The coupling of vibrational relaxation and dissociation

(1)

, I · • • • •

van KARMAN INSTITUTE

FOR FLUID DYNAMICS

- TECHNICAL NOTE

19

THE COUPLING OF VIBRATIONAL RELAXATION AND DISSOCIATION

by

M 0 GREENBLATT

RHODE-SAINT-GENESE, BELGIUM

(2)

von KAR MAN INSTITUTE FOR FLUID DYNAMICS

TECHNICAL NOTE 19

THE COUPLING OF VIBRATIONAL RELAXATION AND DISSOCIATION

by

M. GREENBLATT

(3)

(4)

FOREWORD

The WODk described here in was done in partial

fulfilment of the requirements for receiving the Diploma-of the von Karman Institute for Fluid Dynamics. Mr Greenblatt, an American stud~nt, obtained a Distinction G,rade, and was co-winner of the Belgian Government Prize, awarded annually to the student ranked second in the graduating class, for the year

1962-63

(5)

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ACKNOWLEDGEMENTS

, (

The author would like to thank especially Dr J.J. Smolderen for his help and guidance through all stages of the preparation of this paper.

Professor K.E. Tempelmeyer gave some valuable sugges-tions and insights.

The computing staff, Messrs Chanoine, Otte and

Laurensis, are to be thanked for all their patience in making the calculations.

Mlle Demulder helped in the prep.ration of the manuscript and did and admirable jobo

Credits go to my wonderful wife for her encouragement and understandingo

The author thanks the Republic Aviation . Corporation under whose sponsorship it was possible to attend the Center.

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CONTENTS ABSTRACT o • • • •

· .

. . .

LIST OF SYMBOLS • 0 • • • • • • • • • • • 1. INTRODUCTION • • • • 0 • • • • • 2. COUPLING MODELS • • • • • • 0 0 • • • • •

3.

INCUBATION TIME o • • • • • • • • • • • •

4.

NO COUPLING • • • • o • • • • •

. . .

.

5.

NO DISSOCIATION

· . . .

. . .

.

. . .

6.

INFLUENCE OF THE UNINHIBITED DISSOCIATION

RATE CONSTANT, (KD)VE • • • • •

7.

INFLUENCE OF INITIAL TEMPERATURE • • • • •

8.

FLOW CALCULATIONS • • • • • • 0 • • • • •

9.

RESULTS • • • • • • • • • • • • • • • • • 10. DISCUSSION OF RESULTS

. . .

lL. CONCLUSIONS • • • • • • • 0 • • 0 • • • • 12. RECOMMENDATIONS

· . . .

13. BIBLIOGRAPBY • • •

· . . . .

. . .

.

APPENDIX GRAPHS

. . . .

. . .

.

. .

P. 1 2 5 7 17 20 20 21 23 25 29 32 39 40 42 44

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1.

ABSTRACT

The coupling of vibrational relaxation and dissocia-tion behind strong shock wawes is discussed. The flow for various coupling models is calculated, with the CVDV model of Treanor and Marrone, giving the most reasonable description

of the process. The effects of various parameters is indicated, including the uninhibited dissociation rate, and the upstream Mach number, temperature and pressure.

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C D EV e e h K p K c K M N N o p 20 LIST OF SYMBOLS Latin constant

dissociation energy (ergs/gram) energy of vibration (ergs/gram) energy of vibration in level internal energy (ergs/gram)

base of Nap:~·rl:a-n.l logari thms

unspecified functions

\

/

average energy of vibration of recombin1ng 1molecules

(ergs/gram) Planek's constant

equilibrium constant equilibrium constant

Boltzmann's constant

dissociation

~a~e (cm3/moleo~sec)

· 6 2

reoomb1nation rate (om /moleo sec)

mass of one mole of .molecular oxygen

Mach number

number of lvevels of vibration

Avogra~o's number

pressure (dyne/cm2)

probability of dissociation from

v

th level

(12)

r T m T T v T T v t

u

x

vibration amplitude (cm) Tl

='è--~

(OK)

m v

translational temperature

(OK)

vibrational temperature

(OK)

T/T T /T

v

time (sec)

potential energy of vibration (erg/gram) velocity (cm/sec)

3.

fraction of vibrators in th vibrational level

colliding molecule

Greek

mass degree of dissociation incubation time (sec)

TV/T_f 2270/T 2270/T 2270

v

mean free path (cm) frequency of vibratian

density (gram/cm3)

relaxation time (sec)

-1

(13)

( )00 ( )e ( )VE

[

]

( ) } Subscripts, etc

free stream conditions equilibrium conditions

vibrational equilibrium cond1tions molecules/cm)

(14)

10 INTRODUCTION

The flow of a diatomio gas, sueh as oxygen, through a strong (M~~

6)

sho~k. wave is oharaoterized by downstream

temperatures approximatel y

M!/5

times higher than the free

stream temperature o As ths latter temperature is of the order

of 3000

K for most problems of present interest, the downstream

temperature will be high enough to excite the vibrational

degrees of freedom and eventually oause dissooiationo

Vibrational nonequilibrium oeours when the trans

la-tional degrees of freedom are at a temperature T

(T

'= the usual temperature with which the aerodynamioist is familiar) different

than the temperature T , oorresponding to the amount of energy

v

E , in the vibrational modeo Note that for a harmonio osoillator v E v hl> hl) e k'l\r_l

where h

=

Planckus oonstant, k = Boltzmannus oonstant and

V

is the oharacteristio frequency of vibration for the speoies

oonsideredo It aan be shown (Refo 1) that many more collisions

are neoessary for equilibrium in the vibrational mode than

either the translational or rotational modes, equilibrium ex~~ isting when T = T 0 Relaxation is the approac~ of T to T, the

v v

relaxation time and distance being the time and distanoe from

the onset of the nonequilibrium state, unt!l the equilibrium

one is reaohedo The relaxation is governed by a linear rate

(15)

dE v dt

wh~ah must be integrated with the equations of motion to determine the flow variables of interest.

( 1 )

Dissociation of oxygen will occur for temperatures, T

above 2000c_{K, since} _the _{amplitude of} _the_{vibration will be}

suf-ficiently high for collisions with other energetic moleoules to

impart enough energy to the vibrational mode to overcome the

energy binding the atoms togeth~r. The dissociation process is governed by a rate equation of the form

As for the vibrational mode, a finite t~me is necessary until

the equilibrium dissociation condition obtain~d

Various models have been proposed for the coupling of these prooesses, some of which are studied in this paper. The

equations for the onedimensional flow of initially undissociated

oxygen behind a shock wave are integrated simultaneously with

the rate equations for vibration and dissociation relaxation.

Graphs are shown of the var1ation of T, Tand _v

oG

downstream of the shook for the various modelso In add1tion,

calculations have been carried out w1th several experimentally

found dissociation rates to investigate the dependence of the

results on the rate constan~ used o Also, calculations were made

(16)

7.

2. eOUPLING MODELS

eVD

Hammerling, et al., (Ref. 2) have pointed out that a

gaswhose moillemules do not have the full vibrational energy

corresponding to the translational temperature will not diss~

ciate as readily as predicted bythe dissociation rate for a

gas in vibrational equilibrium, hereafter called

(KD)VE.

Accord-lnly, their expression for the dissociation rate K

D becomes

1)

where f3 :;J;ncreases from 0 to 1 as TV~ T • The analytical

expresslon for f3 i~ derived as follows:

We assume that dis~ociation may proceed with equal

probabi11ty from any level of vibrational energy in a collision having sufficient translational energy to cause dissociation.

The dissociation rate 1s thus taken proport1onal to the

proba-b1lity of being in a given level E~ multiplled by the fraction

of molecules Xy , whose translational energy 1s equal to or

greater than (D-E_{y ),} all this summed ov~r all the v1brational

l? As with all relaxation processes, TV never equals T, but

rather, approaches i t in the limit as X(distance behind the shock

or t(time),oo; thus, equilibrium is reached only at 00, but a

Tv

quasi-equili.brium can be defined to exist when

T

=

0.95 or

(17)

8.

levels, N. Thus, we obtain an expreBsion of the forms n-E)) X)/exp(-

TT)

N-l

L

)1=0

x

_)I th

where ~ is the fraction of vibrators in the

V

vibrational

level, and the exponential term is a Boltzmann factor. Montr~ll

and Shuler have shown that throughout the relaxation proceBs X~

is a Boltzmann distribution at the temperature TV. Introducing thiB into the above expression gives

(Kn)VE

l-exP[-N(ev-e~

ex p

8

_{v -} l

N exp(6 -8)-loexp9-l

v

e

h»

6

hi> h))

where = kT and v = kTv and

ïk

= 2270oK. It will be noticed

that when there is very li ttl.é vibration, T

«

T,

e

»8,

and

v v

Kn

=

(Kn)VEpe where N

~

1 and

e

is at leaJlt one-sixth, so

Kn is quite smaller than (KD)VE. Also, when vibration is almost

at equilibrium, Tv

~

T,

ev~e

,and Kn

~

(Kn)VE.

The relaxation of vibrational energy is governed by the linear rate equation suggested by Landau and Teller

dE v dt

=

E '-E v v

"t'

(2)

where E' is the energy in vibration if the molecules~ we re

oscil-v

(18)

of Tand Po Note that if T and

'11

are constant 1) , ~ is the necessary for E to become (1-

~)

of ER and is called the

v e v

relaxation time. The experiments of Blackman (Ref.

3)

give

followlng value for ~ , which is used throughout this paper

1 0.00162 exp(101.4T

-3)

=

sec P (p

=

dyne/cm ) time the

In ~he relaxation zone T decreases and P increases very slightly

so ~ls always increasing.

~ith this model for the coupling (hereafter called

the eVD model), we will have less dissociation in the beginning

of the relaxing zone where the vibrational relaxation will be

dominant . Thus, the temperature will be higher than otherwise

calculated as the energy taken by dissooiation will be less.

CVDV

Treanor and Marrone (Ref. 4) have extended the eVD

model to acoount for the faot that the dissociati~g vibrator

has more vibrational energy, on the average, than the other

vibrators. Acoording to their model, hereafter called the CVDV

model, there is a loss of E with each dissociation since

E,

v

the energy in vibration of the average dissociator, is greater

than E • As their first approximation the above authors take

v

1) These oonditions are met in a very highly diluted mixture

(19)

the 1055 te~m proportional to this d1fferenoe (Ë-E ) mult1p11ed

v

by the rate of d1ssociation

or

for a per v1brator bas1s.

Tak1ng aocount of the ga1n 1n v1brat10nal energy for eaoh reoombinat10n (8 small term oompared to the others except near d1socos1ative equilibrium), eq. 2 be~omes:

d'E v dt

=

EI-E v v + R

where G 1s the average energy of the reoomb1nat10n molecules, Ë and Gare expressed w1th the assumptions used 1n the eVD model. The probab11ity of a d1ssooiat10n oom1ng from the Vth level 15

D-~

P~

=

CX~ exp (- kID ) ( 4 )

Nc>l

where C 1s a constant such that

L:

P)1

-

=

1. The faot that .,.

~=O

e

f

C(V)

15 the assumpt10n that the d1ssoc1at10n may proceed w1 th equal probab1..l i ty from any level 1n a 00111s10n w1 th

suff1c1ent energy 1nvolvedo For a Boltzmann d1str1but10n 1n the

translat10nal and v1brational

E~ exp(~ - ) k~v Q(T ) v modes&

(20)

11.

where Q(T ) is the partition function for vibration at

temper-v ature T : v Putting (5) in

(4)

we get E)I C exp(- kT ) exp v Q(T ) v D-E~ (- kT) N-l

Summing over the ~leve1s, we get

L:;

P~ 1 or

1l=o 1 Letting T m = 1 T v EV E)) exp (- kT + kT) v 1 T ' this becomes

Putting

(8)

into

(7),

we get

E~ exp(- ~) Pl>

=~

E! -

=

L...J exp (---) )} kT_m El> exp( - kT ) m Q(T ) m (6)

(8)

(21)

12.

th

The energy removed by a dissociat1on from the

V

level 1s E but the average energy removed by the average (that 1s, we1~hted) d1ssoc1at1on 1s E, where

E P

)} ~

The expression for Q(T

M) depends on the model we have for the

oscillator; a s1mple harmonie oscillator 1s chosen, cut off at the d1ssociat1on energy D. lts potent1al energy curve is thus of the parabo11c form shown belowa

v

D

The number of levels, N, is taken as the Smallest .. D

integer greater than

at ,

the ratio of the characterist1c tempe-ratures of d1ssoc1ation and v1brat1on, the spaèing between levels is taken as ê'k

=

h • For oxygen, D = 59380oK,

$'

=

22700

K

and

N

=

27. T~us at any 'temperature' TM.

N-l

vet

=

I:

exp(-

T)

= ))=0 M

Né

Q l-exp(- - ) TM _._{( 10)}

(22)

Note that tbis reduces to the familiar expression

1

6'

l-exp(- - )

TM

~or N

=

000 Putting (10) into

(9),

we obtain

exp(~t

-1) M N6'k

-

_exp(~

ei

_·

-1) M (11)

The evaluation of G follows immediatelyo .At vibrational

equilibrium,e~QresStQn(3) must be zero o Since TV ~7T, E~

=

EV and

D

sb that G E, no matter what the temperature. Since equilibrium

. 1

means ~~O we can take the limit of the expression for Ë to

M obtain

G.

G

=

l im .l:..~o TM

(12)

This equation ~~d (11) must be multiplied by NoMo2 for an erg/gram (not particle) basis.

Equation (3) may be put in more familiar terms if we use K

D and ~, the ràtes of dissociation and recombination as defined below:

(23)

The oonoentration of oxygen moleoules is, in moleoules/

(13)

Likewise

( 0) _.

=

2 0( 0 _, N /M ₀ ₀2

{14)

Lettlng X = the ool11dlng moleoule for the ool11s1on, that 1s,

e1ther moleoular oxygen or atomio, we get:

It may be assumed that th,at the rate of removal of 02

moleoules b~ dlssoo1atlon is p~oportional to the number dens1ty

of oxygen moleoules availa~le for d1ssoo1atlon, (02)' and to

th~ number density of other moleoules whioh oan partioipate in a

,

oolllslon, (X). Thls lmpllcltly negleots the iaDe three body

oolllsions , as is usually done. We deflne the dissoolation rate

K

D, as the proportionallty faotor suoh that:

(16~ I

D

(24)

15.

Recombination is assumed to be a three-body process: a two-body collision provides nep mechanism for remov'a1 of the

1

excess energy and four-body collisions are very rare compared to three ,.o'ody collisions 0 The reaction is thus represented by

the equation

°

+

°

+ X

~

02 + X

Likewise, the rate of appearance pf 02 due to recombination is proporti'ona1 to

(~)

and' (0

f,

f0110wirlg conventiona1 chemica1 rate

for~Û1ations. The rate of recombination, ~, is defined as the proportionaiity factor in:

R

putting (13) , (14) and {15) in (16) and (17) , we obtain

1 d(02) N (°2) dt _D - KD (1+/X)

r

M 0 ₀₂ {18) 1 d (0 2 ,) 40<2 1+0< N.

2.

- K ( f _ o ) (19) (°2) dt _R

=

R 1-0( M₀₂ 1 '

When a wa11 is the third body, this energy causes the high temperatures associated with recombing surfaces.

(25)

Our expressi on for the vibrational relaxation becomes

putting (18), (19), (11) and (12) into (3)

\

shown that

dE

v

dt (20)

From equilibrium statistical mechanics, it can be

N K

K =~

c R T

o

(21)

where K is the equilibrium -constant. It can be calculated from

c

the equilibrium partition functions of the reacting gases. The use of these K vs T tables is not strictly valid in th~s appl~

p

cation, as pointéd out by Bray (Refo

5).

Various attempts have been made to correct this, but to date none seem completely

. ,

satisfactory (Bray, Ref.' --6) 0 According to Smolderen (Ref. 7),

i t is only a second order effect and should not affect the

results very mucho Use of (21) eliminates KR from equation (20) and K

(26)

17.

30 INCUBATION TIME

In a paper by Wray (Ref o 8), experiments are described revealing the existence., at temperature~aabove 110000K of an

incubation time At,;immediately behind the Shock,wave, during which disso.iation did not take placeo He therefore contends

that dissociation c$nnot proceed until "sufficient" vi~rational equilibrátion has taken place, the incubation time being the amount of time until "sufficient" vibrational energy extsts. If we refer to the terminology of the CVD model, we have here a

model which postulates that

except that here f3 0 f3

=

1

when t

<

I::i

t when t ~~t

whereas in the CVD model f3 is a continuous function of Tv and

T such that

f ₃-- 1

WraiVs experiments showed that ~t, divided by the relaxation time for v-ibration, varied. approximately inversely with the final temperature which a non-dissociating vibrationally relaxing gas would come to at vibrational equilibriumo Calling

this temperature T

f , he was able to derive an ~xp~ession _Tv for the .amount of vibra~ional equilibrium necessary, n

=

, f o r the

I T

(27)

18.

dissociation rate to be uninhibited by the r~~axation of vibra-tion, i.e. for t

=

At and f3

=

1. Although his calculations are for al highly diluted flow of oxygen and for some reason he uses a value of

9'

of 2228~K, the computations ou~lined below give very similar values of

Î

vs T

f as shown in graph 1. The deriva-tion is as follows:

The relaxation of the energy vibration

!p

a non-dis-sociating gas is governed by the equation

E (T ) t

v v

1 - E '(T)

=

exp (- T)

v

(22 )

where. we neglect the energy of vibrationat the beginning of the process, i.e. E (T

=

3000_K)

₌

_{O. From} _{statistical mechanics}

v E v

Ei"

_v . exp

(~{l)

:::

el

exp ('l\r-l)

When the oincubation period has been completed

t

=

Llt, T = T

f

and since _f3

=

1, dissociation is uninhibited

T T

1=

-Y..

=

v T T f (23 ) (24) and

(28)

Combining (22) , (23) , (24) and (25) we obtain

é'

~

-1

1=

~

rOg(l

+ eXP(Tf

~~))

_~26) T f 1-exp(--) 1;'

.

4t

.

The curves in graph 1 are equation

(?6)

with ~ taken from

graph 2, which shows the va1ue óf

~

found by Wray for the

(29)

200

4.

NO COUPLING

The assumption that no coupling takes place is of course a poor one, but nevertheless usefulo lts intèrest~lies in the fact that i t provides us with a lower limit for the temper~

ture drop in the relaxation zone. If we assume that both vibra~

tion and dissociation relaxation take place, each unaffeeted by the non-completion of the other process, the maximum amount of energy will be plaeed in these modea and the translational mode

(the temperature) will be drained more quickly than with any other model.

50

NO DISSOCIATION

This model will be helpful for two reasons. Primarily, _, i t will provide an upper ~imit for the temperature-distance varia-tion, giving, in conjunction with the no-coupling model, 'the e~ velope of the temperature variation. Secàndly, the results of the

,

ealculation are needed lor the eomputation of the incubation time in Wray's model. Speeifiea~ly, the temp~rature which would be ap-proaehed in vibrational equilibrium is T

f • Graph 2 can then be

At

used to find

-:e-

0 We evaluate1: , using for pand T averages over

the firstAX mean free paths, where we guess the value of ÄX. Wi th "C known we have At, and by using average values for the

vel-oeity, ean find a better value for~X. We ean 1~erate for any degree of aeeuracy, but one eorreetion has been found suffieient. Alternatively we can use graph 1 to find ~ for the partieular T

f •

Ax

is found directly as the distanoe whieh is neeessary for T in

v

the no-dissoeiatipn flow te reaDh ~Tf. The values of ~X are some-what different for the two methods.

(30)

6.

INFLUENCE OF THE UNINHIBITED DISSOCIATION RATE CONSTAN~, (K

D)

. VE

21.

Calculations were made for M~

=

14,

T~

=

300oK, for three different experimentally dete~mined values of (K

D) , the

un-, VE

~n.lbited dissociation rate constant, to determine its influence

on the results. Although a poor model for the coupling, the no-coupling one was used, as i t gives the largest variation of the temperature and shows the largest possible effect of the various

(K

D) _VE, a l l of which depend

on

the temperature only. Dissociation rates are of the form

(K D) _VE ~ ~ clT exp(--) T molec-sec where cl and c

2 are constants and D

U _{a dissociation energy} accor-ding to the Arrhenius formulation ~Rèf.

9)

a very good introdu~ tion to chemical kinetics for the aerodynamicist). In Ref. 10, Wood derives an expression for K

D from simple collision theory in which c

2

=

~.

Experimental determinations of (Ki D) VE have given data whose temperature dependence is somewhat masked by the

exponential factor 50 that valu~s of c

2 range from -2.5 to +;5. DI is usually taken as 'the diss~ciation energy, D (117,960 cal/ mole for oxygen). Shexnayder and Evans (Ref. 11) found a bet ter fit to their experimental data with an 'effect1vev _dissociation

(31)

Shexnayder

&

Bvans (Ref. 11)

Matthews tRef:'. 12)

Camac & Vaughan

(Ref.13 ) (K D) . VE (KD)VE (K D) _VE

=

1.5073 x 10-10 exp(-37,378/T)

=

18.6 T- 205 exp (-59380/T)

=

5.98 x 10-6T -l exp(-59,380/T)

These formulae are compared in Fig. 3 where the

quantity KV

10 9 cm~

molse-sec

has been p10tted ver~us temperature to avoid the strong temper~

ture dependence of the exponential termo Although KI d~creases with temperature while (KD)VE increases, the graph is nevertheless

useful in ind~cating the relative magnitudes of th rates at any

temperature. Note the good agreement of Shexnayderts and Matthew's

values for temperatures abuve 45000

Kw Below 45000

K the values

diverge but the dtscrepancy is not important since the major part

of the dissociation will already have taken plap,e, if Moo> 10

and P

<

10- 3 atmospheres v Camac and Vaughants value

i~

usually

00 .

between one half and one order of magnitude smaller than the ot~ers

~ small difference considering the usual range of values offered

for rate constants. We ean expect the highest temperatures and

lowest degree of di so iation from Camac and Vaughan's (K~)VE' w1th Shexnayder and Evans predi6ting values between the a~ove

and those of Matthewso FinallY1 at the lower temperatures the

(32)

70

INFLUENCE OF INITIAL TEMPERATURE

Some calculations were oarried out for M~

₌

15~1 with

T~ of

249°K

and also of

300

o

_K,

in, order to ascertain the

depen-dence on the initial temperatureo According to Ref. 14, free

stream temperatures much below or above these values do not

appear in the atmosphere below

70

kilometers. The lower· value,

although more representative of the atmosphere, is rarely used

by other authors (except Ref~

4).

Since

300

0

_K

is most of ten taken,

it affords a means for comparison and is therefore employed in

all other calculations.

Two mode-ls were chosenl the no-dissociation and the

no-coupling. The former gives the lowest temperature variation

(sinee the large energy drain of dissociation does not take place)

and the latter the highest of the models, so that we can put limits

on the effect o

For the first model we have TJrao)- 6 wi thin one

(supeI'-sonic) mean free path. At temperature levels such that T /T _v _Oe> ~

6

we can use the simple belation

E

=

constant x T

v v

The vibrational relaxation equation becomes, in nondimensional

terms

dT

_v

T-T

_v

(33)

24.

where ~ alone depends on the initial temperature. It is this

dependence that will be interest. During the first mean free

path we should expect differences also since then the full

expression for E and Et must be used f~r the relaxation

v v

process.

For the second model much different results are

expected. There will be an additional relaxation mechanism dissociation, which depends much more strongly on the

(absolute) temperature so that we can expect the comparison

(34)

25

8

0 FLOW CALCULATIONS

The integrated equations of motion for the one d~men

sional 1nv1se1d, non-heat-eondueting flow of in1t1ally

undisso-e1ated oxygen areg

equat1onof state

eonservat1on of mass l' _ p'U e-,Qonstant = p u

- \ \"0 - (28)

eonservation of momentum

:2.~

eonservat1on of energy f~+e (30)

where -eQP

and

R 2 Ro R

e

=

(1-CX)(3

2 M0 _o2 T + - -2 M_o₂ T + E v )+Ot(L 2 Mo -2. T+ D) and 0(.= grams

0:\

atomie oxygen

one gram of mixture of atomie

&

moleeular oxygen

Cbnservation of oxygen atoms

(31) net

wh1eh is s1mply the net SU~ of (16) and

(17)0

Assum1ng that the d1ssoc1at1on takes plaee at constanb dens1ty, wh1eh is val1d as long as

1.

dD(, ~

1.

d

P

(35)

26.

(at the beginn1ng of the relaxationo( is very small and at the

'tnd

~t

is

0 ),

we ean obtain (from

(13),

=

-El~mination of KR from

(31)

by means of

(21)

and use of

(27),

(29),

(30)

and

(31)

in

(32)

gives do(

=

dx ,.-f 4 _{(KD ,}

f'

T , 0<, u)

Vibrational relaxation is represented by an equat10n

of the form dE v dt dE

=

u dXv

=

fs(P,T,«)

where f depends on the model chosen.

s

('34 )

Equations

(27), (28), (29)

and

(30)

can be rearr~nged

to give

P,f,

T, and u each as explicit functions of the other variables and 0( and E • Bquations

(33)

and

(34)

have been

inte-v

grat~d simultaneoualy with the explicit expressions·for P,

f'

T and u on the

1620

IBM calculator at TCEA using the Runge-Kutta

1 .

technique • It was necessary to carry the calculations out for

1

(36)

only a few mean free' paths (about twenty for M~= 14) since the

effect of the vib~ation· energy on the coupling is important

mainly in init~ally 1nhibiting the dissociation rate, which is

the dominant cause of ths temperature dropo All the characteristic

eftects' of the coupling are nevertheless e~idento

The initial pressure ohosen is always ths same, that

corresponding to flight at 200,000 feet (61 km) aecording to the

NACA standard atmosphere, namely 2003 x 10-

3

atm or 1,54 mm HG o

This pressure is sufticiently low tnat the equilibrium value of

~ is one, and thus the equilibrium results are independent of

the initial pressure o As the nmaüm interest of the paper is the early part of the relaxation zone, where ths initial pressure may be still significant, only one value is used tor easier co~ parison of the caleu1ationso

Exoept for th~ two oalculations (Figs o 10 and lil used to inve.tigate the in~luence of (Kn)VE' Shexnayder and Evans! value has been used throughout~ Likewise, except tor Figso 12 and 14, all initial temperatures are 300o_Ko

To simplity the caloulations, certain simplirying

assumptions wer~ ~adeo First, the recombination term in the

correotion to the vibration relaxation equation has

~een

neglected. A term by term analysis by Refo

4

shows th1s te be not justified

only at the very htgh Mach numbers, very close to equilibrium,

conditions not oons1dared in this papero Secondly, electronic

(37)

1

influence is usually not considered large below lO,OOOoK.

Thirdly, translational and ~otational equilibrium are ass~med

but a reeent artiele by Ta~~ot (Ref. 15) suggests that above 70000

K rotational equilibrium may obtain in a distance compar- •

able to the vibratiop and dissooiation relaxation distanoe, Due to the pauoity of information this had to be negleoted. \

Also any ooupling of the rotational and vibrattonäL.lIB.odes due

to large amplitude osoillà~ions was not acoounted for. The

transport terms in the oonservation equations were omitted, and

this ean easily be justified tor the diffusion and viacosity.

The heat eonduction term is probably of seeond order, relative

to the other terms in the energy equation. However, in :the "l'(:,'

region where the temperature gradient is high, i t might be

interesting to include the effect as i t is here also that

vibra-tio~~aissociation eoupling is strongest.

1

Tempelmeyer 5ref. 5) suggests retaining all electronic

excitation terms such that ~~/k <5 T

MAX where ê~ is the energy in ergs of the particular level exoited. Such a procedure was not followed here primarily beoause most of the other authOD~S have not ei ther 0

(38)

9.

RESULTS

The results of the calculations are plotted in F1gs. 4 to 15. Flow variables downstream of the shock are plotted versus ~, the distance in cm ~vided by the (supersonic) mean

~ -4 ~

free path,~~ • For T~ = 3000

K and p~

=

2023 x 10 atm, ~=

0.03 cm. All temperatures have been non dimensionalized by

dividing by T~ and are represented by T

=

T/T and T

=

T /T_.

~ ~ v v

-It was not thought necessary to plot any flow quantities except T, Tv' and

oe

for two reasons. First, relative changes in pand

pare small compared to those of T and~, that is

\~I

so that distinctions could not be made as readily. Secondly, the best experimental techniques depend on the intrinsic variables for the energies of vibration and d1ssoc1ation, T and~. These

v

are the sodium D-line reversal method and the absorption characteristics of the reaoting species.

The translational and vibrational temperatures are drawn in all curves, w1th the degree of dissociat1on shown in some cases for an additional comparison. The key to the graphs is given below.

a. Coupling models

For these curves, M~= 14, T~

=

3000K and (KD)VE is

I as given by Shexnayder and Evanso In the graphs depend1ng on

(39)

30.

the incubation time model, the two ways are used to oompute

~X (the method of graph 1 g1vingÀX for graph

6

and graph 2

for graph

7).

Although Wray did not mention i t in his paper, i t seems w~hile to make the dissooiation oorreotion to the vibrational relaxation model and to oompare the results, w1th and without the EV oorreotion. This is denoted as the INCUBEV modele

In addit10n a oaloulation was made for an inoubation model suoh that (KD)VE was in effect only af ter TV beoame one half on the looal translational temperature. Although th1s model has no theoret10al or experimental justifioation, the author

was not aware of Wray's analysis yet and 1t seemed an interesting case. Fortuitously, the

Ax

so obtained is close to the values found by Wray's two methods o

Number 4

5

6

7

8 9 Coupling model no d1ssociation CVD, CVDV

INCUB (Graph 1) and INCUBEV

INCUB

&

INCUBEV (Graph 2)

INCUB (TV

=

.5T) No ooup11ng

b. Influence of the un1nhibited dissociation rate constant (KD)VE

The Maoh number and init1al temperature are 14 and 300oK, respectively, w1th the no-ooupling model

(40)

Number 9 10 11 31.

(Kn)VE

Shexnayder and Evans Mat.thews

Camac and Vaughan

c. Influence of initial temperature

The Mach number is 15.1 and the models as shown below

Number T Cou}21inS model

12 249 No dissoc1ation

13 300 No dissoc1ation

14 249 No coupling

(41)

32.

10. DISCUSSION OF RESULTS

A. COUPLING MODELS

Figurre 4 through

9

show the curves for the varlous coupllng models. One result 15 obvious and most lnteresting. Every model wlth only the 11near rate equation for vlbrational relaxatlon (that is, no correction for the dlssoclation draln) shows the "overshoot" of the translatlonal temperature by the vibrational temperature. The mathematical explanation for this physical impossibility (for flow behlnd a normal shock) is that the coupling model used is not "strong" enough and the two

relaxation processes are s t i l l somewhat independ~nt. Af ter the relaxation has proceeded to a certaln distance (between eight an~ fifteen mean free paths), Tand TV may be sufficlently close that the temperature drop AT, due to the increases in dissocl&-tion and vibradissocl&-tional energy over a small distance

Ax

may be greater than the temperature difference T-T at x,

v

The important effect is that of dissociation on the vibrational relaxation. If we ta~' the ratio of the first two terms in the second membel' of equatlon (3), we get

N E (TM)-E (T ) o ( ..1) V V v -KD

P

M

1+11\ _{E (T) -E (T ) "'-KD} 02 v v v T 2 v - R

T:"T

_v

=

where all vibrational energies are denoted as functlons of their respective temperatures and taken proportional tothem

(thls is not justified in the first mean free path but the results are the same). During the flrst few mean free paths

(42)

33.

linear rate equation for vibrational relaxation (1) is sufficient.

As to

T 7 T according to (1), dissociation will v

a considerable extent, although f3

=

00313

begin to proceed 1

only • T~us there

will be large temperature drops due to dissociation and R will

increase not only because KD ~ (K~)VE an~ Tv

be ca use T --:;;. T • For large enough val ues { of

increases but also

v dEy

K

D, \RI can become greater than one, ~ is

T, and therefore negative and T

v

falls, giving curves of the following form

""--<=---'>~ X

This overshoot by T of its equilibrium value may seem peculiar

v

physically but is easily ascertained experimentally (see

Recommendations). Again the formulation for (KD)VE is critical.

In the light of the above remarks, graph

5,

shàwing

the curves predicted by the eVD and the CVDV models, is seen

to give the most reasonable results o The E correction eliminates

v

the T

>

T overshoot and the temperature drop is smooth. The

v 1

Note that Camac and Vaughan give a (KD)VE which is ab out 0.3

of Shexnayder and Evans'. This makes all these corrections

somewhat meaningless unless the value df (KD)VE is more accurately known.

2

Treanor and Marrone have shown that there is no such overshoot

for M.

=

801 and T.

=

249°~ . which· gives a temperatu~e immediately

(43)

34.

TV >(TV)EQ overshoot is present. Both temperatures deqrease continuously, as would be expected, eeaching an equilibrium value, T

EQ on1y at infinity. The value of Te can be computed

independent of the various relaxation processes, and computa-tions by Treanor and Marrone suggest that wellover one hundred mean free paths may be necessary for T to be within 5% of T •

e

The most outstanding feature of the INCUB (Graphs

6

and 7) model is the prediction of a shapp kink in the T curve.

Howeve~, we expect this to be discontinuous as the amount of energy absorbed by dissociation is much greater than that by

vibration. There being no .dissociation for X

<Llx,

the T curve

is higher than either the CVD of CVDV curves. We find also that even though the INCUB models have temperatures which are 10000K greater than those of the CVD at four mean free paths, only

five mean free paths later they:: are lower. This is because

d~ssociation is uninhibited in tne INCUB model, once started.

This gives ~emperature drops large enough to cause the T ~ T

v overshoot characterlstic of all models with no dissociation correction in the vibrationa1 relaxation equation.

The tab1e below shóws some interesting points. . \' ,. ,

' . ' ~! • 1 ',' Distance (mean free paths)

Model 9 18

CVD 25.4 18.1

INCUB (1) 25.1 18.8

INCUB (2) 23.7 18.4

(44)

35.

First, the different INCUB models will give different

results near

Ax

of course, but little distance is needed for the

temperatures to come quite close together. This is due to the

very large temperature drop associated with the uninhibited

dissociation rate which wipes out the effect of different ÄX.

The CVD model approaches the no coupling model more quickly

than the others primarily because its f3 is increasing.

Making the correction to the INCUB model corresponding

to the CVDV improvement on the CVD model gives interesting results. When K

D becomes (KD)VE'

IRI

is not necessarily close to zero.

Graphs 6 and 7 (dotted lines~ show that Tincreases for some

v

short distance and then falls. This occurs because of the sudden

dissociation caused in T o. Therefore

IRI

ro,.> (K

D )VE

~~'J:v

incr.eases

abruptly. As R passes -1, there is the drain on the vibratbnal

energy and T falls. The INCUBEV model will therefore give a

v T

v «Tv)INCUB just as (Tv)CVDV

<

(Tv)CVD' Likèwise, each model

with a corrected E

v has a temperature greater than a non-corrected

model. However, the CVDV model has less {)( at any point than

the CVD where as just the opposite is true for Wray's model~ even

though higher temperatures result from the E correction, ~ is

v

higher. There seems to be only one expl~nationo Making the E v

correction to Wray's model gives a T which is l7000

K lower at

v

X=ll (for both graph

6

and 7) so that there is a good deal of

energy available to the translational modeL Hawever, the latter

increases by only 200o

K. Obviously some of the available energy has gone into more dissociation. Note that in the CVDV model at

X=ll, T is 7200_{K lower} _{than in}_{the CVD, but T is 390}0

K higher.

v

The dissociation is less however since the larger (T-T ) causes

v

(45)

36.

In figure 16 is shown the T curves for all the coupling models and in figure 17 T • Figure 18 is the degree(' eH

dissocia-v tion.

B. INFLUENCE OF THE UNINHIBITED DISSOCIATION RATE

Figure 19 shows the temperatures in the relaxation zone for the no coupling model for Moo

=

16 and Toa

=

300oK. As expected, the curves resulting fro. SheKnayder and Evans l

value of (KD)VE and from Matthews agree better than 500oK. However, Camac and Vaughan's curve ~s around 15000

K higher. Figure 20 is the predicted degree of dissociation. Again agreement is good between Matthews and Shexnayder and Evans, with Camac and Vaughan's values being ab out one half. Note that as the tempe-rature decreases K'Shex~ K'Matth and that accordingly TShex~ TMatth and~Shex~ ~Matth. There is only one conclusion: any

experimental determination of incubation times or other coupling effects must involve as well an investigation Df the dissociation rateo Calculations based on experimental data from apparatus

and the dissoc1ation rate from some other ean be highly question-able and would be expected to yield results like those of,

figures 19 and 20.

Shexnayder and Evans' value was used throughout the rest of the .calculations as i t is based on a wider temperature range .

(46)

-37.

C. INFLUENCE OF INITIAL TEMPERATURE

Curves 12 and 13 show the no dissociation "medel" for

T oo = 249°K and 300oK, respectively. As expected the curves are

very similar suggesting that they m~ght be the same if the

ordinate

~

=

-k-

were "stretched" to include a dependence on

';\I"" X

T cP (other than through ~ = ~oo alone). It can be s een tha t

the non dimensional equilibrium temperature will be the same for

each. A "quasi-equilibrium", defined as T

=

.95

T for example,

v

is reached sooner by the higher temperature condition of course.

The no coupling model is shown in graphs 14 and 15.

Now there does not appear to be simple way of making the results

similar. Any stretching of coordinates would necessarily involve

T also. Thus, whereas ,the non-dimensional equilibr.ium conditions

will be the same for all T (as in the nO ,di&sociation model),

the relaxation zone dependence is stronger when dissociation is

involved. However, if we consider the absolute temperatures

rather than the non dimensional, we find results are quite close.

At ~ = 5, they are within 3000

K and at

r

= 10, 100oK. We see

again how quickly the diasociation ean make tne ~eff,ect of '

1

diffdndnt initLalconditions smalle

Calculations were also made with an initial pressure

1

Compare the results fromWray's model for the different

incuba-tion times. Though dissociaincuba-tion began over one mean free path

la ter.- in graph )6 than 'i n graph 7, the flow variables are qui te

(

(47)

38.

of one half the usual one but the results carried to the fourth significant digit showed no change. The effect is not unexpected as pressure is not an important variable, if sufficiently low.

(48)

39.

11. CONCLUSIONS

Of the various coupling models, the CVDV seems the best. All incubation time models give physically impossible kinks in the temperature curves. Any model without the

dissociation correction to the vibrational relaxation equations

gives a T > T overshoot which is equally incorrect for a one

v

dimensional flow behind a shock. The possibility o~ a T ~ (T )E

v v

overshoot is shown.

The effect of the uninhibited dissociation rate is large and (KD)VE must be determined to a greater degree of accuracy.

(49)

120 RECOMMENDATIONS Theoretical

In any further analysis of this type there are certain refinements which might be importanto First, the extent of

electronic. excitation and ionization may be sufficient to lessen the energy available to vibration and dissociation 0

Secondly, there may be some rotational coupling with the vibration and an overlapp~rig of the relaxation zone of each. This would make a large difference in the results, if the shock were :strong o Also, heat conduction could be included as a

per-turbation. All these effects have.!~~en neglected but their omission should be justified by fuller calculations. Certain assumptions would have to be checked by experim~ntal agreement

"

for example that the dissociation can proceed from any level with ~qual probability o~ that unharmonic oscillatiQns can be neglectedo

,

Experimental

Figures 17 and 18 show the variation of T and.~ with

v

~o It is immed~ately obvious that a good test~ng procedure for studying the eoupling would involve a measurement of the degree of dissociation in the first ten mean free paths and T a f ter

v

the eight (for the case of M~

=

l~ and T~

=

3000

K)o Measuring

0<

af ter ~= 10 Qr"T

v before

t>

=

8 would permit little evaluation of the models all of which give somewhat similar results in

(50)

41.

measurements. The sodium D-line reversal tecpnique could be used to measure the Tand a light absorption technique for ~ •

v

A suggested schematic diagram is given on Fig. ~l.

To demonstrate the exiten~e of a Tv. > T

VE overshoot, the sodium temperature can be kept at a temperature between

the oalculated maximum Tand the equilibrium T • The light

v v

received by the photomultipliers is somewhat insensitive to the absolute value,IT -T

I

in the particular mass passing the

s v

sodium source, but gives aocurately the sign of T -T • Thuà

s v

the record shown on the oscilloscope will verify the exi.t~nce

of the Tv> T

VE overshoot by giving a curve of the following type

Accurate determination of the T vs time curve is not

v

possible with this method except if many tests are taken. The reversal method Can narrow down the range of va~ues of T but

v

this requires a high degree of reproductibility of runs.

The determination of ~ by characteristic absorption techniques is well estab11$hed and reliable.

(51)

130 BIBLIOGRAPHY

10 HERZFELD, KoF.g Relaxation phenom~na in gases, in

ftThermodynamics and p~ysics of matter", ed. Rossini, Oxford, 1955.

2. HAMMERLING, Po, TEARE, J.D. & KIVEL, B.I Theory of radiati.on from luminous shock waves in nitrogeh.

The Physics of Fluids, vol. 2, n° 4, po 422, 1959.

3.

BLACKMAN, Vog Vibrational relaxation in oxygen an~ nitrogen. Journalof fluid mechanics, vol. I, n° 1, 1957.

40 TREANOR, CoE.

&

MARRONE, P.V.: The effect of dissooiation on the rate of vibrational relaxation.

Cornel1 Aer. Lab. Rep. QM-1626-A-4, 1962.

50 SMOLDEREN, JoJo, BRAY, KoNoCo, TEMPELME~ER, K.E.: Physics of gas es 0

60 BRAY, K.NoC. g. Private oommtinication.

70 SMOLDEREN, J.Jo& Private oommunication o

8_{0 WRAY, KoLo8 A shock tube study of the coupling of the}

02- Ar rates of dissociation and vibrational relaxation. Avco-Everett.Research Labo, RR 125, 1962.

(52)

90 PENNER, SoSo~ Introducti6n to the study of chemica1 reactions in flow systems.

Agardograph 70

100 WOOD, G oP08 Ca1cu1ations of the r.te of therma1 dissoci~tion

of air behind normal shock waves at Mach numbers of

10, 12,. 14 0

NoAoCoAo TN 3634, 19560

110 SHEXNAYDER, CoJo

&

EVANS, J oS08 Measurements of the

dissocia-tion rates of molecular oxygen o

NASA TR R-108, 19610

120 MATTHEWS, DoLo8 Interferometric measurement in the shock

tube of the dissociation rate of oxygen o

The physics of f1uids, vo1 0 2, n° 2, 1959.

130 CAMAC, Mo

&

VAUGHAN, Aog Oxygen vibration and dissociation

rates in oxygen-argon mixtures o

Journalof Chemica1 Physics, vol. 34, 19610

140 TEMPELMEYER, KoEo8 Introduction to low-density gas dynamics.

TCEA eN 34, 19630

150 TALBOT, Log Survey of the shock structure prob1emo

(53)

APPENDIX 44. "

The flow ahart for the aaleulations 1s g1ven below.

ACCEPT INITIAL VA LUES T ooI MooI (KD)VE' MODEL

COMPUTE OTHER INITIAL VALUES Pm' Ev(Tm), À m

a _m= 0 B = 0 a = a m E a EV(T .. ) V Begin 1terat1on

I

Compute P, T, Tv, T, P

I

l

D1ssoe1at1on be gun yet ?

_J

I

yes no

I

Compute (KD)VE

I

_d8~da 0 1 Wh10h model?1 R a 0 CVD INCUB CVDV INCUBEV NO COUP Compute f 3'KD

I

tKD=(KD)VE

I

I I

Wh1eh mode 1 ?"II Wh1 eh mode 1 ?J CVD CVDV INCUBEV INCUB NO COUP R =

ol

Compute R

~

I I Compute da

·1

dB = KD

.

dE E'-E Compute -...:!.. dB = u ~7À:(l+R)

I

a = a _m+ t da dB _dB dE E = E + ~ -...:!.. dB v vm dB Iterate 3 t1mes(Runge-Kutta) 1,2,3

J

\4

I

Compute x,a ,E~l

I

x= 1nteger ? yes

I

no

(54)

1.0 .' .9

\

; .. FIG. 1

_'1

v s Tf .8

\

II WRAY Dilute O 2 S':2228°K

°

THIS REPORT Pure 02 S':2270"K

.7

\\

Tv ~=-Tt .6

r\

\

.5

\.

.4

"

~

-

--..:

""""

---a .3 6 8 1 0 12 ·14 16 18 20

Vibration Equilibrium Temperature xlO-3

(OK)

Tt

10 FIG.2

-

4t vs Tt 'r 2.6 22 6t 1: 1.8

\

•

1.4

1\

1.0

\

.6

~

I---,2 5 7 9 11 13 15 Tl 19 21

(55)

FIG.3 K' vs T

,i

~~~.~

~.fjX

(

\

\. \.

K'

~, _'1\

'"

~~

'\

~

~ "ilI r"..:---... ""'I!~ I: ~ ...

"è

,...,., ...

r--,

~ ""'-_N ""'"'1 _~

~

N

r--~

_r--

~

_N

10 3 5 6 7 8 9 10 11 12

TEMPERATU RE

x

1Q-3(OK) 40

I

_I

~

FIG. 4

T,

_Tv

vSJ3

NO 0155 M.= 14

r'---r----

---35 30

-

---

~

/

,,/ 25 20

I

V

15

/

10 5

o

2.5 ·5 7.5 10 125 15 17.5 20

(56)

30

~

FIG.5 I

- I _ I

T, Tv vs

J3

,CVO-,CVOV-_·· M..=14

I

"

~

"

-...

~

_--

---40 35 25

V--

_~

-

,..-V---

----

1 - - - -1- _ _ _

-~

/

" . ~

V

20 15

/

10 5

o

25 5 7.5 10 125 15 17.5 20 40

~

FIG.6 T. Tv vs I

-

~

13 I

INCUS -

I

• INCUBEV-·--

J

_M.o=14I

r---.-.

GRAPH 1

r\,

~

35 30

~

...

,

::s

l'~ 25

/ :

-

~

~-

-

r-/

---

--

-

--..:::-

-

--

~

--

---

_--

---

-V

20 1 5 10

!

/

, 5

o

2.5 5 7.5 10 125 15 11.5 20

(57)

40

'

~

I I I I I

. I FIG.7 f I Ty ySP INeUB - ,INCUBEV···· M..=14

~

GRAPH 2 c;

\

3

~

'\

~, 30 25 .~ ...",

~

-20 15

/

---~

---V

/

10 5 25 5 7.5· 10 40 _I _I

~

FIG.8 I

_T

I

T

y yS

.P

~

35 30 25 20 15 10 5

o

I

/

2.5

\

l\.

~

"""

/

5 7.5 10

-...::.::::.:

_~

-

-..

-

-::.:::.-:::::

r=----

-

---

---125 15 17.5 20 I I _I INeUB 'fy =f/2 M..=14

...

---

--

----

-

----125 15 17.5 20

(58)

40 I_- I I I

\

FIG.9 T,Tv vs

ft

NO COUP M..= 14

\

35 30 25 20 15 10

s

o

35 30 25 20 15 10 5

o

\

~

"

_~

I'---

---..---

~ /

V

/

25 5 7.5 10 12.5 15 I

_I

_I I

\

FIG.l0 T,'fvvs

J3

NO COUP

_M..=

14 KOVE

=

MATH

\

"

~

----

_----

r-

--V

V--I

2.5 5 75 10 12.5 15

-17.5 20 17.5 20

(59)

35

'"

! - ""I I I -'

FIG.ll T. TyYS

P

NO COUP M.= 14

' "

Ko\l~

&v ~

~

r---

r--

r--30 25

-,...

--/

~

/

V

20 15

/

10 5 2.5 5 7.5 10 12S IS 17.5 20 40

~

FIG.12

T, T

y YS

J3

I NO 0155 M_ ~15.1

i'---t'..

_-

T_. = 249°K

r--

t--· 30

~

....--~

.

~

/

V

.

V

35 25

/

20 IS 10

/

o

2.5 5 7.5 10 12.5 15 17.5 20

(60)

45 40 35 30 25 20 15 10

o

40 35 30 25 20 15 10 5

o

"'"

I 1 . 1 .

I

FIG.13

T,T

_{v vsfl} NO 0155 M_=15.1 ... T_=300oK

---

~

-/

V--. /

/

25 5 7.5 10 12.5 15 17.5 20

\

FIG.14

1 T.T

_v

V~J3

NO COUP

J

1 M..,= 15.1 I T_ = 249°K

~

"-~

_i"---... -...

---

----

---l---

-V

/'"

V

/

2.5 5 7.5 10 125 15 17.5 20

(61)

40 1- _

I

I 1

35 \

FIG.15 TJT

_{v vSJ3}

NO COUP M-= 15.1

Toe»: 300

0

_K

\

30

\. I

~

r---~

~

---

~

f-,..

/

-/

25

20

15

10 /

o

2.5

5

7.5

10

125

15 17. 5

20

(62)

40 35 FIG. 16 T vsJ3. M.= 14 FIG.17 Tv vsj3 M..=14 35 ₃₀

-/

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NO DI 5 30

/

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55

/

25

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rlNCUB 25

I~

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_____ t

vov

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--,- _{.... - _ .... INCUBEV}

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f

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I

10

o

5 10 15

o

5 10 . 15

(63)

.3S.----,---,---,----,---,---r----,---. FIG18.,(vsj3 M..=14 · ~O~--~---+----~----~----~--~~=-~----~ .15 1---4---/---+---b:1'-~,.-f---~----t----~r----.1 0 1---JL-f----~~d_-1---~----+---+---___1---1 .05 1---.,I~---+_+_--____1iL-._I_-t---+---+---t----___t---1 o~

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o

2..5 5 7.5 10 125 15 17.5 20 35 30 25 20 15 10 5

o

'0

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I

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FI G.19 i' vSJ3 NO COUP M_= 14

~

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t'---~

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r---

---

C &V

t---=

::::::::::::

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SHEX MATT 2.5 5 7.5 10 12.5 15 17.5 20

(64)

~x ~w <{:I: ~tJ)

...:r

-

11

:J

a..

:::>

8

0

z

co..,

ti)

>

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.

(!)

u.

o

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.

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al

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.

lt')

o

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r--!

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r--:

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(65)

)

I

PHOTOMULT-

PHOTOMULT-S

EJ

IPLIER IPLIER T _v a

I

--...

_-_SHOCK r l SODIUM , ~ONOFREQUEN. RESISTANCE CY' THERMOMETER EXCITER EMITTER 'TRn'WER

(66)

~

V.K.I. TN 19

von KARMAN INSTITUTE FOR FLUID DYNAMICS, 1964.

THE COUPLING OF VIBRATIONAL RELAXATION AND

DISSOCIATION, by M. Greenblatt.

The coupling of vibrational relaxation and dissociation

behind strong shock waves is discussed. The flow for

various coupling models is calculated, with the CVDV

model of Treanor and Marrone, giving the most reasonable

description of the process. The effects of various

para-meters is indicated, including the uninhibited

dissocia-tion rate, and the upstream Mach number, temperature and pressure.

V.K.I. TN 19

description of the process. The effects of various para-meters is indicated, including the uninhibited

V.K.I. TN 19

~ISSOCIATION, by M. Greenblatt.

description of the process. The effects of various para-meters is indicated, including the uninhibited

V.K.I. TN 19

description of the process. The effects of various para

-meters is indicated, including the uninhibited