 14 and 7 higher than the rectangular (gray) profile for z = 0.5 and z = 2.0, respectively. The radiative flux Q~ is largest for the Lorentz profile which has largest wings and smallest for the rectangular profile which has no wings at all. As expected, the results show that as the optical thickness increases the wings become more important in the transfer of energy.
480
A. L. CROSBIE a n d R. VISKANTA
TABLE 8. VAI.UES (IF }z(2.0, T()) FOR TIlE VARIOUS I'ROF1LI~S
Y(2.(), r . ) To .01 .02 .04 .06 .08 .10 .20 .30 .40 ,50 ,60 .70 .80 .90 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 60.0 70.0 80,0 90.0 100.0
Rectangular 1.02082 1.(13545 1.05914 1.07894 1.09635 1.11208 1.17537 1.22354 1.26258 1.29510 1.32260 1.34600 1.36596 1.38296 1.39739 1.43962 1.44676 1.43136 1.40121 1.36158 1.31616 1.26757 1.21769 1.11887 1.02592 .94139 .86592 .79916 .56629 .4350(1 .35275 .29662 .25590 .22501 .20077 . 18125 . 15174 .13050 .11447 . lO195 .091895
Triangular 1.01320 1.02215 1.03614 1.04733 1.05681 1.06505 1.09495 1. I 1373 1.12587 1.13341 1.13751 I. 13891 1.13813 I. 13556 l. 13151 1.09583 1.04554 .98922 .93146 .8748(t .82068 .76983 .72257 .(13897 .56889 .51046 .46170 .42082 .29069 .22296 , 18148 . 15332 .13288 . 11734 . 1051 I .095221 .080212 .069340 .052753 .054620 .049401
Exponential 1.00901 1.01478 1.02329 1.02960 1.03456 1.03855 1.04952 1.05174 1.04884 1.04250 1.03369 1.02303 1.01096 .99780 .98380 ,90689 ,82702 .75022 .67907 .61452 .55673 .50539 .46(X12 .38486 .32666 .28141 .24592 .21776 . 13773 . 10133 .080467 .066828 .057179 .049981 .044401 .039947 .033279 .028524 .024960 .022189 .019973
Doppler
Lorentz
1.014(12 1.02355 1.03846 1.05043 1.06057 1.06940 I. 10156 1.12182 I. 13491 1.14299 1.14729 I. 14859 1.14748 1,14436 I. 13955 1.09803 1.03977 .97465 .90805 .84303 .78127 .72365 .67053 .57777 .5(1156 .43936 .38861 .34701 ,22204 . 16244 . 12792 . 10551 ,08973 .07802 .06898 .06180 .05112 .04355 .03791 .03355 .03007
1.00872 1.01419 1.022(/7 1.02774 1.03204 1.03534 1.04269 1.041 (13 1.03410 1.02362 1.01061 .99573 .97946 ,96213 .944(12 .84790 .75180 .66208 . 581 18 .5097 I .44736 .39343 .347(12 .27314 .21897 . 17913 .14955 . 12731 .071394 .049926 .038608 .03153 I .026663 .023105 .020389 .018247 .I) 15(187 .012866 .011222 "009956 .(X18951
481
Nongray radiative transfer in a planar medium exposed to a collimated flux TABLE 9. VALUES OF Qc(0.5, To) FOR THE VARIOUS PROFILES
ro •01 ,02 .04 •06 ,08 ,10 .20 .30 ,40 .50 .60 .70 ,80 .90 1.0 1.5 2.0 2.5 3.0 3.5 4,0 4.5 5.0 6.0 7,0 8.0 9.0 10.0 15,0 20.0 25.0 30.0 35.0 40,0 45.0 50.0 60.0 70.0 80,0 90.0 100,0
Rectangular .99010 .98039 .96152 .94335 .92584 .90896 .83284 .76839 .71324 .66562 .62417 ,58781 .56124 ,52716 ,50163 ,40589 .34280 .29762 .26338 ,23637 .21446 .19631 ,18100 .15660 ,13800 ,12335 .11152 .10175 .07077 ,05425 .04398 .03698 ,03191 .02806 .02507 .02260 ,01892 .01627 ,01427 .01271 .01146
Triangular .99010 .98039 .96153 .94338 ,92590 .90906 .83341 .76981 ,71587 ,66972 .62996 .59542 .56523 .53864 .51507 ,42853 ,37278 .33299 .30257 ,27820 .25809 .24110 .22650 .20261 ,18378 .16850 .15580 .14506 .10900 .08813 .07438 .06456 .05718 .05140 .04674 ,04290 .03694 ,03250 ,02907 .02632 .02407
Exponential .99010 .98039 ,96154 .94340 .92594 .90912 .83378 .77075 .71763 .67255 .63402 .60088 .57217 .54715 .52519 .44674 .39825 .36451 .33902 .31867 ,30181 .28749 .27512 .25463 ,23822 .22468 .21326 .20343 .16910 .14790 .13313 ,12207 .11338 .10633 ,10045 .09545 .08736 .08103 .07590 .07165 .06804
Doppler
Lorentz
.99010 .98039 .96153 .94339 .92590 .90905 .83336 .76971 .71569 ,66947 .62965 .59507 .56485 .53825 .51471 ,42860 .37369 .33500 .30575 .28258 .26360 .24769 .23411 .21203 .19475 .18077 .16918 .15940 ,12644 .10714 .09418 .08477 .07756 .07181 .06710 .06317 .05689 .05209 .04827 ,04514 .04252
.99010 ,98040 ,96154 .94340 ,92594 .90914 .83384 .77093 .71803 .67324 .63509 .60241 .57425 .54984 .52855 .45406 .40997 .38058 .35915 .34251 .32902 .31775 .30812 .29238 .27989 .26963 .26099 .25356 .22729 .21061 .19865 ,18945 .18203 .17586 ,17060 .16604 .15845 .15231 .14719 ,14281 .13900
A. L. CROSBIE and R. VISKANrA
482
TABLE 10. VALUES OF Qc( 1.0, zo) FOR THE VARIOUS PROFILES
rO
Rectangular
Triangular
Exponential
Doppler
gorentz
.01 .02 .04 .06 .08 .10 .20 .30 .40 .50 .60 .70 .80 .90 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 60.0 70.0 80.0 90.0 100.0
.99503 .99010 .98039 .97086 .96150 .95231 .90873 .86868 .83171 .79750 .76572 .73616 .71564 .68280 .65867 .55810 .48249 .42396 .37756 .34000 .30907 .28321 .26128 .22617 .19934 .17820 .16111 .14699 .10224 .07837 .06354 .05343 .04609 .04053 .03617 .03265 .02733 .02351 .02062 .01836 .01655
.99503 .99010 .98039 .97086 .96152 .95234 .90889 .86909 .83251 .79878 .76763 .73876 .71197 .68704 .66382 .56830 .49791 .44426 .40214 .36823 .34034 .31694 .29700 .26470 .23953 .21924 .20249 .18836 .14121 .11406 .09620 .08347 .07390 .06642 .06039 .05543 .04772 .04198 .03754 .03399 .03108
.99503 .99010 .98039 .97086 .96152 .95235 .90899 .86935 .83304 .79968 .76896 .74063 .71444 .69019 .66769 .57648 .51099 .46230 .42492 .39541 .37147 .35161 .33482 .30774 .28664 .26952 .25526 .24310 .20114 .17554 .15780 .14456 .13420 .12579 .11880 .11286 .10325 .09574 .08966 .08462 .08034
.99503 .99010 .98039 .97087 .96152 .95233 .90887 .86906 .83245 .79871 .76753 .73864 .71184 .68690 .66368 .56833 .49839 .44545 .40422 .37133 .34450 .32219 .30334 .27311 .24981 .23121 .21591 .20308 .16032 .13552 .11897 .10698 .09781 .09053 .08456 .07957 .07164 .06557 .06075 .05680 .05349
.99502 .99010 .98040 .97087 .96153 .95236 .9090(/ .86941 .83316 .79990 .76931 .74115 .71518 .69118 .66898 .57979 .51704 .47155 .43761 .41158 .39107 .37452 .36083 .33941 32317 .31024 .29957 .29052 .25921 .23971 .22584 .21522 .20668 .19960 .19358 .18836 .17969 .17268 .16685 .16187 .15753
Nongray radiative transfer in a planar medium exposed to a collimated flux
483
TABLE 11. VALUES OF Qc(2.0, to) FOR THE VARIOUS PROFILES
ro
Rectangular
Triangular
Exponential
.O1 .02 .04 .06 .08 .10 .20 .30 .40 .50 .60 .70 .80 .90 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0 15.0 20.0 25.0 30.0 35.0 40,0 45.0 50.0 60.0 70.0 80.0 90.0 100.0
.99751 .99502 .99010 .98522 .98038 .97559 .95225 .92989 .90842 .88778 .86790 .84876 .83858 .81250 .79531 .71754 .65134 .59450 .54535 .50256 .46513 .43222 .40312 .35427 .31512 .28326 .25695 .23492 .16386 .12564 .10187 .08565 .07390 .06498 .05798 .05234 .04382 .03768 .03306 .02944 .02654
.99751 .99503 .99010 .98521 .98039 .97559 .95229 .93000 .90863 .88813 .86846 .84953 .83133 .81380 .79692 .72110 .65726 .60298 .55645 .51625 .48131 .45072 .42379 .37872 .34264 .31320 .28877 .26816 .19981 .16095 .13556 .11752 .10399 .09342 .08492 .07792 .06705 .05898 .05258 .04774 .04365
.99751 .99502 .99010 .98521 .98039 .97560 .95232 .93007 .90878 .88838 .86884 .85008 .83207 .81478 .79814 .72395 .66226 .61050 .56671 .52940 .49736 .46966 .44557 .40586 .37464 .34951 .32885 .31153 .25400 .22036 .19747 .18056 .16739 .15675 .14792 .14044 .12837 .11895 .11136 .10505 .09972
Doppler
Lorentz
.99751 .99502 .99010 .98522 .98039 .97559 .95229 .92999 .90861 .88811 .86843 .84949 .83130 .81376 .79689 .72111 .65744 .60348 .55739 .51777 .48349 .45366 .42754 .38422 .34994 .32229 .29955 .28056 .21854 .18361 .16063 .14415 .13161 .12168 .11356 .10680 .09606 .08786 .08136 .07604 .07160
.99751 .99503 .99010 .98522 .98039 .97560 .95232 .93009 .90882 .88845 .86893 .85023 .83230 .81508 .79855 .72510 .66458 .61437 .57245 .53725 .50751 .48225 .46064 .42597 .39963 .37910 .36270 .34928 .30632 .28162 .26457 .25170 .24144 .23297 .22580 .21961 .20935 .20108 .19422 .18837 .18328
A. L. CROSBIE and R. VISKANTA
484 ].0
I
'
I
I
I
Z=I
.8
~o a
I
.6
o
LORENTZ .4
DOPPLER
TRIANGULAR O .01
~ .05
.I
.5
I
5
I0
50
I00
re F'I(~. 1 I. Comparison of Q,( I, r,) for the various protiles.
A cknowh, dgenwnts The authors are grateful to Purdue University for providing computer facilities. One of the authors (A.I_C.) acknowledges the financial support received from the National Science Foundation through a traineeship.
REFERENCES I. 2. 3. 4. 5. 6. 7. 8. 9. 10. I 1. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.
A. L. CROSBIEand R. V|SKANTA,J Q S R T 9 , 553 (19691. J. T. JEFFERIES,Spectral Line Formation, Blaisdell, Waltham. Mass. (19681. L. M. BmERMANN, Zh. ~'ksp. teor. Fiz. 17, 416(1947). T. HOLSTEIN, Phys. Rev. 72, 1212 (1947). T. HOLS~rEIN,Phys. Rev. 83, 1159 (19511. A. G. HEARN, Proc. phys. Soc. 81, 648 (19631. A. G. HEARN, Proe. phys. Soc. 84, 11 (19641. D. G. HUMMER, Mort. Not. R. astr. Soc. 125, 21 (19621. D. G. HUMMER, Astrophys. J. 140, 276 (1964). E. H. AVRETT and D. G. HUMMER, Mon. Not. R. astr. Soe. 130, 295 (19651. D. G. HUMMER and J. C. S~rEWART Astrophvs. J. 146, 290 (19661. D. G. HUMMERand G. RYBICKL Methods in Computational Physics, Vol. 7, pp. 53 127. Academic Press, New. York (1967). D. G. HUMMER, Mon. Not. R. astr. Soc. 138, 73 (19681. D. G. HUMMER, J Q S R T 8 , 193 (19681. S. CUPERMAN, F. ENGELMANNand J. OXEN]US, Phys. Fluids 6, 108 (19631. S. CUPERMAN, F. ENGELMANNand J. OXENIUS, Phys. Fluid~ 7, 428 (19641. F. ENGELMANN,J Q S R T 4 , 507 (1964). S. GIUFFRE and F. ENGELMANN,private communication ( 19681. V. V. SOBOLEV,Soviet Astr. 6, 497 (19631. E. H. AVRETTand W. KALKOFEN,J Q S R T 8 , 219 (19681. G. D. FINN, J Q S R T 8 , 251 (1968). V. V. 1VANOV,Soviet Astr. 6, 793 (19631.
Nongray radiative transfer in a planar medium exposed to a collimated flux 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48.
485
V. V. IVANOV, Soviet Astr. 7, 199 (1963). V. V. IVANOV, N A S A T T F-457, 92 (1967). V. V. IVAt~OVand V. T. SHCHERaAKOV,Astrophys. I, 22 (1965). V. V. IVANOV and V. T. SHCHERaAKOV,Astrophys. 2, 31 (1965). V. V. IVANOV and D. I NAGIRNER, Astrophys 1, 143 (1965). D. I. and V. V. IVANOV, Astrophys. 2, 5 (1966). V. V. IVANOV and D. 1. NAGIRNER, Astrophys. 2, 147 (1966). V. V. SOnOLEV, N A S A T T F-457, 75 (1967). V. V. SOnOLEV, A Treatise on Radiative Trans/er, Van Nostrand, Princeton, New Jersey (1963). M . A . HEASLETand R. F. WARMING, JQSRTB, 1101 (1968). N. A. YAKOVKINand M. Yu. ZELDINA, N A S A TT F-455. 1 (1967). R. I. KOST1K,N A S A T T F-455, 65 (1967). A. SCHACK, Industrial Heat Transfer, John Wiley, New York (1927). D. K. EDWARDS and W. A. MENARD, Applied Opt. 3, 621 (1964). D. K. EDWARDS, L. K. GLASSEN,W. C. HAUSER and J. S. TUCHSCHER,.J. Heat Tran,~[br89 (1967). A. C. G. MITCHELL and M. W. ZEMANSKY, Resonance Radiation and Excited Atoms, Cambridge University Press (1961). R. M. GOODY, Atmospheric Radiation L Theoretical Basis, Oxford University Press, New York (1964). S. CHANDRASEKHAR,Radiative TransJbr, Dover, New York (1960). R. BELLMAN,KAGIWADA,R. KALABA and S. UENO, JQSRT6, 479 (1966). S. W. N. STIBBSand R. E. WEIR,Mort. Not. R. astr. Soc. 119, 512 (1959). J. L. CARLSTEDTand T. W. MULLIKIN,Astrophys. J. Suppl. 12, 449 (1966). Y. SOBOUT1,Astrophys. J. Suppl. 7, 411 (1963). A. L. CROSBIE, "Radiation Heat Transfer in a Nongray Planar M e d i u m , " Ph.D. Thesis, Purdue University (1969). F. B. FULLER, personal communication (1968). F. B. FULLER and B. J. HVETX, NASA TN-4712 (1968). A. L. CROSmE and R. VlSKANTA,J Q S R T 10, 465 (1970).
NONGRAY RADIATIVE TRANSFER IN A PLANAR MEDIUM EXPOSED TO A COLLIMATED FLUX A. L.
CROSBIE
Department of Mechanical and Aerospace Engineering, University of Missouri at Rolla, Rolla, Missouri 65401
and R . VISKANTA School of Mechanical Engineering, Purdue University, Lafayette, Indiana 47907 (Received 2 September 1969) Abstract -The problem of energy transfer through a plane layer of an absorbing and emitting medium exposed on one side to a monochromatic collimated radiation flux is analyzed to determine the effect of line or band shape on radiative transfer. The analysis is restricted to a medium in local thermodynamic equilibrium and an absorption coefficient xv of the Milne-Eddington type, i.e. rcv(T) = ct(v)fl(T). For the case when ~¢~consists of an array of equal intensity, nonoverlapping bands or lines, the solution for the source function and the radiative flux are expressed in terms of universal functions. Numerical solutions for the Chandrasekhar X and Y-functions are presented in graphical and tabular form for the rectangular, triangular, exponential, Lorentz and Doppler profiles.
INTRODUCTION
BECAUSEof the mathematical complexity of the nongray analysis, the gray approximation is usually employed in most non-isothermal radiative transfer studies. While the use of a mean absorption coefficient appears to be a logical compromise to the nongray problem, the validity of this approach has yet to be established. Even when the medium is isothermal, the analysis is complicated by the lack of detail information on the spectral absorption coefficient. In this paper we investigate the effect of line or band shape on the radiative transfer. Because of its simplicity and its relation to other problems, nongray radiative transfer in a plane layer exposed to a collimated flux is studied. The medium can absorb and emit but is unable to scatter thermal radiation. The other modes of heat transfer, i.e., conduction and convection have been neglected to simplify the analysis. A review of the analytical investigations of radiative transfer in a nongray absorbing and emitting media is found in Ref. 1. Since the integral equations describing noncoherent scattering are similar to those of radiative transfer in a nongray absorbing and emitting medium under the conditions of radiative equilibrium, a discussion and review of noncoherent scattering is appropriate. In the theory of spectral line formulation by noncoherent scattering the simplest model is that of complete redistribution. In coherent scattering, the absorbed photon is assumed 465
466
A . L . CROSBIEand R. VISKANTA
to be reemitted with no change in frequency, while with complete redistribution the frequencies of the absorbed and reemitted photons are completely unrelated. The work of JEEFERIES(2) provides a comprehensive review of the physical phenomena involved. The formulation of the complete redistribution model often results in a linear integral equation for the source function similar to that derived for the coherent scattering case. Implicit in such a formulation is the assumption that the source function is independent of frequency. The line shape is accounted for in the kernel and the nonhomogeneous term. BIBERMAN (3) and HOLSTEIN (4 5) were the first to obtain solutions for the source function. While Biberman has replaced the integral equation for a planar medium by a set of linear equations, Holstein has employed a variational approach for planar and cylindrical geometries. More recent work includes that of HEARN,"~ 7) HUMMER et al. (s ~ and ('UPERMAN et al., (ts ~6) ENGELMANN(17) and GIUFFRE and ENGELMANN. (18} SOBOLEV,(l'~i AVRETT and KALKOFEN(2°) and FINN (2~) have studied line radiation in multilevel atoms. IVANOV(22 2,*) IVANOV and SHEHERBAKOV, (25 26) IVANOV and NAGIRNER (27 29) and SOBOLEV(3°) have used the analytical methods of SOBOLEV(3~) for resonance radiation in finite and semi-infinite atmospheres, lvanov and associates have developed generalized 1t, X and Y functions for a single line. These results are summarized and extended by HEASLET and WARMING. (32) A couple of additional references worth noting are those of YAKOVKIN and ZELDINA (33) and KOSTIC. (34) ANALYSIS
Physical model and energy equation The physical system considered consists of a plane layer of absorbing and emitting medium of thickness L. One side of the layer is exposed to a collimated monochromatic flux Fo of frequency vo and direction po(COS0o), while on the other side there is no incident radiation. The absorption coefficient ~ is of the Milne Eddington type, i.e. ~ = ~(v)fl(T). The dimensionless function ~(v) can range from zero to unity. The index of refraction of the medium is considered to be unity, and the medium is assumed to be in local thermodynamic equilibrium. For the physical model and form of spectral absorption coefficient considered, the local radiative flux becomes ,~-(z) = Fol~oexp[-~(Vo)Z/t~o]+2iiE~v(t)sign(z-t)~(v)E2[ct(v),z-tl]dtdv
(1)
0 0
where Ebb(T) = 7tlb,(T) is the Planck function, E,(t) is the exponential function of order n, and r = J'~ fl(T)dx and ro = f~6 fl(T)dx are respectively the optical depth and optical thickness. The conservation of radiant energy equation under the conditions of radiative equilibrium becomes
f ~(v)Ehv(T) dv = ~Fo~(Vo) 1 e x p [ - ~(Vo)ro//Lo]+ 12if Eb~(t)~2(v)El[~(v)lz- t]] dt dr.
0
(2)
o 0
Five different models for ~(v) have been selected for investigation to determine the effect of ~(v) on the radiative transfer. The rectangular, triangular (3s) and exponential (36 371
N o n g r a y r a d i a t i v e transfer in a p l a n a r m e d i u m exposed to a c o l l i m a t e d flux
467
shapes are usually associated with bands, while the D O P P L E R (38) and L O R E N T Z ( 3 8 - 3 9 ) profiles characterize spectral lines. These models have been selected because they approximate the absorption characteristics of a wide range-of substances. The mathematical definition of the models is given in Table 1. Each model is the superposition of m symmetric lines or bands. In Table 1, c~i is the amplitude, v~ is the center, di is the damping factor and Av~ is the width of the i-th band or line. The unit step function is defined as u(t) = 0 for t < 0 and u(t) = 1 for t > 0. In all models frequency vi+ 1 is larger than frequency v~. T A B L E 1. M O D E L S FOR THE ABSORPTION COEFFICIENT ct(v)
Model
M a t h e m a t i c a l definition ra
Rectangular
~Zu(v - v, + ½av,) - u(v - vl - ½Av,)]
ct(v) = i=1
Triangular
ct(v) = ~, i=1
Exponential
ct(v) =
t
t
~ 1- - - I v - v~l Eu(v-vi+½Av~)-u(v- vi- ½A~'i)] Av i
cti exp[ - dl[ v - vii ] i=l
Doppler
cti e x p [ - d,(v
~(v) =
-
vi)2]
i=1
Lorentz
l
ct(v) = i=1
o~] + d~v- vl) 2
Narrow band approximation The narrow band approximation assumes that Planck's function does not vary greatly across the band or line. Mathematically, this approximation can be stated as follows
f Eb~(t)f(v, t) dv ~- Ebi(t) f f ( v , t) dv R
(3)
R
where R is the region of the spectrum over which the integration is performed and Ebi is the function Ebv evaluated at the center of the band or line. The narrow band approximation is obviously more applicable to lines than to bands. Since many bands are too broad for this approximation to apply, an alternate procedure must be worked out. A broad band can be divided into smaller bands which can be considered narrow. The narrow band approximation may be applied to the individual lines which compose the band if there are not too many important lines. Dividing the portion of the spectrum which is capable of absorbing radiation [ct(v) ~ 0] into m regions and applying the narrow band approximation to each region, the local radiative flux becomes
mf ~o
~(z) = Fop 0 e x p [ - ct(Vo)Z/po] + 2 ,~=1=Vi
0
Eb,(t) sign(z-- t)K/2([z-- tl) dt.
(4)
468
A . L . CROSB1Eand R. VISKANTA
The kernels K~ and quantity 7~ are defined as follows K~(r) = ~ ~(V)Ez[0~(vjz ] dv/"7;
(5)
Ri
~(v) dv
Yi
(6)
,d
Ri
where R i is the i-th region of the spectrum. Note that K~(0) ~ 1. Similarly, the energy equation (2) becomes 7~Eb~[T(z)] =
Fo,:~(vo) exp[ -:~(Vo)r/po] + 2
i=1
,,~Eb~[r(t)lK~(lr - t]) dt 0
(7)
i=
where
K](z) = [ 0~2(V)El[~(v)r] dv/Ti. Ri
If the absorption coefficient in each region is identical in shape [Av~ = Av 2 . . . . . Av., = Av or dl = d2 . . . . . d,. = d] and intensity [e 1 = c~2 . . . . . ~., = 1] and the profiles are nonoverlapping, the quantities K](t), K~2(t) and ),~ are independent of i [K ~(t) = Ki~(t), K d t ) = K~(t) and 7 = 7i where i = 1, 2 . . . . , m]. Thus, equation t8) reduces to the linear integral equation
J('r,Z, Zo) = e ~":+2
J(t'z'z°)Kx(lr-t[)dt
(9)
0
where z = po/~(Vo) and m
J(z, z, z o) = 47 ~ Ehi[T(r)]/'~(vo)Fo.
t l0)
i=1
Equation (9) represents the dimensionless energy equation. The function J(r, z, to) is a universal function of the optical depth, the shape of the absorption coefficient and z. The case ofz > 1 for the rectangular profile has no physical meaning, since Po < 1 and c~(vo) - I. However, for the other profiles the values o f z greater than unity imply that ,:~(v0) < 1. Once J(z, z, %) is known, the t e m p e r a t u r e distribution in the m e d i u m can be obtained by solving the transcendental equation (10). The assumption that the a b s o r p t i o n coefficient in each spectral region is identical in shape and intensity reduces equation (4) for the radiative flux to ,~(z) = Fo/~oQc(r) wh'ere ro o
ro) s i g n ( r - t)K2([z- t[) dt.
(1 1)
o
It can readily be shown from the conservation of energy that Q~(z) is a constant, i.e. independent of r.
Nongray radiative transfer in a planar medium exposed to a collimated flux
469
Representation of Kernels K.(t) Substitution of the definition of El(t) into expression for Kl(t) with the assumption that the absorption coefficient is symmetric about v = vc yields a
1/0tIy)
K,(t) = 2_f f ~Z(y)exp(-t/x) dx dy 7
x
0
(12)
0
where 7 = 2.fo c~(y)dy with y = (v-vc)/D. The constants a and D for the various profiles are defined in Table 1. Letting x = la/c~(y),the function Kl(t) can be expressed as follows a
1
Kl(t) = 2 f f ct2(y)exp[_ct(y)t/it]d#dy. 0
(13)
0
Changing the order of integration requires that the integral be broken into two parts
KI(t)--i[i ~2(y)dy]exp(-t/x)d~Xx+ i [ 0
0
1
f =2(y)dy] exp(-t/x)d;.
(14)
f(x)
Rewriting equation (14) in a more compact form yields
t/x)G(x)dXx
Kl(t) = i exp( -
(15)
0
where a
G(x) = G1 = 7/f=2(y) dy
0 < x < 1
(16a)
0
G(x) =
2i
G2 = ~
0~2(y)dy
x > 1.
(16b)
f(x)
The functions G(x) and f(x) for the various profiles are summarized in Table 2. Similarly, the function K2(t) can be expressed as
Kz(t) =
j exp(-
t/x)G(x) dx.
(17)
0
X and Y functions Following closely the procedure outlined by SOBOLEV,~31) the integrodifferential equations for J(z, z, To) are obtained. Replacing z by Zo - z , equation (9) becomes ro
J(ro-r. z. to) = e
f J ( r o - t, z, zo)Kl([z-t])dt. 0
118)
A. L. CROSBIE and R. VISKANTA
470
TABLE 2. FUNCTIONS G(x) AND f(x) FOR THE VARIOUS PROFILES Profile
:¢(y)(0 _< y _< a)
a
D
7
Gl
G2
Rectangular
1
Av
Av
1
Triangular
1 -.i,
1
Av f
Av 2
2 3
2 1 3 x3
~
Exponential
e x p ( - y)
~;
d
2 ~i
1 2
I1 2 ,¢2
In
Doppler
exp( - >,2)
~
,/d
d
Lorentz
1/(1 ky 2
cf~
,/d
~[)d
0
I
erli'[x/(2 In ~:j]l'x/2
\/2
/r
1
.'~
/(x)
I
I
-'~-/z tan
I ' "v/x-I
1 V'(x-]) n x
ll,,x
\,,lln ~-}
x
1
Differentiating equations (9) and (18) with respect to to and superimposing solutions results in the integrodifferential equations
~J (¢, z, To) ~ro (~J (T O - "C, Z, tO)
= J(zo, z, To)~(Zo-T, To)
J ( t 0 - t, z, To)
=
Z
~-J(0, z, ro)@(Zo-r, to)
(19a) (19b)
where ~(T, To) is defined by O(z, to) = ~ K l ( r ) + ~
~(t, r o ) K d l z - t l ) dr.
{201
0
Multiplying equation (9) by ½G(z) dz/z and integrating from 0 to vo yields
J(z,Z, zo)G(z)dz/z.
~(r, to) = ~
(21)
0
The initial conditions for equations (19) are determined by equation (9) and are J(t, z, 0) = 1 f o r z > 0. The X and Y functions correspond to the values of J(r, z, To) at z = 0 and r = r o, respectively, X(z, To) = J(0, z, to) and Y(z, To) = J(zo, z, to). The X and Y functions for the rectangular profile are identical to Chandrasekhar's X and Y functions. ~4°1 Setting z = 0, the integrodifferential equations (19) b e c o m e
OX (z,
1 Y(z, z o) [ Y(z', ro)G(z') dz'/z' 2
"CO)
OTo Y (z, zo) ~"C0
(22a)
0
-
g(z, Zo) 1 Z
(
~-~X(z, to) j Y(z', zo)G(z') dz'/z' 0
{22b)
Nongray radiative transfer in a planar medium exposed to a collimated flux
471
with the initial conditions, X(z, 0) = 1 and Y(z, 0) = 1 for z > 0. Using the definition of qb(z, "Co)and Y(z, "Co),one can show that the function ~("Co,"Co)has the form
q~("Co,"Co)= ~
Y(z', "Co)G(z')dz'/z'.
(23)
o Equations (22) were derived in a somewhat different manner by HEASLETand WARMING.(a2)
Calculation of X and Y functions In summary of the mathematical development, the values of J("C, z, "Co)at the boundaries can be obtained by solving integrodifferential equations (22). The integrodifferential equations are nonlinear, and the possibility of an analytical solution is remote. Since modern digital computers possess the ability to solve large numbers of ordinary differential equations, equations (22) are reduced to a system of ordinary differential equations in a manner similar to that of BELLMANet al. (41) Breaking the integral term ~("Co,Zo) into two integrals and applying the transform z I = 1/z to the second integral, equation (23) becomes 1
[Gl Y(zl, "Co)+ Y(1/zl, "Co)G2(1/zO] dzl/z 1.
O("Co, "Co)= ~
(24)
0
The integral (24) is evaluated by using Gaussian Quadrature 1
"
qb("Co,"Co)= ~ k=~l ~OR[G1Y(Zk, "CO)+Y(1/ZR, "Co)Gz(I/Zk)]/ZR
(25)
where ~ok and z k are Gaussian weights and abscissas, respectively. The evaluation of the integrodifferential equations (22) at discrete values of z yields dXi d"Co
-- ~ Yi ~ O)k[GiYk q- Yk+nG2(l/Zk)]/Zk k 1
dY~ dzo
(26a)
=
Y~ 1
--
.L
"~i 'l'~-'Jt-~Xi kL 1= 09k[Gl Yk-t- Yk+nG2(1/Zk)]/Z k
(265)
where
Xl = X(zl, to)
Xn+ i = X(1/zi, "Co) i = 1, 2 . . . . . n
(27)
= r(zi, "Co)
Y,+i = Y(1/zl, Zo). The system of ordinary differential equations was solved numerically by a fourth-order Runge-Kutta method. The step size used depends on n and Zo. In general a step size of Azo = 0.005 was used for n < 20, while Az o = 0.002 for n = 40. When the argument "Covanishes the function *("Co, %) is undefined. The derivatives of X and Y at z o = 0 are therefore infinite. While this result would discourage the use of
472
A.L.
CROSB1E and R. VISKANTA
equations (26), actually the influence of the singularity decreases quickly. The results for a given fixed step size may be in error for small values of %, but for larger values of ~o the accuracy improves quickly. In actual computations, a smaller step size was used for small values of Zo. The X and Y functions at equally spaced values of z are determined by interpolation and extrapolation. RESULTS AND DISCUSSION X and Y junctions Since a large number of results have been obtained only some representative ones will be presented in this section, and general trends nointed out. The X and Y functions for the various profiles are presented graphically in Figs. 1 10. The abscissa in the figures consists of two different scales to cover the complete range of z from 0 to ~ . For z _< 1 the scale is linear, and for z > 1 the scale is the inverse of z. The curves corresponding to r0 -" ~v in Figs. 1 and 4 are taken from Refs. 42 and 27, respectively. Numerical values of the X and Y functions at z = 0.5, 1.0 and 2.0 are tabulated in Tables 3 8. These results are obtained by solving equations (26). The computations for n = 40 were terminated at ro = 10 due to the excessive computer time required. The values in tables correspond to n=40forro< 10 and n = 20 for r0 > 10. It is seen from the graphs that the X and Y functions for the various profiles all exhibit similar trends. For a given optical thickness X and Y are monotonically increasing functions of z. However, for a fixed value of z, X and Y functions behave quite differently. The X function is monotonically increasing with ro for all z, while the Y function is monotonically decreasing with ro for only z < 1. In the region z > 1 no general trends are noted in the t ^
40
"
3.0
5 N X
X
2.0 I
I-~z
vz-~
FIG. l. X(z, "Co)for the r e c t a n g u l a r profile.
)
IlO
12
--z
14
16
,8
IlO .8
16
.4
12
,/z~-~
FIG. 2. S{z, r o) for the t r i a n g u l a r profile.
Nongray radiative transfer in a planar medium exposed to a collimated flux
473
4.0
4oI .................. 30
3C
% =loo
N X
v
N
20 20
2.(
3 2 I
5 I.C . . . . . . . . . . .
'
.2 .4 6
.8 1,0 .8 6
'
'
.4 .2 0
Z
I/Z
I.C
F'lc~.3. X(z, %) for the exponential profile.
4.0
I
'
I
I
i
,
i
,
l
,
i
,
i
,
i
,
i
,
j
"~"-~
i
FIG. 4. X(z, r0) for the Doppler profile.
•
3.0 @ i ~
t
5
I 20
To=IO0 ~/ N
/
X
>- .05
50
iO
2.0
5 3 2
,
I
,
I
1.0~ .2 .4
l--z
,
I
.
i
.
.
.
i
,
i
o21 .
.6 .8 II0 .8 .6 .4 .2 0
,,,-I
FIG. 5. X(z, z0) for the Lorentz profile.
• 001
o
.2
F'-z
.4
.6
.8
to
.8
.6
.4
.z
o
,~z--I
FIG. 6. Y(z, %) for the rectangular profile.
474
A. L. CROSBIE and R. VISKANTA
-% N v N ),-
.I - ' 0
~--z
.2
.4
.6
.8
1.0
8
.6
4
.2
0
vzM
i:I(;. 7. Y(Z, ~o) for the triangular profile.
FI(;. 8. Y(z, ~0) for the exponential profile.
I0 5
N v >--
>-
.05
.or
O(
005
00[0 .2 ~,--Z
4
6
8
I0
.8
6
.4
2 0 ,lZ-~
FIG. 9. Y(z, %) for the Doppler profile.
"Ot"O
.2
4
.6
.8
LO .8
6
.4
.2
0
,,z-4
l-'lo. I0. Y(z, r0) for the Lorentz profile.
Nongray radiative transfer in a planar medium exposed to a collimated flux
475
TABLE 3. VALUESOF X(0.5, "t-0)FOR THE VARIOUSPROFILES
X(0.5, r0) to
Rectangular
.01 .02 .04 .06 .08 .10 .20 .30 .40 .50 .60 .70 .80 .90 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 60.0 70.0 80.0 90.0 100.0 oc
1.02564 1.04485 1.07711 1.10484 1,12966 1.15232 1,24479 1.31548 1.37252 1.41989 1.46000 1.49446 1.52442 1.55073 1.57403 1.65982 1.71535 1.75480 1.78459 1.80803 1.82702 1.84276 1.85602 1.87716 1.89327 1.90595 1.91620 1.92466 1.95149 1.96580 1.97469 1.98075 1.98515 1.98848 1.99110 1.99321 1.99639 1.99869 2.00042 2.00177 2.00286 2.01278
Triangular 1.01807 1.03171 1.05462 1.07426 1.09175 1.10766 1.17160 1.21920 1.25660 1.28687 1.31186 1.33283 1.35063 1.36592 1.37919 1.42561 1.45367 1.47281 1.48693 1.49791 1.50677 1.51409 1.52026 1.53013 1.53770 1.54370 1.54859 1.55266 1.56582 1.57307 1.57769 1.58089 1.58326 1.58508 1.58652 1.58769 1.58949 1.59080 1.59180 1.59259 1.59323
Exponential
Doppler
Lorentz
1.01391 1.02443 1.04206 1.05712 1.07047 1.08256 1.13049 1.16533 1.19207 1.21320 1.23024 1.24420 1.25578 1.26550 1.27374 1.30092 1.31587 1.32535 1.33199 1.33695 1.34083 1.34396 1.34653 1.35055 1.35354 1.35585 1.35770 1.35920 1.36390 1.36636 1.36788 1.36892 1.36966 1.37023 1.37067 1.37103 1.37157 1.37196 1.37225 1.37248 1.37266
1.01889 1.03309 1.05690 1.07726 1.09537 1.11182 1.17773 1.22658 1.26481 1,29561 1,32093 1,34207 1.35994 1.3752l 1.38839 1.43375 1.46028 1.47777 1.49026 1.49968 •.50706 1.51301 1.51790 1.52549 1.53110 1.53540 1.53880 •.54156 1.5500 1.5543 1.5569 1.5586 1.5599 1.5608 1.5615 1.5621 1.5629 1.5635 1.5640 1.5643 1.5646 1.5671
1.01362 1.02385 1.04089 1.05534 1.06809 1.07957 1.12455 1.15658 1.18069 1.19939 1.21419 1.22607 1.23574 1.24370 1.25030 1.27091 1.28109 1.28695 1.29070 1.29331 1.29522 1.29668 1.29784 !.29953 1.30072 1.30159 1.30226 1.30279 1.30434 1.30509 1.30553 1.30582 1.30603 1.30618 1.30630 1.30640 1.30654 1.30664 1.30672 1.30678 1,30683
behavior of the Y function. At one limit z = 1 the Y function is monotonically increasing with to- In general for a fixed ro and z, the X function for the rectangular profile has the largest numerical value, followed by the triangular, Doppler, exponential and Lorentz profiles in that order. The same behavior is true for the Y function in the region 0 < z < 1. Comparison of the X and Y functions for the rectangular profile obtained in the present study with those reported by CARLSTEDT and MULLIKIN, (43) BELLMAN e t a/. (41) and SOBOUTI (44~ s h o w s excellent agreement. (45) The X and Y functions for the Doppler profile are compared with those of FULLER (46) in Ref. 45 and are found in excellent agreement except for large and small to. The calculation procedure of FULLER and HYETT (46 47)
476
A. L. CROSBIE a n d R. VISKANTA
TABLE 4. VAI.UI~SOF X( 1.0, r{}) FOR tHE VARIOUS PROFILFS
X( I.t/, to} ro .01 .02 .04 .06 .08 .10 .20 .30 .40 .50 .60 .70 .80 .90 1.0 1.5 2.0 2.5 3.1} 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9,0 I 0.0 15.{1 20.(/ 25.0 30.0 35.0 40.0 45.(I 50.0 60.0 70.(I 80.0 90.0 100.0
Rectangular 1.02576 1.04525 1.07846 I. 10754 1.13406 1.15872 1.26452 1.35233 1.42876 1.49682 1.55830 1.61435 1.66582 1.71333 1.75737 1.93756 2.07065 2.17286 2.25358 2.31878 2.37242 . 2.41725 2.45525 2.51606 2.56253 2.59916 2.62878 2,65320 2,73074 2.77207 2.79776 2.81527 2.82798 2.83761 2.84517 2.85127 2.86047 2.86710 2.87210 2.87601 2.87914 2.90781
Triangular
Exponential
Doppler
Lorenlz
1.01816 1.03199 1.05558 1.07619 1.09490 1.11224 1.18561 1.24515 1.29584 1.34007 1.37923 1.41427 1.44586 1.47452 1.50(t66 1.60305 1.67387 1.72547 1.76456 1.79513 1.81968 1.83984 1.85671 1.88339 1.90363 1.91957 1.93248 1.94318 1.97762 1.99647 2.00846 2.01678 2.02291 2.[)2761 2.03135 2.03438 2.03903 2.04242 2.04501 2.04704 2.04869
1.01397 1.02465 1.04281 1.05861 1.07290 1.08608 1.14119 1.18497 i.22151 1.25277 1.27993 1.30379 1.32492 1.34375 1.36063 1.4237(/ 1.46404 1.49141 1.51087 I. 52527 1.53628 1.54496 1.55196 l. 56257 1.57026 1.57611 1.58071 1.58444 1.59594 1.60189 1.60554 1.60800 1.6(1979 1.61114 !.61219 1.61304 1.61432 1.61524 1.61594 1.61648 1.61692
1.01898 1.03339 1.05790 1.07926 1.09864 1.11655 1.19217 1.25324 1.30501 1.34998 1.38963 1.42494 1.45664 1.48526 1.51124 1.61156 1.679 [ I 1.72695 1.76215 1.78890 1.80977 1.82645 1.84(/05 I. 86083 1.87593 1.88739 1,89638 1.90362 1.9256 1.9366 1.9432 1.9476 1.9508 1.9531 1.9549 1.9564 1.9585 1.9601 1.9612 1.9621 1.9628 1.9691
I.{) 1368 1.02406 1.04161 1.05678 1.07043 1.08295 1.13466 I. 17495 1.21)797 1.23573 1.25944 1.27992 1.29775 1.31339 1.32717 1.37{,33 1.4(}528 1,42338 1.43527 1.44343 1.44925 1.45355 1.45683 1.46145 1.46453 1.46673 1.46839 1.46967 1.47337 1.47513 1.47615 1.47683 1.47730 1.47766 1.47793 1.47815 1.47848 1.4787 I 1.47888 1.479(t2 1.47013
differs from the one used in the present analysis in several ways. A special quadrature formula was developed by Fuller to evaluate the integral term in equations (22), and the X and Y functions at regularly spaced values of z were obtained by solving the differential equations for these values along with those required to evaluate the integral term. The differential equations were solved with a variable step size Adams Moulton method, and values of the X and Y functions at regularly spaced values of To were obtained by cubic interpolation.
Nongray radiative transfer in a planar medium exposed to a collimated flux
477
TABLE 5. VALUES OF X(2.0, l"o) FOR THE VARIOUS PROFILES
X(2.0, ~o) ro
Rectangular
Triangular
Exponential
Doppler
Lorentz
.01 .02 .04 .06 .08 .[0 .20 .30 .40 .50 .60 .70 .80 .90 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 60.0 70.0 80.0 90.0 100.0
1.02582 1.04545 1.07914 1.10893 1.13635 1.16207 1.27531 1.37335 1.46213 1.54424 1.62113 1.69370 1.76259 1.82823 1.89099 2.16956 2.40257 2.60109 2.77211 2.92064 3,05047 3,16459 3.26541 3.43467 3.57029 3.68068 3.77183 3,84810 4.09418 4.22653 4.30887 4.36500 4.40572 4.43661 4.46085 4.48037 4.50988 4.53112 4.54715 4.55967 4.56972
1.01820 1.03214 1.05607 1.07718 1.09654 1.11463 1.19328 1.25996 1.31916 1.37292 1.42239 1.46832 1.51124 1.55155 1.58953 1.75194 1.88035 1.98471 2.07108 2.14356 2.20507 2.25777 2.30332 2.37779 2.43579 2.48206 2.51972 2.55093 2,65070 2,70467 2.73875 2.76232 2.77965 2.79294 2.80347 2.81203 2.82512 2.83466 2.84193 2.84767 2.85231
1.01401 1.02476 1.04319 1.05938 1.07416 1.08793 1.14704 1.19618 1.23901 1.27722 1.31179 1.34337 1.37242 1.39929 1.42424 1.52668 1.60258 1.66073 1.70632 1.74270 1.77217 1.79636 1.81645 1.84758 1.87033 1.88750 1.90084 1.91147 1.94298 1.95862 1.96803 1.97434 1.97887 1.98229 1.98495 1.98709 1,99031 1,99262 1.99436 1.99571 1.99680
1.01902 1.03354 1.05841 1.08029 1.10033 1.I1903 1.20007 1.26845 1.32889 1.38353 1.43360 1.47987 1.52292 1.56317 1.60092 1.76018 1.88309 1.98055 2.05923 2.12364 2.17697 2.22159 2.25925 2.31881 2.36325 2.39731 2.42404 2.44547 2.5092 2.5403 2.5585 2.5707 2.5793 2.5857 2.5906 2.5945 2.6004 2.6045 2.6076 2.6100 2.6119 2.6288
1.01372 1.02417 1.04197 1.05752 1.07164 1.08472 1.14020 1.18544 1.22417 1.25815 1.28840 1.31559 1.34022 1.36265 1.38316 1.46392 1.51964 1.55956 1.58891 1.61096 1.62783 1.64095 1.65129 1.66624 1.67625 1.68324 1.68832 1.69213 [.70229 1.70675 1.70927 1.71091 1.71205 1.71290 1.71355 1.71407 1.71484 1.71539 1.71580 1.71612 1.71637
Dimensionless radiative flux Qc(z, %) The dimensionless
r a d i a t i v e f l u x c a n b e e x p r e s s e d a s (32)
9.c(z, ~o) = ½flo[X(z, ~o)+ Y(z, ~o)] where the zeroth moment
(28}
o f t h e Y f u n c t i o n is d e f i n e d a s
Bo(~o) = r Y(z, ~o)G(z) dz. o
(29)
A. L. CROSBIE and R. VISKANTA
478
TABLE 6. VALUES OF Y(0.5, r o) FOR THE VARIOUS PROFILES Y(0.5, to) ~o .01 .02 .04 .06 .08 • 10 .20 .30 .40 •50 .60 .70 .80 .911 1,0 1.5 2.(t 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0
9.11 I0.0 15.0 20.0 25.0 311.0 35,0 411.0 45.0 50.11 60.1/ 70.0 80.0 90.0 100.0
Rectangular 1.00579 1.00544 .99946 .99006 .97884 .96651 .89858 .83039 .76676 .70895 .65704 .61070 .56944 .53273 .50005 .38176 .31056 .26393 .23091 .20601 . 18635 • 17031 . 15690 • 13566 . 11952
Triangular
Exponential
Doppler
Loremz
.99824 .99237 .97723 .96005 .94193 .92338 .83098 .74564 .66972 .6/1310 .5451/3 .49455 .45071 .41262 .37948 .26675 .20582 • 16953 • 14569 . 12867 . 11573 • 10543 ./196970 .083795 .073930
.99408 .98512 .96480 .94319 ,92114 .89903 .79266 .69753 .61457 .54294 .48135 .42849 .38315 .34422 .31078 .20081 . 14517 • 11417 .(194999 .081956 .072397 .065002 .059062 .050032 .043449
.99905 .99374 .97947 .96298 .94543 .92735 •83647 .75173 .67585 .60893 .55033 .49917 .45456 .41565 .38168 .26504 .20109 . 16268 • 13739 • 11939 . 10579 .095079 .086372 .073012 .063212
.99380 .98454 .96362 .9414 I .91876 .89606 .78676 .68891 .60349 .52966 .46612 .41156 36474 .32455 .29002 . 17677 • 12013 .089272 .070785 .1/58672 .050132 .043765 .038822 .031630 .02665 I
. 10683
.066233
.038422
.055709
.023006
•096575 .088119 .061287 .046982 .038091 ,032029 .027632 .024297 .024925 .019572 .016385 ,014091 .012360 .011008 .009923
.060045 .054952 .038792 .030108 .024656 .020904 .018159 .(tl 6061 .014404 .013061 .011017 .009532 .008404 .007517 .006801
.034450 .031232 .021330 .016219 .013092 .010980 .009457 X~)8305 .(107405 .I106681 .005589 .004804 .004213 .003752 .003381
.049780 .(144978 .030265 ,022747 .01818 .01513 .01294 ,01130 .010113 ,00901 .00748 ,00639 •00557 .(10494 .00443
.020227 .018041 .0l 1696 .008643 .006852 .005675 .004843 .004224 ,003746 .003366 .002798 .002396 ,1/112096 ,001863 .001678
The values of flo are tabulated in Refs. 45 and 48. The dimensionless radiative flux for tho various profiles is compared graphically in Fig. 11 for z = 1.0. Numerical results are given in Tables 9, 10 and 11. All the profiles show similar trends. The radiative flux Qc for the triangular, exponential and Doppler profiles is bounded by the results for the rectangular and Lorentz profiles. The variation of Qc with the profiles increases with z0 and decreases with z. The results for the various profiles collapse to the same curve as Zo --" 0. The radiative flux vanishes as Zo --~ x:~. From Tables 9 and 11 the
479
Nongray radiative transfer in a planar medium exposed to a collimated flux TABLE 7. VALUES OF Y(1.0, z o) FOR THE VARIOUS PROFILES Y(1.O, ~o) •0 .01 .02 .04 .06 •08 .10 •20 .30 .40 .50 .60 .70 .80 .90 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 60.0 70.0 80.0 90.0 100.0
Rectangular
Triangular
Exponential
Doppler
1.01578 1.02535 1.03886 1.04843 1.05564 1.06118 1.07415 1.07360 1.06588 1.05379 1.03886 1.02208 1.00407 .98529 .96606 .86966 .78075 .70281 .63571 .57828 .52911 .48687 .45039 .39099 .34501 .30855 .27900 .25459 .17707 .13574 .11005 .092540 .079836 .070199 .062638 .056547 .047341 .040712 .035712 .031806 .028670
1.00819 1.01213 1.016!I 1.01737 1.01700 1.01549 .99832 .97307 .94434 .91412 .88347 .85301 .82312 .79403 .76589 .64127 .54265 .46563 .40538 .35784 .31988 .28915 .26391 .22514 .19683 .17524 .15818 .14431 .10110 .078234 .063965 .054177 .047030 .041576 .037274 .033790 .028489 .024642 .021721 .019426 .017574
1.00401 1.00480 1.00340 .99994 .99525 .98973 .95543 .91621 .87566 .83532 .79593 .75792 .72150 .68679 .65382 .51416 .41059 .33426 .27770 .23534 .20313 .17824 .15866 .13017 .11064 .096440 .085623 .077077 .051716 .039028 .031370 .026237 .022553 .019780 .017615 .015879 .013267 .011393 .009985 .008886 .008006
1.00901 1.01352 1.01841 1.02041 1.02067 1.01971 1.00452 .98041 .95226 .92220 .89139 .86053 .83003 .80018 .77115 .64109 .53666 .45439 .38974 .33870 .29804 .26529 .23859 .19812 .16921 .14764 .13095 .11766 .07800 .05823 .04636 .03848 .03287 .02867 .02541 .02280 .01891 .01614 .01407 .01246 .01118
Lorentz 1.00372 1.00421 1,00220 .99811 .99278 .98661 .94893 .90627 .86231 .81861 .77599 .73487 .69550 .65800 .62241 .47219 .36189 .28181 .22368 .18120 .14983 .12635 .10850 .083849 .068087 .057325 .049550 .043670 .027521 .020111 .015845 .013073 .011127 .009687 .008577 .007697 .006388 .005462 .004774 .004241 .003818
maximum difference between the flux Qc for the rectangular and Lorentz profiles is less than 6 per cent for Zo < l and z = 0.5 and 0.04 per cent for Zo < 1 and z = 2. At an optical thickness o f z o = 100 the flux Qc for the Lorentz profile is about a factor of 14 and 7 higher than the rectangular (gray) profile for z = 0.5 and z = 2.0, respectively. The radiative flux Q~ is largest for the Lorentz profile which has largest wings and smallest for the rectangular profile which has no wings at all. As expected, the results show that as the optical thickness increases the wings become more important in the transfer of energy.
480
A. L. CROSBIE a n d R. VISKANTA
TABLE 8. VAI.UES (IF }z(2.0, T()) FOR TIlE VARIOUS I'ROF1LI~S
Y(2.(), r . ) To .01 .02 .04 .06 .08 .10 .20 .30 .40 ,50 ,60 .70 .80 .90 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 60.0 70.0 80,0 90.0 100.0
Rectangular 1.02082 1.(13545 1.05914 1.07894 1.09635 1.11208 1.17537 1.22354 1.26258 1.29510 1.32260 1.34600 1.36596 1.38296 1.39739 1.43962 1.44676 1.43136 1.40121 1.36158 1.31616 1.26757 1.21769 1.11887 1.02592 .94139 .86592 .79916 .56629 .4350(1 .35275 .29662 .25590 .22501 .20077 . 18125 . 15174 .13050 .11447 . lO195 .091895
Triangular 1.01320 1.02215 1.03614 1.04733 1.05681 1.06505 1.09495 1. I 1373 1.12587 1.13341 1.13751 I. 13891 1.13813 I. 13556 l. 13151 1.09583 1.04554 .98922 .93146 .8748(t .82068 .76983 .72257 .(13897 .56889 .51046 .46170 .42082 .29069 .22296 , 18148 . 15332 .13288 . 11734 . 1051 I .095221 .080212 .069340 .052753 .054620 .049401
Exponential 1.00901 1.01478 1.02329 1.02960 1.03456 1.03855 1.04952 1.05174 1.04884 1.04250 1.03369 1.02303 1.01096 .99780 .98380 ,90689 ,82702 .75022 .67907 .61452 .55673 .50539 .46(X12 .38486 .32666 .28141 .24592 .21776 . 13773 . 10133 .080467 .066828 .057179 .049981 .044401 .039947 .033279 .028524 .024960 .022189 .019973
Doppler
Lorentz
1.014(12 1.02355 1.03846 1.05043 1.06057 1.06940 I. 10156 1.12182 I. 13491 1.14299 1.14729 I. 14859 1.14748 1,14436 I. 13955 1.09803 1.03977 .97465 .90805 .84303 .78127 .72365 .67053 .57777 .5(1156 .43936 .38861 .34701 ,22204 . 16244 . 12792 . 10551 ,08973 .07802 .06898 .06180 .05112 .04355 .03791 .03355 .03007
1.00872 1.01419 1.022(/7 1.02774 1.03204 1.03534 1.04269 1.041 (13 1.03410 1.02362 1.01061 .99573 .97946 ,96213 .944(12 .84790 .75180 .66208 . 581 18 .5097 I .44736 .39343 .347(12 .27314 .21897 . 17913 .14955 . 12731 .071394 .049926 .038608 .03153 I .026663 .023105 .020389 .018247 .I) 15(187 .012866 .011222 "009956 .(X18951
481
Nongray radiative transfer in a planar medium exposed to a collimated flux TABLE 9. VALUES OF Qc(0.5, To) FOR THE VARIOUS PROFILES
ro •01 ,02 .04 •06 ,08 ,10 .20 .30 ,40 .50 .60 .70 ,80 .90 1.0 1.5 2.0 2.5 3.0 3.5 4,0 4.5 5.0 6.0 7,0 8.0 9.0 10.0 15,0 20.0 25.0 30.0 35.0 40,0 45.0 50.0 60.0 70.0 80,0 90.0 100,0
Rectangular .99010 .98039 .96152 .94335 .92584 .90896 .83284 .76839 .71324 .66562 .62417 ,58781 .56124 ,52716 ,50163 ,40589 .34280 .29762 .26338 ,23637 .21446 .19631 ,18100 .15660 ,13800 ,12335 .11152 .10175 .07077 ,05425 .04398 .03698 ,03191 .02806 .02507 .02260 ,01892 .01627 ,01427 .01271 .01146
Triangular .99010 .98039 .96153 .94338 ,92590 .90906 .83341 .76981 ,71587 ,66972 .62996 .59542 .56523 .53864 .51507 ,42853 ,37278 .33299 .30257 ,27820 .25809 .24110 .22650 .20261 ,18378 .16850 .15580 .14506 .10900 .08813 .07438 .06456 .05718 .05140 .04674 ,04290 .03694 ,03250 ,02907 .02632 .02407
Exponential .99010 .98039 ,96154 .94340 .92594 .90912 .83378 .77075 .71763 .67255 .63402 .60088 .57217 .54715 .52519 .44674 .39825 .36451 .33902 .31867 ,30181 .28749 .27512 .25463 ,23822 .22468 .21326 .20343 .16910 .14790 .13313 ,12207 .11338 .10633 ,10045 .09545 .08736 .08103 .07590 .07165 .06804
Doppler
Lorentz
.99010 .98039 .96153 .94339 .92590 .90905 .83336 .76971 .71569 ,66947 .62965 .59507 .56485 .53825 .51471 ,42860 .37369 .33500 .30575 .28258 .26360 .24769 .23411 .21203 .19475 .18077 .16918 .15940 ,12644 .10714 .09418 .08477 .07756 .07181 .06710 .06317 .05689 .05209 .04827 ,04514 .04252
.99010 ,98040 ,96154 .94340 ,92594 .90914 .83384 .77093 .71803 .67324 .63509 .60241 .57425 .54984 .52855 .45406 .40997 .38058 .35915 .34251 .32902 .31775 .30812 .29238 .27989 .26963 .26099 .25356 .22729 .21061 .19865 ,18945 .18203 .17586 ,17060 .16604 .15845 .15231 .14719 ,14281 .13900
A. L. CROSBIE and R. VISKANrA
482
TABLE 10. VALUES OF Qc( 1.0, zo) FOR THE VARIOUS PROFILES
rO
Rectangular
Triangular
Exponential
Doppler
gorentz
.01 .02 .04 .06 .08 .10 .20 .30 .40 .50 .60 .70 .80 .90 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 60.0 70.0 80.0 90.0 100.0
.99503 .99010 .98039 .97086 .96150 .95231 .90873 .86868 .83171 .79750 .76572 .73616 .71564 .68280 .65867 .55810 .48249 .42396 .37756 .34000 .30907 .28321 .26128 .22617 .19934 .17820 .16111 .14699 .10224 .07837 .06354 .05343 .04609 .04053 .03617 .03265 .02733 .02351 .02062 .01836 .01655
.99503 .99010 .98039 .97086 .96152 .95234 .90889 .86909 .83251 .79878 .76763 .73876 .71197 .68704 .66382 .56830 .49791 .44426 .40214 .36823 .34034 .31694 .29700 .26470 .23953 .21924 .20249 .18836 .14121 .11406 .09620 .08347 .07390 .06642 .06039 .05543 .04772 .04198 .03754 .03399 .03108
.99503 .99010 .98039 .97086 .96152 .95235 .90899 .86935 .83304 .79968 .76896 .74063 .71444 .69019 .66769 .57648 .51099 .46230 .42492 .39541 .37147 .35161 .33482 .30774 .28664 .26952 .25526 .24310 .20114 .17554 .15780 .14456 .13420 .12579 .11880 .11286 .10325 .09574 .08966 .08462 .08034
.99503 .99010 .98039 .97087 .96152 .95233 .90887 .86906 .83245 .79871 .76753 .73864 .71184 .68690 .66368 .56833 .49839 .44545 .40422 .37133 .34450 .32219 .30334 .27311 .24981 .23121 .21591 .20308 .16032 .13552 .11897 .10698 .09781 .09053 .08456 .07957 .07164 .06557 .06075 .05680 .05349
.99502 .99010 .98040 .97087 .96153 .95236 .9090(/ .86941 .83316 .79990 .76931 .74115 .71518 .69118 .66898 .57979 .51704 .47155 .43761 .41158 .39107 .37452 .36083 .33941 32317 .31024 .29957 .29052 .25921 .23971 .22584 .21522 .20668 .19960 .19358 .18836 .17969 .17268 .16685 .16187 .15753
Nongray radiative transfer in a planar medium exposed to a collimated flux
483
TABLE 11. VALUES OF Qc(2.0, to) FOR THE VARIOUS PROFILES
ro
Rectangular
Triangular
Exponential
.O1 .02 .04 .06 .08 .10 .20 .30 .40 .50 .60 .70 .80 .90 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0 15.0 20.0 25.0 30.0 35.0 40,0 45.0 50.0 60.0 70.0 80.0 90.0 100.0
.99751 .99502 .99010 .98522 .98038 .97559 .95225 .92989 .90842 .88778 .86790 .84876 .83858 .81250 .79531 .71754 .65134 .59450 .54535 .50256 .46513 .43222 .40312 .35427 .31512 .28326 .25695 .23492 .16386 .12564 .10187 .08565 .07390 .06498 .05798 .05234 .04382 .03768 .03306 .02944 .02654
.99751 .99503 .99010 .98521 .98039 .97559 .95229 .93000 .90863 .88813 .86846 .84953 .83133 .81380 .79692 .72110 .65726 .60298 .55645 .51625 .48131 .45072 .42379 .37872 .34264 .31320 .28877 .26816 .19981 .16095 .13556 .11752 .10399 .09342 .08492 .07792 .06705 .05898 .05258 .04774 .04365
.99751 .99502 .99010 .98521 .98039 .97560 .95232 .93007 .90878 .88838 .86884 .85008 .83207 .81478 .79814 .72395 .66226 .61050 .56671 .52940 .49736 .46966 .44557 .40586 .37464 .34951 .32885 .31153 .25400 .22036 .19747 .18056 .16739 .15675 .14792 .14044 .12837 .11895 .11136 .10505 .09972
Doppler
Lorentz
.99751 .99502 .99010 .98522 .98039 .97559 .95229 .92999 .90861 .88811 .86843 .84949 .83130 .81376 .79689 .72111 .65744 .60348 .55739 .51777 .48349 .45366 .42754 .38422 .34994 .32229 .29955 .28056 .21854 .18361 .16063 .14415 .13161 .12168 .11356 .10680 .09606 .08786 .08136 .07604 .07160
.99751 .99503 .99010 .98522 .98039 .97560 .95232 .93009 .90882 .88845 .86893 .85023 .83230 .81508 .79855 .72510 .66458 .61437 .57245 .53725 .50751 .48225 .46064 .42597 .39963 .37910 .36270 .34928 .30632 .28162 .26457 .25170 .24144 .23297 .22580 .21961 .20935 .20108 .19422 .18837 .18328
A. L. CROSBIE and R. VISKANTA
484 ].0
I
'
I
I
I
Z=I
.8
~o a
I
.6
o
LORENTZ .4
DOPPLER
TRIANGULAR O .01
~ .05
.I
.5
I
5
I0
50
I00
re F'I(~. 1 I. Comparison of Q,( I, r,) for the various protiles.
A cknowh, dgenwnts The authors are grateful to Purdue University for providing computer facilities. One of the authors (A.I_C.) acknowledges the financial support received from the National Science Foundation through a traineeship.
REFERENCES I. 2. 3. 4. 5. 6. 7. 8. 9. 10. I 1. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.
A. L. CROSBIEand R. V|SKANTA,J Q S R T 9 , 553 (19691. J. T. JEFFERIES,Spectral Line Formation, Blaisdell, Waltham. Mass. (19681. L. M. BmERMANN, Zh. ~'ksp. teor. Fiz. 17, 416(1947). T. HOLSTEIN, Phys. Rev. 72, 1212 (1947). T. HOLS~rEIN,Phys. Rev. 83, 1159 (19511. A. G. HEARN, Proc. phys. Soc. 81, 648 (19631. A. G. HEARN, Proe. phys. Soc. 84, 11 (19641. D. G. HUMMER, Mort. Not. R. astr. Soc. 125, 21 (19621. D. G. HUMMER, Astrophys. J. 140, 276 (1964). E. H. AVRETT and D. G. HUMMER, Mon. Not. R. astr. Soe. 130, 295 (19651. D. G. HUMMER and J. C. S~rEWART Astrophvs. J. 146, 290 (19661. D. G. HUMMERand G. RYBICKL Methods in Computational Physics, Vol. 7, pp. 53 127. Academic Press, New. York (1967). D. G. HUMMER, Mon. Not. R. astr. Soc. 138, 73 (19681. D. G. HUMMER, J Q S R T 8 , 193 (19681. S. CUPERMAN, F. ENGELMANNand J. OXEN]US, Phys. Fluids 6, 108 (19631. S. CUPERMAN, F. ENGELMANNand J. OXENIUS, Phys. Fluid~ 7, 428 (19641. F. ENGELMANN,J Q S R T 4 , 507 (1964). S. GIUFFRE and F. ENGELMANN,private communication ( 19681. V. V. SOBOLEV,Soviet Astr. 6, 497 (19631. E. H. AVRETTand W. KALKOFEN,J Q S R T 8 , 219 (19681. G. D. FINN, J Q S R T 8 , 251 (1968). V. V. 1VANOV,Soviet Astr. 6, 793 (19631.
Nongray radiative transfer in a planar medium exposed to a collimated flux 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48.
485
V. V. IVANOV, Soviet Astr. 7, 199 (1963). V. V. IVANOV, N A S A T T F-457, 92 (1967). V. V. IVAt~OVand V. T. SHCHERaAKOV,Astrophys. I, 22 (1965). V. V. IVANOV and V. T. SHCHERaAKOV,Astrophys. 2, 31 (1965). V. V. IVANOV and D. I NAGIRNER, Astrophys 1, 143 (1965). D. I. and V. V. IVANOV, Astrophys. 2, 5 (1966). V. V. IVANOV and D. 1. NAGIRNER, Astrophys. 2, 147 (1966). V. V. SOnOLEV, N A S A T T F-457, 75 (1967). V. V. SOnOLEV, A Treatise on Radiative Trans/er, Van Nostrand, Princeton, New Jersey (1963). M . A . HEASLETand R. F. WARMING, JQSRTB, 1101 (1968). N. A. YAKOVKINand M. Yu. ZELDINA, N A S A TT F-455. 1 (1967). R. I. KOST1K,N A S A T T F-455, 65 (1967). A. SCHACK, Industrial Heat Transfer, John Wiley, New York (1927). D. K. EDWARDS and W. A. MENARD, Applied Opt. 3, 621 (1964). D. K. EDWARDS, L. K. GLASSEN,W. C. HAUSER and J. S. TUCHSCHER,.J. Heat Tran,~[br89 (1967). A. C. G. MITCHELL and M. W. ZEMANSKY, Resonance Radiation and Excited Atoms, Cambridge University Press (1961). R. M. GOODY, Atmospheric Radiation L Theoretical Basis, Oxford University Press, New York (1964). S. CHANDRASEKHAR,Radiative TransJbr, Dover, New York (1960). R. BELLMAN,KAGIWADA,R. KALABA and S. UENO, JQSRT6, 479 (1966). S. W. N. STIBBSand R. E. WEIR,Mort. Not. R. astr. Soc. 119, 512 (1959). J. L. CARLSTEDTand T. W. MULLIKIN,Astrophys. J. Suppl. 12, 449 (1966). Y. SOBOUT1,Astrophys. J. Suppl. 7, 411 (1963). A. L. CROSBIE, "Radiation Heat Transfer in a Nongray Planar M e d i u m , " Ph.D. Thesis, Purdue University (1969). F. B. FULLER, personal communication (1968). F. B. FULLER and B. J. HVETX, NASA TN-4712 (1968). A. L. CROSmE and R. VlSKANTA,J Q S R T 10, 465 (1970).