Approximation to the radiative transport of energy in an optically thick line

,

Ir

2

[)

SEP, 1970

rECHN!S(I.,r:

H

.

VUE,;' : ..

~~~'i;~100l

DELFT

APPROXIMATION TO THE RADIATIVE . "-IJ JOE

I

Llvliln'

TRANSPORT OF ENERGY IN AN OPl'ICALLY

TRICK LlNE

by

APPEO;KIMATION TO THE RADIATIVE

TRANSPORT OF ENERGY IN AN OPTICALLY

TRICK LINE

by

R. R. Gilpin

Manuscript received January 1970

ACKNOWLEDGEMENT

I would like to thank ~r. S.

J.

·Townsend of the Institute for Aerospace Studies whose interest in MHD led to the problem investigated here.

SUMMARY

The radiative transport of energy in an optically thick line is studied. First an asymptotic expansion for the radiative kernal is obtained for a Lorentz line shape. This result is used to calculate the radiative transport in some problems of interest in MBD gases. The limits of applicability of this asymp-totic expansion are then determined by considering such effects as ltne overlap and statistical broadening.

1. 2.

3.

4.

5.

6.

7.

8.

9.

TABLE OF CONTENTS SUMMARY NOTATION INTRODUCTION

DERIVATION OF THE EQ,UATION OF RADIATIVE TRANSPORT

RADIATIVE TRANSPORT IN A LINE WITH LORENTZIAN PROFILE '.

3.3

Asymptotic Expansion of the Radiation Kernal ·

3.4

One-dimensional Radiative Transport

3.5

Effect of Geometry on Radiative Loss

3.6

Effect of Geometry on Radiative Damping :

3.6.1

Kernal for Radiative Damping in~a Cylinder

3.6.2

Damping of a Sine Wave in a Cylinder -:.

THE LINDHOLM LINE PROFILE

THE EFFECT OF OVERLAPPING LINES

CALCULATION OF THE RADIATION LOSS FOR A POTASSlUM SEEDED MHD, PLASMA

MEASURED LINE PROFILES

RADIATION KERNAL INCLUDING STATISTICAL EFFECTS AND LINE OVERLAP

RESULTS AND CONCLUSIONS REFERENCES APPENDIX FIGURES 1 1 2 2

5

6 6 6

7

8

9

10 11 11

13

14

NOTATION a Radius of a Cylinder

b Coefficient in Vander Waals equation d Diameter of a Cylinder

f. Oscillator Strength

1

k Absorptivity of the gas at frequency

m Reciprocal photon mean free path at line centre

o

mv

Reciprocal photon mean free path at frequency

v

r Radial distance from point at which radiatiye loss is being calculated v Mean velocity of atoms

w m/m

v 0

w

R Radiative damping coefficient for a sine wave

~ Black body intensity at line centre

~v Black body intensity at frequency v

N Number density of perturbing atoms a R Radiative loss S' Surface area at v o V Volume

V~

Volume normalized by À

3

E Parameter ~v/(2

v )

p Radius normalized by À

Characteristic length for nonuniformities in ~

Parameter

(v-v )/v

o p

v Frequency of radiation

v Frequency at line centre o

v Frequency at which statistical effects become important

p

6v Line width

1. INTRODUCTION

In some gases a large amount of energy is transferred as radiation in aspectral line. For example in a nonequilibrium MHD gas which consists of an alkali metal seeded rare gas a major part of the energy transfer t~t)determines

the electron energy occurs in the resonant line of the alkali metal(l. Usualky the gas is optically thick at the line centre. That is the mean free path of

the photons at line centre is much shorter than the characteristic length of

any nonuniformities in the gas. On either side of the line centre the mean free path of the photons increases and far from the l~ne centre becomes much larger than the characteristic dimension of the gas. Because the photon mean free path has such a large range of values the commonly used diffusion approximation-cannot be applied. However some simplification of the radiative transfer equation can be obtained using the observation that the photons which are likely to be most effective in radiative transfer will be those with a mean free path of the same order of magnitude as the characteristiclength of nonuniformities in the gas. Since in an optical~y thick line this occurs for a part of the line far away from the line centre only the shape of the wings of the line is important in

de~ermining the radiative transport. In this work a simplified expression for the radiative transport is first determined for a Lorentz profile and some appli-cations of this expression are made. Then deviations fr om the Lorentz profile due to line overlap and statistical effects are considered.

2. :QERIVATION OF THE EQUATION OF RADIA'IlIVE TRANSPORT

In the derivation of an expression for the net amount of radiative lost from some point r in the gas, the coordinate system in figure 1 will be used.

o

First -the amount of radiation emitted from dV in the frequence interval

dv and in the solid angle dS'/r2 is:

p k B (r) dv dV dS'/r2

v v _

(1)

Bv(r) is the black body intensity per unit solid angle at the temperature of the gas in dV. NormallY B does not change much over the width of a line

v

-therefore, k is the

v

B will be set equal to B, the intensity at the centre of the line. atsorptivity of the gas at the frequency v. In the following p

and kv will be assumed constant over the volume occupied by the gas and the product pk will be set equal to m •

v v

path of photon with frequency v.

m is the reciprocal of -the mean free

v

Because of absorption-between dV and dV' the amount of radiation -mvr

reaching dV' from dV is e of the amount in eq. 1. Of this radiation m _v dr'

is absorbed in dV' • at r is then:

o

R( :ab.s-.:;)

where r

I

r

I.

The total rumount of radiation absorbed per unit volume

00

J

mv 2

J

B(r) -m r _dV d v (2)

=

e

_2"

v r 0 v

The amount of radiation emitted at r i s : o 00 R(emit.) 47T

f

mv

d

,

~

.. B (r 0) o I

-The net radiative energy loss from the gas at rö can then be written as:

00 R (r ) =

f

m ;2 _dy,f -m r dV [B(r ) - B(r + r)] e v

(4)

o v 0 0

2'

r 0 v

3. RADIATIVE TRANSPORT IN A LINE WITH LORENTZIAN PROFI;LE For a Lorentzian line profile

"'v

= ma

[1

+

G

::a)2

-1

]

where m is the reciprocal of the photon mean free path at the line centre,

o

v , and ~v is the line width.

o From eq.

5:

dv - - ~v

_4"

1 m2 o ( 6)

Using eq.

6

the integral over v in eq.

4

can be converted to an integral over

Ir), giving:

} ~v

R(r =

-o 2

No~e that since the line is symmetrical the integral over mv has been taken over only one half of the line and is multiplied by two.

In eq.

7

lengths and volume will be nondimensionalized by introducing a characteristic length, À, for changes in B. Setting t

=

v/À and

dV~ = dV/À3 and setting w = m /m gives:

~ v 0 1 R(t ) =

~v

m2

Àfw~

o 2 0

(l-W)-~

dw

xJ

[B(~ )-B(~

o 0 + o

v

t

3. 3 ASYMPI'OTIC EXPANSION OF RADIATION KERNAL

_ -w~(m

À)

t)]e 0 (8)

In practice eq.

8

is difficult to use because of the integral form of the kernal. However in the case of large m À , that is when the character-istic length is much larger than the mean freeOpath at the line centre, a simplified asymptotic expansion can be obtained for the kernal.

For large m À most of the heat transfer occurs in the wings of the

o

line, that is for small w. Expanding the line shape in terms of small w gives:

Substituting (9) in (8) and integrating term by term gives the desired asymptotic

expansion.

The first term, Rl' is: 1

rrl-

À

J

wl/2dw

J

[Ba )-Ba +

~)]

0 0 0

o v

To carry out the integration over w let 1 0 0 0 0

J=J-J

_{o o 1} e -w{;(m

À)

o (10)

The first and second integrals on the right hand side will be called R

la and Rlb respectively. R1a can be carried out directly to give:

Rl

=

-è-

7Tl/26.vm (m À)-1/2 J {;

-7/2[B(~ )-B(~

+

~)]

dV".

a 0 0 0 0 ~ (11)

v

In Rlb_ only small values of {; contribute thus

B(~o+ ~)

can be expanded locally

about {; • The volume integral can then be done first. To carry out this inte-o

gration it is convenient to write Rlb in spherical coordanates as:

00 7T/2 27T

Rlb =

6.vm~À

J

wl/2 dw

J

sine de

J

drj)

1 0 0

o

(12)

where the coordinates are as in Fig. 3a. Expanding B in coordinates x. gives:

1 [ -

B(~

+

~) ~ B(~

-{;) ] B({; ) _ 0 0 o 2 1 2"

dx.dx.

1 J X.X. 1 J

I

x.x.xkxo {;. 1 J k !D ( 13)

Setting xl = scosS , x

2 = ssinS cos, and x3

=

Ssin8sin~, eq. 13 can be substi-tuted into eq.ll and the volume integration carried out to give:

=

27T llvm

f

[~l

d

2 B

I

+

~

1

~

I ---

J" o

L

9

2 -2 35

4 "

<+ • . 1 (m À) dX . l' (m À) ox . l' l= 0 l ~o 0 l ~o

(14)

In the same way the second term in the expansion in w can be divided into an integral part, R

2a, and a differential part, R~b' to give: R_2a =

ft;

7Tl/2 llvm o(mo

l\)-3/

2

J

ç9/

2

[B(~o)-

Ba) ] dV S and

~,

'

[1

=

27T _llvmo

L

'3

i=l v + 1 25 1 (15) (16 )

If the third term is divided in the way the first and second terms were it is found that the integral over w from one to infinite is nat fini te. However, since this part of the integral is negative and the whole term must be positive it follows that:

Since

R =

45

7Tl/2 llvm (m À)-5/2,( çll/f2[B(j: ) -

B(~o+ ~)

dVJ') (17)

3a 128 0 0

J

~ 0 ~

v

the order Of" magnitude of R3 is

(mol\)-5/

2•

Col~ecting terms and arranging them in increasing order of ma.gni tude gives:

:?(~

) = llvm

{-è-

7Tl/2(m 1\) -1/2

J

ç7

/2[B(~

,-Ba +

ö]

o 0 0 0 0 v +

ïÎ

7Tl/2(moÀ)-3/2

J

(9/2[B(~o)-Bao

+

~)]

dV S v 3 d2 B 10 7T (m 1\)-,2

L

"

-

_{9 0 2} dx. i=l l _So

31+.

ONE-DlMENS IONAL RADIATIVE TRANSPORT

, Equation 18 forms a series in powers of m À. Also note that as the order of the term increases the range 'of the kerna~ decreases. This can best be illustrated for a one-dimensional variati on of B. For a variation of B in a direction, x, where x is a nondimensional distance measured from a point xo' the volume integrals in eq. 18 can be expanded in coordinates x, 8 and ~

as: v TT/2 2

J

sin8d B(x +x)+ B(x -x) o 2 0 ] dx. o o o ( 19) In eq. 19 the 8 and ~ integrals can be done directly and integrating the x

integral by parts twice gives:

I ==

16

3

TT Ix 11/2 d B

J

oo dx2 2 -00 X +x o dx.

Integrating the second term in the same way the expansion for the one-dimensional case then becomes:

R(x ) o 6vmo {

~

2

-5 10

9

00 d2B

,

~/2(moÀ)

-1/2J Ixll/2 dx2 -00 00 :2 TT3/2(m À)-3/2J Ix 1-1/2 d B o dx2 -00 x o dx x +x 0 dx x + x 0 (20 )

Here the progression from a long range to short range kernal can clearly be seen. In the first term in fact the kernal is zero at x . That is the value of the integr~l does not depend onthe local value of the ~econd derivative. This result is in complete contradiction to the diffusion approxi-mation in which the integral is replaced by a term depending on the focal value of dÇB/d2x. It can be seen that the ker~als of higher order terms be-come more localized until the third term is a function of the local value of

d2B/dx2 • This term, however, is of order (m À)-2 as compared to the largest

term which is of order (m À)-1/2. The firstOterm of eq. 20 is the same a~

3.5. EFFECT OF GEOlw1ETRY ON RADIATIVE LOOS

The À-l/2 dependence of

th~ ~adiation

loss on characteristic length was calculated by Mitchell and Zemansky\3) for the one-dimensional case. To illustrate that this radiative loss dependance is independent of geometry, the loss was

calculated for a slab, a cylinder and a sphere using the first term of eq. 18. For an infinite slab of gas with a uniform black body intensity, B and

o thickness, L, the radiation loss from the centre of the slab is:

R

_(slab)

= - -

4J2

3

~/2~vm

o 1/2 B 0

(~.O) ~1/2.

For an infinite cylinder with B = Bo and diameter,

D:

and for a sphere wi th B

=

Bo and diameter, D:

R

_(Sphere)

4J2

3

(21)

(22)

(23)

Thus the radiation loss from the centres of these three geometers differs at the most by 50%.

The average energy ~oss from the gas is always greater than the ~o~s

from the centre because of the increased loss near the surface. Hiramoto(4) has calculated the ratio of the average loss to the loss from the centre of these three geometers. The ratios he found were 1.70, 1.66 and 1.41 for the sp here , the cylinder and the slab respectively. Thus the variation in average loss between the three geometries is approximately 70%. In some problems this may be a significant variation however in most problems where only an estimated of the order of magnitude of the radiation loss is required the specific

geometry will not be important.

3.6 Effect of Geometry on Radiative Darnwing

Radiative transport, in addition to producing a net loss of energy from a gas, can also act as adamping mechanism for nonuniformities in~the

gas temperature. That is transfer of energy o~curs between hot and cold regions inside the gas.

To illustrate the effect that geometry has on the nature of this damping, first, the kernal for radiative transfer along a cylinder will be derived, then this kernal will be applied to calculating the damping occurring for a sine wave nonuniformity along the cylinder.

3.6.1 Kernal for Radiative Transport in a Cylinder

angles to the direction óf variation of B. The effect of a three dimensional

geometry on the kernal will be calculated for radiative transfer along the

centre line of a cylinder of gas. B will be assumed constant over any cross

section or the cylinder but will be allowed to vary along the cylilinder length.

Thus let

B [(Z + Z), ~, rJ

=

B(Z + Z), r

=

0 to a

o 0

O r > a.

Where a is the radius of the cylinder normalized by À. Inserting this into the first term of eq. 18 written in cylindrical coordinates, eq. A2, and carrying

out a number of integrations gives:

R 2

3

-00 1 + 2"

r

(~)

r(-IT-)

(m Àaf l / 2B(Z

)}

<

r

U/4)

0 0 (24)

In eq. 24 the first term gives the radiative transfer inside the gas and the

second term is just the radiative loss from a cylinder of radius, a. Note that

the second term depends only on the local value of B. In the traïsfer term

the kernal which for ~he infinite one-dimensional case, was Ixl- 3 2 is now

[lzl- 3

7

2- (Z2 + a 2 )-3/

4

J. The kernal for the three-dimensional problem decreases

much more rapidly than that for the one-dimensional case for values of Z

>

a.

Thus in the three dimensional problem the range of the kernal has been limited

to the order of the characteristic dimension of the gas.

3.6.2 Damping of a Sine Wave in a Cylinder

As a fina~ illustration of the use of this approximation to radiative

transfer, the damping of a sine wave disturbance along the axis of a cylinder

will be calculated. This calculation has re~evance to the stability of a

non-equilibrium gas discharge(5). Here as in the previous section, B is assumed

constant over the cross-section of the cylinder but varies along its length.

Thus substi tutJng

B(Z + Z) o in eq. 24 gives: B + ~ sin 2~ (Z + Z) o 0 00

R

=}

~/2~vmo{(moÀfl/2

lili sin 27T Zo! [ Izl- 3/2 _ (Z2+ a2)-3/4J(1_COS27T Z)dZ

o

(25)

The second term is again the energy lost from the gas. It does provide some

damping of the sine wave as any energy loss would do. However since aÀ is

The first term of eq. 25 is the damping which results from transfer of energy from crest to trough of the sine wave disturbance. It is this part of

the damping that is similar to that resulting fr om a dispersive mechanism at work in the gas.

The first term of eq. 25 was calculated numerical~ and is plotted in Fig. 2 (dashed curve). The value of the damping coefficient, w was normalized by the damping of an infinite one-dimensional sine wave of

wave~ength,

À. For _o À greater than the diameter of the cylinder; d, the dispersive type of damping decreases rapidly.

In Fig. 2 the total damping for several values of d/À are also shown. o

By comparing the curves for d/Ào

=

00 and d/Ào

=

1 it can be seen that the damping

of a finite dis\urbance, d/Ào = 1, is very nearly equal to that for an infinite disturbance, d/Ào

=

00 , for À

<

~ d. However for large À the damping of the

finite disturbance asymptotes to the value for damping due to energy loss alone, second term of eq. 25.

The most important observation to be made in Fig. 2 is that for a given À an increase in d always decreases the total damping. This would imply that a discharge that is unstable in a small test apparatus will necessarily be unstable in a large apparatus. I Also the larger apparatus will be unstable to disturbances

of larger wavelength.

4 •

THE LINDHOLM LINE PHOFILE

In the preceeding section the radiating line was assumed to have a

-Lorentzian profile. Using this assumption it was shown that the main contri-bution to the radiative transfer comes from the far wings of the line. The far wings however correspond to large energy of interaction and thus close approach of the perturbing and emitting atoms during the radiation process. In this case the impact theory of line broadening which gives the Lorentzian profile may not be adequate to describe the broadening process. In the impact theory the time required for a collision is assumed to be negligible compared to the time required for the radiation process. The opposite limit is described by the statistical theory which assumes that the atoms are stationary during the radiation process. The transition between these two limits should occur when the collision time is of the order of the reciprocal of the frequency shift. That is, 6wr/v ~ 1 where r is the effective radius of the atom for an optical broadening collision, v is the mean thermal speed of the catoms, and 6w is the frequency difference between the emitted photon and tpe line centre. For the inverse sixth power Van der Waal interaction, Goody(6) gives an expansion for the Lindholm theory. This expansion is fitted to the Lorentz line in the centre of the line and to the statistical theory in the wings of the line. This shape is:

E

2

(1 -

0.286

Il);

_{- 2.5}

_1.5

(26a)

m

₌

m

_<

_Il

_<

v 0

11

2

+ E

2

E

2

(0.638 ll-l/3)

_1.5

_<

Il

(26b)

=

m

2

2 0 Il + E

where mo

"""'":2:::-E_2_=2 (0.

932/1-L

/1/2

+

O.

319/1-L

r

1/3);

I-L

<

-2.5

I-L + E v -v o v p and E 6.v 2v P (26c)

This is just the Lorentzian line profile for v close to the line centre v.

How-ever for frequencies differing from the line centre frequency by some cri€ical

amount v the statistical effects become important. This produces an asymmetry

p

of the line toward the red end of the spectrum. Statistical effects also produce

a shift of the line, however, this does not effect the radiation calculation

and v is taken to be the centre of the shifted line.

o

Very far in the wings of the line the contribution of the red wing~

dominates giving

m =

0.932

m v 0

2 ,,,

,-3/

2,.

E,.... I-L

»

1 . (27)

Using this line shape and expanding eq.

4

as was done for the Lorentzian line

profile, the asymtotic form of the radiative transfer now becomes

R(v

_o

)

J

çlO/3[B(~0)

- Ba

o

+

s)]

dV

s .

(28)

v

Since this term decreases as the

-1/3

power of m À and the dominant term in

eq.

18

decreases as the

-1/2

power, eq.

28

will ~ominate for a sufficiently thick gas. The transition point occurs when

2

m À

=

15

(~)

o 6.v

5 .

THE EFFECT OF OVERLAPPING LINES

In all the preceeding sections it was assumed that the radiative

transfer was occuring in an isolated spectral line. In reality spectral lines more commonly appear in groups. For a MBD plasma the main radiative energy loss

occurs in the resonant doublet of the alkali metal seed.

It was noted in section

3.3

that the maximum contribution to the

energy transfer occurred when the optical depth m À was of order one. In particular

for an optically thick Lorentz line the maximum o} the frequency integral, dm

v

2

m À~

=

1/2. Since for an optically thick line m ~ m

[6v/4(v-v )] ,

the maximum

v

v o o

contribution to the energy transfer occurs where

(v-v)

= o max 6v

J2

../m

À~' o

Thus the maximum moves away from the line centre as the optical thickness in-creases. If

(v-v)

is much less than the difference in frequence between

o max

two lines they act as isolated lines which from eq. 18 would give a total energy loss proportional to ~~ +

.JID

2; ~ and m2 are the reciprocal photon mean free paths at the centres of the two lines. Alternatively if

(v-v)

is much greater

o max

than the line spacing the pair of lines would act as one line giving a radiation loss proportional to ../~ +~. Since the ratio of optical strengths of the two lines of the doublet is 1:2, the ratio of radiation loss between the two cases described is 1.4:1.

The effect of the overlap of such a pair of lines for separations be-tween 0 and 00 was calculated numerically and is shown in Fig.4. Plotted is the

variation of frequence integral, K

À, with line separation vl -v2• The line

separation was normalized by the frequency difference for maximum energy transfer in the stro~ger of the two lines. As the two lines are moved apart or alterna-tively as the optical thickness is decreased, the integral first drops below that expected for the superimposed lines. The integral then increases rapidly passing through one approximately when

(v-v)

is equal to the line separation.

o max

The integral increases above the value for two isolated lines, 1.4, then returns asymtotically to it. As aresult the radiative kernal varies by a factor of

1.69

between its maximum and minimum values.

6.

CALCULATION OF THE RADIATION LOSS FOR A POTASS;III1M!,SEEDED MBD PLASMA

In a nonequilibrium MBD device the gas mixture used is usually a rare gas seeded with an alkali metal vapour. The following calculations were done for argon at atmospheric pressure and 20000K seeded with 0.4 molar percent of potassium vapour. Under these conditions the main contribution the ~ine

broadening occurs through collisions between argon and potassium atoms.

by:

The reciprocal mean free path of photons at the line centre is given

m.

1

where f

i is the oscillator strength of the line, N

K

is the number density of ground state potassium atoms in cm-

3,

and 6v is the line width in cm-l

For a Van der Waal interaction the line width given by the Lindholm theory is

where v is the mean thermal speed of t he colliding atoms in cm/sec, b is the constant in the Van der Waal's interaction potential such that 6E

=

hb/r6 and NA is the number density of argon atoms in cm-3. Finally the frequency at which statistical effects become important is

~

p

=

0.036 (v)6/ 5 b-l/ 5 •

To do the radiation calculations values of band f must be known.

values of the oscillator strengths were tak~n to be 0.33 and 0.67 for the

lines of the potassium doublet~~). Mahan(7) obtained a theoretical value

of approximately 3.8 x 10- 31 cm /sec.

For the gas conditions described above these values give:

6v 0.108 cm -1 v p 3.3 -1 cm 2.7 x 104 -1 ~ cm aT\~ _~ 5.4 x 104 cm -1 The two of b where lines ~ and m

2 are the reciprocal mean free paths for the 7699K~ and the

7665K

respecti ve ly •

Ch'en and Takeo(8) give an experimental value of 0.15 cm-l for 6v.

7. MEASUR.ED. LINE PROFILES

The radiation emitted from a potassium seeded argon plasma under the

condition used in MHD devices has been analyzed spectroscopically. The radiation

emitted from a depth of plasma, L, has an intensity profile given by

-m L

( v) •

I = B 1 - e

v 0

A typical result' is shown in Fig. 5(9). The black body intensity, B , was

de-termined by reversal of the line. The centres of the lines have beeR absorbed

by cool gas in the boundary layer. A ~orentz profile is shown fitted to the

measured profile; however, because of an asymmetrie shift to the red in the

measured profile this does not produce a good fit. A very good fit was obtained

by using a Lindholm profile with v and 6v as adjustable parameters. The values

of v and 6v which fit the measur~d profile were 17 and 0.079 cm- l

respecti-p

vely. The resulting profile is shown in Fig. 5. t o agree very ëlosely with

the measured line.

The value obtained for v differs substancially from the theoretical

value. However, since the Lindho~m profile is a fit between the impact theory

and the statistical theory it cannot be expected to give quantitatively

accurate values.

8. RADIATION KERNAL INCLUDING STATISTICAL EFFECTS AND LINE OVERLAP

= llvm _o !K(m

;\1;)[Ba ) -

B(~

+

~)]

0 0 0 V

dV

.. ~

7

the radiation kernal K(m

;\1;)

is given by the frequence integral o

For the Lorentz and the statistical limits analytic expressions have been found for this kernal. To determine the region of validity of these approximate

expressions the kernal was calculated numerically for the Lindholm profile for a large range of optical depths. The resulting kernal was normalized by the kernal, ~Z' for the ~rentz limit such that

K = 7jJ ~z

Log 7jJ is plotted in Fig.6 as a function of log (m

;\1;).

o

The shor~ dashed line shows the result for a single line of strength

m

=

~ + m

2• The solid curve shows the result for the potassium doublet with tRe normal separation between the two lines. Also shown are the Lorentz limit for two separate lines and the statistical limit for superimposed lines.

Below an optical depth of 10 the kernal diverges ffom the Lorentz limit due to higher order te~ms in t~e expansion of the Lorentzian line shape. Between optical depths of 10 and 10 the line undergoes a transition from sep-arate to single line behaviour as wgl1 as a transition from the Lorentz limit to the statistical limit. Above 10 the line fol~ows the statistical limit for superimposed lines.

In argon at atmospheric pressure and 20000K seeded with 0.4% potassium an optical depth of 10

5

corresponds to a physical depth,

;\1;,

of 1 cm.

At

this point the real kernal deviates from the Lorentz limit for separate line by 10%. To an accuracy of 10% the Lorentz approximation is good to an optical depth of 10 or, in this gas, to physical dimensions of 10-4 cm. This means that the Lorentz approximation is good even for boundary layer calculations.

An approximate kern al which might be used in this gas is the Lorentz kernal for separate lines below an optical depth of 10

7

and the kernal for the statistical limit of superimposed lines above 10

7.

This kernals differs from the real kernal by at most 20% for all opticaL depths above 10.

As an illustration of the use of this approximate kernal the radia-tive loss from the centre of a slab was calculated. The solid line in Fig.

7

shows the result. Numerical calculations using the exact kernal in Fig. 6 were within

5%

of the approximate result.

It can be seen however that for optical depths greater than 106 the radiative loss does deviate substantial~y from that predicted by the Lorentz limit. As stated previously an optical depth of 10

5

corresponds to a phy-sical dimensions of 1 cm for typical MBD gas conditions. Thus in sealing the radiation losses from a device with a characteristic dimension of 1 cm to one with a characteristic dimension of 10 cm, the ;\-1/2 relationship for

the Lorentz limit prediets a radiation loss which is 20% too small. Since the uncertainty in the line width produces an uncertainty of the same size in the radiation loss this error is not significant. However, for larger devices or for a greater seed density the effect of statistical broadening on the radiative loss may be important.

9~ RESULTS AND CONCLUSIONS

An expansion of the integral for radiative transport of energy in an

optically thick line was obtained. The expansion was obtained by using the

assumption that the photons most effective for energy transfer would be those

wi th a mean free path comparable to the characteristi c length of nonuniformi ties

in the gas. Since the gas was assumed optically thick this implies that the

wings of the line are the most important for energy transfer. The expression obtained by expansion of the shape of the wings of the line was a converging

asymtotic expansion thus verifying that the assumption made was a correct one.

This asymtotic expansion a1so showed that the kernal of the first

order term had the greatest range and 1arger order terms were more 10ca1ized. In the one-dimensional case i t was explici tly shown that the integra1 could

not be approximated by the local value of the second derivative as is done

in the diffusion approximation.

The first term in the expansion was used to calculate radiation

108s fr om severalj~eometrics. The loss fr om a slab, a cylinder, and a sphere

had the same )...-1 dependence on characteristic length wi th the numerical

factor differing by at most 50%. Thus i t was cOl1cluded that radiation 10ss

was not strongly dependent on geometry.

However the radiation also contributes to damping of nonuniformities

in the gas, The damping of nonuniformities along the length of a cylinder

was considered. Here i t was found that the range of the radiation kernal

inside the cylinder 1fas less than that for an infinite gas. A calculation

of the damping of a sine wave, inside a cylinder showed that the damping due

to energy transfer inside the gas was much smaller than that ca1culated fon

the infinite case when the wavelength was greater than the diameter of the

cylinder. However the total damping which includes a contribution from the

radiation loss from the gas, was always larger for the sine wave in the cylinder

than for the infinite sine wave.

Statistical effects and effects of line overlap which cause a

devia-tion from the predicdevia-tions based. on the Lorentz profile were a1so considered.

By using a Lindholm profile which includes statistical effects i t was found

that for large optical depths statistical broadening became important. jAlSO

for sufflcient1y large opt~cal depths the radiation loss scales as )...-1 3 as

opposed to sealing as )...-1/2 which it does when the Lorentz profile exists.

In a potassium seeded argon plasma of the type used in MBD devices

the overlap of' the two lines of the resonant doublet also occurs for large

optical depths. This results in a transition from behaviour of the doublet

as two separate lines tobehaviour as a single line of greater strength.

The radiation 10ss decreases in this transition.

It was found however that for optical depths between 10 and 10

6

the

first term in the expansion of the Lorentz profile gave results within 20% of

l . Cool, T. A. Zukoski, E. E. 2. Solbes , A. 3. Mitchell, A. C. Zemansky, M. W. 4. Hiramoto, T. 5. Gilpin, R. R. , Zukoski; .E. E. 6. Goody, R. M. 7 . Mahan, G. D. 8. Ch 'en, S. Takeo, M.

9.

G. REFERENCES

Recombination, Ionization and Nonequilibrium Electrical Conductivity in Seeded Plasmas; Phys. Fluids 4, 780 (1966).

Radiation Operator in Nonequilibrium Plasmas; AIAA 4, 737 (1968).

Resonance Radiation and Excited Atoms; Cambridge University Press, Cambridge (1961)

Rates of Total and 10cal RadiativeEnergy 10ss in Nonequilibrium Plasmas; Journalof the Physical Society of Japan, No.3, 785 (1969)

Experimental and Theoretical Studies of Electro-thermal Waves; AIAA 7, 1438, (1969)

Atmospheric Radiation:

I: Theoretical Basis, Oxford (1964)

Van der Waals Constants between Alkali and Noble-Gas Atoms. 11: Alkali Atoms in Excited States; Journalof Chem. Phys. No.6, 2755, (1969)

Broadening and Shift of Spectral 1ines Due to the Presence of Foreign Gases; Rev. Mod. Phys. No.l. 20, (1957)

Unpublished Measurements by R.R. Gilpin, J. K. Keoster, and E. E. Zukoski

APPENDIX

Different.Coordinate Systems

The first term in the expansion eg. 19 is written out in various coordinate systems. Spherica1 o o o Cy1indrica1 2~ 00 00

J

_d~

J

rdr _{2 2}

_7/4

J -) (- -

_{[B(r -B r + r)] dZ} (z t r ) 0 0 o 0 .. -00 (A2) Cortesian (A3) -00 -00 -00

dIJ

1.0

FIG.2.

DAMPING OF

A

SINE

WAVE

DISTRIBUTION

"

...

...

...

"

...

"

_"

,

_,

"

_"

"

,

_,

, ,

_,

,

_,

,

_,

\

/ ~

/

/

/

/

I

/

/

I

/.1 X

_____ __ ..v

y

a)

SPHERICAL

b) CYLINDRICAL

c)

CARTESIAN

1.5

K

Ko

As

ymp

tote

---

---

---1.4

for separate

.

lines

1.3

1.2

1.1

2

3

4

./2

Cv,-Jl

2

)

~JI~m25\

t

1.0

111

80

.8

.6

4

.2

-_

.... .,....,...-' I I , , / ~,.

7620

7640

~ I, I

I

,---,

I

I , I

I

I I I

r

I

I

~

, I

~

I

i

I I

,

I

,

\ \

,

\

7660

\ \ \ \ \ \

,

\

_{,_./}

VI

7680

FIG

.

5

.

LlNE

PROFILES

KC 4P-4S>

,...."",,,

r

I

J

I

I

I I \ , I I I I : I 1 I I I _I I

,

I I I I I

,

_,

I I

,

_,

,

_,

,

\ \ \ \ \ \

7700

\ \

,

,

meosured 'ine

Lorentz theory

o

Lindholm theory

"-

_...

...

_---7720

7740

7760

o

XCA)

.

5

log

t.V

.

4

.3

.2

FIG.6

.

VARIATION

OF

KERNAL

WITH

- - - - superimposed

lines

- - - - potassium

doubl et

- - -

:::::::-=---====-==-=

Lorentz limit separate lines

OPTICAL

DEPTH

f

~~

/

.t;:-fb

'"

# Ifb

b

.I. /

o~

I{"

j;#

/,

/

~'

;:

~ / c.; / /

/

...

/

...

1:~~

\ / \ /

,I

~

\

/ /

\

.,...'"

,

'

_'

_{- -}

.,...

_,

₇

' e

'

-

]

I/II/.f

!

_~

'

~

o

/:'

"

--* - - - ; ,

6

.

À

... -

- ,. -

I

n

~(m

+ m)

~

- - . , - 0"

r

I "'2"

0'

2

3

4

4~

a:

(/) (/) 0 _lLI U- _Z ~ (I)

u

en

-0

:t

...

.J Z lL

0 0

_I

-t-

I

4

_Ol

ëS

Cl

4

.J

a:

en

.

.-...

~

-LL

/

I

I

I

ft)

•

CD

.-...

~

-

N

E

•

-E

""'"

0 0

UTIAS lEPOlT .0. 1'0

Institute for Aerospace Studies, University of T oronto

Approximation to th. ladiatiT. Tran.port ot En.rsy in an Optically

Thick Lin.

01lpin, R. l. 111 pa, . . T tl,ur . .

1. ladiatiTe Tran.port 2. Pota •• ium Line Radiation 3. .on-Equilibriua

MBD Pover Gen.ration

I . Gilpin, 11. 11. 11. UTIAS Report .0. 150

~

The radiatiTe tran.port ot en.rsy in an optically thlck line 1 • • t.udled. Flr.t

an asymptotlc expan.lon tor th. radl,atlv. kernal 11 obtalned tor aLorent. 11n.

• hape. Thi. r •• ult 1. uled to calculat. th. radlative tran.port ln .ome·probl.m.

ot lntere.t in MRD ,a •••• The 11mitl ot appllcabillty ot thl. alymptotlc

.x-panslon are then determlned by conald.rlns .uch .ttect, a. 11n. overlap and

.tatistical broadening.

UTIAI REPOlT .0. 1'0

Institute for Aerospace Stud., University of T oronto

Approxlmatlon to th. RadlatlT. Tranaport ot InerlY ln an Optlcal1y Thlck Llne

GllpiD, R. R. 14 pasea T ~lSl1r . .

1. Radiativ. Tran.port 2. Pota •• iu. Line Radiation 3. .on-Sql1ilibriua

MBD Pover Generation

I. Gllpin, R. R. 11. UTIAS R.port Io. lSO

~

Th. radiative tran.port ot enersy ln aD optlcally thick 11n. i • • tudled. Firat

an a.ymptotic expanslon tor th. radiativ. k.rnal 1. obtaln.d tor a Lorent~ 11n •

ahap.. Thl. r •• ult i . u •• d to calculat. th. radiatlTe tran.port in 80me probl •••

ot inter •• t in MBD sa.e •. Th. limit. ot applicability ot thi. a.ymptotic .x-paD.ion are th.n determlne4 by eon.lderlD& .uch e~t.ct. al liDe overlap .n4 .tati.tical broadenlns.

AVllilllble copies of th is report lire limitecf. Return this cllrd to UTlAS, if you require 11 CQPY. AVllilllble copies of this report lire limited. Return this CIIrd to UTIAS, if you require 11 copy.

UTIAS REPORT .0. 150

Institute for Aerospace Studies, University of T oronto

Approximation to the Radiative Transport ot Energy in an Optically

Thick Line

Gilpin, R. R. 14 pages T figurea

1. Radiative Transport 2. Potaaaium Line Radiation 3. Ron-Equilibrium

MHD Pover Generation

I. Gilpin, R. R. 11. UT lAS Report No. 150

~

The radiative transport ot energy in an opticn1ly thick line is studied. First

an asymptotlc expanalon for the radiative kernal ia obtained for aLorents 11ne

shape. This result 1s used to calculate the radiat1ve transport in some probleml

of interest in MHD gases. The limits ot applicability ot this aaymptotic

ex-pansion are then determlned by cODsiderlng auch effect. &8 11ne overlap &Dd statistical broadenlng.

UTIAS REPORT .0. 150

Institute 'for Aerospace Studies, University of T oronto

Approximation to th. Radiativ. Transport ot Energy in an Optically

Thick Line

Gilpin, R. R. 14 pas . . 7 tigures

1. Radiative Transport 2. Potasaium Line Radiation 3. .on-Equilibrium

MHD Pover Generation

I. Gilpin, R. R. 11. UTIAS Report Ro. 150

~

The radiative tran8port ot energy in an optica1ly thick line is studied. First

an asymptotlc expanalon tor tbe radiative kern al is obtalned tor aLorents line

.hape. Thi_ result 1. u.ed to calculate tbe radiatlve tr&nsport in some problems

ot interest in MBD gases. The limits ot applicabi1ity ot this asymptotic

ex-paDalon are tben deteralned by cODalderlng 8uch .~t.ct. a. line overlap &Gd