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Transmission functions in the numerical solution of radiative heat transfer problems
f. Quanr.
Spectrosc.
Radiar.
Transfer.
Vol.
II, pp.
1741-1748.
Pergamon
Press 1971. Printed in Great
Britain
NOTE TRANSMISSION FUNCTIONS IN THE NUMERICAL SOLUTION OF RADIATIVE HEAT TRANSFER PROBLEMS* D. A. MANDELL College of Engineering, Washington State University, Pullman, Wash. 99163, U.S.A. (Received 26 February 1911)
Abstract-The purpose of the present work is to study the use of high-temperature, nongray kernel functions in radiative heat-transfer problems which are solved numerically by the method of undetermined parameters. The first seven moments of the kernels (transmission functions) are presented for a hydrogen plasma at 10,000 K and 20,000 K and electron densities of lOi cmw3 and 10” cm-s. These moments are used in solving the problem of a hydrogen plasma bounded by two parallel black boundaries. Within the plasma is a uniform heat source (per unit volume). It is shown that, by using the moments, the computer time required for the solution is greatly reduced compared to previously reported direct calculations. By using these moments, solutions of radiative heat transfer problems including nongray affects may be obtained economically and compared with results from approximate methods, such as those involving use of the differential approximation. l.INTRODUCTION
of undetermined parameters has been used by a number of authors; but, in general, details of the method have not been given!iP4’ In the solution of radiative heat-transfer problems by the method of undetermined parameters, a polynomial solution is assumed for the unknown temperature distribution. This polynomial is substituted into the governing integro-differential equation and the resulting equation is evaluated at a sufficient number of points to yield one equation for each unknown coefficient in the polynomial. Moments of the transmission functions must then be calculated for use in the resulting matrix equation. The numerical integrations involved in calculating these moments are very time-consuming. This note contains tables of the moments needed for the solution of radiative heat-transfer problems in a hydrogen plasma. The computer time needed to solve problems by using these tables is significantly shorter than the times needed in direct calculations. THE METHOD
2. ANALYSIS
For nongray linearized radiation, the radiative flux, transmission functions,‘5’ viz.
qR,
has been given in terms of
(1)
* This research was supported by the National Science Foundation through Grant Number GK-5276. 1741
1742
D. A. MANDELL
(2)
(3)
The linearized radiative flux and its derivative become qR = 8aT;( T1- T,)F,(u, - u)
+8aT:
s
‘[T(u’)-T,][F,(u-u’)du’
(4)
0
-80T$T(u’)-TJ&(u’-u)du’ u and dq, = 8oT:(T,
- T,)F,(u, - u)
du
+ 16oT:[T(u)-
T&c&,
(5) -80T;
UIT(u’)-Tl]F,(u-u’)du’ s
0
-8aT:
1 [T(u’)-
T,]F,(u’-u)du’,
where rcL, is the linear Planck mean absorption coefficient. As an example, we consider the problem of radiation and conduction in a hydrogen plasma bounded by two black boundaries at the same uniform temperature. It is assumed that a heat source exists within the plasma and that the plasma is in local thermodynamic equilibrium. The energy equation isC4’
u,$+;-;
= (7)
{ 1 &uf)F2(u-uU))duf
(6) “”
-
s ”
&u’)F,(u’ -u) du’ , I
where 4 = L(T - T,)/QL? and the boundary condition is r#~(0)= 0.
Transmission functions in the numerical solution of radiative heat transfer problems 3. UNDETERMINED
1743
PARAMETERS
The solution of equation (6) by the method of undetermined parameters is obtained by assuming a polynomial solution. Since the problem is symmetrical, a polynomial of the following form may be assumed :
When equation (7) is substituted into equation (6), integrals of the following form must be evaluated :
j[(u-$)-Z]2nP2(Z)dZ
(8)
0
and (9)
where the substitution Z = u-u’ was made in the first integral and Z = u’- u in the second integral. When the brackets in equations (8) and (9) are expanded, moments of the second transmission function of the following form are needed : ”
M, =
I
Z”F,(Z) dZ,
(10)
0
where n varies from 0 to 6. Substitution of equation (7) into equation (6) results in an algebraic equation, once the moments are known ; thus
io{F[
j (u-p-Z)2nF2(Z)dZ-
‘5’ (u-$+Z)2”F2(Z)dZ]
(11) -+;j2n-1uo~
d, = t-f.
’
Equation (11) contains four unknowns, do, d, , d,, d3, and these unknown polynomial coefficients are determined by evaluating equation (11) at three u values and, in addition, using the boundary condition 4(O) = 0, i.e. 3
2n uo
n=O Id(
"1
= I
0.
(12)
1744
D. A. MANDELL
Since the problem is symmetric, only half the distance between the boundaries need be considered and, for equally-spaced increments, u = u,/8, u,/4, and 3u,/8 are used. A 4 x 4 matrix equation results,
:
a11
al2
al3
al4
ho’
a21
az2
a23
az4
d,
0
-
- 0.375 =
a31
a32
a33
a34
4
-0.25
a41
a42
a43
a44
4
-0.125
(13)
The Uij are functions of the U, MO, and M, values, and the thermodynamic conditions. In order to evaluate aij, it is only necessary to interpolate the moments, and this was done by using linear interpolation on a log-log scale. The matrix equation may then be solved by using a standard IBM program. (W The accuracy of the solution is checked by using both fourth- and sixth-order polynomials.
4.
TRANSMISSION
FUNCTIONS
AND
MOMENTS
If equation (10) is repeatedly integrated by parts, in a manner similar to that done by for the exponential integral functions, (‘) the moments can be expressed in terms of higher transmission functions ; thus, CHANDRA~EKHAR
s ”
Z”&(Z)dZ
= n!FK+n+,(0)-n!F,+,,+,(u)
0
(14) Since a recursion formula exists for the transmission functions,“’ the higher transmission functions may be calculated from the previously reported F2 values.(4) Unfortunately, it was found, at least for a hydrogen plasma, that the higher F values were nearly constant to within the number of significant figures used in the input. Thus nearly equal numbers were subtracted in evaluating the moments by using equation (14). For this reason, the moments were calculated directly from equation (10). The first seven moments of F, are given in Tables 1, 2, and 3 for temperatures and electron densities of 10,000 K and 10’6cm-3 (P = 0.43atm); 10,OOOK and 10” cmP3 (P = 37.5atm); and 20,000K and 10” cm- 3 (P - 0.57 atm), respectively. Solutions of equation (6) by using Tables 1, 2, and 3 are identical with previously reported direct calculations, as would be expected.(4) In general, the conservation-of-energy equation includes the divergence of the radiative flux, dq,/dy, and thus moments of F, are needed. These moments can be expressed in terms of the moments of F,,
s
”
”
Z”F,(Z) dZ = n
0
5
0
Z”- ‘F,(Z) dZ - u”F,(u).
(15)
1745
Transmission functions in the numerical solution of radiative heat transfer problems TABLE 1. MOMENTS OF F2 FOR T, = 10,000 K AND
u(cm-atm) 0 5xlo-5 1 x 1o-4 5 x lo-4 1 x 10-Z 5x1o-3 1 x 10-Z 5 x lo-2 1x10-r 5x10-r 1 5 1 x 10’ 5 x 10’ 1 x lo2 5 x lo2 1 x lo3 2.5 x lo3 5 x lo3 1 x lo4
M0 0 2.93 x 5.02 x 1.30 x 1.84 x 3.96 x 5.62 x 1.41 x 2.10 x 4.74 x 6.70 x 1.94 x 3.35 x 1.30 x 2.36 x 8.54 x 1.40 x 2.37 x 3.20 x 3.96 x
MI
1O-6 lO-6 lo-5 lo-’ lo-5 lO-5 10-b 10-4 lo-4 1O-4 10-J lo-’ lo-2 1O-2 lo-’ 10-r 10-r lo- ’ IO- 1
0 6.75 x 2.21 x 2.38 x 6.26 x 6.44 x 1.88 x 2.57 x 7.71 x 7.84 x 2.24 x 3.95 x 1.45 x 2.98 x 1.08 1.88 x 5.90 x 2.21 x 5.21 x 1.06 x
lo-” lo-lo lO-9 lo- 9 10-s lo-’ lO-6 lO-6 lo-’ lO-4 1O-3 lo-2 lo- 1 10’ 10’ lo2 10’ lo3
M, 0 2.19 x lo- ” 1.39 x lo-r4 7.02 x lo-l3 3.65 x 10-r’ 1.92 x 10-r’ 1.14x lO-9 7.97 x lo-” 4.72 x lo-’ 2.29 x 1O-5 1.35 x 1O-4 1.28 x lO-2 9.43 x 1o-2 9.73 7.06 x 10’ 5.99 x lo3 3.67 x lo4 3.25 x lo5 1.45 x 106 5.45 x lo6
Ne = lOr6 cmw3
M, 0 8.11 x lo-*” 1.01 x lo-‘” 2.48 x lo- I6 2.57 x IO-r5 6.83 x lo-” 8.21 x 10-r’ 2.89 x lo-’ 3.39 x 10-s 8.08 x 1O-6 9.72 x 1O-5 4.74 x lo-’ 7.00 x io-’ 3.61 x lo2 5.23 x 10’ 2.19 x lo6 2.65 x 10’ 5.70 x 10s 4.94 x lo9 3.47 x 10”
M, 0
3.22 x lO-24 7.88 x lO-23 9.58 x lo-” 1.98 x 10-r’ 2.65 x lo-r5 6.43 x lo-l4 1.13 x 10-r’ 2.64 x lo-’ 3.12 x lO-6 7.63 x lO-5 1.88 x 10-r 5.57 1.44 x lo4 4.15 x 105 8.64 x lo8 2.07 x 10” 1.10 x 1or2 1.87 x lOr3 2.64 x lOI
M5
M,
0 0 1.33 x lo-** 5.70x 1O-33 6.46 x lo-” 5.47 x lO-3’ 3.90x lO-23 1.64 x 1O-26 1.61 x lo-” 1.36 x lO-24 1.08 x 10-r’ 4.59 x lo-” 5.29 x lo-l6 4.49 x 10-r” 4.67 x lo-r2 1.98 x lo-l3 2.16 x 10-r’ 1 83 x lo-” 1.27 x 1O-6 5:35x lo-’ 6.28 x 1O-5 5.34 x lO-5 7.81 x 10-r 3.34 4.62 x 10’ 3.95 x lo* 5.95 x lo5 2.54 x 10’ 3.44 x 10’ 2.94 x 10’ 3.57 x 10” 1.52 x lOr4 1.69 x 1Or3 1.44 x lOr6 2.20 x lOr5 4.66 x 10’s 7.49 x 10’6 3.12 x 10zo 2.09 x 10’” 1.72 x 10”
For flow over surfaces, integrals of F, other than those involving moments are needed.“) In order to calculate these integrals, values of F, are needed, and these are given in Table 4 for the same thermodynamic conditions as was used in the previous tables. The absorption coefficients, rc,, were calculated by using previously reported methods.@) As an internal check of the consistency of the calculations, the F, values were integrated to show that TABLE2. MOMENTSOF F2 FOR T, = 10,000 K u(cm-atm) 0 5x10-5 1 x 10-4 5 x lo-4 1 x 10-a 5 x lo-3 1 x 1o-2 5 x lo-2 1x10-r 5x10-r 1 5 10 50
1 x 102 5x 1x 5x 1x
102 10” lo3 lo4
MO 0 5.27 x lo- 6 9.45 x 10-e 2.81 x 1O-5 4.20x lO-5 9.38 x lo-’ 1.28 x 1O-4 2.82 x lO-4 4.08x 1O-4 1.09x lo--’ 1.78 x lO-3 6.80x 10-j 1.26 x lo-’ 4.92 x lo-’ 8.36 x lo-’ 1.98 x 10-r 2.54 x 10-r 3.70 x 10-r 4.16 x 10-r
MI 0 1.27 x 10-r’ 4.35 x lo-r0 5.58 x lO-9 1.58 x 1O-8 1.55 x lo-’ 4.10x lo-’ 4.67 x lO-6 1.40x lO-5 2.09x lO-4 7.24 x lO-4 1.56x lo-’ 5.89 x lo-’ 1.12 3.66 3.33 x 10’ 7.31 x 10’ 3.59 x 102 6.91 x lo2
M, 0 4.18 x lo-l5 2 78 x lo-r4 1:70x 10-r’ 9.49 x 10-r’ 4.51 x lo-” 2.40x lO-9 1.42 x lo-’ 8.59x lo-’ 6.57x 1O-5 4.65 x lO-4 5.14x lo-* 3.87 x lo-’ 3.60 x 10’ 2.31 x 10’ 9.42 x lo3 3.91 x lo4 8.92 x 10’ 3.37 x 106
M, 0 1.56 x 2.03 x 6.10x 6.76 x 159x 1:69x 5.09 x 6.20x 2.40x 3.44 x 1.92x 2.88 1.33 x 1.68 x 3.25 x 2.62 x 2.95 x 2.21 x
lo-r9 lo-‘s lo-l6 lo-r5 10-r’ lo-” lo-’ 10-s lO-5 lo-“ lo-’ 10” lo4 lo6 10’ lo9 1O’O
10” cmm3
AND
M4
0 6.20 x 1.59 x 2.38x 5.24x 6.14x 1.31 x 1.99 x 4.86x 9.46x 2.74 x 7.64x 2.29 x 5.25 x 1.32 x 1.24 x 1.96 x 1.10 x 1.64 x
M5
0 lO-24 2.58 x lo-*’ 1.31 x lo-r9 9.73x 4.28 x lo-” lo-l5 2.50x lo-r3 1.07 x 8.15 x 10-r’ 1O-9 3.99x lO-6 3.90x 2.27 x lO-4 10-r 3.81 1.90 x 10’ 2.17 x lo4 1.08 x lo6 5.00 x lo9 1.56 x 10” 4.40 x lOI 1.30 x lOr4
lO-28 lo-26 1O-23 lo-” 10-r’ lo-r5 lo-r2 10-r’ 1O-6 lO-4 lo2 lo6 10s 10” 10” lOl6 10”
M6
0 1.10 x 1.11 x 4.12x 3.61 x 1.05x 9.03 x 3.46 x 339x 1:66x 1.94 x 1.36 x 1.63 x 9.26 x 9.18 x 2.10 x 1.30 x 1.83 x 1.08 x
IO-= lo-“O lo-= lO-24 lo-l9 lo-r8 lo-r3 lo-‘! lO-6 1O-4 10’ lo3 10’ lo9 1014 1Ol6 10” lo=
1146
D. A. MANDELL
TABLE 3. MOMENTSOF Fz FOR T, = 20,000 K AND Ne = 10” cmm3 u(cm-atm)
*,
0
0
5x10-5 1 x 1o-4 5 x 1om4 1x10-” 5x1o-3 1 x 10-2 5x1o-2 1x10-l 5x10-’
4.09 7.76 2.92 5.01 1.68 2.80 9.15 1.51 3.97 5.45 9.86 1.28 2.47 3.16 4.40 4.67 4.80 4.80
5 1 x 10’ 5 x 10’ 1 x 102 5 x 102 1 x10’ 5 x lo3 1 x 104
0
0
x x x x x x x x x x x x x x x x x x
lo- 5 lo- 5 lo-“ 1om4 10-3 1O-3 10-3 1o-2 10-Z 1o-2 lo-’ 10-l lo- 1 10-l 10-l lo- I 10-l 10-l
1.01 3.73 6.55 2.20 3.59 1.19 1.94 6.33 7.30 1.89 1.34 3.54 3.53 8.56 3.69 5.62 7.59 7.62
x x x x x x x x x x x x
x x x x
1o-9 1o-9 lo-” 10-7 10-C lo- 5 10-4 lo-“ 10-j 10-I lo- 1 lo- 1
10’ 10’ 10’ 10’
3.34 x 2.44 x 2.09 x 1.40 x 1.14x 7.55 x 6.15 x 3.99 x 2.17 x 1.02 x 3.74 x 2.06 1.03 x 4.81 x 8.47 x 2.23 x 5.80 x 5.95 x
0 10-l“ lo-l3 lo-‘* 10-l’ 1o-8 1o-8 10mb 1O-5 10m3 lo-* lo- ’ lo2 10’ lo3 lo4 lo4 lo4
0
1.25 x 1.80 x 7.67 x 1.03 x 4.16 x 5.53 x 2.25 x 2.91 x 7.70x 7.02 x 1.30 1.47 x 3.65 x 3.30 x 2.66 x 1.30 x 8.97 x 9.86 x
lo-‘” lo-” lo-l5 10-l’ IO-” lo-” lo-’ 10-h 1O-4 lo--’ 10’ lo3 lo4 10’ 10’ 10’ 10’
4.99 1.43 3.03 8.12 1.64 4.36 8.88 2.29 2.98 5.33 4.98 1.14 1.41 2.50 9.67 9.03 2.05 2.61
0 x x x x x x x x x x
10mz3 lo-” lo-‘” 10-l’ lo-l3 lO_” 1O-9 lo-’ lo-“ lo-’
x x x x x x x
lo2 105 lo6 10’ lo9 10” 10”
TABLE 4. FIRSITRANSMISSIONFUNCTIONS FOR HYDROGEN,F,(U~ cm-’
u(cm-atml
Tl = 10,000 K Ne = 10”’ cms3
T, = lO,O+lOK Ne = 10” crne3
0
2.08 x 1.18 x 1.25 x 6.70 x 6.77 x 3.60 x 3.66x 1.88 x 1.22x 4.28 x 2.02 x 9.40 x 5.73 x 2.01 x 3.79 x 6.86 x 5.92 x 9.48 x
atm-’
TI = 20,000 K Ne = 10” crne3
0
2x10-: 4 x lo-’ 6 x lo-’ 8x10_’ 1 x lUh 1 x 10-S 5x 1o-5 1 x 1o-4 5 x 10-4 1x10-J 5x10-3 1 x 10-l 5 x 1om2 1x10-l 5x10-’ 1 5 1 x 10’ 5 x 10’ 1 x lo2 5 x lo2 1 x lo3 5 x lo3 1 x lo4
6.74 :104 5.22 x lo4 4.36 x lo4 3.77 x lo4 3.33 x lo4 3.51 x lo3 4.50 x lo* 1.90x lo* 1.73 x 10’ 5.42 2.96 x 10-l 9.88 x lo-* 1.46 x 10m2 6.84x 10-j 3.86 x 10 ~’ 6.35 x 1O-5 6.34 x 10m6 2.96 x 10-h 6.30 x lo-’ 3.51 x lo-’ 1.04x lo.-’ 5.45 x lo- 8 5.38 x 1O-9 1.47 x 1om9
1.32 1.14 1.03 9.58 9.00 3.30 7.70 3.09 3.70 1.38 1.04 3.03 2.17 8.21 3.97 7.87 2.16 1.40 5.26 3.08 3.60 8.60 2.89 8.42 -_
:104 x lo4 x lo4 x 10’ x 10’ x 10) x lo2 x lo2 x 10’ x 10’ x x x x x x x x x x x x x
lo- ’ lo-* lo-’ lo-& lo- 5 1O-5 1o-5 lo-’ 10mh lo-’ 1o-8 lo-’ lo- lo
lo-” lo-= lO_” lo-” lo-l6 lo-l4 10-l’ lo-” 1O-4 10m3 10’ lo2 lo6 lo8 10’ ’ 1OlZ lOI 1Ol4
2.20 1.95 1.80 1.70 1.61 7.84 3.03 1.61 2.32 1.02 1.34 5.46 7.84 4.02 1.11 2.25 4.96 2.15 3.36 1.07 3.96 6.47 7.37 2.49
:104 x lo4 x lo4 x lo4 x 10“ x lo3 x lo3 x lo3 x 10’ x lo2
x x x x x x x x x x x x
lo- 1 10-l 10-Z 10-l 1O-4 1O-4 1O-5 1o-5 lo-’ lo-” lo- ’ ’ lo-l3
8.89 x 1.01 x 5.32 x 5.71 x 2.88 x 3.06 x 156x 1:60x 5.13 x 3.58x 8.49 x 7.97 x 2.42 x 1.68 x 1.56 x 5.50 x 1.97 x 4.35 x
lo-= lo-l9 lo-” 1Om23 lo-‘” 10-lh lo-” 1O-9 lo-’ lo-” 10’ 10) 10’ 10’” lOI lOI 10” 10’”
Transmission functions in the numerical solution of radiative heat transfer problems
1747
TABLE4.-continued
t&m-atm)
Tl = 10,000 K
TI = 10,000 K
Ne = lOI cm-s
Ne = 10” cm-s
5x lo4
1x 5x 1x 5x 1x 5x
lo5 lo5 lo6 lo6 10’ 10’
1.77 x 1.49 x 1.15 x 1.06 x 5.34 x 1.00 x 0
10-i’ 10-i* 10-15 10-i’ 10-29 1o-41
1.48 x lo-” 1.45 X lo-‘2 1.10 x lo-i5 9.75 x lo-‘s 4.70 x lo- *g 9.33 x 1o-4* 0
TI = 20,OMIK Ne = 10” crnd3
2.98 x lo-=’ 5.72 x lo-“* 0
the following result is obtained : m I
F,(Z) dZ = F,(O).
(16)
0
Since F,(u) has a singularity at u = 0, a six-point Newton-Cotes open formula was used to integrate F, near the singularity. (‘) A Romberg method was used for the remainder of the integration. For a Au of 2 x 10P6, the errors in equation (16) JimaXF,(Z) dZ--F,(O))/ F,(O), were 11.8, 2.8, and 11.1 per cent for TI = lo4 K, Ne = 1016 cme3; Tl = lo4 K, Ne = 10” crnw3., and TI = 2 x lo4 K, Ne = 10” cme3, respectively. When Au was changed to 2 x lo-‘, the errors became 8.5, 2.8, and 11.0 per cent for the three cases. These errors are within the accuracy of the line-absorption coefficient data used in calculating F, JIO) and thus it is not worthwhile to attempt to obtain greater accuracy. NOTATION
e, E” F” L M” Ne P ii 9R T
TI T2 3
u UO Y
Planck’s function exponential integral functions transmission functions boundary spacing, cm moments of F,, (cm-atm) electron density, cmm3 total pressure, atm partial pressure of atomic hydrogen, atm heat source per unit volume radiative heat flux temperature, “K boundary temperatures, “K pressure path length, u = P,,y, cm-atm total pressure path length, u0 = PHL, cm-atm
distance measured from lower boundary, cm
Greek symbols
K” % /I CT w
spectral absorption coefficient, linear Planck mean coefficient, thermal conductivity Stefan-Boltzmann constant wave number, cm
cm-’ cm-’
1748
D. A. MANDELL REFERENCES
I. 2. 3. 4. 5. 6. 7. 8. 9. 10.
E. M. SPARROWand R. D. Cass, Radiation Heat Transfer. Books/Cole, Belmont, Calif. (1966). R. D. Cuss, P. MIGHDOLLand S. N. TIWARI, In?. J. Heat Mass Transfer, 10, 1521 (1967). D. A. MANDELLand R. D. CESS,Znt. J. Heat Muss Transfer, 13, 1 (1970). D. A. MANDELL,AIAA Jnl. 8, 1510 (1970). S. E. GILLES,A. C. COGLEY and W. G. VINCENT],Int. J. Heat Mass Transfer, 12,445 (1969). IBM System/360 Scienrific Subroutine Package, (360A-CM-03X) Version III. IBM Corporation, White Plains, New York (1969). S. CHANDRASEKHAR, Radiafiue Transfer. Dover, New York (1960). D. A. MANDELLand R. D. C&s, JQSRT 9,981 (1969). M. ABRAMOWITZand I. A. STEGUN,Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Natn. Bur. Stand. Applied Mathematics Series 55 (1964). H. GRIEM,Plasma Spectroscopy. McGraw-Hill, New York (1964).
Spectrosc.
Radiar.
Transfer.
Vol.
II, pp.
1741-1748.
Pergamon
Press 1971. Printed in Great
Britain
NOTE TRANSMISSION FUNCTIONS IN THE NUMERICAL SOLUTION OF RADIATIVE HEAT TRANSFER PROBLEMS* D. A. MANDELL College of Engineering, Washington State University, Pullman, Wash. 99163, U.S.A. (Received 26 February 1911)
Abstract-The purpose of the present work is to study the use of high-temperature, nongray kernel functions in radiative heat-transfer problems which are solved numerically by the method of undetermined parameters. The first seven moments of the kernels (transmission functions) are presented for a hydrogen plasma at 10,000 K and 20,000 K and electron densities of lOi cmw3 and 10” cm-s. These moments are used in solving the problem of a hydrogen plasma bounded by two parallel black boundaries. Within the plasma is a uniform heat source (per unit volume). It is shown that, by using the moments, the computer time required for the solution is greatly reduced compared to previously reported direct calculations. By using these moments, solutions of radiative heat transfer problems including nongray affects may be obtained economically and compared with results from approximate methods, such as those involving use of the differential approximation. l.INTRODUCTION
of undetermined parameters has been used by a number of authors; but, in general, details of the method have not been given!iP4’ In the solution of radiative heat-transfer problems by the method of undetermined parameters, a polynomial solution is assumed for the unknown temperature distribution. This polynomial is substituted into the governing integro-differential equation and the resulting equation is evaluated at a sufficient number of points to yield one equation for each unknown coefficient in the polynomial. Moments of the transmission functions must then be calculated for use in the resulting matrix equation. The numerical integrations involved in calculating these moments are very time-consuming. This note contains tables of the moments needed for the solution of radiative heat-transfer problems in a hydrogen plasma. The computer time needed to solve problems by using these tables is significantly shorter than the times needed in direct calculations. THE METHOD
2. ANALYSIS
For nongray linearized radiation, the radiative flux, transmission functions,‘5’ viz.
qR,
has been given in terms of
(1)
* This research was supported by the National Science Foundation through Grant Number GK-5276. 1741
1742
D. A. MANDELL
(2)
(3)
The linearized radiative flux and its derivative become qR = 8aT;( T1- T,)F,(u, - u)
+8aT:
s
‘[T(u’)-T,][F,(u-u’)du’
(4)
0
-80T$T(u’)-TJ&(u’-u)du’ u and dq, = 8oT:(T,
- T,)F,(u, - u)
du
+ 16oT:[T(u)-
T&c&,
(5) -80T;
UIT(u’)-Tl]F,(u-u’)du’ s
0
-8aT:
1 [T(u’)-
T,]F,(u’-u)du’,
where rcL, is the linear Planck mean absorption coefficient. As an example, we consider the problem of radiation and conduction in a hydrogen plasma bounded by two black boundaries at the same uniform temperature. It is assumed that a heat source exists within the plasma and that the plasma is in local thermodynamic equilibrium. The energy equation isC4’
u,$+;-;
= (7)
{ 1 &uf)F2(u-uU))duf
(6) “”
-
s ”
&u’)F,(u’ -u) du’ , I
where 4 = L(T - T,)/QL? and the boundary condition is r#~(0)= 0.
Transmission functions in the numerical solution of radiative heat transfer problems 3. UNDETERMINED
1743
PARAMETERS
The solution of equation (6) by the method of undetermined parameters is obtained by assuming a polynomial solution. Since the problem is symmetrical, a polynomial of the following form may be assumed :
When equation (7) is substituted into equation (6), integrals of the following form must be evaluated :
j[(u-$)-Z]2nP2(Z)dZ
(8)
0
and (9)
where the substitution Z = u-u’ was made in the first integral and Z = u’- u in the second integral. When the brackets in equations (8) and (9) are expanded, moments of the second transmission function of the following form are needed : ”
M, =
I
Z”F,(Z) dZ,
(10)
0
where n varies from 0 to 6. Substitution of equation (7) into equation (6) results in an algebraic equation, once the moments are known ; thus
io{F[
j (u-p-Z)2nF2(Z)dZ-
‘5’ (u-$+Z)2”F2(Z)dZ]
(11) -+;j2n-1uo~
d, = t-f.
’
Equation (11) contains four unknowns, do, d, , d,, d3, and these unknown polynomial coefficients are determined by evaluating equation (11) at three u values and, in addition, using the boundary condition 4(O) = 0, i.e. 3
2n uo
n=O Id(
"1
= I
0.
(12)
1744
D. A. MANDELL
Since the problem is symmetric, only half the distance between the boundaries need be considered and, for equally-spaced increments, u = u,/8, u,/4, and 3u,/8 are used. A 4 x 4 matrix equation results,
:
a11
al2
al3
al4
ho’
a21
az2
a23
az4
d,
0
-
- 0.375 =
a31
a32
a33
a34
4
-0.25
a41
a42
a43
a44
4
-0.125
(13)
The Uij are functions of the U, MO, and M, values, and the thermodynamic conditions. In order to evaluate aij, it is only necessary to interpolate the moments, and this was done by using linear interpolation on a log-log scale. The matrix equation may then be solved by using a standard IBM program. (W The accuracy of the solution is checked by using both fourth- and sixth-order polynomials.
4.
TRANSMISSION
FUNCTIONS
AND
MOMENTS
If equation (10) is repeatedly integrated by parts, in a manner similar to that done by for the exponential integral functions, (‘) the moments can be expressed in terms of higher transmission functions ; thus, CHANDRA~EKHAR
s ”
Z”&(Z)dZ
= n!FK+n+,(0)-n!F,+,,+,(u)
0
(14) Since a recursion formula exists for the transmission functions,“’ the higher transmission functions may be calculated from the previously reported F2 values.(4) Unfortunately, it was found, at least for a hydrogen plasma, that the higher F values were nearly constant to within the number of significant figures used in the input. Thus nearly equal numbers were subtracted in evaluating the moments by using equation (14). For this reason, the moments were calculated directly from equation (10). The first seven moments of F, are given in Tables 1, 2, and 3 for temperatures and electron densities of 10,000 K and 10’6cm-3 (P = 0.43atm); 10,OOOK and 10” cmP3 (P = 37.5atm); and 20,000K and 10” cm- 3 (P - 0.57 atm), respectively. Solutions of equation (6) by using Tables 1, 2, and 3 are identical with previously reported direct calculations, as would be expected.(4) In general, the conservation-of-energy equation includes the divergence of the radiative flux, dq,/dy, and thus moments of F, are needed. These moments can be expressed in terms of the moments of F,,
s
”
”
Z”F,(Z) dZ = n
0
5
0
Z”- ‘F,(Z) dZ - u”F,(u).
(15)
1745
Transmission functions in the numerical solution of radiative heat transfer problems TABLE 1. MOMENTS OF F2 FOR T, = 10,000 K AND
u(cm-atm) 0 5xlo-5 1 x 1o-4 5 x lo-4 1 x 10-Z 5x1o-3 1 x 10-Z 5 x lo-2 1x10-r 5x10-r 1 5 1 x 10’ 5 x 10’ 1 x lo2 5 x lo2 1 x lo3 2.5 x lo3 5 x lo3 1 x lo4
M0 0 2.93 x 5.02 x 1.30 x 1.84 x 3.96 x 5.62 x 1.41 x 2.10 x 4.74 x 6.70 x 1.94 x 3.35 x 1.30 x 2.36 x 8.54 x 1.40 x 2.37 x 3.20 x 3.96 x
MI
1O-6 lO-6 lo-5 lo-’ lo-5 lO-5 10-b 10-4 lo-4 1O-4 10-J lo-’ lo-2 1O-2 lo-’ 10-r 10-r lo- ’ IO- 1
0 6.75 x 2.21 x 2.38 x 6.26 x 6.44 x 1.88 x 2.57 x 7.71 x 7.84 x 2.24 x 3.95 x 1.45 x 2.98 x 1.08 1.88 x 5.90 x 2.21 x 5.21 x 1.06 x
lo-” lo-lo lO-9 lo- 9 10-s lo-’ lO-6 lO-6 lo-’ lO-4 1O-3 lo-2 lo- 1 10’ 10’ lo2 10’ lo3
M, 0 2.19 x lo- ” 1.39 x lo-r4 7.02 x lo-l3 3.65 x 10-r’ 1.92 x 10-r’ 1.14x lO-9 7.97 x lo-” 4.72 x lo-’ 2.29 x 1O-5 1.35 x 1O-4 1.28 x lO-2 9.43 x 1o-2 9.73 7.06 x 10’ 5.99 x lo3 3.67 x lo4 3.25 x lo5 1.45 x 106 5.45 x lo6
Ne = lOr6 cmw3
M, 0 8.11 x lo-*” 1.01 x lo-‘” 2.48 x lo- I6 2.57 x IO-r5 6.83 x lo-” 8.21 x 10-r’ 2.89 x lo-’ 3.39 x 10-s 8.08 x 1O-6 9.72 x 1O-5 4.74 x lo-’ 7.00 x io-’ 3.61 x lo2 5.23 x 10’ 2.19 x lo6 2.65 x 10’ 5.70 x 10s 4.94 x lo9 3.47 x 10”
M, 0
3.22 x lO-24 7.88 x lO-23 9.58 x lo-” 1.98 x 10-r’ 2.65 x lo-r5 6.43 x lo-l4 1.13 x 10-r’ 2.64 x lo-’ 3.12 x lO-6 7.63 x lO-5 1.88 x 10-r 5.57 1.44 x lo4 4.15 x 105 8.64 x lo8 2.07 x 10” 1.10 x 1or2 1.87 x lOr3 2.64 x lOI
M5
M,
0 0 1.33 x lo-** 5.70x 1O-33 6.46 x lo-” 5.47 x lO-3’ 3.90x lO-23 1.64 x 1O-26 1.61 x lo-” 1.36 x lO-24 1.08 x 10-r’ 4.59 x lo-” 5.29 x lo-l6 4.49 x 10-r” 4.67 x lo-r2 1.98 x lo-l3 2.16 x 10-r’ 1 83 x lo-” 1.27 x 1O-6 5:35x lo-’ 6.28 x 1O-5 5.34 x lO-5 7.81 x 10-r 3.34 4.62 x 10’ 3.95 x lo* 5.95 x lo5 2.54 x 10’ 3.44 x 10’ 2.94 x 10’ 3.57 x 10” 1.52 x lOr4 1.69 x 1Or3 1.44 x lOr6 2.20 x lOr5 4.66 x 10’s 7.49 x 10’6 3.12 x 10zo 2.09 x 10’” 1.72 x 10”
For flow over surfaces, integrals of F, other than those involving moments are needed.“) In order to calculate these integrals, values of F, are needed, and these are given in Table 4 for the same thermodynamic conditions as was used in the previous tables. The absorption coefficients, rc,, were calculated by using previously reported methods.@) As an internal check of the consistency of the calculations, the F, values were integrated to show that TABLE2. MOMENTSOF F2 FOR T, = 10,000 K u(cm-atm) 0 5x10-5 1 x 10-4 5 x lo-4 1 x 10-a 5 x lo-3 1 x 1o-2 5 x lo-2 1x10-r 5x10-r 1 5 10 50
1 x 102 5x 1x 5x 1x
102 10” lo3 lo4
MO 0 5.27 x lo- 6 9.45 x 10-e 2.81 x 1O-5 4.20x lO-5 9.38 x lo-’ 1.28 x 1O-4 2.82 x lO-4 4.08x 1O-4 1.09x lo--’ 1.78 x lO-3 6.80x 10-j 1.26 x lo-’ 4.92 x lo-’ 8.36 x lo-’ 1.98 x 10-r 2.54 x 10-r 3.70 x 10-r 4.16 x 10-r
MI 0 1.27 x 10-r’ 4.35 x lo-r0 5.58 x lO-9 1.58 x 1O-8 1.55 x lo-’ 4.10x lo-’ 4.67 x lO-6 1.40x lO-5 2.09x lO-4 7.24 x lO-4 1.56x lo-’ 5.89 x lo-’ 1.12 3.66 3.33 x 10’ 7.31 x 10’ 3.59 x 102 6.91 x lo2
M, 0 4.18 x lo-l5 2 78 x lo-r4 1:70x 10-r’ 9.49 x 10-r’ 4.51 x lo-” 2.40x lO-9 1.42 x lo-’ 8.59x lo-’ 6.57x 1O-5 4.65 x lO-4 5.14x lo-* 3.87 x lo-’ 3.60 x 10’ 2.31 x 10’ 9.42 x lo3 3.91 x lo4 8.92 x 10’ 3.37 x 106
M, 0 1.56 x 2.03 x 6.10x 6.76 x 159x 1:69x 5.09 x 6.20x 2.40x 3.44 x 1.92x 2.88 1.33 x 1.68 x 3.25 x 2.62 x 2.95 x 2.21 x
lo-r9 lo-‘s lo-l6 lo-r5 10-r’ lo-” lo-’ 10-s lO-5 lo-“ lo-’ 10” lo4 lo6 10’ lo9 1O’O
10” cmm3
AND
M4
0 6.20 x 1.59 x 2.38x 5.24x 6.14x 1.31 x 1.99 x 4.86x 9.46x 2.74 x 7.64x 2.29 x 5.25 x 1.32 x 1.24 x 1.96 x 1.10 x 1.64 x
M5
0 lO-24 2.58 x lo-*’ 1.31 x lo-r9 9.73x 4.28 x lo-” lo-l5 2.50x lo-r3 1.07 x 8.15 x 10-r’ 1O-9 3.99x lO-6 3.90x 2.27 x lO-4 10-r 3.81 1.90 x 10’ 2.17 x lo4 1.08 x lo6 5.00 x lo9 1.56 x 10” 4.40 x lOI 1.30 x lOr4
lO-28 lo-26 1O-23 lo-” 10-r’ lo-r5 lo-r2 10-r’ 1O-6 lO-4 lo2 lo6 10s 10” 10” lOl6 10”
M6
0 1.10 x 1.11 x 4.12x 3.61 x 1.05x 9.03 x 3.46 x 339x 1:66x 1.94 x 1.36 x 1.63 x 9.26 x 9.18 x 2.10 x 1.30 x 1.83 x 1.08 x
IO-= lo-“O lo-= lO-24 lo-l9 lo-r8 lo-r3 lo-‘! lO-6 1O-4 10’ lo3 10’ lo9 1014 1Ol6 10” lo=
1146
D. A. MANDELL
TABLE 3. MOMENTSOF Fz FOR T, = 20,000 K AND Ne = 10” cmm3 u(cm-atm)
*,
0
0
5x10-5 1 x 1o-4 5 x 1om4 1x10-” 5x1o-3 1 x 10-2 5x1o-2 1x10-l 5x10-’
4.09 7.76 2.92 5.01 1.68 2.80 9.15 1.51 3.97 5.45 9.86 1.28 2.47 3.16 4.40 4.67 4.80 4.80
5 1 x 10’ 5 x 10’ 1 x 102 5 x 102 1 x10’ 5 x lo3 1 x 104
0
0
x x x x x x x x x x x x x x x x x x
lo- 5 lo- 5 lo-“ 1om4 10-3 1O-3 10-3 1o-2 10-Z 1o-2 lo-’ 10-l lo- 1 10-l 10-l lo- I 10-l 10-l
1.01 3.73 6.55 2.20 3.59 1.19 1.94 6.33 7.30 1.89 1.34 3.54 3.53 8.56 3.69 5.62 7.59 7.62
x x x x x x x x x x x x
x x x x
1o-9 1o-9 lo-” 10-7 10-C lo- 5 10-4 lo-“ 10-j 10-I lo- 1 lo- 1
10’ 10’ 10’ 10’
3.34 x 2.44 x 2.09 x 1.40 x 1.14x 7.55 x 6.15 x 3.99 x 2.17 x 1.02 x 3.74 x 2.06 1.03 x 4.81 x 8.47 x 2.23 x 5.80 x 5.95 x
0 10-l“ lo-l3 lo-‘* 10-l’ 1o-8 1o-8 10mb 1O-5 10m3 lo-* lo- ’ lo2 10’ lo3 lo4 lo4 lo4
0
1.25 x 1.80 x 7.67 x 1.03 x 4.16 x 5.53 x 2.25 x 2.91 x 7.70x 7.02 x 1.30 1.47 x 3.65 x 3.30 x 2.66 x 1.30 x 8.97 x 9.86 x
lo-‘” lo-” lo-l5 10-l’ IO-” lo-” lo-’ 10-h 1O-4 lo--’ 10’ lo3 lo4 10’ 10’ 10’ 10’
4.99 1.43 3.03 8.12 1.64 4.36 8.88 2.29 2.98 5.33 4.98 1.14 1.41 2.50 9.67 9.03 2.05 2.61
0 x x x x x x x x x x
10mz3 lo-” lo-‘” 10-l’ lo-l3 lO_” 1O-9 lo-’ lo-“ lo-’
x x x x x x x
lo2 105 lo6 10’ lo9 10” 10”
TABLE 4. FIRSITRANSMISSIONFUNCTIONS FOR HYDROGEN,F,(U~ cm-’
u(cm-atml
Tl = 10,000 K Ne = 10”’ cms3
T, = lO,O+lOK Ne = 10” crne3
0
2.08 x 1.18 x 1.25 x 6.70 x 6.77 x 3.60 x 3.66x 1.88 x 1.22x 4.28 x 2.02 x 9.40 x 5.73 x 2.01 x 3.79 x 6.86 x 5.92 x 9.48 x
atm-’
TI = 20,000 K Ne = 10” crne3
0
2x10-: 4 x lo-’ 6 x lo-’ 8x10_’ 1 x lUh 1 x 10-S 5x 1o-5 1 x 1o-4 5 x 10-4 1x10-J 5x10-3 1 x 10-l 5 x 1om2 1x10-l 5x10-’ 1 5 1 x 10’ 5 x 10’ 1 x lo2 5 x lo2 1 x lo3 5 x lo3 1 x lo4
6.74 :104 5.22 x lo4 4.36 x lo4 3.77 x lo4 3.33 x lo4 3.51 x lo3 4.50 x lo* 1.90x lo* 1.73 x 10’ 5.42 2.96 x 10-l 9.88 x lo-* 1.46 x 10m2 6.84x 10-j 3.86 x 10 ~’ 6.35 x 1O-5 6.34 x 10m6 2.96 x 10-h 6.30 x lo-’ 3.51 x lo-’ 1.04x lo.-’ 5.45 x lo- 8 5.38 x 1O-9 1.47 x 1om9
1.32 1.14 1.03 9.58 9.00 3.30 7.70 3.09 3.70 1.38 1.04 3.03 2.17 8.21 3.97 7.87 2.16 1.40 5.26 3.08 3.60 8.60 2.89 8.42 -_
:104 x lo4 x lo4 x 10’ x 10’ x 10) x lo2 x lo2 x 10’ x 10’ x x x x x x x x x x x x x
lo- ’ lo-* lo-’ lo-& lo- 5 1O-5 1o-5 lo-’ 10mh lo-’ 1o-8 lo-’ lo- lo
lo-” lo-= lO_” lo-” lo-l6 lo-l4 10-l’ lo-” 1O-4 10m3 10’ lo2 lo6 lo8 10’ ’ 1OlZ lOI 1Ol4
2.20 1.95 1.80 1.70 1.61 7.84 3.03 1.61 2.32 1.02 1.34 5.46 7.84 4.02 1.11 2.25 4.96 2.15 3.36 1.07 3.96 6.47 7.37 2.49
:104 x lo4 x lo4 x lo4 x 10“ x lo3 x lo3 x lo3 x 10’ x lo2
x x x x x x x x x x x x
lo- 1 10-l 10-Z 10-l 1O-4 1O-4 1O-5 1o-5 lo-’ lo-” lo- ’ ’ lo-l3
8.89 x 1.01 x 5.32 x 5.71 x 2.88 x 3.06 x 156x 1:60x 5.13 x 3.58x 8.49 x 7.97 x 2.42 x 1.68 x 1.56 x 5.50 x 1.97 x 4.35 x
lo-= lo-l9 lo-” 1Om23 lo-‘” 10-lh lo-” 1O-9 lo-’ lo-” 10’ 10) 10’ 10’” lOI lOI 10” 10’”
Transmission functions in the numerical solution of radiative heat transfer problems
1747
TABLE4.-continued
t&m-atm)
Tl = 10,000 K
TI = 10,000 K
Ne = lOI cm-s
Ne = 10” cm-s
5x lo4
1x 5x 1x 5x 1x 5x
lo5 lo5 lo6 lo6 10’ 10’
1.77 x 1.49 x 1.15 x 1.06 x 5.34 x 1.00 x 0
10-i’ 10-i* 10-15 10-i’ 10-29 1o-41
1.48 x lo-” 1.45 X lo-‘2 1.10 x lo-i5 9.75 x lo-‘s 4.70 x lo- *g 9.33 x 1o-4* 0
TI = 20,OMIK Ne = 10” crnd3
2.98 x lo-=’ 5.72 x lo-“* 0
the following result is obtained : m I
F,(Z) dZ = F,(O).
(16)
0
Since F,(u) has a singularity at u = 0, a six-point Newton-Cotes open formula was used to integrate F, near the singularity. (‘) A Romberg method was used for the remainder of the integration. For a Au of 2 x 10P6, the errors in equation (16) JimaXF,(Z) dZ--F,(O))/ F,(O), were 11.8, 2.8, and 11.1 per cent for TI = lo4 K, Ne = 1016 cme3; Tl = lo4 K, Ne = 10” crnw3., and TI = 2 x lo4 K, Ne = 10” cme3, respectively. When Au was changed to 2 x lo-‘, the errors became 8.5, 2.8, and 11.0 per cent for the three cases. These errors are within the accuracy of the line-absorption coefficient data used in calculating F, JIO) and thus it is not worthwhile to attempt to obtain greater accuracy. NOTATION
e, E” F” L M” Ne P ii 9R T
TI T2 3
u UO Y
Planck’s function exponential integral functions transmission functions boundary spacing, cm moments of F,, (cm-atm) electron density, cmm3 total pressure, atm partial pressure of atomic hydrogen, atm heat source per unit volume radiative heat flux temperature, “K boundary temperatures, “K pressure path length, u = P,,y, cm-atm total pressure path length, u0 = PHL, cm-atm
distance measured from lower boundary, cm
Greek symbols
K” % /I CT w
spectral absorption coefficient, linear Planck mean coefficient, thermal conductivity Stefan-Boltzmann constant wave number, cm
cm-’ cm-’
1748
D. A. MANDELL REFERENCES
I. 2. 3. 4. 5. 6. 7. 8. 9. 10.
E. M. SPARROWand R. D. Cass, Radiation Heat Transfer. Books/Cole, Belmont, Calif. (1966). R. D. Cuss, P. MIGHDOLLand S. N. TIWARI, In?. J. Heat Mass Transfer, 10, 1521 (1967). D. A. MANDELLand R. D. CESS,Znt. J. Heat Muss Transfer, 13, 1 (1970). D. A. MANDELL,AIAA Jnl. 8, 1510 (1970). S. E. GILLES,A. C. COGLEY and W. G. VINCENT],Int. J. Heat Mass Transfer, 12,445 (1969). IBM System/360 Scienrific Subroutine Package, (360A-CM-03X) Version III. IBM Corporation, White Plains, New York (1969). S. CHANDRASEKHAR, Radiafiue Transfer. Dover, New York (1960). D. A. MANDELLand R. D. C&s, JQSRT 9,981 (1969). M. ABRAMOWITZand I. A. STEGUN,Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Natn. Bur. Stand. Applied Mathematics Series 55 (1964). H. GRIEM,Plasma Spectroscopy. McGraw-Hill, New York (1964).