COMBUSTION, EXPLOSION, AND SHOCK WAVES

T H E R M A L DISSOCIATION MECHANISM OF DIATOMIC M O L E C U L E S

S. A. L o s e v and O. P. Sha ta lov

F i z i k a G o r e n i y a i V z r y v a , Vol. 6, No. 1, pp. 3 0 - 3 4 , 1970

UDC 539.196.6+536.46

One m e t h o d of f o r m a t i o n of the a c t i v e r a d i c a l s in i t i a t ing the d e v e l o p m e n t of cha in c o m b u s t i o n p r o c e s s e s is t h e r m a l d i s s o c i a t i o n of the m o l e c u l e s . T h e r m a l d i s s o c i a t i o n of d i a tomie m o l e c u l e s usua l ly p r o c e e d s by way of a m o I e c u l a r t r a n s i t i o n f r o m exc i t ed v i b r a t i o n a l l e v e l s to a con t inuous s p e c t r u m of e n e r g y va lue s , s i n c e the d i s s o c i a t i o n of u n e x c i t e d m o l e c u l e s is e x t r e m e l y un l ike ly [1]. Thus , d i s s o c i a t i o n is d i r e c t l y connec ted wi th the p r o c e s s of e x c i t a t i o n of m o l e c u l a r v i b r a t i o n s ; a c c o r d i n g l y , in ana lyz ing the t h e r m a l d i s s o c i a t i o n m e c h a n i s m of d i a t o m i c m o l e c u l e s we wi l l c o n s i d e r t h e s e q u e s t i o n s t o g e t h e r . In this c a s e it is d e s i r a b l e to obta in p h e n o m e n o l o g i c a l r e l a t i o n s that would enab le us to d e s c r i b e the changes in the m a c r o s c o p i c c h a r a c t e r i s t i c s of the s y s t e m .

A n u m b e r of s tud ies in th is a r e a [1] have the d i s a d v a n t a g e of a s s u m i n g that the p r o c e s s of d i s s o c i a t i o n does not d i s t u r b the B o l t z m a n n d i s t r i b u t i o n of the m o l e c u l e s o v e r a l l the v i b r a t i o n a l l e v e l s , w h e r e a s it is we l l known that d i s s o c i a t i o n l e ads p r e c i s e l y to a s i gn i f i c an t d i s t u r b a n c e of the Bo l t zmarm d i s t r i b u t i o n at the upper l e v e l s . A p h y s i c a l l y sound p i c t u r e of the d i s s o c i a t i o n p r o c e s s can be c o n s t r u c t e d only if the p o s s i b i l i t y of a s i gn i f i c an t d i s t u r b a n c e of the B o l t z m a n n d i s t r i b u t i o n at the l e v e l s f r o m which d i s s o c i a t i o n p r o c e e d s is admi t t ed .

L e t us c o n s i d e r the p r o c e s s of d i s s o c i a t i o n of m o l e c u l e s p r e s e n t in low c o n c e n t r a t i o n in an i n e r t d i luent . F o r this p u r p o s e we e m p l o y the d i f fus ion a p p r o x i m a t i o n of r e a c t i o n theory ; then fo r the d i s t r i bu t i on func t ion f in v i b r a t i o n a l e n e r g y s p a c e we h a v e the F o k k e r - P l a n c k equa t ion (see, f o r e x a m p l e , [21)

0~ 0r 0-: / j ' (1)

where

e - s]kT f~ =, g ~ - - (2) Z(T)

is the e q u i l i b r i u m d i s t r i bu t i on funct ion , n o is the n u m b e r of m o l e c u l e s in e q u i l i b r i u m , Z(T) is the p a r t i t i o n funct ion , and ga is a m u l t i p l i e r c h a r a c t e r i z i n g the s t a t i s t i c a l weight of the s t a t e s with e n e r g y s. We a l so m a k e a n u m b e r of a s s u m p t i o n s that do not l ead to s e r i o u s d i s t o r t i o n s of the p h y s i c a l p i c t u r e of the d i s s o c i a t i o n p r o c e s s , but m a k e it p o s s i b l e to ob ta in c l e a r and s i m p l e m a c r o s c o p i c r e l a t i o n s f r o m (1). We take ga = 1 ( h a r m o n i c o s c i l l a t o r model) and r e p r e s e n t the unknown d i s t r i b u t i o n func t ion f ( s , t) in the f o r m

f = / B (1 + ~), (3)

w h e r e f B is the B o l t z m a n n d i s t r i b u t i o n with v a r i a b l e v i b r a t i o n a l t e m p e r a t u r e Tv:

a

n (t) ~ r v -fB - z (T v) e , (4)

D *

j ' f ( ~ , t l d E . ~ (t) = (5) 0

As the boundary cond i t ions we a s s u m e t h a t f ~ f B a s s ~ 0 a n d f ~ 0 as ~ D*, i . e . , the e x i s t e n c e of a B o l t z m a n n d i s t r i b u t i o n in the r e g i o n of s m a l l ~ and the a b s e n c e of m o l e c u l e s n e a r the d i s s o c i a t i o n t h r e s h o l d D* ( s ign i f i can t ly d i s t u r b e d B o l t z m a n n d i s t r ibu t ion) . We a s s u m e that t h e r e is no f low of m o l e c u l e s into the r e g i o n e < 0:

~, oZ-~ - / ~ ~=o o. (~)

We f u r t h e r a s s u m e tha t at l e a s t f o r s m a l l e the d i f fus ion c o e f f i c i e n t is p r o p o r t i o n a l to e: B = be, w h e r e b is a c o n s t a n t [3]. F i n a l l y , we a s s u m e that go v a r i e s in t i m e m u c h m o r e s lowly than f B , i . e . , (p depends only on e. Then

25

COMBUSTION, EXPLOSION, AND SHOCK WAVES

integrat ing (1) with respect to e f rom 0 to D* gives an express ion for the var ia t ion of the number of d issocia t ing molecules

an(t)dt - f B ( D * ' t) B ( D * ) - - ~ - ~ =/9* ; (7)

hence for the d issocia t ion ra te constant we immedia te ly obtain

D*

k T v B(D*~ [ 0 ~ ,~ e (S)

(n A is the number densi ty of the pa r t i c l e s of iner t diha~ent). Multiplying (1) by e and again integrat ing, we obtain the re laxat ion equation for the un i t -vo lume vibra t ional energy Ev:

d Ev _ _ E~ E v - - + D * a n ( t ) , (9) dt • v d t

where E~ is the equi l ibr ium value of the v ibra t ional energy at the t empera tu re T of the ine r t -gas di luent and z v = b /kT is the v ibra t ional re laxat ion t ime. The second t e rm on the r ight-hand side takes into account the change in E v resu l t ing f rom dissocia t ion and is close to zero only in the two ext reme cases: in the absence of d issocia t ion and in the s tate of total s ta t i s t ica l equi l ibr ium.*

F r o m (8) it is c l ea r that the ratio of the va r i ab le value of the dissocia t ion ra te constant Kg to the value of the same constant K} in the s tate of total s ta t i s t ica l equi l ibr ium is equal to**

D* I '~

~o z(rv)

Under the conditions cha rac te r i s t i c of the s ta te of the gas behind a shock front it is poss ib le to dis t inguish three s ta tes of the d issocia t ion process [4, 5]. Init ial ly, at t-< T V vibra t ions are vigorously excited and T V increases ; in this case the f i r s t t e rm on the r ight-hand side of Eq. (9) is important . Then, as all the higher vibrat ional levels are excited, the ro le of d issocia t ion [and the second t e rm on the r ight-hand side of (9)] inc reases and the quas i - s t a t iona ry state, in which the expenditure of v ibra t ional energy on dissocia t ion is compensated by vibra t ional excitation, is reached; in this case the two t e rms on the r ight-hand side of Eq. (9) are equal in absolute magnitude and opposite in sign. In this state the v ibra t ional energy E* and the v ibra t ional t empera tu re T~ change only slightly, but may differ s ignif icant ly f rom E~ V and T. F ina l ly , as the recombinat ion p rocess develops, we get total s ta t i s t ica l equi l ib r ium (T V = T). We note that, in accordance with (10), in the quas i - s t a t iona ry state the dissocia t ion ra te constant K~ is

re la ted to Kg and K~ by the express ions :

- - e ; ~'~, z(rv)

~ ( ' - - + ) , Kg _ Z fT) e

~ z(r~,) Kg (11)

Thus, the dissocia t ion ra te constant Kg actually va r i e s continuously, at f i r s t approaching K~ and then K~

(as T v ~ T).

*Equation (9) for v ibra t ional re laxa t ion in the absence of d issocia t ion was obtained f rom the F okke r - P l anck

equation in [3]. **Relations (9) and (10) coincide with the known re la t ions [4, 5]; the agreement with [4] indicates the s imi l a r i t y

of the diffusion model (with our assumptions) and the one-quantum t rans i t ion model employed in [4].

26

COMBUSTION, EXPLOSION, AND SHOCK WAVES

The poss ib i l i ty of using the model developed can only be evaluated exper imenta l ly . Accordingly, we conducted shock tube exper iments , in which we obtain a r e c o r d of the absorpt iv i ty of v ib ra t iona l ly -exe i t ed m o l e c u l a r oxygen

behind the shock f ron t in a 10% 02 + 90% Ar mixture . Thus, we studied the var ia t ion of the population of a s e r i e s of v ibra t iona l l eve ls (up to fifth) of the e lec t ron ic ground s ta te of 02 contributing to absorpt ion in the bands of the Schumann-Runge sys tem (3. = 1850-2245.~). The s ignals were r eco rded pho toe lec t r i ca l ly with a reso lu t ion not in fe r io r to 0.1 psec . A typical o sc i l l og ram is shown in Fig. 1. The descending branch A of the curve cor responds to the exci ta t ion of v ibra t ions ( increase in absorption); the i nc r ea se in the signal (branch B) is assoc ia ted with a d e c r e a s e in the total number of molecu les , i . e . , with d issocia t ion (in the quas i - s t a t i ona ry per iod of the p rocess ) . By analyzing the A branches of the cu rves we w e r e able to obtain values of ~-v on the assumption that in f i r s t approximat ion the re is no d i ssoc ia t ion in this region. An examinat ion of the B branches enabled us to obtain values of K*-- the quas i - s t a t iona ry d i ssoc ia t ion r a t e constant. In this case we took into account the change m the d t s somahon ra te upon the appearance of f r ee oxygen a toms and the d i f ferent (and var iable) e f fec t iveness of O and Ar atoms and 02 molecu les in the d i s so - c ia t ion of 02~ In this way we obtained the dependence of r v and K~ on the t e m p e r a t u r e T~

Fig. 1. O s c i l l o g r a m of absorpt iv i ty (• = = 1850 ~) in oxygen behind a shock f ront propagat ing in a 10% 02 + 90% Ar mix tu re at a veloci ty V = 2.94 mm/t~sec (initial

p r e s s u r e 6.6 mm Hg).

To desc r ibe the en t i re s imul taneous p r o c e s s of exci tat ion of v ibra t ions and mo lecu l a r d issocia t ion behind the shock f ront we use re la t ions (9) and (11), taking intb account the values of ~ and K* obtained together with the

v m a s s , momentum, and energy conserva t ion laws, the equation of state, and Dalton s law. Solving this probIem numer i ca l l y on a compute r we can easi ly find the dis t r ibut ion of all the gas c h a r a c t e r i s t i c s in the flow behind the shock f ront and compute va lues of the absorpt iv i ty fo r compar i son with experiment �9 The quas i - s t a t iona ry value of the v ibra t ional t e m p e r a t u r e T* in exp re s s ion (11) is de te rmined f rom the quas i - s t a t iona r i ty condition [5]

= 0 ; e~ e v _ D , _ --,dn(t)" (12) dEv dt ~v dt

hence for T > 0 (0 is the c h a r a c t e r i s t i c v ibra t iona l t empera ture )

Tv l ~, n% n A D* ~ - = -K~ ~ k r (13)

(n is the total number of p a r t i c l e s pe r unit volume).

In compar ing with expe r imen t s , it is n e c e s s a r y to take two things into account�9 F i r s t , our model does not allow for ro ta t ion of the molecu les or the effect of additional d issocia t ion channels due to the p r e s e n c e of excited e lec t ron ic t e r m s . Taking these f ac to r s into account [6] leads to an inc rease in K~ (cor rec t ion fac to rs g rge l ) . In this connection, on subst i tut ing into (9), (12), and (13), the exper imenta l ly m e a s u r e d values of K~ should be reduced by dividing by grgel- F o r 02 at T --< 10 000 ~ K grge l ~ 7. Secondly, it should be kept t n m l n d t h a t D di f fers f r o m the d lssoe ia t ion energy D by approximate ly kT (D* = D-kT) [6].

A compar i son of the expe r imen ta l and calculated curves indicates sa t i s fac to ry ag reemen t over the en t i re nonequi l ibr ium flow reg ion (Fig. 2a). We have also shown the calculated t ime dependence of the t rans la t ional v ibra t iona l t e m p e r a t u r e s (Fig. 2b) and the oxygen atom concent ra t ion (Fig. 2c) in this region. We note that in this case it p roved n e c e s s a r y to take the f ac to r s g rge l into account, s ince o therwise ag reemen t could not be achieved

27

COMBUSTION, EXPLOSION, AND SHOCK WAVES

O, ! 20O6

0 0 0 2 • 6

- f ' I b

_

I , i O 2 ~ t , psec

Fig. 2. Time plot of absorpt ion s ignal r eco rded behind shock f ront (~ = 1900/~, V = 3 k m / s e c , Pl = 3.71 mm Hg) and calcula ted values of absorpt ion intensi ty I, t rans la t ional and vibra t ional t e m p e r a t u r e s T and T v, and oxygen atom

concent ra t ion 7 d = (nd/n0)(p0/p), where n o is the pa r t i c l e number densi ty in f ront of the shock wave, n d is the atom number density behind the shock f ront , P0 is the gas density ahead of the wave, and

p the gas densi ty behind the shock front .

28

COMBUSTION, EXPLOSION, AND SHOCK WAVES

(dash-dot l ine in Fig . 2).

REFERENCES

1. E. V. Stupochenko, S. A. Losev , and A. L Osipov, Relaxat ion P r o c e s s e s in Shock Waves [in Russ ian] , Nauka, Moscow, 1965.

2. E. V. Stupoehenko and M. N. Safaryan, Teor . i eksp. khimiya , 2, 649, 1966. 3. A. I. Os ipov and N. A. Genera lov , FGV [Combustion, Explosion, and Shock Waves], 2, 2, 83, 1966. 4. A. I. Osipov, Teor . i eksp. khimiya , 2, 649, 1966. 5. N. M. Kuznetsov, DAN SSSR, 164, 1097, 1965. 6. E. E. Nikit in, Modern Theor ies of The rma l Decomposi t ion and I somer i za t i on of Molecules in the Gas P h a s e

[in Russ ian] , Nauka, Moscow, 1964.

20March 1969

]Yioscow

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