K I N E T I C S OF H E A T I N G OF

DUE TO R A D I A T I V E HEAT

L I G H T - S C A T T E R I N G

T R A N S F E R

L A Y E R

A . P . I v a n o v UDC 536.2/3

OT

Ot

Here T is the t empera tu re , ~ t is the t ime

It was shown in [1] that for a gray l ight -sca t ter ing p lane-para l le l l aye r enclosed between two ho r i - zontal plates heated to t r e m p e r a t u r e Tn the thermal dfffusivity equation with only radiat ive heat t r ans fe r taken into account has the form

2(I --A)e (E~ + E~--2(~T*). c9

i , I

2(i--A)(~ {[I + r(%--~j)l Tat(~-- ~)[i -}- r (~)law E~ = 1--r(~i)r(%--xJ) 1 -- r (~) r (xs -- x)

E~ (~T~ (I + R)(1--Re)-2~"L)(e-2~c + e -~(~-~)~ ) = - - " 1 - - R~e -4~.L

a r e the sums of the i l luminances f rom above and below of a horizontal a rea within the layer due to the thermal background of the considered medium at optical depth r j and due to the thermal emiss ion of the pla tes ;

(1)

I " T~t_(~__ - %.)[I -}- r (% -- x)ldx I . (2) -t-I1 +r(~s)] 1--r('%--'Or('~--'~j) I '

~ 4 T'/O, deg a

/,8

b c

i ,

q./g-1 t, tl"

(3)

[ /

ZO, " : q./o'~ ./~rr /Oo t, h

Fig. 1. E / e as function of t for different A: a) 0; b) 0.7; c) 0.9; d) 0.97; the f igures beside * The values of x l for par t ic les of different s ize a re the curves a re the values of Tj. given in [2].

Trans la ted f rom Zhurnal Pr i tdadnoi Spektroskopii, Vol. 17, No. 3, pp. 507-512, September, 1972. Original a r t i c le submitted May 12, 1971.

t(z)= (1 - - R 2) e -2~L 1 - - e -dzL

1 - - R 2 e -dzL , r(z) = R l _ R ~ e _ ~ , L ,

(4) R = I + 1- -A V ( 1 - - i / u l - - A

Aq) ~ - - - - ~ ] + 2 A----~

a r e the t ransmiss ion and ref lec t ion coefficients , r e - spect ively, of a l aye r of finite and infinite thickness; A = s / r is the photon survival probabil i ty, where is the attenuation index, equal to the sum of the sca t - te r ing (s) and absorpt ion (k) indices; T O = ex 0 is the optical thickness of the layer (x 0 is the geometr ic thickness); the p a r a m e t e r ~ = (3 -Xl ) /8 , where

1

3 ~ x (~?) cos ~d (cos ~) X 1 ~ - ' ~ - Lt --1

is the f i rs t coefficient in the expansion of the s ca t t e r - ing indicatr ix x (7) of an e lementary volume in Legendre polynomials*; the p a r am e te r L = ~/(1-A) 2 + 2 ( 1 - A)Ar

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a /

3

2

2

/

b

/

Y I

c /

0 0,5 ~o ~5 ~"

Fig. 2. (E~-)/E(T = T0/2 ) as function of ~- for different A: a) 0; b) 0.7; c) 0.9; d) 0.97. Values of t: 1) 0; 2) 5.6 �9 10-2; 3) 7-10-2;4) 0.11; 5) 0.13; 6) 0.28; 7) 0.36; 8) 0.84; 9) 15.

c and p a re the specific heat and density of an e lementary volume of the medium; cr is the Stefan-Bol tzmann constant.

In the present paper, using the solution of equation (I) by the finite-difference method [3-51, we ana- lyze the kinetics of heating of a plane-parallel layer.

Since the temperature is determined by the radiation density at each point of the layer, we begin with a consideration of the relationship between the total flluminance E = E i + E e and the time of heating of the medium.

Figure i shows E/~ as a function of t at the surface and center of the layer for different A. In this and subsequent figures ~ = 2a/cp = i0 -? h -i .deg -3, T n = 600~ ~'0 = 2, and ~ = 0.375, which corresponds to a fairly elongate scattering indicatrix in the forward direction. The whole process is analyzed from the

T, deg

o I o I ~ S

y 400

,~,-r ~ . / ~

Fig. 3.

~0~

f~

T as function of ~" for different A: a) 0; b) 0.7; c) 0.9; d) 0.97; the figures beside the curves are the values of T].

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T, deg I

a 5 J

0 ' , , (

C

, I , I I

/

$

d 8

Y

5

/,0 /,5 v

Fig. 4. T as function of T for di f ferent values of A: a) O; b) 0.7; c) 0.9; d) 0.97; and t: 1) O; 2) 10-2; 3) 3 .3 .10 -2 4) 5 .7-10-2; 5) 0.1; 6) 0.23; 7) 0.25; 8) 0.63; and 9) 1 h.

t ime t = 0, when the t e m p e r a t u r e of the medium is zero , to a t ime cor responding to complete heating of the l aye r to 600~ Figure 1 shows that th ree cha rac t e r i s t i c regions can be dist inguished on the cu rves of E / a = f(t). The light f ield in the medium is unal tered init ially. This is due to the fact that the t e m p e r a -

0,75

~55

0,25

Jl

/ / //

Plo t s of t, co r responding to heat ing

t,h

0 Fig. 5. of d i f ferent points of l ayer to T = 300~ (solid l ines) and 580~ (dashed l ines) , aga ins t A. The f igures beside the curves a r e the values of Tj.

ture of the medium is low and the intr insic t he rma l background is much l e s s than the radia t ion level due to the p la tes . This is followed by a sharp inc rease in i l luminance. The ra te of i nc rea se is g rea te r wi th- in the l aye r than on the sur face . I t a lso i nc rea se s with reduct ion of the photon surviva l probabi l i ty . At subsequent t imes the inc rease in i l luminance slows down and E / ~ a t any point tends asympto t ica l ly to 2T~n. The t ime for the medium to at ta in the s teady the rma l r eg ime is g r ea t e r , the g r ea t e r A.

We consider how the radia t ion is d is t r ibuted within the l a y e r . I t is obvious that the dis tr ibut ion is l e a s t uniform init ial ly, when T = 0, E i = 0, and E = E e. In this case , accord ing to (3), the ra t io of the i l luminances a t the edges and center of the l aye r is

E ('~ = 0) ch %L. (5) E (~ --- ~Col2)

The subsequent var ia t ion of the d is t r ibut ion of i l luminance in the l aye r during its heat ing is r evea led by Fig. 2. The data for di f ferent photon surviva l p rob- abi l i t ies a re given. It: is an in te res t ing fact that the inc rease in i l luminance towards the edges of the

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medimn at d i f ferent t imes is gradual and s y m m e t r i c . With inc rease in t the d i f ference between the i l - luminances a t d i f ferent points in the l a y e r is reduced. The g r ea t e r A, the m o r e uni form the radia t ion dis t r ibut ion.

We turn to an ana lys i s of the va r i a t ion of the l aye r t e m p e r a t u r e with t ime. P lo t s of this re la t ionship for di f ferent A, and Tj = 0 and 1, a r e shown in Fig. 3. Accord ing to (1), when t = 0 the r a t e of t e m p e r a t u r e i nc rea se is g r e a t e s t and is [ 2 ( 1 - A ) e / c p ](E i + Ee). As Fig. 3 shows, however , this r a t e is constant until the t e m p e r a t u r e in the med ium at ta ins a value of app rox ima te ly 0.75 T n. Af te r this the heating s lows down cons iderab ly . Since E i + E e a t the su r f ace is g r e a t e r than within the medium, the s lope of the cu rves is g r e a t e s t when Tj = 0 (and, hence , a lso when Tj = 2). The s teady the rma l r eg ime se ts in e a r l i e r , the lower the photon surv iva l probabi l i ty .

F igu re 4 i l l u s t r a t e s the t e m p e r a t u r e d is t r ibut ion in the l aye r a t d i f ferent t imes for d i f ferent A. In i - t ial ly T = 0 eve rywhe re . The t e m p e r a t u r e then begins to i n c r e a s e , m o r e rapidly a t the edges than in the center of t he l a y e r . At a c e r t a i n t ime , which depends on the photon surv iva l probabi l i ty , the t e m p e r a t u r e d i f fe rence for d i f ferent Tj a t ta ins a m ax i m um. The dis t r ibut ion T(I-) then begins to level out and tends to T = 600~ The t e m p e r a t u r e d is t r ibut ion is l e a s t uniform when A = 0.

I t is impor tan t to find out how the t ime in which the l aye r is heated to a p a r t i c u l a r t e m p e r a t u r e v a r i e s with the photon surv iva l probabi l i ty . The plots in Fig. 5 co r respond to heat ing of the edge and cen te r of the l a y e r to T = 300 and 580~ As Fig. 5 shows, for A = 0-0.7 the heat ing t ime does not depend grea t ly on the photon surv iva l probabi l i ty . Subsequently t i n c r e a s e s sharp ly with i nc rea se in A and tends to infinity.

1. 2.

3o 4. 5.

LITERATURE CITED

A. P . Ivanov, Zh. P r i l d . Spektroskopi i , 16, 709 (1972). C. M. Chu, G. C. C l a rk , and S. W. Churchi l l , Tab les of Angular Dis t r ibut ion Coefficients for L~ght Scat ter ing by Spheres , Eng. Res . Ins t . , Univ. of Michigan (1957). O. V. Lokuts ievsk i i , Usp. Mat. Nauk, 11, 224 (1956). A. P . Ivanov and E. B. Khodos, Opt. Spektrosk. , 8, 556 (1960). A. P . Ivanov, Opt. Spekt rosk . , 11, 654 (1961).

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