L A R G E - D I A M E T E R S H O C K T U B E S T U D Y

O F T H E P R O C E S S E S IN A G A S - D Y N A M I C

S. A . L o s e r , V. N. M a k a r o v , V. A . P a v l o v , a n d O. P . S h a t a l o v

L A S E R

UDC 662.373.8 +533.6.911

D e s c r i p t i o n o f t h e A p p a r a t u s

The p e r f o r m a n c e of e x p e r i m e n t s involving p r o c e s s e s in carbon dioxide g a s - d y n a m i c l a s e r s (GDL) r e - qu i res a l a r g e un i fo rmly heated volume of the inves t iga ted gas mix ture main ta ined at a p r e s s u r e of s e v e r a l a t m o s p h e r e s and at t e m p e r a t u r e s up to two thousand or more d e g r e e s Kelvin. The bes t method for r e a l i z - ing the s t a ted condit ions is heat ing of the gas by shock waves gene ra t ed in shock tubes. The spec i f i c s of the opera t ion ca l l for the p roduc t ion of g a s - d y n a m i c flows having a l a r g e t r a n s v e r s e d imension. A l a r g e - d i a m e t e r shock tube is p r a c t i c a l for such e x p e r i m e n t s . In the p r e s e n t s tudy we used a shock tube c r ea t i ng a flow d i a m e t e r of about 500 m m in the working sec t ion . The shock tube, which has an inside d i a m e t e r of 90 mm, has a h i g h - p r e s s u r e c h a m b e r 3 m long. The d i aph ragm s e p a r a t i n g the high- and l o w - p r e s s u r e c h a m b e r s is mounted in a spec i a l a s s e m b l y by means of a c a s s e t t e . A conical t r a n s i t i on sec t ion with a ha l f -angle of 15 ~ is loca ted between the two c h a m b e r s . The l o w - p r e s s u r e c h a m b e r has a length of 13 m0 a d i a m e t e r of 493 mm, and s t a i n l e s s s t ee l wa l l s . F o r e x p e r i m e n t a l work with a cooled gas mix ture a p la te wi th a f lat wedge - shaped nozzle having a c r i t i c a l c r o s s sec t ion in the f o r m of a s lo t (height 2 mm; width 453 ram) is i n s e r t e d in the end sec t ion of the shock tube t r a n s v e r s e l y to the flow. The subsonic nozzle con- tour i s c i r c u l a r with a r ad ius of 4 ram; the ha l f -ang le of the nozzle in the s u p e r s o n i c flow reg ion is 15 ~ The nozzle mounted on the p la te can be moved along the tube axis on guide rods and is s e c u r e d in a def ini te pos i t ion r e l a t i ve to the viewing windows of the tube. This f ea tu re makes it pos s ib l e to conduct e x p e r i m e n t s in d i f ferent c r o s s sec t ions of the nozzle .

So that the gas wi l l have the n e c e s s a r y p a r a m e t e r s p r i o r to en t ry into the nozzle the h i g h - p r e s s u r e c h a m b e r (30 atm) is f i l l ed p r i o r to the e x p e r i m e n t with a s t o i e h i o m e t r i c mix tu re of hydrogen and oxygen di luted with an exces s of hydrogen o r hel ium. The mix ture is igni ted at eight points s imu l t aneous ly along the c h a m b e r axis by a s y s t e m of s e r i e s - c o n n e c t e d d i s c h a r g e r s . F o r our work we used b r a s s d iaphragms with a th ickness of 1 to 1.5 m m and s ix notches cut the re in to a depth of 0.5 to 0.7 mm.

The l o w - p r e s s u r e chambe r is evacua ted to a p r e s s u r e of 1 .10 -2 t o r r , and the r a t e of gas leakage due to inadequate sea l ing and deso rp t ion f rom the wa l l s is 1" 10 -4 to 5 �9 10 -5 t o r r / m i n . P r i o r to the e xp e r imen t the c h a m b e r is "f lushed" with the inves t iga ted gas mix ture . With fo rced c i r cu la t ion the m i x t u r e occupies a volume of ~. 1 m 3 in the individual tank. P a r t i c u l a r c a r e is taken to r e m o v e f rom the tube any w a t e r fo rmed a f te r ignit ion of the detonat ing mix tu re by pumping and f reez ing out. The r e s i d u a l w a t e r vapor in the tube is moni to red by r e c o r d i n g the i n f r a r ed r ad ia t ion of the w a t e r vapor heated in a shock wave p r o p - agating in argon o r oxygen, with which the l o w - p r e s s u r e chamber is s p e c i a l l y f i l led. The i n f r a r ed r a d i a - t ion de t ec to r is c a l i b r a t e d by r e c o r d i n g the r ad i a t i on f r o m the shock wave in w a t e r vapor in jec ted in the tube at a known p r e s s u r e . The f lushing p r o c e d u r e used in the s tudy reduced the w a t e r vapor impur i ty con- tent to a l eve l of < 10 -2 t o r t , i .e . , l e s s than 0.1% of the ini t ia l p r e s s u r e of the inves t iga ted mix tu re s .

An ana lys i s of the r a d i a t i o n f r o m the shock-hea ted gas showed that the d imens ions of the "plug" b e - tween the wave f ront and the contact su r f ace a r e cons i s t en t with p red i c t i ons b a s e d on ca lcu la t ions wi th a l lowance fo r the boundary l a y e r at the tube wall . The l i f e t ime AT 0 of the un i formly hea ted gas column b e - hind the r e f l e c t e d shock p r i o r to nozz le e n t r y is about 1 msec under the given ope ra t ing condit ions of the appara tus .

Moscow. T r a n s l a t e d f r o m F i z i k a Goren iya i Vzryva, Vol. 9, No. 4, pp. 463-473, Ju ly -Augus t , 1973. Or ig ina l a r t i c l e submi t t ed March 16, 1973.

�9 19 75 Plenum Publishing Corporation, 22 7 West 17th Street, New York, N. Y. 10011. No part o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission o f the publisher. A copy o f this article is available from the publisher for $15.00.

401

Fig. 1. OscUlogram of the absorp t ion of u l t ravio le t r a d i a - t ion by heated 02 in flow through the shock tube nozzle; t ime m a r k e r : 10 psec .

Fig. 2. Sys tem for m e a s u r i n g Iight ampl i f ica t ion in a gas mixture .

Upon s t a r t ing of the undiaphragmed superson ic nozzle mounted in the end of the shock tube a p r i m a r y shock wave propaga tes through the gas in the nozzle , followed by the contact su r face , behind which flows the gas enter ing the nozzle , and a s econda ry shock wave moving against the flow (and ent ra ined by it). Af ter pa s sage of the r a r e f ac t ion wave a quas i - s t eady flow is es tab l i shed [1], forming the main object of investigation. Its durat ion is l imi ted e i ther by the t ime AT 0 or by the inc reased p r e s s u r e in the c h a m b e r behind the nozzle and the t r ans i t of the contact shock inside the nozzle.

The nozzle s t a r t ing t r ans ien t was invest igated in the given study by record ing the absorpt ion of u l t raviole t radia t ion in the flow through the nozzle [2]; the t e s t medium was mo lecu l a r oxygen. The absorp t ion- record ing network includes a pulsed u l t ravio le t radia t ion counter , which p ropaga tes its l ight ac ros s the flow in the nozzle; a VM-1 vacuum monochromato r ; a photomul t ip l ie r with a terphenyl radia t ion conver t e r ; and a pulse osc i l loscope . A typical osc i l loscope plot of a signal ob- tained in the invest igated r eg imes is given in Fig. 1 (initial p r e s s u r e

pi = 15 to 40 to r r ; wave veloci ty in the l o w - p r e s s u r e c h a m b e r v =1.4 to 1.6 km/ sec ) . The c h a r a c t e r of the s ignals is cons is tent with the gene ra l notions concerning the p r o c e s s e s in such nozzles ; a f t e r en t ry of the shock wave into the nozzle a reduct ion of the s ignal is obse rved in connection with the appearance in the in- s t rumen t field of view of the f i r s t - s h o c k - h e a t e d oxygen, which absorbs u l t ravio le t radiat ion. Af ter pa s sage of the second shock the signal grows (the absorp t ion diminishes) , and a f t e r a few tens of mic roseconds a p rac t i ca l l y constant s ignal level is es tabl ished, co r respond ing to quas i - s t eady flow of the gas. The l a t e r signal decay obse rved in the o s c i l l o g r a m is a t t r ibutable to a reduct ion of the level of the i l lumination s ig - nal due to the insufficient width of the light source pulse . As shown by expe r imen t s on the ampl i f ica t ion and absorpt ion of l ight in ca rbon dioxide and CO2-N2-He(H20) mix tu res and on las ing (see below), the dura - tion of the quas i - s t eady gas flow in the given nozzle is 0.8 to 1 msec .

, . V I e a s u r e m e n t o f I n f r a r e d A m p l i f i c a t i o n

a n d A b s o r p t i o n

The bulk of our invest igat ion was a imed at the absorp t ion and ampl i f ica t ion of the emi s s ion of a CO 2 l a s e r (wavelength 10.6 p) by CO 2 -N2-He(H20) mix tu res flowing in a nozzle of the type desc r ibed above. To m e a s u r e the gain (absorpt ion coefficient) we a s s e m b l e d and made the n e c e s s a r y ad jus tments of a s y s t e m (Fig. 2) incorpora t ing the method desc r ibed in [3]. The emis s ion f r o m an industr ia l d i s cha rge - type CO 2 in- f r a r e d l a s e r is d i rec ted by means of a s e m i t r a n s p a r e n t KBr plate into the shock tube arid into the divergent p a r t of the wedge-shaped nozzle; then the b e a m p a s s e s through a f i l te r , which s e p a r a t e s out a wavelength of 10.6 #, and is focused by a s y s t e m of m i r r o r s onto a G e - A u photodetector . The e l ec t r i c a l s ignal f r o m the de tec to r is sent through a cathode fol lower to one input of the osc i l loscope . The peak sens i t iv i ty of the in- put b lock a t ta ins 50 p V / c m . The re la t ive intensi ty of the b e a m t r ansmi t t ed through the nozzle is moni tored by m e a s u r i n g the level of a r e f e r e n c e modulated signal f r o m the s a m e l a s e r ; that s ignal is r e c o r d e d in the

402

Fig. 3. O s c i l l o g r a m s of the gain. a) Mixture of 10% CO 2 + 40% N2 +50gc He, v=1.67 k m / s e c , T0=2080~ P0=5.85 a tm; b) mix ture of 15g CO 2 +83~ N 2 +2% H20, v =1.34 k m / s e c , T0=1860~K, P0=6.5 a tm.

a,%/era e,4

o,2

o 2 -0,e

-0,4

Fig. 4.

'] 4 # 8 ,x.cm

.!

Measured gain values in a mix ture of 15gc CO 2 + 83gc N2+2% H20 ve r sus dis tance f r o m the c r i t i ca l c r o s s sec t ion of the nozzle. 1) T0=lT00~ p0=5.9 a tm; 2) T0=2060~ p0 = 6.8 a tm; 3) T0=lS00~ p0 = 5.9 a tm; 4) T0=2040~ p0=6.9 aim; 5) T0=1650"K, P0=4.8 a im; 6) T0=1860~ P0=26.5 a im; 7) To= 1840~ p0=5.4 atm; 8) T0=1940~ p0=6.3 a tm.

second channel of the s a m e osci l loscope. The re la t ive sens i t iv i ty of the channels is moni tored by s imul taneous modulation of the s ignals . The expres s ion used to de te rmine the gain has the f o r m

1 czl~ : l , , K = -7-1n -af~ " '

in which l is the optical path of the light b e a m in the nozzle, I 1 is the r e f e r ence signal level , 12 is the signal level co r respond ing to the intensi ty var ia t ion of the l a s e r b e a m through the nozzle , and

is the intensi ty ra t io of the t r ansmi t t ed b e a m and the b e a m r e - f lected f r o m the s e m i t r a n s p a r e n t plate.

Typical o se i l l og rams of the gain a r e given in Fig. 3. F o r the l a s e r mix tu res , spikes a re f i r s t obse rved due to sch l i e ren ef- fects induced by the optical tnhomogenet t ies a s soc ia ted with s t a r t - ing of the nozzle. Then an a lmos t - cons t an t signal level appea r s , co r respond ing to quas i - s t eady flow.* in many of the t e s t s initial gain peaks w e r e observed , exceeding the constant l eve l by 10 to 50gc. Af te r 0.8 to 1 msec the signal gradual ly subsides to ze ro , ampl i f ica t ion does not take place for about 1 msec , and then it r e a p p e a r s . Af te r a ce r ta in per iod of t ime a third, weake r ampl i - f icat ion signal appears . Fo r flow of undiluted CO 2 in the nozz le such "pulsa t ions" of the absorpt ion signal do not occur , and the flow is c h a r a c t e r i z e d by a s table quas i - s t eady s ta te l as t ing s e v -

e ra l mi l l i seconds . This d i spa r i ty is c l e a r l y a t t r ibutable to the nature of the in terac t ion of the r e f l ec ted shocks with the contact su r face of the detonating gas in the region ahead of the nozzle entry.

*In some tes t s the gain did not a t ta in a constant level (the signal had an a lmos t t r i angu la r waveform) ; we were unable to de te rmine the dependence of the w a v e f o r m on the initial expe r imen ta l conditions.

403

T A B L E 1

x, cm r0, ~ po, atmlcK~O i , ( -

2,2 4,2 4,2 4,2 6,2 6,2 6,2 7,0 7,0 7,0

2300 1610 1780 2300 1640 1780 2600 1700 2100 2540

4,9 2,6 4,0 4,7 2,8 4,0 5,6 3,0 5,3 5,5

--0,48 0,33 0,54 0,40 0,40 0,56 0,43 0.,17 0156 0,40

x, ClTI To, CK P0 aDn K.10', I (cm ~

m

5,6 2,9 5,3 5,3 2,8 .3,1 4,0 5,2 4,8

7,0 8,2 8,2 8,2

10,2 10,2 10.2 10,2 10,2

2600 1700 2080 2500 1640 1740 1740 2080 2350

0,58 0,50 0,53 0, 43 0,45 0,41 0,44 0,50 0,48

Note : x is the d i s t a n c e f r o m the c r i t i c a l s e c t i o n of the nozz l e .

T A B L E 2

i l g ~' l g ~" In Q,~

1--3 4 5

17,42 x -- 7,85 I5,55 x - - 8,06 5,72 x --7,306

--21 x -- 5,46

104 x -- 11,2 95,5 x - - I0,76 56,2 x -- 9,854

36,8 x -- 9,812

I --.10,684+5,93. I02'-x--4,76 �9 l03 x2+ 1,22 �9 104 x 3 --4,884+-{-5,08.102 x--4,58.103 x'2-} - 1,27:l04 x a 31,216--2,65,102 x+7,62.102 x2--7,13.102 x a

F / - "6--%81

19,066--18,39x+ln/ ' l -e o:8! [

L l _ e - r J

Note : x=T-1/31

The e x p e r i m e n t s in a m i x t u r e of 10% CO 2 +40% N 2 +50% He w e r e conduc t ed in the r a n g e of s h o c k v e - l o c i t i e s v =1.43 to 1.9 k m / s e c at i n i t i a l p r e s s u r e s p1=32 .4 to 44.0 t o r r , and in the m i x t u r e of 15% CO 2 + 83~ N2+2% H20 a t v = 1 . 3 to 1.43 k m / s e c and p1=38 to 42 t o r r . The r e s u l t s of the ga in m e a s u r e m e n t s in t h e s e m i x t u r e s a r e s u m m a r i z e d in* T a b l e 1 and in F i g . 4. I t was a s s u m e d tha t the p a r a m e t e r s of the gas in f ron t of the n o z z l e e n t r y (To, P0) had the v a l u e s o b t a i n e d f r o m the l aws of m a s s , m o m e n t u m , and e n e r g y c o n s e r v a t i o n and the equa t ion of s t a t e fo r the gas m i x t u r e beh ind the r e f l e c t e d s h o c k f r o n t in the c o m p l e t e v i b r a t i o n a l e q u i l i b r i u m s t a t e wi thou t r e g a r d fo r d i s s o c i a t i o n . r At d i s t a n c e s up to 3 e m f r o m the c r i t i c a l s e c t i o n of the n o z z l e (the r a t i o of the f low a r e a A to the c r i t i c a l c r o s s s e c t i o n A , is l e s s than 9) the a b s o r p - t ion of l igh t is o b s e r v e d in the i n d i c a t e d m i x t u r e s . F u r t h e r a long the n o z z l e the ga in b e c o m e s p o s i t i v e and i n c r e a s e s to d i s t a n c e s of 7 o r 8 c m f r o m the c r i t i c a I s e c t i o n ( A / A , =20 to 22), a t t a i n i n g v a l u e s of 0.5 to 0 .6~c /cm. D o w n s t r e a m , d e s p i t e f u r t h e r e x p a n s i o n (to A / A , =31), the ga in v a r i e s on ly i n s i g n i f i c a n t l y .

The l a r g e s t ga ins r e c o r d e d h e r e c o i n c i d e wi th the w e l l - k n o w n m a x i m u m v a l u e s o b t a i n e d in n o z z l e s of d i f f e r e n t c o n f i g u r a t i o n s a t ana logous in i t i a l p r e s s u r e s fo r d i f f e r e n t C O 2 - N 2 - H e ( H 2 0 ) m i x t u r e r a t i o s and a t d i f f e r e n t t e m p e r a t u r e s in hea t i ng of the gas due to c o m b u s t i o n of C O - H 2 - O 2 ( O 3 ) m i x t u r e s w i th a n i t r o g e n a d d i t i v e [4, 5] b y an e l e c t r i c a l d i s c h a r g e [6, 7] and in s h o c k w a v e s [3, 8]. We p o i n t out t ha t w i th an i n c r e a s e in p r e s s u r e i t has b e e n p o s s i b l e in e x p e r i m e n t s w i t h a s l o t to ob t a in h i g h e r ga ins (up to 1%) [9-11]. By us ing n o z z l e s wi th a c o r n e r t u r n i n g po in t in the c r i t i c a l s e c t i o n and r a i s i n g the p r e s s u r e a p p r e c i a b l y (with a s i m u l t a n e o u s i n c r e a s e in the t e m p e r a t u r e ) , a c c o r d i n g to [12], i t is p o s s i b l e to i n c r e a s e the s p e c i f i c p o w e r l e v e l of the output e n e r g y . Thus , the r e c o r d e d p e a k v a l u e s of the ga in a r e s i m p l y o b t a i n e d in d i f f e r e n t d e - v i c e s . The g e n e r a t i o n of h i g h e r v a l u e s of K r e q u i r e s a c o n s i d e r a b l e i n c r e a s e in the gas p r e s s u r e and o p t i - m i z a t i o n of the n o z z l e c o n t o u r s .

The r a p i d g r o w t h of K in o u r e x p e r i m e n t s f o r A / A , > 9 is a s s o c i a t e d w i th the i ncep t i on and g r o w t h of i n v e r s i o n b e t w e e n the v i b r a t i o n a l s t a t e s (001) and (100) of the CO 2 m o l e c u l e . The s l i g h t v a r i a t i o n of the

* M i x t u r e of 10% CO s +40% N 2 +50% He. § F o r the h i g h e s t v a l u e s of the t e m p e r a t u r e in the r e f l e c t e d s h o c k the t e m p e r a t u r e r e d u c t i o ~ of the g a s due to d i s s o c i a t i o n a t t a i n e d 150~

404

a

1ooo

6oo ~.. ~.." b " ; 5

o

o 2 4 ~ ~ /o /s

5~ C

t I I I . _ _ ~ d m

t I ~ 'a

7 -i J/} i , , : I!

2 4 6 ~ I0 ~v, c r n

Fig. 5. Dis t r ibu t ion of t e m p e r a t u r e s T, T i and p r e s s u r e p (a, c) and of absolu te i nve r s ion AN and gain K (b, d). Solid c u r v e s give va lues of K fo r Ac N (1/4"-T), dashed c u r v e s f o r Z~ c ~ 1 /T . a, b) Mixture of 10% CO 2 +40~c N 2 + 50% He, T0=2540~ p0=5.5 a tm; c, d) mix tu re of 15% CO2+83~c N2+2~c H20 , T0=1700~ p0=5.15 a tm.

gain n e a r the end of the nozz le is expla ined by the fac t tha t its r educ t ion due to the d e c r e a s e of the flow dens i ty is l a r g e l y offset by n a r r o w i n g of the s p e c t r a l l ines due to a r e d u c t i o n in the p r e s s u r e (see below),

C a l c u l a t i o n o f I n v e r s i o n a n d G a i n i n t h e G D L

F o r the ana lys i s of the p r o c e s s e s in a g a s - d y n a m i c l a s e r it is n e c e s s a r y to solve the p r o b l e m of a r e l ax ing gas flow in a nozz le . A n u m e r i c a l so lu t ion of the p r o b l e m fo r s o m e spec i f i c GDL nozz le c o n f i g u r a - t ions may be found in [13-16] and in s e v e r a l o the r p a p e r s . In p lanning the p r e s e n t s tudy it s e e m e d log ica l to develop a p r o g r a m fo r ca l cu la t ing the flow in a GDL with a l lowance fo r the m o s t r e c e n t advances in this area.

For the quantitative description of the cooling of the investigated gas mixtures in the nozzle we use the ordinary gas-dynamical equations for a one-dimensional steady flow of a relaxing gas. The relaxation equations for the vibrational energy are written in the simplest form consistent with the current state of knowledge about the kinetics of vibrational energy exchange in the system CO2-N 2- He(H20), i.e., on the assumption of equal vibrational temperatures T i and T 2 for the v I and v 2 vibrational modes of CO 2 and sep- aration of the principal channels (001)~-(030) and (001)~-~(II0) for energy exchange with the ~3 mode of CO 2. It is assumed that the water molecules are in vibrational equilibrium with the translational degrees of free- dom. For the i-th mode vibrational energy ei expressed in terms of the average number of quanta per molecule, e i = [exp (0i/Ti)- i] -I (0 i is the characteristic vibrational temperature of the i-th mode, and Ti is the vibrational temperature*), we obtain the equations

0 ~ , ~ ' P (eo eo/; - - v ( p . 3 -,,- -77- . - - . ~ e 3

07 = - - q~-3 - - ~4" %4;

in which

0,-r30~ 1 %~=pQ~.a[e3(e2-1- l)3--e~(e3-~-1)e- j; 0:l--0~ e 2

(~4=pQ~a[ea(e4+ 1) e --f- - - ea (ea-)- I)]; e l ~ l + 2 e 2 , Q_~ = ~ ~ $i~23,e ~(i). i

i i

*The s u b s c r i p t s a r e i n t e r p r e t e d as fol lows: 1-3) t h r ee modes of the CO 2 m o l e c u l e s ; 4) the N 2 molecu le ; 5) the he l ium a tom; 6) the w a t e r molecu le .

405

}i is the mole f rac t ion of the i - th spec ies , v is a constant f a c to r of the o r d e r of unity;* T' and T" a r e the values of the v ibra t ional re laxa t ion t imes of CO 2 and N 2 for v ib ra t iona l - t r ans la t iona l exchange, in s e c . a tm;

3 and } 4 a re the mole f rac t ions of the spec ies ; and Q23 and Q34 a re the probabi l i t i es of v ibra t ional exchange fo r the cor responding t r ans i t ions , in 1 / s e e . a tm. The t e m p e r a t u r e dependences adopted fo r T ' , ~" , and Q23 a re taken from~f [17-19] and a r e given in Table 2. It i s a s sumed for N2-CO2(v 3) exchange, accord ing to [17], that log Q34[1/sec" atm] = 7 . 4 2 - 1 . 6 5 " 10-3T +5.7 " 10-7T2.

The p r o b l e m of the flow of a re laxing gas in a nozzle is solved numer i ca l ly by an implici t scheme; the computat ion t ime for one var ian t on the B]~SM-6 compu te r i s about 30 see. The solution yields the d i s - t r ibution of the values of T, Ti, and p along the flow for var ious initial p a r a m e t e r s (composit ion of the gas, t e m p e r a t u r e , and p r e s s u r e ) before the nozzle entry .

As we know, the gain (absorpt ion coefficient) at the cen te r of the line broadened s imul taneous ly due to the Doppler effect and the influence of col l is ions can be wr i t ten in the f o r m [20]

K = s~ g.7~ N . - - N,. �9 ~ H (a, o) ore-'.

Here Nn, Nm, gn, and gm a re the populations (cm -3) and s ta t i s t ica l weights of the upper and lower level , r e spec t ive ly ; Anm is the Einstein coefficient for the spontaneous t rans i t ion n - ~ m (see-l); X is the t rans i t ion wavelength (cm); c is the speed of light (cm/see) ; h c is the half-width of the col l is ion l ine;

Ac V i ~ a AD '

where AD=(2RT ln2/p)~/2(1/Xc) is the Doppler half-width; R is the universa l gas constant; p is the molec - ular weight of CO2; and H(a, o) is the value of the Voigt function at the cen te r of the line. It is cu s tomar i l y a s sumed that the shock half-width of the line is inve r se ly propor t iona l to the square root of the t e m p e r a - ture , so that

300 ~ - Ac ---- P )// T" ~ ui~i cm'l,

where a i is the shock half-width of the CO 2 l ines at T =300~ and a tmosphe r i c p r e s s u r e of the i - th gas and i is the mole f rac t ion of that gas. However, m e a s u r e m e n t s [21] in the t e m p e r a t u r e range f r o m 300 to

420~ have shown that A c is i nve r se ly propor t ional to the t e m p e r a t u r e :

300 ~ -

M e a s u r e m e n t s of A c at higher values of the t e m p e r a t u r e have not yet been p e r f o r m e d . Both expres s ions for A c were subsequent ly used for compar i son with the exper imenta l r esu l t s . The values of Anm and a t have been m e a s u r e d in s e v e r a l pape r s : Anm=0.21 sec -1 [22], ~CO2=0.098 1 / c m . a tm [23, 24], ~N2=0.0704 1 / c m . a tm [24, 25], a H e =0.0547 1 /cm" a tm [23], and for ~H2 O, because of the l ack of data, we can use the data on the broadening of hydrogen [23], but for molecules with the m a s s of H 2 0 : ~ H 2 0 =0.0423 1 / c m . a tm.

Abiding by the usual assumpt ions , we cons ider that the main f ac to r contr ibuting to ampl i f ica t ion (ab- sorption) is the t rans i t ion for J =20 in the p - th b ranch of the (001-100) band of the CO 2 molecule , while the shift of the l ines genera ted in the l a s e r p robe and obse rved in the invest igated gas is zero . Then for the population of the upper level we can use the exp re s s ion (s t ronges t - l ine approximat ion)

2hc /Vn = ~ - B0olgnN001 exp [-- hcBool J,~ (J,~ + l)/kT],

in which B001 is the rotat ional constant (B001=0.3866 cm-~), N001 is the population of the CO 2 (001) v ibra t iona l level , hc /k=1.43883 c m . ~ Jn=20 , and gn=2Jn +1. An analogous exp re s s ion can be wri t ten fo r N m (B100 = 0.3897 cm -1, J m = 1 9 ) . The function H(a, o) can be expanded f o r a _ 0.2into the s e r i e s [26] H(a, o) = 1 -1 .12838a +a2-O.752a3+O.5a4+ .... and for a > 1.4 into the s e r i e s [27]

1 ~_ 3

* The p r e s e n c e of this f ac to r is a t t r ibutable to the s imul taneous inclusion of the two ene rgy-exchange chan- nels (001)~---(030) and (001)---~(II0).

~The authors a r e deeply indebted to R. I.. Ser ikov fo rb r ing ing to the i r at tention the content and r e su l t s o~ [19].

406

T a

r

c i

T

Y d

Fig. 6. Compar i son of experimental and calculated gain values. Mixture of 15~ CO2+83~c N2+2~c H20: a) T0=1800~ p0=5.92 atm, x = 5.9 cm; b) T o = 1860~ P0 = 6.5 aim, x = 8.5 cm; mix- ture of 10% CO 2 + 40% N 2 + 5 0 ~ He: c) T o = 2300~ P0 = 4.7 atm, x = 4.2 cm; d) T O = 2100~ P0 = 5.6 aim, x = 7 .0cm. Dashed l ines: calculated values.

Fig. 7. Osc i l lograms of l a s e r signals for a mixture of 10% CO 2 +40% N2+50~ He (the ver t ical line gives the initial inten- si ty level of the probe signal), a) 200 psec sweep; b) 10 psec sweep.

The intermediate values for 0.2 < a _< 1.4 can be approximated with acceptable accuracy by the s imple r e - lation

H ( a , o) ~ 1.067/(1+1,5a).

The populations N i of the vibrational levels are determined by the vibrational and translat ional t empera - tures and part ia l p r e s s u r e s of the gas, and the absolute value of the inversion between the given vibrational levels is AN=Nool-Nlo o.

Typical solutions obtained for the investigated mixtures and nozzle configurat ion used in the exper i - ments descr ibed above are given in Fig. 5. It is evident f rom the calculations that the vibrational t empe r - ature of the v3 mode of CO 2 increases due to exchange with ni trogen molecules in the expanded flow. The absolute value of the inversion increases rapidly at f i rs t , attains a maximum, and then gradually dec reases toward the nozzle exit. We note that the gain in this case can increase even fur ther in connection with the p r e s s u r e reduction and narrowing of the lines.

C o m p a r i s o n o f E x p e r i m e n t a l a n d C a l c u l a t e d G a i n s

The compar i son of the calculated and experimental resul ts is complicated by the t ime variat ion of the gain level r ecorded in the experiments . To facil i tate the compar ison the calculated values are plotted

407

di rec t ly on the gain osc i l lograms for the flow (Fig. 6). It may be stated by and l a rge that sa t i s fac tory agreement obtains between the calculated and exper imenta l resu l t s fo r the l a s e r mixtures . Agreement has also been noted in many o ther papers .

F o r a quantitative es t imate of the degree of agreement we r e f e r to the average gain level r eco rded in the quas i -s teady stage of the p roces s . On account of the dispar i t ies in the initial conditions it is con- venient to normal ize the d i sc repancy between the analyt ical and exper imenta l in each tes t to the value ob- tained in that test . It turns out in this case that the re la t ive discrepancy, averaged over fourteen exper i - ments (six for 10~ CO 2 +40~c N2+50~ He and eight for 154 CO s § N 2 +2~ H20) can be cha rac t e r i zed by the following indices. The calculat ion pe r fo rmed for Ac ~ 1/4-T yields resu l t s 9% lower than the exper i - mental. If we assume that A e ~ l / T , the agreement improves ; the calculated data a re , on the average, only l g higher than the exper imenta l . However, the s ca t t e r of the data is r a the r large; it can only be stated with 0.95 confidence l imits that the d iscrepancy between the exper imental and theore t ica l is less than 23% in the f i r s t case and 144 in the second case. This compar i son not only conf i rms the overal l agreement of the calculated and exper imenta l , it also indicates that the extrapolat ion of the values of the shock half-width as inverse ly propor t ional to the t empera tu re is p re fe rab le to its extrapolat ion as inverse ly proport ional to the square root of the t empera tu re , even though the l a t t e r is f requent ly assumed.

P r o b e L a s i n g

For probe las ing the optical windows in the side. walls of the nozzle a re rep laced by two m i r r o r s 30 mm in d iamete r with an aluminum coating (one m i r r o r flat, and the other concave with a radius of curva ture of about 1300 ram). The beam is taken out through a single aper tu re 2 mm in d iamete r in the concave mi r - r o r , which is covered with a potass ium bromide plate, The generated energy is r eco rded by a radiat ive heat de tec tor and amounts to 0.1 to 0.2 J, and its t ime pa t te rn is r eco rded by means of the Ge-Au photo- de tec tor used ea r l i e r . Typical lasing osc i l lograms a re shown in Fig. 7. In the quas i -s teady per iod of the gas flow in the nozzle the lasing level is fa i r ly stable; then a lasing cutoff sets in, and af ter about 1 msec las ing r e sumes . Then a third, weaker spike is observed. These fluctuations are consis tent with the p r e - viously noted gain var ia t ions in the investigated mixtures .

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