ROTATIONAL NONEQUILIBRIUM EFFECTS IN CW CHEMICAL LASERS

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ROTATIONAL NONEQUILIBRIUM EFFECTS IN CW

CHEMICAL LASERS

L. Sentman

To cite this version:

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L.H. Sentman

AeronauticaZ and AstronauticaZ Engineering Dept., University o f IZZinois, Urbana,

IL 61801,

U. S.A.,

.

Abstract.- Theoccurrence of rotational nonequilibrium in cw chemical lasers is reviewed. The influence of this phenomenon on the power spectral output of the laser for both Fabry-Perot and confocal unstable resonators is summarized. The results of an investigation to construct an efficient, rotational nonequilibrium model which will give quantitative predictions of power and power spectral distribution is described. The essential elements of the model and their influence on laser performance are shown by detailed comparison with a model which contains boundary layer flow striations, F atom wall recombination and detailed, finite rate kinetics (typically 21 species and 143 reactions). Rota- tional equilibrium and nonequilibrium species profiles, saturated gains, intensity distributions, and power spectral distributions are compared for several different cases. Good agreement with experimental power spectral distributions is obtained.

I.

Introduction

The performance of a cw chemical laser is de- the pumping distributions and the observed power

termined by the nonlinear interactions which occur spectral distributions is striking.

between the fluid dynamics, the chemical kinetics

C 2 1

i

V . 2

and the optical resonator. Of all the phenomena that occur in chemical lasers, rotational nonequl- librium is the mechanism responsible for the power spectral output. Since rotational relaxation 5 s one of the fastest collisional relaxation mecha- nisms, rotational nonequilibrium would not be expected to have such a significant effect on laser performance. The reason for this anomaly is the fact that the pumping reaction produces the excited species in a rotational nonequilibrium distribu-

1

tion

.

Since the pumping reaction operates throughout the laser cavity, it acts as a continuous

1 source generating rotational nonequi librium through- Figure 1. Comparison of the pumping distribution

and two cw

HF

laser power spectral dis-

out the cavity. Thus, even though rotational re- tributions. Open bar is from Ref. 11;

hatched bar is from Ref. 17.

laxation is very fast, the deactivation of the ex- The significant effects of rotational nonequi-

cited species by laser action is faster and the librium on the pecformance of chemical 'lasers have

8

C O ~ ~ ~ ~ U O U S generation of rotational nonequilibrium been studied in both the C W ~ *and pulsed ~ , ~ ~ ~ by the pumping reaction results in this phenomenon cases6". These studies have demonstrated that:

playing the major role in determining'the power 1. Rotational nonequilibrium is the mechanism re-

sponsible for the power spectral distribution

spectral performance of the laser. This fact is of the laser;

2. Rotational equilibrium models over the

illustrated in Fig. 1 where the similarity between total power and maximum intensity;

3. For rotational nonequilibrium, the ~ab$y-~erot

*

This work was supported by an AFOSR Grant and unstable resonator power spectral

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C9-18 JOURNAL DE PHYSIQUE

d i s t r i b u t i o n s a r e n e a r l y i d e n t i c a l ;

4 . R o t a t i o n a l nonequilibrium i s t h e mechanism r e - s p o n s i b l e f o r a nonlinear coupling which may occur between t h e s a t u r a t e d g a i n and t h e r e s o - n a t o r geometry r e s u l t i n g i n l a r g e amplitude f l u c t u a t i o n s on a given l i n e .

These r e s u l t s show t h a t f o r a model t o p r e d i c t t h e power s p e c t r a l d i s t r i b u t i o n , r o t a t i o n a l nonequilib- rim must be included. However, when t h i s phenomenon i s included i n t h e l a s e r models, computer times tend t o become u n d e s i r a b l y l a r g e because t h e number of dependent v a r i a b l e s i n c r e a s e s from about 30 t o 80 o r more. The p r e s e n t work was d i r e c t e d towards overcoming t h i s problem by c o n s t r u c t i n g a model

(denoted MNORO) which would have reasonable r u n times and s t i l l g i v e q u a n t i t a t i v e answers.

1 1 . The Model

Since computational e f f i c i e n c y r e q u i r e s t h a t t h e number o f dependent v a r i a b l e s be k e p t t o a mini- mum, t h e approach used was t o i n c l u d e t h e e s s e n t i a l chemical k i n e t i c s

-

t h e pumping r e a c t i o n s , major d e a c t i v a t i o n r e a c t i o n s and r o t a t i o n a l r e l a x a t i o n r e a c t i o n s

-,

and t o i n c l u d e t h e f l u i d dynamic e f - f e c t s o f mixing by i n p u t t i n g t h e p r e s s u r e , tempera- t u r e , v e l o c i t y , mass flow r a t e remaining i n t h e primary, d mass flow r a t e remaining i n t h e secon-

P'

dary, d and t h e r a t i o o f t h e t h i c k n e s s of t h e S'

mixed stream t o t h e geometric t h i c k n e s s of t h e flow, L ~ / L ~ , a s f u n c t i o n s o f x which a r e obtained from one o f t h e a v a i l a b l e quasi-2D, 2D o r 3D f l u i d dynamic, r o t a t i o n a l equilibrium, chemical k i n e t i c

$

l a s e r models 8y9'10. ? h i s approach was motivated by t h e f a c t tha-t changes i n chemical k i n e t i c s , which s i g n i f i c a n t l y a f f e c t t h e power s p e c t r a l performance, i n most c a s e s , have o n l y a minor e f f e c t on t h e flow f i e l d ' i n which t h e k i n e t i c s t a k e s p l a c e .

S i n t e t h e p r e s e n t model w i l l reproduce t h e r o - t a t i o n a l equilibrium r e s u l t s when it i s r u n using a v a l u e o f t h e r o t a t i o n a l r e l a x a t i o n r a t e constant.

5

which i s 10 times i t s measured v a l u e , t h e model h a s evolved from d e t a i l e d compmisons. with, t h e

8

Blaze I1 model f o r two d i f f e r e n t cases: an Aero- space a r c d r i v e n HF power s p e c t r a l experiment1' and t h e Bell Aerospace Textron CL X I l a s i n g l i n e s

17 experiment

.

For purposes of developing t h e k i n e t i c - f l u i d dynamic model, t h e o p t i c a l c a v i t y was modeled a s a Fabry-Perot r e s o n a t o r , Refs. 2, 12. With t h i s mod- e l , t h e

had

terms a r e simply nonlinear terms on t h e r i g h t hand s i d e s of t h e s p e c i e s equations. Thresholds and c u t o f f s a r e obtained a s p a r t o f t h e s o l u t i o n o f t h e d i f f e r e n t i a l equations f o r t h e flow v a r i a b l e s

To o b t a i n q u a n t i t a t i v e p r e d i c t i o n s , t h e c o r r e c t pumping and c o l l i s i o n a l d e a c t i v a t i o n r e a c t i o n s i n - volving t h e primary d e a c t i v a t o r s HF, F , H and H2 have been included. The model comprises t h e f o l - lowing r e a c t i o n s e t : Pumping Reactions F + H2 + HF(1,J) + H F + Hz + HF(2,Jt) + H F + H2 +HF(3) + H F2 + H -+ HF(3)

+

F C o l l i s i o n a l Deactivation Reactions HF(2,Jt) + M

+ HF(1,J)

+ M (5) HF(1,J) + M

+ HF(0)

+ M (6) HF(3) + M

+

HF(2,Jt) + M ( 7 ) where M = HF, F, H and H 2' where M = HF, F and H. R o t a t i o n a l Relaxation Reactions HF(2,J1) + M

+

HF(2,J) + M (11) HF(1,J) + M

+

HF(1,JIt) + M (12) where M = HF, F, H, Ha, Fa, He, A r .

The s p e c i e s denoted by Ar i s included t o t a k e ac- count o f any o t h e r combustion o r d i s s o c i a t i o n pro- d u c t s t h a t may be p r e s e n t i n t h e mixture and which would c o n t r i b u t e t o t h e r o t a t i o n a l r e l a x a t i o n but n o t t o t h e c o l l i s i o n a l d e a c t i v a t i o n of t h e l a s i n g s p e c i e s . The above model, while r e t a i n i n g t h e key chemical k i n e t i c s , i s q u i t e concise compared t o t h e

8

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constants f o r the above r e a c t i o n s a r e developed i n Refs. 2 and 12.

The f l u i d dynamic variables P(x)

,

T(x)

,

u(x)

,

inp (x)

,

ms

(x) and Le/Lg, a r e input a s polynomials i n x where t h e c o e f f i c i e n t s a r e determined by a l e a s t squares f i t t o t h e p r o f i l e s obtained from a d e t a i l - ed r o t a t i o n a l equilibrium model such a s Blaze 1 1 . Experience has shown t h a t P , T and u a r e best f i t with 9th order polynomials, m and Bs with ax+bx 1/ 2

P

+mio and Le/Lg with ax 3/4+bx1/2+cx. The coef f i - c i e n t s of these polynomials are input data. The mixing term which appears i n the species equations i s given i n Ref. 12.

The model c o n s i s t s of the species equations 12 f o r HF(l,J), HF(2,Jt), HF(O), HF(3), F, F2, H, H2, He and A r . With J M = 20, t h e r e a r e 50 d i f f e r e n - t i a l equations f o r t h e mole mass r a t i o s of t h e various species i n t h e mixture. The i n i t i a l values of the mole mass r a t i o s of the species a r e calcu- lated'from t h e i n i t i a l flow r a t e s of t h e speci.es i n the primary and secondary streams.

F atom recombination a t the wall i s included by varying t h e F atom concentration i n the primary stream from zero a t t h e wall t o i t s value i n the core of t h e primary flow. This r e s u l t s i n a v a r i - a t i o n of nF

,

the value of nF a t the edge of the

PO

mixing layer which appears i n t h e mixing terms, with x. The boundary layer p r o f i l e i s used t o pro- vide t h e v a r i a t i o n of nF from t h e wall t o t h e core. Since t h e present model i s a two l e v e l model, t h e e f f e c t of l a s i n g on t h e 1 t o 0 band was i n - cluded a s a r a t e constant kd!: which was determined

10

from t h e Blaze I 1 value of q a d ( x ) by s e t t i n g

10 10 2

&ad = krad p n ~ ~ " ~ ~ ( 1 ) . The r a t e constant f o r t h e deactivation of HF (1)

bL

HF i s then w r i t t e n a s

HF

k10 = k10 + kizd and t h e e f f e c t of l a s i n g on t h e depopulation of HF(1) i s taken i n t o account i n t h e

111. Results and Discussion

To e s t a b l i s h t h e r o l e of various aspects of t h e k i n e t i c s model, MNORO run i n t h e equilibrium mode has been compared with t h e Blaze I 1 r e s u l t s f o r t h e CL XI case. Several Blaze I1 runs were made t o determine t h e e f f e c t of t h e v i b r a t i o n a l bands which a r e allowed t o l a s e on t h e f l u i d dynamic v a r i a b l e s . The hot r e a c t i o n bands have a neg- l i g i b l e e f f e c t on t h e species and f l u i d dynamic pro- f i l e s 1 2 . Not allowing lasing t o occur on 3+2 r e - s u l t s i n a 10% increase i n the peak pressure and temperature and a n e g l i g i b l e e f f e c t on t h e other p r o f i l e s except f o r t h e HF(3) profile12. Thus, one s e t of f l u i d p r o f i l e s was used throughout t h i s study.

The hot r e a c t i o n bands have a n e g l i g i b l e i n - fluence on t h e -1 power s p e c t r a l di-stribution. However, lasing on J-t2 has a s i g n i f i c a n t e f f e c t not only on the t o t a l power i n t h e 2+l band, Table I , but a l s o on t h e 2+1 power s p e c t r a l d i s t r i b u t i o n . W E!I CL LL B ~ t j s r L a r t w ~ Tors i,,,,, r,,, p , , , L E N G ~ ~ ~

P&;j

EL:]

l~~zk:nj

1 I t s ~ w u : . 1 1 ,417 .U22 ,113 .W19 .025 .ti148 0 52. 21. I0 1 ,956 . I .443 .I22 (.'1]7) c.923) ( . I l l ) 1 1 . 1 0 .XI9 7 .518 .U&? (:>MI (.3%)

Table I . BLAZE I1 CL X I Results when Various Bands a r e Allowed t o Lase. (Relative powers and l a s i n g zone lengths a r e shown).

The MNORO r e s u l t s when t h e hot r e a c t i o n i s i n - cluded by pumping only t o HF(3) and when multiquantum VT deactivation by HF, F and H a r e :included a r e

b

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C9-20 JOURNAL DE PHYSIQUE

p r o f i l e s , Fig. 2. As might be expected, t h e

B L U E R C L I SPECIES 'SOFaLES - B L A Z E 1 LASING 2 - I , ] - 0 X X X M N O R O HOT REACTION AND MOVT 0 0 0 MNORO HOT REACTION.NO M O V i

646 MNORO !I0 H O T IEACTION,NO MOVT

Figure 2. Comparison of Blaze I1 and MNORO H, F, H and F2 p r o f i l e s .

2

multiquantum deactivation has a negligible e f f e c t on t h e F, H, F2 and H2 species p r o f i l e s . However, both t h e hot r e a c t i o n and multiquantum deactivation a r e required t o obtain good agreement with t h e Blaze I1 HF(v) p r o f i l e s . The agreement i s not a s good f o r HF(3) because t h e hot r e a c t i o n pumps com- p l e t e l y t o HF(3) r a t h e r than t o t h e upper l e v e l s with c o l l i s i o n a l cascade t o HF(3). When hot reac- t i o n pumping t o HF (3) and HF (4) i s included, t h e r e i s no change i n t h e HF(v) species p r o f i l e s f o r v = 0, 1, 2 and only a s l i g h t improvement i n HF(3). To obtain b e t t e r agreement with HF(3) would require including a t l e a s t upsto HF (5) because, based on the magnitude of the r a t e constants involved, t h e r e i s a b o t t l e neck a t HF(4) which would r e s u l t i n de- creased HF (3). Since t h e e f f e c t on the 2+1 power and power s p e c t r a l d i s t r i b u t i o n would be small, t h e add'itional complexity i s not warranted. Thus, the f i n a l mohel includes t h e e f f e c t of t h e hot reac- t i o n by allowing it t o pump t o HF(3) only. From

of t h e s e r e a c t i o n s i s t h e 2+0 deactivation by a f a c t o r of about two t o one.

The r o t a t i o n a l nonequilibrium species p r o f i l e s when t h e hot r e a c t i o n pumps only t o HF(3) and multiquantum deactivations a r e included a r e compared t o t h e Blaze I1 p r o f i l e s i n Fig. 3. The nonequilib-

rim

species p r o f i l e s agree as well with t h e Blaze I1 species p r o f i l e s a s t h e equilibrium ones do.

i L P SPECIES PRORLES

-

8 L A i E P LASING ON 2'1.i-0 X X X MmRO ROTATICNAL NONEWILIBRIUM

HOT REACTION P U M R H F I P , M Q V T LASING I N 2 9 . 1 - 0

Figure 3. Comparison of Blaze I1 and r o t a t i o n a l nonequilibrium MNORO HF(v) p r o f i l e s . However, it i s seen from Fig. 4 t h a t t h e nonequilibrium power s p e c t r a l d i s t r i b u t i o n i s consider- ably d i f f e r e n t than t h e equilibrium power s p e c t r a l

runs made with only t h e 2+0 multiquantum deactiva- J

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a t J=8 i s i n d i c a t i v e of the r e l a t i v e l y l a r g e T(Tm = 585'~); since t h e experimental data peaks a t J=6, Blaze I 1 may be overpredicting t h e tempera- t u r e . The J - s h i f t i n g i s evident i n the gain and i n t e n s i t y d i s t r i b u t i o n s f o r t h e equilibrium case 12 whereas simultaneous lasing on many l i n e s occurs i n t h e nonequilibrium case, Fig. 5.

predicted i n t h e upper v i b r a t i o n a l bands. I t has a l s o been determined on the b a s i s of comparison with zero power gain data t h a t t h e W t r a n s f e r reac- t i o n

H2(1) + HF(v)

+

H2(0) + HF (v+1) ₍₁₄₎ i s about a f a c t o r of four f a s t e r than previously thought14. Both of these modifications have been included i n t h e Blaze I 1 calculations and i n the MNORO model. The r a t e data f o r (13) is obtained from Bartoszek, e t a l l 5 and Bott e t a l l 6 . Table I 1 shows t h a t t h e c o l l i s i o n a l decomposition reaction e f f e c t i v e l y eliminates the lasing on the upper v i - b r a t i o n a l bands and reduces the 3+2 l a s i n g t o a t most 2% of t h e t o t a l power and r e s u l t s i n an increase i n the length of t h e lasing zone. The addi-

MNt I I CL X I

7 w a ~ lrncm ur

POWLH L 1 5 1 1 1 !WE P;

p:

%

1::

tc

F G ~

I r

A ) 1 1 .'I17 .W ,113 ,006 .ulS .ill%

Figure 5. MNORO r o t a t i o n a l nonequilibrium P branch 8 ) LO,L, P I O N A L w,,Sl,lUII

saturated gains. ob IK(V). v13 .a71 1.E ,423 .417 ,023 0 .W'6 0

From t h e Blaze I1 r e s u l t s , Table I , it i s seen t h a t t h e occurrence of l a s i n g on 3+2 r e s u l t s i n a 21% increase i n the power on 2+l; MNORO i n d i c a t e s a 24% increase when l a s i n g on 3+2 occurs12 ( l a s i n g on 3+2 i s simulated i n MNORO by s e t t i n g k 2 2 3

P

= k~ + k ~ ) '

Since t h i s i s a s i g n i f i c a n t e f f e c t , which was a l s o found i n t h e Aerospace l a s e r case, it may be worth- while promoting 3+2 l a s i n g t o obtain the increased performance on 2+1 and l+O.

The prediction by t h e very complete l a s e r mod- e l s a y 9y10 of lasing on the upper v i b r a t i o n a l bands and on 3 2 f o r cases i n which t h e experimental d a t a i n d i c a t e s no lasing above 2+1 suggests a fundamen- t a l problem with t h e HF r a t e package. Recently it has been suggested t h a t t h e c o l l i s i o n a l decomposi- t i o n r e a c t i o n 1 3

H + HF(v) + H2 + F, v

2

3 (13)

Table 11. Blaze I 1 CL X I Results when t h e C o l l i - s i o n a l Decomposition of HFCv), v

2

3 and t h e F a s t

V V

Rates a r e Included.

t i o n of the W t r a n s f e r reaction decreases t h e pow- e r by about 10% and decreases t h e length of t h e l a s - ing zone. The reason f o r t h i s i s t h a t reaction

(14) runs i n t h e d i r e c t i o n t o depopulate HF(v+l) because t h e concentration of H2(1)

i s

two t o three orders of magnitude smaller than t h e concentration of H2(0).. The Blaze I1 and MNORO r o t a t i o n a l nonequilibrium species p r o f i l e s when r e a c t i o n s (13)

and (14) a r e included i n MNORO a r e compared i n ~ i ~ ; 6. Again, t h e r e i s very good agreement between the

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JOURNAL DE PHYSIQUE

Figure 6. Comparison of Blase I 1 and r o t a t i o n a l nonequilibrium MNORO HF (v) p r o f i l e s

.

From t h e s e comparisons, i t is seen t h a t t h e MNORO model i n c l u d e s t h e e s s e n t i a l elements r e q u i r - -

ed t o g i v e good agreement w i t h t h e very d e t a i l e d Blaze I 1 model. These elements a r e , i n a d d i t i o n t o t h e pumping, c o l l i s i o n a l d e a c t i v a t i o n and r o t a - t i o n a l r e l a x a t i o n r e a c t i o n s : l a s i n g on 1+0 and 3+2 i f it occurs, F atom w a l l recombination, t h e hot r e a c t i o n , t h e c o l l i s i o n a l decomposition o f HF(v) f o r v , 3 and t h e HF-H VV t r a n s f e r r e a c t i o n . The

2

importance of t h e multiquantum d e a c t i v a t i o n depends upon t h e temperature. A t low temperatures, m u l t i - quantum e f f e c t s were l e s s than 3%; however, f o r t h e CL XI nozzle, i n which t h e temperature was q u i t e high (%58S0K), multiquantum d e a c t i v a t i o n r e s u l t s i n about a 29% decrease i n power and has a s i g n i f i c a n t e f f e c t on t h e l e n g t h o f t h e l a s i n g zone, Table 111.

Figures 7 and 8 compare t h e MhQRO r o t a t i o n a l nonequilibriwn power s p e c t r a l d i s t r i b u t i o n s with

Table 111. The E f f e c t o f t h e Hot Reaction and Multiquantwn

VT

Deactivation on t h e 2+1 Power and t h e Length of t h e Lasing Zone f o r CL XI.

r;rr Aff:.1SP4CF 1 A S t A DATA

LY- MfiOnO R O T A T I , J N A L

> , O N F ~ V I L I L R I U h l

Figure 7 . Comparison of MNORO r o t a t i o n a l nonequi- l i b r i u m PSD with t h e d a t a of Ref. 11.

-rz C L X l U A 7 A

--

MNORO /~O).ATI@I\IAL

NONEr?l111 IARItrM

t h e experimental d a t a f o r two c a s e s . I n both c a s e s ,

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z . c * <

Figure 9. Comparison o f MNORO r o t a t i o n a l nonequilib- rim i n t e n s i t y d i s t r i b u t i o n s with t h e CL XI d a t a . S o l i d l i n e : MNORO; dashed l i n e : d a t a .

( a l l times a r e on t h e CYBER 175). This d i f f e r e n c e i n run time i s t h e r e s u l t of t h e d i f f e r e n t p r e s s u r e s i n t h e two c a s e s . When t h e C L XI i s r u n with t h e p r e s s u r e reduced by a f a c t o r of 10, t h e execution time i s 66 s e c and when t h e Aerospace l a s e r is r u n w i t h t h e p r e s s u r e increased by a f a c t o r o f 10, t h e r u n time i s 203 s e c . I t should be noted t h a t v a r i a -

t i o n i n o t h e r parameters such as t h e temperature p r o f i l e , mass flow r a t e s , mirror r e f l e c t i v i t i e s , had a t most a 10% e f f e c t on r u n time.

The p h y s i c a l reason t h e p r e s s u r e has such an e f f e c t i s t h a t t h e k i n e t i c terms depend upon p" and

2

hence P ; t h u s a f a c t o r o f 10 change i n t h e p r e s s u r e changes t h e magnitude of t h e k i n e t i c terms by a f a c - t o r of 100 which s i g n i f i c a n t l y i n f l u e n c e s t h e s t i f f - ness of t h e equation system. This i s manifested i n t h e g a i n curves; t h e high p r e s s u r e c a s e reaches s a t u - r a t i o n a t one order o f magnitude smaller value of x than t h e low p r e s s u r e case.

The MNORO model has been coupled t o a s t r i p , wave-optics model of t h e r e s o n a t o r . While t h e c a l - c u l a t i o n s have not y e t been r u n t o convergence, sev- e r a l i t e r a t i o n s have been r u n using d i f f e r e n t num- b e r s of p o i n t s i n t h e c a v i t y . For t h e low p r e s s u r e case, t h e r u n time p e r i t e r a t i o n v a r i e d l i n e a r l y

p o i n t s . The c a l c u l a t i o n s were performed f o r a 50% geometric outcoupled, confocal, u n s t a b l e r e s o n a t o r . From t h e d e t a i l e d comparison o f MNORO w i t h t h e Blaze I 1 code and t h e experimental power s p e c t r a l d a t a , t h e o b j e c t i v e of developing an e f f i c i e n t , r o - t a t i o n a l nonequilibrium model o f a cw chemical l a - s e r which i s capable of giving q u a n t i t a t i v e p r e d i c - t i o n s of t h e power s p e c t r a l d i s t r i b u t i o n has been achieved.

References

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