Nonlinear radiation analysis in hydrogen plasma

J. Quanr. Spectrosc. Radiat.

Tramjer.

Vol.

I I. pp.1163-l

170. Pergamon

Press 1971. Printed in Great Britain

NONLINEAR RADIATION ANALYSIS IN HYDROGEN PLASMA M. Polytechnic

M. ABU-ROMIA*and R. PYARE~ Institute

of Brooklyn,

Brooklyn,

(Received 7 December

New York

11201

1970)

Abstract-This paper deals with the radiation energy transport within a plasma gas enclosed between two parallel black plates, together with the interaction of radiation and conduction under uniform heat generation, An expression for the radiation heat flux is being obtained in terms of the modified emissivity without resorting to the assumption of linearized radiation previously used. Results for the temperature and radiation heat flux distributions are obtained for various values of heat generation and for different spacings between the plates. The results presented are for hydrogen plasma gas at pressures of 0.43 and 37.5 atm with wall temperatures of 10,OOtPK. The present results for the centerline temperatures are compared with those utilizing linearized radiation, and it was found that the linearized radiation assumption overestimates the temperature values especially for large heat generations. I. INTRODUCTION

As HEAT transfer investigations of coupled radiation-conduction or convection in nongray media requires lengthy numerical computations, several attempts to simplify the analysis have recently been reported. Of these is the use of modified emissivity in radiation transport calculations.“*2’ The modified emissivity takes into account the nongray behavior of the radiating gas and therefore enables one to avoid any description of the microscopic properties of the gas in the governing energy equation. Up to the present, the use of modified emissivity has been limited to linearized radiation. (2*3)Since in most high temperature applications where radiation is the dominant transport mechanism, the temperature varies within more than several orders of magnitudes, it is expected that in some cases the assumption of linearized radiation may not yield results of acceptable accuracy. In the present analysis, the assumption of linearized radiation is being avoided and .instead a more general expression for the radiation flux is obtained. The analysis, although based on the assumption that the absorption coefficient is not a strong function of temperature, was found to yield the correct functional dependence on temperature in both the optically thick and optically thin approximations. To illustrate the importance of accounting for the nonlinearity of radiation transport, the derived expression for the radiation flux is being utilized in the governing energy equation which includes in addition heat transfer due to conduction and internal heat generation. With the constant property assumption used in obtaining the results, values of the modified emissivity were taken from the calculations of MANDELL and CESS(‘)for hydrogen plasma at temperatures 10,OOO”K and electron densities of 1016 and 1Or7cmm3 which correspond to gas pressures of 0.43 * Associate

Professor

and tTeaching

Fellow, respectively. 1163

1164

M. M. ABU-ROMIA and R. PYARE

and 37.5 atm, respectively. Results for the temperature distribution, conduction and radiation heat fluxes for hydrogen gas boundary by two parallel black plates are obtained for various values of spacing between the plates. The centerline temperature results are compared with those obtained by the linearized solution for various values of heat generation. The comparison indicates that the linearized solution overestimates the centerline temperatures especially for large values of heat generation. Il. THE

RADIATION

HEAT

FLUX

Consider the case of an absorbing-emitting gas enclosed between two parallel black plates, with one plate (y = 0) at temperature 7’i and the other (y = 2L) at temperature T2. The monochromatic radiation heat flux is given byc4’

4,&y) = 2r,.,&(

i Uy’) dy’) -2e&(

7 Uy’) dy’)

+ 2 [ ~~O~‘)eJy’)EZ[ j k,(y”) di]

dy’

- 2 j k,,(y’)e,(y’)E, ( i k,,(y”) d y .j dy’. s

(1)

Y

Integrating equation (1) by parts and applying the condition of continuity of temperature at both walls, the result is

(2) The total radiation heat flux is obtained by integrating equation (2) over all wave numbers. After re-arrangement, one obtains

+(e, -e2)

(3)

Nonlinear radiation analysis in hydrogen plasma

1165

where ?J

=$

e =

oT4.

(4)

When the monochromatic absorption coefficient is not a strong function of temperature, then the integration over the variable q“ can be approximated by tl s

k,(V) d?” = k,(rl’) . [v -

~‘1.

(5)

T’

Utilizing this approximation in equation (3) and performing the integration over the wave number, the radiation heat flux can be written as 1

Ml) =

2

%[T’,L(q-tf)] drj’+

f d?’

%[T’,L(tl’-q)] s drl’

d$+(e,

-e2)

L

=

s

%[T’.LIq-q’l]

dq’+(e, -e,)

d,,’

(6)

0

where &(TI, X) =

(1 0

#

- 2E,[k,(

T’)X]} do

(7)

and T’ = T(q’). Equation (7) indicates that the modified emissivity is a function of both the temperature and path length for a fixed gas pressure. It is to be noted, that the approximation utilized in equation (5) is based on the fact that the local radiation heat flux is influenced not only by the local absorption coefficient but also by the absorption coefficients of other elements. In both the optically thick and thin limits, the resulting expression for the radiation flux and its derivative yield the correct functional dependence on temperature as shown in the appendix. For the optically thick limit, the radiation heat flux reduces to qR(Y) = -___

4

de

-

3K,(T) dy

(9

while in the optically thin limit, the derivative of the radiation flux is given by %(Y)

~

dY

= 4&e-2K,,,(T

T&e, -2K,,,(‘I: T2)e2

(10)

where K,, K,, and K,,, are the Rosseland, Planck and modified Planck mean absorption coefficients, respectively.“)

M. M. ABU-ROMIA and R. PYARE

1166

111. COUPLED RADIATION~CONDUCTION UNIFORM HEAT GENERATION The governing energy equation uniform heat generation is

for a medium

influenced

WITH by radiation,

conduction

and

(11) Utilizing

the assumption

of constant

properties

in which both the thermal

conductivity

K

and the modified emissivity are evaluated at a certain appropriate temperature taken here as Ti and using the terminology for K, = K(T,) and i:,(X) = i:(T]. X), equation (11) can be written in dimensionless

form as d20 d$ p-Mdrl+S

= 0

where

With

the

utilization

of the exponential

MANDELL and CESS”) have calculated

kernel

approximation

the modified

emissivity

for E3(t) = 1~)_ 3,(Zr)T

for hydrogen

plasma

at

various temperatures and electron densities. The dimensionless radiation flux $ in terms of their calculated modified emissivity takes the form (for the case 7” = 7, )

(14)

where (15)

The solution conditions

of equation

(12) is obtained

by integrating

it once and using the boundary

This yields the equation

d’~ - MI) +S(rl - 1) = dy which is subject

to the boundary

0

condition O(0) = 1.

Equations (14) and (17) are solved simultaneously, in the results section.

and the method

(1x1 of solution

is indicated

Nonlinear radiation analysis in hydrogen plasma IV. RESULTS

1167

AND DISCUSSION

The results for the temperature distribution of a hydrogen plasma gas enclosed between two parallel black plates at a distance 2L apart are shown in Fig. 1. These results are obtained by simultaneous solution of equations (14) and (17) utilizing an iterative scheme in which an 8th degree polynomial for the fourth power of temperature is assumed. The coefficients of the polynomial were determined by satisfying equation (17) at eight interior points of equal intervals apart and equation (18) for the boundary condition. Initial values of the coefficients were obtained simply by linearizing the radiation flux, and subsequent values were determined by iteration taking into account the nonlinearity of radiation transport until convergence is assured. The temperature results obtained by the 8th degree polynomial were found to compare reasonably well with those obtained by a 6th degree polynomial, and therefore no higher degree polynomial was attempted. Values of the modified emissivity utilized in the analysis are those calculated by MANDELL and CESS(‘)for hydrogen plasma at a temperature 10,OOO”Kand gas pressures of 0.43 and 37.5 atm. As indicated in Fig. 1, the temperature distribution at low gas pressure deviates slightly from that obtained by neglecting radiation transport. At 37.5 atm gas pressure radiation transport is the dominant mechanism, and as can be noticed from the figure ten times the value of heat generation is required to achieve heating the gas to the same temperature level as that of low pressure. Figure 2 shows the distribution of the radiation heat flux for various values of the heat generation. As indicated, the radiation heat flux attains its maximum value near the boundaries. Comparing the results of 0.43 atm with those of 37.5 atm for one cm spacing between the plates indicates that the radiation heat flux increases by one hundred times its value at low pressure for a corresponding increase in heat generation of ten times. Selective values of the radiation heat flux at the wall are shown in Table 1 for various values of heat generation and spacing. Values of the conduction heat flux can be obtained directly from equation (17) and the reported values of radiation heat flux. It is of interest to compare the present results with those obtained by linearizing the radiation heat flux in the governing equations. U) Shown in Fig. 3 are the results for the

FIG. 1. The temperature distribution for hydrogen plasma gas.

1168

M. M. ABU-ROMIA and R. PYARE -2

XI0

IO

IO Tz=T,

= 10,OOOeK

P = 0.43

otm

T2=T, = 10,OOOeK

. L=O.Scm

P = 37.5atm M = 160.0

.L=O.Scm

L

-8

6-

-6

FIG. 2. The radiation

flux distribution for hydrogen plasma gas enclosed for various values of heat generation.

between

two black plates

TABLE 1. EFFECT OF HEAT GENERATION“S” AND SPACING “L” ON THE RADIATIVEHEAT TRANSFERAT ‘THE WALL “i)(oy’ L ,cm

0.1 0.5 1.0 0.1 0.5 1.0

Gas pressure

0.25

0.50

2.5

0.138 x 1O-4 0.850 x 2.165 x 3.60 x 1o-4 18.35 x 37.50 x

0.407 x IO_‘+ 2.460 x 6.070 x 4.86 x 1O-4 24.35 x 48.80 x

2.502 x 1O-4 13.240 x 28.030 x 6.02 x lo-=’ 30.50 x 60.15 x

s

(atm) 0.43 0.43 0.43 37.5 37.5 37.5

IO T2= T, 2 10,000

OK

P = 37 5 ahl ----

Lmearued

-

Present Re*“l+S

Solvt,on [3]

o.8_

----

Lmeor,zed Soluhon[3]

-

Present Results

/ I 06-

/ I

04-

05

IO

2.0

0.1

S

FIG. 3. Centerline

temperatures

IO

0.5

5.0

s

as function of heat generation linear radiation transport.

for both cases of linear and non-

Nonlinear radiation analysis in hydrogen plasma

1169

centerline temperatures for various values of the parameter S and spacing L. As indicated in the figure, the results of the linearized radiation heat flux and those of the present analysis are in agreement for small values of heat generation and for low gas pressure. At high gas pressures and large values of heat generation, where radiation transport is the dominant mechanism, linearization of the radiation heat flux overestimates the centerline temperature. NOMENCLATURE

e E CT 4+ w

black body emissive power, e = uT4 Planck’s function monochromatic absorption coefficient thermal conductivity Planck and modified Planck absorption coefficient, respectively half-spacing between plates dimensionless parameter, M = (uT:L)/K, radiation heat flux heat generation per unit volume dimensionless parameter, S = (QL2)/(K1 TI) absolute temperature physical coordinate dimensionless y-coordinate, tl = y/L dimensionless temperature, 0 = T/T, _ modified emissivity Stefan-Boltzmann constant dimensionless radiation flux, + = q&T:) wave number

Subscripts 1 2 C

property evaluated at plate y = 0 property evaluated at plate y = 2L property evaluated at the centerline y = L

e e, k, K

K,, Km L M 4R f T Y tl

REFERENCES 1. D. K. MANDELL and R. D. Ces~, JQSRT 9,981 (1969). 2. S. E. GILLS, A. C. CbGLEY and W. G. VINCENT,Int. J. Hear Muss Transf. 12,445 (1969). 3. D. K. MANDELL and R. D. Cuss, ht. J. Heat Mass TransS. 13, 1 (1970). 4. E. M. SPARROW and R. D. Cms, Radiation Heat Transfer. Brooks/Cole, Belmont, Calif. (1966).

APPENDIX

A-THE

OPTICALLY

THICK

LIMIT

The expression for radiation heat flux in terms of the modified emissivity given in equation (6) can be written as 41(‘1)= -

1 -&[I-‘, L(q-q’)]

dr/’

dn’.

(1-A)

In the optically thick approximation, the integral is weighted about n’ = u. The result with de/d$ and Eevaluated at 0 is

M. M. ABU-ROMIA and R. PYARE

1170

with -7 = L(n - 9’) and Z’ = L(s’ - rl) equation

(2-A) becomes XL-y

qR(y)

=

d’

-

{I-c(T,z))dz

+

When y is removed takes the form

sufficiently

the definition

far from the boundaries,

for the modified

1

0

= -2d;

4&) Utilizing

(3-A)

{I-E(T,Z’))dZ’

s

dy

emissivity

so that y --t x. and (2L-,‘)

ls 0

{ 1 --~(7; z”)} d:“.

given in equation

s

( 1 - E(T, ?‘)I dz” = 2

(3-A)

(4-A)

(7), the integral

in equation

(4-A) reduces to

E,[k,,,( T);“] d-” dc, 0

”

+ x, then equation

0 (S-A)

The resulting integral absorption coefficient,

in equation (5-A) is recognized as equal to l/K,(7J where K,(T) is the local Rosseland The final expression for the radiation flux, in the optically thick limit, is 4 %(?.I = -32,(T)

APPENDIX In the optically

B-THE

de d;,,

OPTICALLY

thin limit X << 1 and therefore

THIN

the exponential

integral

LIMIT

E, can be approximated

E, =f-x-t. as a result, the expression

for the modified

emissivity

for small x by (l-B1

given by equation

(7) reduces

to

k,,( 7’) dtu.

(2-B)

”

After substituting

equation

(2-B) into equation

(6), the radiation

I

heat flux in the optically

thin limit takes the form

I

G(J)) = 2 2 1. ,_

and the derivative

of the radiation

s ,

.’

1

dtu+(e, -@,I

1

dg

2 1.

I

JU 00

3

de,,, d’-k,,(T’) d_r’~ dik,O(T’) .I’ ds’

that k, is not a strong function of temperature T instead of T’, the expression for the radiation

z 4K,r-2K,(7; where K, and K, are the Planck

(3-B)

Rux is

dq, -_=2

When the assumption the local temperature,

de, $, -y)k,,(T’)d~’

and modified

Planck

T,)e, -2K,(T, absorption

d$

1

dto.

(4-B)

is made such that k, can be evaluated flux reduces to

T&J, coefficients.

respectively.

at