SMALL SIGNAL GAIN COEFFICIENT AND POWER OUTPUT CALCULATIONS OF HIGH TEMPERATURE CO2 GASDYNAMIC LASER WITHOUT FLOW STAGNATIONS

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SMALL SIGNAL GAIN COEFFICIENT AND POWER

OUTPUT CALCULATIONS OF HIGH

TEMPERATURE CO2 GASDYNAMIC LASER

WITHOUT FLOW STAGNATIONS

K. Kasuya, M. Minami, K. Horioka, K. Niu

To cite this version:

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JOURNAL DE PHYSIQUE ColZoque C9, suppt&nent au nQ1l, Tome </I, nc?vemBre 2980, page (29-175

SMALL SIGNAL GAIN COEFFICIENT AND POWER OUTPUT CALCHLATIQN$ QF HI'GH,

TEMPERATURE C02 GASDYNAMIC LASER WITHOUT FLOW STAGNATIBQq

K. Kasuya, Y. Minami, K. Horioka and K. Niu.

Department of Energy Sciences, Graduate SchooZ, Tokyo I n s t i t u t e of T e c b l o g y , Nagatsuta, Midori-ku, Yokohama, Japan.

Abstract.- This paper deals with numerical analyses of the flows in a bow shock wave (B.S.W.)- nozzle system which is applied to the gasdynamic laser (G.D.L.). In the B.S.W.- nozzle system, the dissociation of C02 or N2 molecules of the gas is suppressed even if the gas behind the B.S.W. is heated up to 2,000-5,000 K, since the gas is cooled rapidly, expanding through the nozzle,

Numerical calculations for such flow fields lead us to the estimation of the capability and the optimum conditions of such G.D.Ls.. It is turned out in our system that the mole-fractions of C02 can be elevated to 202, and the small signal gain coefficients increase more than four or five times

(3.8 m-l) than those of conventional G.D.Ls..

Numerical estimations of the laser power output are also examined for both cases of CW and Q-switch operations.

1.Introduction

The reservoir temperature of a conventional upstream mixing laser has an upper limit, since the flow stagnates in the reservoir and the gas is dissociated

under the high temperature. In view of

this disadvantage, an alternative ~ e t h o d is proposed, which utilizes an obstatle nozzle in supersonic flows. With this

arrangement, the gas is heated up to high temperature by a bow shock wave and cooled rapidly in a subsequent expanding nozzle before the dissociation proceeds1.

In this paper, numerical calculations are carried out for such flow field. Esti- mations are performed for the small signal gain coefficient and the power output, to clarify the capability and the optimum conditions for such gasdynamic lasers (G.D. Ls.).

2.Basic Equations and Numerical Methods A two-dimensional obstacle nozzle is considered to be in a supersonic CO*-N2-He

flow, as is shown in Fig.1 and 2. For sim-

plicity, we fix the steady state distance

between the bow shock wave and the nozzle throat to be 1 cm. After the assumption

Fig.1 Bow shock wave-nozzle system.

A*: Cross Section of Throat

---

-

- - - -

x ( c m )

Fig.2 Nozzle canfsur w e d

Oar

numerical

calculatigna,

of initial gas cnad$tion

and

the macb

, number g f

the

'lncid,e;nt:

shock

wave, we can

calcuZate

the

vibratinn@$Iy

frozen gas

condtpi~n just behind

ghs

brow wave with

the Rank$ ns

-Hugonia%

aquat$ona.

The small part of the bow wave, the downstream of

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C4-176 JOURNAL DE PHYSIQUE

which flows into the nozzle is assumed to be plane, because it is restricted only near to the flow axis.

The basic equations f o ~ numerical analyses of the small signal gain coefficient are quasi-one dimensional mass, momentum and energy conservation equations as follows***, 8 ~ - 1 ~ ( P u A )

----

-

at^ ax '

G = - l

d p _ au at p ax E*

aE- P (&+U

alnA)

- U

2E

at p a x ax ax7 E=e+e v ~ b ~ + ~ v i b = . '

To calculate the small signal gain coefficient, we adopt here the Anderson's model2 for C02-N2 vibrational modes (2 mode model, rd=O in Fig.3). The cross section A for the nozzle suction between the bow

mode- I

Fig. 3 Vibrational kinetic model for' C02 - N 2

.mixture.

.

shock wave and the nozzle throat (the tip of the obstacle nozzle) is an unknown

***See symbols' section.

parameter. Assuming that the stream line is straight in this region, we can deter- mine the cross section, because the flow rate must be conserved.

After we know the cross section of all over the flow fields, it is not difficult to solve eqs. (l)%(d). Numerical analyses are performed by using the time dependent method2%". Initial distributions of various parameters are assumed as in Fig.4. The mesh distance (Ax) in our case is shown in Fig.2.

Fi,q. 4 Initial distributions. The difference in the basic equations from the above eqs. is as follows, for the CW power extraction with a laser cavity in steady state. The extraction of laser light deactivates v mode of CO molecules, so that an inequilibrium between N2 (v=1 mode) and C O Z molecules must be taken into ac- count (3 mode model, mode-I1 is devided ' into mode 3 and 4 in Fig.3, ~ ~ f 0 . ) . The energy loss by the extraction must also be considered in the energy eq.,

a ~ _ - p au aE

at (, E+U =)-QR, ( 9 )

where

E=e+El 2+E3+Es, (10)

QR=GgI/p. (11)

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modes are referred to Ref. 2. After Lees, the following assumptions are made,

GZ=- (ln rlr2)/(2ZW) (1 3)

P w =Gz*I.V. (14)

Eqs. (11, ( 2 1 , (51, ( 8 1 , (9)%(14) and the rate equations for the 3 vibrational modes are solved with the time dependent method under the boundary conditions at the entrance of the laser cavity.

Another power output is calculated under the Q-switch extraction, during which the

motion of the fluid and the pumping by N2

molecules are neglected. Equations to be solved here are6

We assume here an initially constant small signal gain coefficient along the cavity axis

,'

and numerically solve eqs

.

(15) and

(16) by the Runge-Kutta-Gill method. 3.Results and Discussions

Examples of numerical calculations (2 mode case, estimation of G) are shown in

Fig.5 and 6. Fig.5 shows the temperature

distribution along the nozzle axis. As the

Fig.6 Small signal gain coefficient along nozzle axis with various conditions. are very high, a vibrational inequilibrium state just behind the B.S.W. relaxes within a few mm, and a flow in vibrational equilibrium state enters into the throat. Fig.

6 shows the distributions of small signal

gain coefficient G o along the nozzle axis for various conditions. Under one con- dition Go has a maximum (Max.Go) along x. As XCOz becomes larger, we can obtain

larger Max.Go, and as pl or X decreases,

C02

x becomes larger. In the same manner

max

we can shod the dependence of Max. Go on XHe

or XN ,' respectively. There are optimum

-11nditions in regard to M I , XCO2 and

respectively.

The distributions of maximum available

energy (EM) (See Ref. 2 about the defini-

tion) along the nozzle axis are shown in

Fig. 7. Fig. 8 (a)%(c) show the maximums of

Fig.5 Temperature distribution along nozzle axis.

pressure and the temperature in the region between the bow shock wave and the throat

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C9- 178 JOURNAL DE PHYSIQUE

(Max.Em) a l o n g t h e x - a ~ i p f o r v a r i o u s T"FZ,~QO K ,

p a r a q ~ e t e r s . A s t h e flqw r a t e F changes p " ~ 4 . 2 atm,

w i t h , b o t h t h e p r e s s u r e . a n d t h e mach number, Max.Go[x)=3.75 m - l ,

t h e 6:valuations a r e performed by t h e F ?V@x. Bm - 2 0 0 1(.T/m2s, p r o d u c t of e n e r g y and f l o w r a t e , xrnay=6Q

w e

F i g . 8 (a) F j g . 8 ( b ) The d i s t a n c e d

-!I-

",',

PI= P (krr) where t h e maximum 11=29B(K)

5

$ m a l l s i g n a l g a i q

-

p o e f f i c i e n t i s ob- e a i n k d i s shown Fig.{J. I t i s 6 nece:jsary t o c h o s ~ s e

-

o u r t;ondi t i o n s t q

I

be s l ~ f f i c i e ~ t t o

d

_& _{a d i i d a} 69

J

lo laalce xmax l e s s F i g . 4 ( c ) t h a n . 1 m f r p m thql F i g . 8 Mqxinvm of nlaxi- rnum a v a i l a b f e e n e r g y p o i n f o f view of _{p e r u n i t t i m e and} a r e a Y S . M I , p l {md boun$ary l a y e r @ i x i n g r a t i o . development. With t h i s r e s t r i c t i o n , t h e optimum c o n d i t i o n f o r t h i s k i n d of gasdynamic l a s e r i s g i v e n a s f o l l o w s , M.1 = 5 , p i 130 t o r r , The CW power under t h e c o n d i t i o n o f t p f 5 c m , Z w = 1 2 g cm, ZM=10,2Q,50 C m , and t h e l a s e r

$

c a v i t y i s p l a c e d just, a t t h e e x i t of the n o z z J e ( o n e of $he caviTy F i g . 9 Length needed

edgq comes J u s t t o g e t Max. q O ( x )

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Table 1 Parameters for CW laser power output calculations.

energy.

p " , atm

Fig.11 Threshold gain coefficient v s . radiation flux density. ,

Fig. 12 shows the

profile of the t o tempdrature and the

Values at the entrahce of the laser cavity. 0.00$'?

that CW power out-

I

put depends on the

li-5F-l

0.0058

small signal gain o

5 10

x(cm)

0.0077

,-coefficient

Fin.12 Flow variables _" rather than the along flow axis under

CW laser power ex- maximum available traction.

The laser pbwer output of Q-switch Bx- traction is estimated as :Eollows. A1t:hough

G varies along x as in ~ i B . 6 , averaged, con3tant G a 1 . o ~ ~ x is assbmed. An ex~lmple of the results is shown ih Fig.13 (I?f1*5,

p l = 3 0 torr, x ~ ~:XHe=b.2:0,1:0.7, ~ ; x ~ ~

G t = l . 07 n ' l )

.

~ ii 4 ~is the Ppeak ve1:sus .

1 4 1 , where the parameter is p l .

. , . * . .

t GnS)

~ i g ' . l - 3

,:Gain

~p&ffi~i$nt

'in4

P ~ Y ~ S

9iSt8uf:

vs. tlme under Q swltch operation.

4.Conclusions

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C9- 180 JOURNAL DE PHYSIQUE

in mole-fraction of CO2 makes the small signal gain coefficient four or five times

larger than that of

t

'

I

conventional G.D. ; 0

3 4 5 6 7

Ls.. The higher MI

laser power out- Fig.14 Peak power

output vs. mach

put is also calcu- number

.

lated under CW or Q-switch operations.

Murasaki: Third Generation Gasdynamic Lasers Driven by Bow Shock Waves, An Overview, Gas-Flow and Chemical Lasers, editted by J. F. Wendt (Hemisphere Publishing Co., 1979) p. 359

J. D. Anderson, Jr.: Gasdynamic Lasers: An Introduction, (Academic Press, 1976). J. D. Anderson, Jr.: NOLTR 69-200

(1969).

'

W. J. Glowacki and J. D. Anderson, Jr. :

NOLTR 71-210 (1971).

".

Lee: Phys. Fluids - 17-3 (1974) 644. Acknowledgements

1"1. G. Wagner and B. A. Lengyel: J. Appl.

The authors thank Prof .H.Oertel ,Sr. ,Prof.

Phys. - 34 (1962) 2040.

J.Zierep, Prof. B. Schmidt, Dr. H. Oertel,

'

K. Kasuva et al.: YCTAM-29. to be

Jr. in Karlsruhe University and the late Prof. T. Murasaki (Osaka University) for the initiation of our research. This work was partially supported by the Humboldt Foundation in West Germany and the Grant- in-Aid for Scientific Research of Ministry of Education, Science and Culture in Japan. Numerical calculations have been performed with the HITAC-M180/M160 of the Tokyo

Institute,of Technology, the ACOS 900 of the Osaka University, and the HITAC-8700/ 8800 of Tokyo University.

References

K.Kasuya, K.Horioka, H.Oertel,Jr. and B.Schmidt: Numerical Estimations of Third Generation Gasdynamic Lasers by Bow Shock Waves, Recent Developments in Theoretical and Experimental Fluid

Mechanics, .editted by U. ~{ller, K. G.

t

Roesner and B. Schmidt (Springer-Verlag, 1979) p.166.

K.Kasuya, Y. Minami, K. Niu, H. Oertel,

Jr., B. Schmidt, K. Horioka and T.

published. Symbols

A : cross section of the

flow m2

b : constant

c : light speed m/ s

(Cv) : specific heat at con-

stant volume of j -species

including translational

and rotational modes kJ/kg.K

e : sum of translational

and rotational energy kJ/kg

evibi : vibrational energy of

i level or mode kJ/kg E : internal energy kJ/kg Em : maximum available energy kJ/kg F : flow rate kg/m2 .s G : gain coefficient 1 /m

G o : small signal gain co-

efficient l/m

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coefficient 1 /m

z

: threshold gain co-

efficient 1 /m

X

max

X

: radiation flux density !v/m2 : height of active medium

in laser cavity m

: width of active mecliurn

in laser cavity m

: length of active medium

in laser cavity

m

: pressure ~ / m

: laser power output W :-peak value of laser

power output

: energy loss by extraction of laser

power output kJ/kg - s

: respective mirror re- flectivity of laser cavity : gas constant : gas constant of j species kJ/kg O K : time s : translational temperature K : vibrational temperature of i level or mode K : flow speed m / s : volume of active medium = Z H ~ L M - Z w PI

: distance along the

nozzle axis T? : x of maximum G o P! : mole-fraction of j species : flow variable in general d, : photon density l/kg P : density kg/m

Y : ratio of specific heat

'c

-

1 : relaxation time of i

level or mode Suffixes

1 : state of low pressure

section of shock tube

2 : v2 mode of COz

1 2 : v l and v2 mode of COZ (See Fig.3

for below)

3 : v3 mode of C02

+ : v=l mode of N 2 I : 1st vibrational m.ode

II : 2nd vibrational mode vib : vibrational mode or level eq : equilibrium state

I I : state just behind bow shock

wave (vibrationally frozen)