Integrations over solid angle and frequency are performed in the expressions for the radiant heat flux and local energy loss of a line in a region of strong variations of the source function in one direction. Approximations are given for coefficients and kernels in the resulting forms which involve integrals over the physical coordinate

Harstad, K. G.

English

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N A T I O N A L A E R O N A U T I C S A N D S P A C E A D M I N I S T R A T I O N
Technical Memorandum 33-538
One Dimensional Line Radiative Transfer
K. G. Harstad
J E T P R O P U L S I O N L A B O R A T O R Y
C A L I F O R N I A I N S T I T U T E O F T E C H N O L O G Y
P A S A D E N A , C A L I F O R N I A
April 1, 1972
https://ntrs.nasa.gov/search.jsp?R=19720018275 2020-03-11T18:08:12+00:00Z
N A T I O N A L A E R O N A U T I C S A N D S P A C E A D M I N I S T R A T I O N
Technical Memorandum 33-538
One Dimensional Line Radiative Transfer
K. G. Harstad
J E T P R O P U L S I O N L A B O R A T O R Y
C A I I F O R N I A I N S T I T U T E O F T E C H N O L O G Y
P A S A D E N A , C A I I F O R N I A
April 1, 1972

PREFACE
The work described in this report was performed by the Propulsion
Division of the Jet Propulsion Laboratory.
JPL, Technical Memorandum 33-538 iii
CONTENTS
Introduction 1
Approximations 5
References 10
TABLES
I. Maximum percentage deviations of approximate F. (T, a)
from exact values . 11
II. Maximum percentage deviations of approximate heat
fluxes from exact values for given T and a 11
III. Maximum percentage deviations of approximate local
heat losses from exact values for given TS and a
(m * 0.5) 11
FIGURES
1. Coefficient and kernel functions in the form F. vs. T in
the limit a -» o» 12
2. Ratio of function F, for finite a to that given in Fig. 1 13
3. Ratio of function F^ for finite a to that given in Fig. 1 . . . . . . . 14
4. Ratio of function F, for finite a to that given in Fig. 1 15
5. The end conditions p. (1, a) for the profile functions vs. a" . . . . 16
J
6. Assumed line source distribution for the test model
problems 16
7. Comparison of the exact local energy loss in nondimen-
sional form to approximate losses '. . . 17
8. Comparison of local energy losses vs. normalized
distance y for m = 2, T = 100 18
s
JPL Technical Memorandum 33-538
ABSTRACT
Integrations over solid angle and frequency are performed in the expressions
for the radiant heat flux and local energy loss of a line in a region of strong
variations of the source function in one direction. Approximations are given
for coefficients and kernels in the resulting forms which involve integrals
over the physical coordinate.
vi JPL Technical Memorandum 33-538
ONE DIMENSIONAL LINE RADIATIVE TRANSFER
INTRODUCTION
In a previous paper the author developed an approximate expression for
the local radiant energy loss of a slightly nonuniform, multidimensional, non-
gray, nonequilibrium plasma. The radiation source function was expanded in a
Taylor series in optical depth; this series was truncated. Application of the
expression developed to shock and boundary layers is therefore prohibited.
The present work is complementary in that consideration is given to a narrow
boundary layer region with a strong variation of the source function through
the layer. The plasma core is assumed to be fairly uniform with variations
taking place over lengths greater than or equal to a characteristic plasma
container dimension. Its properties are assumed known; these can be obtained
with aid of the expressions in the paper mentioned. Due to the physical cir-
cumstances, transverse variations in the layer can be neglected and the problem
is treated as one dimensional. Although a boundary layer is examined, a shock
layer would be described by similar equations.
JPL Technical Memorandum 33-538
Continuum radiation and radiation from weak lines are thin in the narrow
regions considered, only strong, low-level lines are likely to have thick
line centers. Thus, this work is concerned only with isolated, non-thin lines.
The following assumptions are utilized:
i) The radiation source function and absorption coefficient are isotropic.
ii) The container surface has diffuse emission and reflection. This
assumption could be relaxed and partial specular reflection also
(2)
could be considered; for many surfaces this is not necessary.
iii) The intensity is isotropic at the thick line center in the plasma
core. If the line center is non-thin in the layer region, it is
very thick in the core. Transfer is by diffusion, gradients are
not large, and this assumption should be nearly realized. It is
to be noted that intensity directions near the normal to the layer
are weighted favorably in the integration over solid angle,
iv) The source function and wall intensity are independent of frequency
at the line center over an interval that contains the line. See
references 1 and 2.
v) The impact broadening semi-half width is greater than or approxi-
mately equal to the Doppler width. This must be satisfied to use
the approximations presented. Different approximations must be
developed for lower pressure plasmas; this is a matter of further
algebraic and computational effort. It is also assumed that the
widths vary weakly through the layer.
Fluxes and the heat loss are calculated in terms of a relative normal
coordinate x, x = 0 being the field point being considered. Subscript "b"
will denote a boundary--either the wall or plasma core. As is customary, the
positive (away from the wall) and negative (towards the wall) directions are
JPL Technical Memorandum 33-538
considered separately. Two optical depths are germane to the problem: the
ordinary spectral optical depth "t" and one descriptive of the line center
"T". These are given by (for either direction)
x
t - / « dx (1)
o
1 x
and T = - i - j J* K dx, (2)
W co
O
Volumetric absorption coefficient a includes the effect of induced emission
v
and w is the impact semi-half width. The radiative heat flux, q , and volume
V R.
energy loss, Q , are split into their backward and forward parts:
R
ID = ^ n - % >
Let I. denote the intensity at a boundary and S the source function, then
under the listed assumptions i) through iv) ,
"
 Wvlx=0 [(SJX=0 - C
~ S - S _, v is the frequency at the line center,
v v vlx=:0' o M J '
I. is the core intensity in the normal direction,
JPL Technical Memorandum 33-538
A q,, = IT (I -1^ l u d Q d v i s a correction for deviations ofR «J.J \ Vc b / ^
the core intensity I in the wings from being isotropic (|j, is the direction
cosine, Q the solid angle), A q =0 by assumption ii). The coefficient
K
functions C. and kernel functions G. are given by
2
P p ft-\ T> /\A p (y) H v (8}
x=0 a v ' ' ^ '
TT W
,f El (t) Pa(v)V
G9 (T) = TVT'A""J E, (t) PQ (v) d v, (9)2
v
00
(t) P (v)
x=0
o
and
_n d V, dO)
C,
where E. (t) are the exponential integrals and P (v) is the absorption pro-
J a
file, assumed to have the Voigt form. In the limit of a very thick line
center (T -* °°) ,
Gi w -77/2 •
(12)
s (v - 1
The functions C. and G. are in general quite complex. Using the limits
(12) , they are written as
Gi s
3 T
C2 = 3 V ^ <
C3 S I ^  F3
JPL Technical Memorandum 33-538
For constant line widths, F7 - F . Given in Figure 1 are the functions F.
for w = constant, the Doppler width w = 0. For w = constant ^0, graphs
of these functions relative to those of Figure 1 are presented in Figures 2
to 4. Integration over frequency was carried out numerically on a UNIVAC 1108
computer. Curves were in part verified by certain asymptotic forms and ap-
proximations. The Voigt profile is given by a rational expression with about
five place or better accuracy in Ref. 3. It is seen that under assumption
v), a = w /w s 0(1), the functions depend weakly on w, and then only for
smaller T. Note the following relations between the coefficients and kernels:
C, (T ) = TT - f b G. (T) d T,i b . ' I
o
T (14)
C3 (y = J b G2 (T) d T.
o
APPROXIMATIONS
Integrals (8) through (11) converge fairly slowly. Attempting numerical
computer evaluation of all the integrals involved in the problem is prohibited
by reasonable constraints on computer time. Quite accurate approximations in
the case of constant line widths can be generated, however. Due to the rela-
tively weak dependence on the width parameter "a", these approximations are
considered useful for many applications.
For constant line widths, after integration by parts, the coefficient
and kernel integrals can be rewritten as
Gi (T) = 5 Ai Ei (ST) + I §3/2 wi <ST> »
G2 (T) = TT E2 (|T) + j A2 T E^gf) + j rg W2 (gr) ,
(15)
82 3 /2 , ,_
 N
C9(T ) = G (T.) ,i b 2. b
C, CO = n T, E (gTJ b b 2 = 1
x j. 3 2
 v) + T TT AQ T. E3 4 3 b 1
JPL Technical Memorandvim 33-538
v^ *?
Define the function D (T|) = /n" T] exp (f| ) erfc (7]) . In the above formulas
(15), £ = t/T| = = D(a), A, = D C/2~a), A2 = 2§ - A^ A3 = (^ + 2 A2>/3, and
d Z
r - - ,-
o
where Z = t/(gr) is a frequency parameter and functions p. depend on the shape
of the line profile. These functions have the following 'end conditions':
Pj (0,a) = 1 and Pj (l,a) = Bj ^ jf'2 ,
0 = 3 T T 0 = 3 n =_9TT <17>
Graphs of the variable end conditions are given in Figure 5. It is to be
noted that the functions p. are not independent, but are related by Eqns. (14)
Assuming a series form for the p.:
p = * Cj Zn , (18)
J
 n=0
equations (14) are satisfied if
C1 = (1 + 2n) (1 - 2n) b ,
(£«(!- 2n) bn, (19)
C3 = b .
n n
The consequent series for the W. are
E Cn In <T1) . (20)
n=0
where I (T|) = J Zn exp (- ZT]) dZ. Let p = 3/2 + n. A series expansion
for I , especially useful for Tl ^ 1, is
j (k
 + pn) '
JPL Technical Memorandum 33-538
For larger 7], a recursion relation can be used:
cn> + e*p (- u) = pn in en),
with I
 l (71) = ~W ^  erf
The approximation to be used for the functions (15) is to truncate the series
(20). In the limit of small T, the terms proportional to the exponential in-
tegrals dominate over the W. terms in Eqns. (15); vice versa in the limit of
large T. The behavior of the functions p. near Z = 0 (the line wings) is thus
most important. Using an asymptotic form for the line wings, the slopes are
determined:
bi = - To 5 (1 - A) - <21)/a
The end conditions at Z = 0 give b =1; the other end conditions are used to
determine b , b and b, with b. assumed null for j 2 5. Results are
K _ , , , . , 99 63
•
 b2 ' 6 + 3 bl * 64 YI - 32 Y2 >
b - 8 + 3 b + II
 v *| (22)j 1 jf. 1 ID i
K •* j. K j. 63 35
- \ ' 3 + bl + 64 Y! - 32 V2 '
where -y = —- and
 Y /v = A,/2?.
2 / f 1 2 1
Table I lists the maximum deviations of this approximation from the exact func-
tions. It should be possible to improve the accuracy by calculating b by use
of exact numerical values of functions (15); this is not really necessary.
Consideration was given to a model problem to test the approximation
developed above and also the "differential approximation" of Ref. 1 for the
local heat loss. A slab of radiating medium, bounded by cold walls, is assumed
to have a source distribution
JPL, Technical Memorandum 33-538
1 I I1 4- ± m mT
 m
 v (1-y)S = S • 2 / VJ" 1'
v c 1^ !_
ff l
 / n Nmy + (i-y)
where S is the source strength at the slab center, y = T /Ts , and T is the slab
optical thickness. See Fig. 6. All necessary integrations were performed num-
erically on a UNIVAC 1108 computer utilizing a Romberg quadrature library sub-
routine. Numerical interpolation from a table of exact numerical values was
used to determine the 'exact' kernels for intermediate values of T; asymptotic
expansions were used for large and small T. The relative error control on inte-
-4grations was set for a value of 10 . Library subroutines existed for the ex-
ponential integrals and error function. Calculations were performed for m = 0.5
(bell shaped distribution), m = 1 (quadratic), m = 2 (nearly flat in the slab
core with a steep square root decay near a wall), and m = 4 (flat core with
steeper fourth root decay near a wall); T = 0.1, 1, 10, 100; a = 0.3, 3.0; and
S
y = 0(0.1)0.5. Table II summarizes the results of the heat flux calculations
(test of G™) . The approximation gives very good results, particularly for larger a.
Function G is approximated with the least accuracy. For the cases m ^ 0.5,
results of heat loss calculations are given in Table III. Again, results are
good. Largest relative errors occur for small losses and the absolute error
is therefore smaller. The entries of Table III for T = 0.1 also hold for
S
m = 0.5. For the case m = 0.5, at the points a = 0.3, T » 1, y = 0.2 the rela-
S
tive error is an order of magnitude larger than that of Table III, also it is twice
as large for the point a = 0.3, T = 100, y = 0.1 and for points a = 3.0, j =1, 10.
S S
A graph for m = 0.5, T = 10 is given in Fig. 7; this is the case of largest
S
deviations. It is seen that the absolute error is small. In general, the
relative error for a = 3.0 is less than or equal to one quarter of that for
a = 0.3.
8 JPL Technical Memorandum 33-538
The approximation of Ref. 1, as expected, fails to hold for j = 0.1 or
s
near a wall except for the case tn = 1, T » 1. Assuming the line is totally
s
thin is fairly accurate for r =0.1 except near a wall. Overestimation of the
S
loss in the slab core by this assumption is about 5 to 10%. Significant absorp-
tion occurs near a wall and the assumption gives large error. For m = 0.5,
T » 1, the approximation of Ref. 1 gives only order of magnitude (factor of
£5
three) agreement with exact results. This can be attributed to the importance
of higher derivatives (change of curvature) of S . For m = 1, the source func-
v
tion is exactly that of the truncated Taylor series and agreement is fair even
for T =1 where the maximum error is about 10%, except near a wall where order
S
of magnitude agreement results. For 7 = 10, maximum error is 30% (a = 0.3) and
S
10% (a = 3.0); for
 T = 100: 3% (a = 0.3, y ± 0) and 1% (a = 3.0). Comparisons
S
of the approximation to exact results for m = 2,4 are similar - it holds in the
slab core where gradients are small (Fig. 8). For T = 1, 10 the error in the
S
core is as large as 25% (a = 0.3) and 10% (a = 3.0). Largest error in the core
is about 5% if T = 100.
s
Finally, it is to be noted that differences between the exact values of
flux or loss for T = 100 and different a are at most about 15%.
S
It is suggested that an average value of a be used in problems with varia-
tions in line widths. The problem of selecting average absorption line widths
is discussed in Ref. 4.
JPL Technical Memorandum 33-538
REFERENCES
1. K. G. Harstad, JQSRT 9, 1273 (1969).
2. T. R. Harrison, Radiation Pyrometry and its Underlying Principles of
Radiant Heat Transfer. John Wiley & Sons, New York (1960), p. 31.
3. K. G. Harstad, submitted to J. Opt. Soc. Am.
4. K. H. Wilson and R. Greif, JQSRT II, 1245 (1971).
10 JPL Technical Memorandum 33-538
TABLE I. Maximum percentage deviations of approximate F.(T,a)
from exact values. Largest deviations occur for 1 ^  T
**
Occurred at point y = 0.1 for m = 1, all other calculations
give an error of 0.3% or less.
Occurred at point y = 0.1 for m = 1, all other calculations
give an error of 2.0% or less.
10.
a
3 to oo
1.0
0.6
0.3
TABLE II. Maximum
fluxes f
Ts
0.1
1.0
10.0
100.0
* Occurred at pom
give an error of
** Occurs for m = 0
m give an error
TABLE III. Maximum
heat lo
T
S
0.1
1.0
10.0
100.0
0.8 0.3 0.2
1.3 0.5 0.3
2.3 0.8 0.5
5 2.7 1.4
percentage deviations of approximate heat
rom exact values for given T and a.
S
a = 3.0 a = 0.3
0.05 0.1
0.2* 0.2
0.2 2.1
**
0.1 0.7
t y = 0.4 for m = 2, all other calculations
0.1% or less.
.5 only, all other calculations at different
of 0.4% or less.
percentage deviations of approximate local
sses from exact values for given T and a (m ^  0.5).
s
a = 3.0 a = 0.3
0.1 0.4
0.2 0.7
0.5* 5.0**
0.15 1.8
JPL Technical Memorandum 33-538 11
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12 JPL Technical Memorandum 33-538
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JPL Technical Memorandum 33-538 13
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14 JPL Technical Memorandum 33-538
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JPL Technical Memorandum 33-538 15
-1
Fig. 5. The end conditions p. ( l , a ) for theJ i
profile functions vs. a"1
u
X&
0.1 0.2 0.3 0.4 0.5
Fig. 6. Assumed line source distribution for the
test model problems
16 JPL Technical Memorandum 33-538
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NASA - in - Coml., L.A., Calif.


On dimensional line radiative transfer

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