Radiative and convective heat transfer between two parallel streams of absorbing and emitting radiating gase

Whang, Y. C.

English

NASA Technical Reports Server

RADIATIVE TRANSFER FOR PARALLEL 
S T R E A M S  OF RADIATING GASES 
by 
Y. C .  Whang 
Report No .  66-001 January 1966 
Department of Space Science and Applied Physics 
The Catholic University of America 
Washing ton, D .  C .  2001 7 
“This work was supported by the National Aeronautics and Space Administration 
under NASA Research Grant NsG-586. 
https://ntrs.nasa.gov/search.jsp?R=19660018644 2020-03-16T19:34:55+00:00Z
. . 
ABSTRACT 
The radiation heat transfer between two parallel streams of absorbing and emitting radiating 
t 
gases is studied. The problem i s  described by the Rossefand diffiision approximation together 
wi th the radiation s l ip  bundary condition. Considering that the flow i s  incompressible and 
invidcid, the exact solution of the problem i s  obtained; The s!tltinr! shows I? sn?ooth tmncition 
1 
regime between the optically thin and the optically thick regions. The thickness of radiation 
layers i s  calculated. 
- I  - 
1. INTRODUCTlON 
This paper presents an exact solution for the radiative and convective heat transfer 
between two parallel streams of absorbing and emitting radiating gases. The Rosseland 
diffusion approximation and the radiation slip boundary condition are used to describe the 
1 
1 problem. Probstein calculated the radiative heat transfer between two parallel plates 
containing a radiating as in  the gap by this method. The calculated results based on the 
radiation slip boundary condition are i n  good agreement with the numerical results 
2-4 
obtained 
by solving the governing integro-differential equations. Probstein suggested that the difficult 
problems of solving the integro-differential equations may be circumvented by the use of the 
"radiation slip" method and that this method may be applied to other gas flow problems where radiation 
heat transfer i s  of importance. Some applications of the radiation slip boundary condition to gas 
flow problems have been carried out in references 5-7. 
In this paper the radiation slip boundary condition i s  used to study the radiative transfer 
between two streams of incompressible inviscid flow. 
of the radiation transfer i n  the transition regime, we consider that the conductive heat flux i s  
neglegible compared with the radiative flux. The exact solution of the problem in  a closed 
form i s  obtained. A special case of this solution gives the exact solution for the radiative 
transfer in  an incompressible inviscid flow over a flat plate. The physical significance of the 
solution i s  discussed and the thickness of  radiation layers i s  calculated. 
In order to investigate the main features 
- 1 -  
4 
2. RADIATIVE TRANSFER IN TWO PARALLEL STREAMS 
We shall study the radiative transfer between two parallel streams of incompressible invicid 
gases, moving horizontally in  the same direction. The x-axis i s  &awn horizontally i n  the direction 
of motion and y - axis vertically upwards. At x = 0 the two .flows come into contact and have 
different temperatures, densities and velocities. The energy equation i s  
where the suffix i = 1 refers to the upper flow and i = 2 to the lower. Using the Rosseland 
diffusion approximation, the radiative flux gR may be expressed as the product of the 
temperature gradient and the radiative thermal conductivity, namely, 
where T i s  the Stefan - Boltzmann constant and k. the volumetric absorption coefficient. 
Assume that kR 
R 
varies as a third power of the temperature, then we can write (1) as 
The radiation slip boundary condition wi l l  be applied to the interface between the two streams. 
L On the interface there i s  a temperature jump that i s  proportional to tbe radiation temperature 
gradient. Thus we can write the boundary conditions 
T =T.  for all y 
I 
(;)As x = O  
(ii) As x 1 0 T =  T. for I Y  1-00 
I 
- 2 -  
and 
where the constants c, and c1 are to be determine# later. 
9 
We introdiice the %!lowing dherisionless quantities: 
Q =(T - T2) / (TI - T2) 
3 = 16 rT,3 x / 3  cf  k, p\ U\%, 
I= Y / C \  
d= e, u\ % I  / e, u2 CPZ 
and 
In terms of the dimensionless quantities, we can .write equation (2) and the boundary 
conditions as 
0 AS 3 = 0  
and 
} (3) for 7 7 0  
for '1 < o 
The exact solution ot this problem can now be obtained by means of the Laplace transformation 
- 3 -  
method. Let f(s ,v  ) be the Laplace transform of Q( 3, q), then we can find the Laplace 
transformation of the solution of (3) subject to the given boundary conditions 
f=,- I e x p  (- 3") 
5 ( I  +q t s " a )  
an d 
The inverse transform of f( s)? ) gives 
I for 7 .o 
for 7 
From (5) we can obtain the radiative heat flux across the interface. 
Now we can determine the two constants C, and C, by requiring that the solution 
reduces to the black-body result in  the l i m i t  of the optically thin regime, namely 
(4) 
- 4 -  
This condition yieids 
and 
Therefore we obtain the exact solution for the radiative flux between the two streams of 
incompressible, inviscid gases as 
(7)  
The temperature changes on the two sides o f  the interface are related by 
Some solution for 0 and (-qR) 
and R. Figure 1 shows that the exact solution processes a smooth transition between the black 
are plotted i n  Figures 1 - 7 for various values of a Y = o  
body formulation for the optically thin regime and the Rosseland diffusion formulation. 
2 -5 
When 0 S (1 + mR) 3' 7.85~ 10 , equation (7) gives O.W< (-qR) &(T,~-T>* 1 
Y= 
Physically this result means that the accuracy of the black body formulation i s  within one percent 
for the region 
-4 2 
x < 4 . 2 ~  10 6, LRl /(1 + OCR) 
where B 1 = e, u\cpr / 0 7 - \ ~  i s  the Boltzrnann number, and 1 = I /  k, i s  
the photon mean free path. If the "radiation slip" boundary condition i s  not used, the heat flux 
across the interface calculated by the Rosseland approximation i s  
- 5 -  
2 2 
Equation (7) approaches to (8) asymptotically as (1 +do) 3-00 . 48.5 
the error introduced by equation (8) i s  less than one percent. This means the no-slip solution (8) i s  
When (1 + oc R) 3 > 
accurate to within one percent in  the region 
- 2 
X > 260 B \  fL~ , / ( l  + O( R) 
In this region the influence of radiation slip can be neglected. 
3 .  RADIATIVE TRANSFER FOR A N  INCOMPRESSIBLE INVISCID FLOW OVER A FLAT PLATE 
The radiative heat transfer for an incompressible inviscid flow over a plate of constant 
8 5 
temperature has been studied by Goulard and Tien and Greif . I ts  exact solution can ac tw l l y  
be obtained directly from (5) and (7) by taking kRz+o  or R = 0 , namely I 
a n d  
The exact solution (10) can be expanded as 
While the solution by Tien and Greif can be expressed as 
These solutions are also plotted in figures 1 and 2. For this case the black body solution i s  accurate 
- 6 -  
to within one percent i n  the region 
x < 4.2 x \o-+ B, &, 
and the effect of the radiation slip condition can be neglected i n  the region 
4. THICKNESS OF RADIATION LAYERS 
The results obtained in  section 2 show that the temperature changes rapidly from the 
temperature at y = 0, (say T, ) to the main stream temperature at y = 00 , T, within 
a layer beyond which the temperature changes asymptotically and the influence of radiation i s  
imperceptible. If we define the radation layer thickness as the distance 6 from the interface 
of two streams(or from the surface of a flat plate) for which 
T - To = 0.99(T, - To) 
then from (5) we obtain that when y = 6 
~ 
where F i s  defined by equation (6) 7 
(when 3 tends to infinity, equation (10) reduces to 
__- 
or 
- 7 -  
when 3 tenus to zero, equation ( S  1 j reduces to 
0 . 0  \ = e x p (- r2/4 3 ) - T * [ 7 /(2 3 y2 >3 er f c [ ’1 / ( 2 3 ‘/z )] 
7/5‘‘2 = 2 . 2 7  or 
From (12) and (13), we can express ihe ihickness of radiation layers as 
6 = E(D,, x /  L 4 ,  y2 
or 
3 
where D 
value of E i s  between 2.27 and 2.58, E equals to 2.27 at the optically thin limit and 
= (16 Q T  )/(3 F C p  k, ) may be called the radiation diffusivity, and the R 
2.58 the optically thick limit. The variation of E shows that the thickness of radiation layer 
i s  slightly reduced near the leading edge of the plate due to the effect of the radiation s l i p .  
This phenomenon i s  quite similar to the effect of the mass velocity slip on the momentum thick- 
9 
ness of boundary layers . In terms of the Boltzmann number and the radiation mean free path, 
the thickness of radiation layers becomes 
where E’ = 4~ (35‘’* = 5.25-5.95. The thickness ra.io of 
the two parallel streams of incompressible inviscid gases is 
or 
he two radiation layers for 
References 
1 .  
2. 
Pr~stejn, R F., "Ra.iiatic?r! r!Ip,".A.!,4,4 Jcgrna! 1, !202-!29433963) 
Usiskin, C. M. and Sparrow, E.M. , "Thermal radiation between parallel plates separated by 
an cbsorbing-emitting nm-isothermal QGS, I '  Intern. J. Heat E v k ~ s s  Transfer 1 , 28-36( 1960) 
3. Sarofim, A.F. , "Radiation heat transmission in  enclosures, 'I  Sc. D. Thesis, Dept . Chem. Eng . 
- 
- 
Mass ~ l n s t  ~ Tech (1  942) 
4. Viskanta, R.  and Gross, R. J .  , "Heat transfer by simultaneous condition and radiation," 
J . Heat Transfer C84, 63-72(1962) 
5. Tien, C.L. and Greif, R., "On the transition from black-body to Rosseland formulations i n  
optically thick flows," Intern. J. Heat Mass Transfer 7, 1145-1 146(1964) - 
6. Greif, R .  , "Couette flow of a radiating and conducting gas," AIAA Journal 3, 98-102(1965) 
7. Deissler, R. G. , "Diffusion Approximation for thermal radiation in  gases with iump boundary 
- 
condition," J. Heat Transfer (to be published). 
8. Goulard, G. , "The transition from black body to Rosseland formulations in  optically 
thick flows," Intern. J. Heat Mass Transfer 6, - 927-930(1963). 
Lin, T. C. and Schaaf, S.A.  , "Effect of slip on flow near a stagnation point and in  a 9. 
boundary layer," NACA Technical Note 2568(1951). 
Table 1, Radiation Heat Flux calculated by equation (7 ) 
approximate so lu t ion  by Tien and Greif  f o r  8 = o a  
compared with t h e  
0.01 
0.02 
0.05 
0.10 
0.20 
0.50 
1.00 
2.00 
5.00 
10.00 
20 * 00 
50.00 
100 . 00 
Equation ( 7)  1 
0 e 89646 
0 e 85848 
0 s 79038 
0 o 72358 
o e 64379 
o 52316 
0 42758 
0 e23233 
O e 33620 
0 17058 
0 a 12321 
0 07901 
0,05614 
Table 2,  Calculated temperature for = o 
I 3 = 001 7 i 
100 
5J 
30 
20 
10 
3 
3 
1.00000 
i 0 99655 
0.5 0,95059 
Oe3 0.89084 
0.2 0.846?0 
0.05 0,75841 
0,03 0 e 74480 
0.02 0 73784 
- 
n 
001 0 e 79049 
0001 0 73076 
0 0 72358 
3 = 1  
1 . 00000 
0 . 99989 
0.98831 
o e 93666 
o (I 62186 
o a 46964 
0,43611. 
0.77095 
0 a 54915 
0 e 51020 
0 44879 
0,44035 
0,43185 
0 42758 
5 = 10 
10 00000 
0.98421 
0 e 81060 
o . 61732 
0.48660 
0 0 33599 
0.25473 
0 e 22137 
0.20453 
0 17910 
0 e 17570 
0 e 17399 
0 17229 
0 18760 
0 e 17058 
~ ~~ 
Tien and Creif, g= o 
o e 98464 
o . 87161 
o e 62367 
0 s 97003 
0 92885 
0 o 78370 
0 49138 
0 37054 
0.24391 
0.17564 
0.12516 
0 05624 
0 e 07954 
f = 100 
1 e 00000 
0 6 99968 
0 . 97126 
0 e 86149 
o . 56216 
0 32779 
0.22208 
o 16750 
0.11210 
0.08417 
0 07297 
o e 06736 
0 a 06175 
o e 05726 
0 05670 
o 05614 
0.05895 
0 * 05783 
- 10 - 
Table 3. Calculated temperature f o r  o( = 1, != 1. 
7 
100 
50 
30 
20 
10 
5 
3 
2 
1 
0.5 
0 * 3  
0.2 
011 
0.05 
0.03 
0.02 
0.01 
0 
-0 
-0.01 
-0 . 02 
-0 03 
- 0 e 0 5  
-0.1 
-0.2 
-0.3 
-0.5 
-1 
-2 
-3 
-5 
-10 
-20 
-30 
-50 
-10 
3 = 0,1 
1,00000 
0 99695 
0 95779 
0,90857 
0 e 82858 
0.87262 
o 80361 
0.78231 
0.77680 
0,21769 
0.21226 
Q 20690 
o 19639 
0 ~ 2 7 3 8  
0 e 79310 
0 78774 
0 e 22320 
0 a 17142 
0 09143 
0 04221 
0 e 00305 
0 
1 0  00000 
0 9 99991 
0 e 99124 
0 ., 95428 
0 84230 
0 .?4671 
0 e 70146 
0.67757 
0 I 65295 
0.64040 
01 63534 
o 63280 
o 63025 
o 62770 
o e 36720 
o . 35960 
0 37230 
0,36975 
0 36466 
0 34704 
0 e 32243 
0 29854 
0 e 2 5 329 
0 e 15770 
0 e 04 572 
0,00876 
0 e 00009 
0 
3 = 10 
1~00000 
0 99023 
0 88924 
0 e 78162 
0,71065 
0 e 63042 
o e 58777 
0 e 55287 
o . 54671 
0 o 57039 
0.56164 
0 54847 
0.54583 
0.54495 
0 e 54407 
0.45505 
0.45417 
0 e 45329 
0.45153 
0.44713 
0.45593 
0,43836 
0 e 42961 
0.36958 
0.21838 
0 11076 
0,41223 
0.28935 
0 00977 
0 
3 = 100 
I. 00000 
0 a 99982 
0.98443 
o . 92629 
0 * 77095 
0.65123 
0 * 59767 
0 e 57011 
0,54220 
o 52816 
0 . 52254 
0,51972 
o 51690 
0 e 51550 
0 51493 
0 51465 
0 51437 
0 51409 
0.48591 
0.48563 
0.48535 
0 . 48507 
0.48450 
0.48310 
0.48028 
0 e 47746 
0,47184 
0.45780 
0.40233 
0.34877 
0.22905 
0.42989 
0 07371 
0 01557 
0.00018 
0 
- 1 1  - 
Table 4, Calcu la t ed  temperature for c>(= 0,1, Q =  1 
. 
I 
s 
100 
50 
30 
20 
10 
5 
3 
2 
1 
Or5 
0: 3 
0 , 2  
Ot1 
0 ,05  
0,03 
0,02 
0,Ol 
0 
-0 
-0.01 
-0,02 
-0c03 
-0,l 
-0.2 
-0e3 
-0.5 
-1 
-2 
-3 
-5 
-10 
-20 
- 30 
-50 
-100 
- 0 ~ 0 5  
3 = c,1 
1,00000 
o r  99660 
0 e 95141 
0 89288 
0 84928 
0,79496 
0 - 76374 
0 73686 
o 02701 
0~02050 
0 e 75051 
0,74373 
0 a 72988 
0,01507 
0.01071 
0,00486 
0 * 00034 
0 
i 
s = i  
1 e 00000 
0 ". 99990 
0 98867 
0 93898 
0 78071 
0 63928 
0 57058 
0 53385 
0 a 49566 
0,47605 
o 4681.2 
0 e 46413 
0 46013 
0 45612 
0,050413 
0.04662 
0 e 04294 
o 03607 
0 0219 3 
o 00610 
0 00113 
0 : 00001 
0 
0 2 05439 
3 = 10 
10 00000 
0.98512 
0 82308 
0 64390 
0.52323 
O D 3 8 U G 2  
0 31002 
0 27942 
0 e 26400 
0 24848 
0 24070 
0 2 37 58 
0 e 23446 
0,23602 
o e 07671 
o e 07360 
0,07206 
0 06900 
0 06154 
0 04768 
0, 03561 
0.01769 
0 c 23290 
0.07515 
0 00149 
0 
1 e 00000 
6 r 99976 
0 0 97350 
o 87267 
o e 38478 
0 59870 
0,28843 
0 E 23871 
0 - 18827 
0,16285 
0,15266 
0,14756 
0 L) 14245 
0 13888 
0 c 13990 
o 13837 
0 13786 
0 ., 08627 
0 08575 
o 08371 
0 07613 
o 07116 
0 e 06152 
0 00265 
0.00003 
0 
0 0 13735 
0.08524 
0 08473 
0 08117 
0 04013 
0 s 01273 
- 12 - 
.* 
Table 5. Calculated temperature for rn = 10, 
7 
100 
50 
30 
20 
10 
5 
3 
2 
1 
Oe5 
0.3 
0,2 
0,1 
0.05 
0*03 
0,02 
0.01 
0 
-0 
-0 * 01 
-0.02 
-0 e 03 
-0 0 05 
-0.1 
-0.2 
-0e3 
-0e5 
-1 
-2 
-3 
-5 
-10 
-20 
-30 
-50 
-100 
3 = on1 
1 * 00000 
0 e 99851 
0 a 98231 
0 o 96439 
0 95232 
0 e 93846 
0.93100 
0 92794 
0,92640 
0.92485 
0 0 92329 
0,76554 
0476398 
0,75930 
0 * 75152 
o e 76710 
0 e 76242 
0 73601 
0.72058 
0,61538 
0 35610 
0.17692 
0.01488 
0 
0 e 68998 
0 e 47677 
3 = l  
1 m 00000 
0 a 9997 
0 e 99735 
0 e 98726 
0 3 95987 
0 Q 93848 
0 92884 
0 4 92387 
0,91883 
0 91527 
0,91425 
0 e 91373 
0,86214 
o 91629 
o a 91476 
o e 86265 
0.86163 
o 86112 
0 86010 
0 e 85755 
0 85244 
0,84734 
0 e 83715 
0 81173 
o e 61522 
o 02650 
0 00030 
0 
O e 76129 
0 0 71157 
0 e 40130 
0 12733 
g =  1 
3 = 10 j 3 = 100 
1 e 00000 
0 99781 
0 97681 
0 955119 
0.94180 
o 92661 
0 a 91865 
0.91380 
o e 91138 
0.91089 
0.91056 
0 91542 
0 e 91219 
0 a91105 
0 e 91073 
0.89435 
0,89419 
0 89403 
0 89354 
0 a 89111 
0 88949 
0 87814 
0 e 89387 
0 89273 
0,88625 
o 86195 
0.81355 
0 e 84578 
0 o 73393 
0.58198 
0 . 44506 
0 23188 
0 e 02189 
1 * 00000 
0.99996 
0 e 99697 
0 * 98587 
0 * 95677 
0 a 93465 
0 e 91978 
0 e 92482 
0 . 91468 
0 91212 
0 g1no 
0.91058 
0 e 91007 
0 e 90981 
0 90971 
0 * 90966 
0 . 90961 
0.90956 
0.90443 
0 e 90438 
0,90433 
0 e 90427 
0.90417 
0 e 90340 
0 0 90392 
o e 90289 
o . 90186 
0 e 89930 
0 . 89417 
0,88904 
0 e 87879 
01 85319 
0 b 80224 
0 e 75181 
0 43229 
0,65351 
Table 6. Calculated temperature f o r  d = f i  s g = u n  
100 
50 
30 
20 
10 
5 
3 
2 
1 
O a 5  
0.3 
0.2 
0.1 
0.05 
0.03 
0002 
0001 
0 
-0 
-0 01  
-0.02 
-0 03 
-0.05 
-0.1 
-0.2 
-0.3 
-0.5 
-1 
-2 
-3 
-5 
-10 
-20 
-30 
-50 
-100 
1 = 0.1 
1.00000 
0 99695 
0 . 95779 
0,90856 
o . 87262 
0,82858 
o . 80361 
0.79310 
0.78774 
0 . 78231 
0.22320 
0 . 77 680 
0 22264 
o ,22209 
o e 21769 
o . 21226 
0 20690 
0,19639 
0 e 22154 
0.22044 
0 . 17142 
0 e 09143 
0 04221 
0.00305 
0 
O a 12738 
i 
? = 1  q ? = l o  
3 
1 b 00000 
0 a 99991 
0 . 99124 
0 . 84233 0 0 95269 
o e 74671 
o 70146 
0.65295 
0 e 63280 
0 63025 
o 62770 
0 0 67757 
0.64040 
0,63534 
0 e 37230 
0 o 37205 
0 37179 
0 37154 
0 a 37102 
0 e 36975 
o e 36720 
o . 36466 
os 35960 
o e 32263 
0 34705 
0.29854 
0 25329 
0 15767 
0 e 04731 
o 00876 
0 . 00009 
0 
- 14- 
3 
1 . 00000 
0 99023 
0 . 88924 
o . 78162 
0 . 71065 
0 63042 
0 a 58777 
0 0 57039 
0,55287 
0.56164 
0 . 54847 
o . 54671 
0,54583 
O o  54495 
0.54407 
0.45593 
0 45585 
0.45576 
0.45567 
0.45549 
0.45505 
0.49417 
0 a 45329 
0.45153 
0.44713 
0.43836 
o . 42961 
0.41223 
0 . 36958 
0 21838 
o e 11076 
0 o 28935 
0 00977 
3 = 100 
i,ooooo 
0. 99982 
0 . 98443 
o . 92629 
0 . 59767 
0,52816 
o . 51690 
0 77095 
0.65123 
0 57011 
0 . 54220 
0.52254 
0 e 51972 
0.51550 
0 . 51493 
0 . 51465 
0 51437 
0 . 51409 
0 . 48591 
0 . 48588 
0.48586 
0.48583 
0.48577 
0.48563 
0.48507 
0 . 48450 
0 . 48310 
0.48028 
0 , 47746 
0 . 47184 
0 . 45780 
0 . 42989 
0.40233 
0 . 34877 
0 48535 
0.22905 
Table 7. Calculated temperature for OC = m, = fl 
1 
100 
50 
30 
20 
10 
5 
3 
2 
1 
0.5 
0.3 
0.2 
0 , l  
0.05 
0.03 
0.02 
0.01 
0 
-0 
-0101 
-0.02 
-0.03 
-0 05 
-0.1 
-0.2 
-0.3 
-0.5 
-1 
-2 
-3 
-5 
-10 
-20 
-30 
-50 
-100 
3 = o s 1  
1 d 00000 
0 99851 
0 . 98231 
0 96439 
0.95232 
0.93846 
0 93100 
0 o 92794 
0 e 92640 
o , 76710 
0.73601 
o . 72058 
0.61538 
0.47677 
0.35610 
0 e 17692 
0 92485 
0 92329 
0 75152 
0 b 68998 
0 e 01488 
0 
3 = 1  
1 4  00000 
0 8 99997 
0,99735 
0 * 98727 
0 * 95987 
0.93848 
0 . 92884 
0 e 923 87 
o 91883 
o . 91629 
0 91527 
0 91476 
0.91425 
0 e 91373 
0 o 85755 
0 84734 
0 e 83715 
0 a 86265 
0,85244 
0 e 81173 
o a 76129 
0 e 61522 
0.02650 
0 e 71157 
0 e 40130 
0 a 12733 
0 . 00030 
0 
3 = 10 
1.00000 
0 . 99781 
0.97681 
0 e 95549 
0.92661 
0 91865 
o , 91380 
0.94180 
0 e 91542 
0 91219 
0.91138 
0 91105 
0.91089 
0.91056 
0.891~ 
0.88949 
0.88625 
0.87814 
0.86195 
0.84578 
0 91073 
0.89435 
0 e 89273 
0.81355 
0,73393 
0 58198 
0 44506 
0 23188 
0 . 02189 
0 
3 = 100 
1,00000 
0.99996 
0.99697 
0 . 9 85 87 
0 e 95677 
0 . 93465 
0.924 82 
0 e 91978 
0 e 91468 
0 e 91212 
0 . 91110 
0.91058 
0 91007 
0.90981 
0.90966 
0.90961 
0 90971 
0 . 90956 
0 , 90443 
0.90340 
0 . 90289 
0.89930 
0 . 89417 
0,88904 
0.90392 
0.90186 
0,87879 
0 e 85319 
0.80224 
0.75181 
0.43229 
0.14130 
0.65351 
0 03033 
0 
o ,00036 
- 15 - 
Figure Captions 
Figure 1 Radiant heat flux between two parallel streams (B = o for flow past a flat plate) 
Figure 2 Temperature profile for 13 = o 
- 
Figure 3 Temperature profile for d = 1, B = 1 
* Figure 4 Temperature profile for &= 0.1, R = 1 
Figure 5 Temperature profile for CA = 10, R = 1 
Figure 6 
Figure 7 
Temperature profile for o(= (\  o)”, B = (\o>-’” 
Temperature profile for oC= (10) B = (lOf’z H 
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Radiative transfer for parallel streams of radiating gases

Abstract

Similar works

Full text

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