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Report No. 123

L

THE COLLEGE OF AERONAUTICS

CRANFIELD

THE TWO-DIMENSIONAL FLOW OF AN

IDEAL DISSOCIATING GAS

by

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.

:

:

L

T H E C O L L E G E O F A E R O N A U T I C S

C R A N F I E L D

The Tvvo-Dimensional Flow of an Ideal

Dissociating Gas

b y

-J. W. Clcever, B.A., D.CAe.

SUMMAEg

By neglecting viscosity, heat conduction and diffusion, a method for investigating the effect of dissociation on the two dimensional flow of a high temperature supersonic gas stream has been examined. The ideal

•oxygen-like» gas introduced by Lighthill (1957) has been used and in all cases the internal modes of the molecules are assumed to be instantaneously adjusted to be in equilibrium with each other.

A brief introduction to the ideal dissociating gas and the rate equation is given and then the partial differential equations governing the motion of this ideal gas are treated by a standard characteristic method. Due to the entropy production associated with the chemical reactions, analytical

solutions are not possible, and a numerical step-by-step method is used to obteiin a solution.

As an application of the method developed the flovT field around a sharp comer of an ideal dissociating gas is examined and a limited investigation of the free stream conditions and expansion angle on the resulting relaxation zone has been given.

Based on a thesis submitted in partial fulfilment of the requirements

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GOIjggffTS

Suramaiy

Contents

List of Symbols

Introduction 1

The Ideal Dissociating Gas 1

The Rate Equation 2

The Two Dimensional Flow of an Ideal

Dissociating Gas 4

4.1. Integration of the characteristic

equations 7

The Application of the Two Dimensional Theory

t o the expansion of an I d e a l D i s s o c i a t i n g Gas 13

Conclusions 14

Acknov/ledgement 15

References 13

Ajrpendix A 17

Appendix B 19

Figures

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LIST OF S::]iBOLS

a» Frozen speed of sound

a Equilibrivim speed of sound

A Atom

A_ Molecule

O Concentration of atomic species in mass fractions

D Dissociation energy per unit mass

e Internal energy per 'unit mass

h Enthalpy per unit mass of molecule

k_ Specific reaction rate coefficient for dissociation

kp Specific reaction rate coefficient for recombination

K Equilibrium constant

K

Net mass rate of production of atomic species

m Mass of atomic species

M Equilibrium Mach Number

M|p Frozen Mach Number

n Normal co-ordinate in natural co-ordinate system

p Pressure

R Gas constant for molecular species

s Streamwiso co-ordinate in nat^ural co-ordinate system

S Entropy per unit mass

T Temperature

GL Characteristic temperature

t Time

X Space co-ordinate

(5)

L i s t jaf ggnbols (Continued

p Density

P_^ Characteristic density for dissociation

T

Characteristic chemical time

6 Direction of velocity in natural co-ordinate system

IJ

Chemical potential of atomic species

U

Chemical potential of molecular species

S Parameter of char'acteristic curve measxared along the curve

A^i Increments of length along chara-cteristic lines

o* Thermodynamic variable defined in equation (2l)

1 Distance measured from comer of the v/all

P P Distance between the r and s points

Subscriipts

o Free stream conditions

e Equilibriijm conditions

(i

gg.

^ e t o ) Points in flov; field

W Conditions at the wall

a atomic species

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1

-1 , ^Introduction

In the hjrpersonic flight regime high stagnation enthalpies are realised which are sufficient to dissociate the diatomic constituents of air. Generally the problem of including the higli energy changes associated with dissociation into aerodynamic problems is difficult owing to tlie

con^jlexity of the thermod;ynamics. However by by postulating an idealised diatomic gas, Lighthill v/as able to formulate the equations govsming such a gas in a simple form, and yet still retain the main characteristics which a real diatomic gas would exhibit at high temperatures. This ideal gas provides an excellent means of investigating the effect of dissociation in high temperature gas flows,

If the 2reaction rate of tlie chemical process is finite then for any deviation from a state of equilibrium there will elapse a finite period of time before a new state of equilibrium is achieved, Y/hen the time to reach equilibrium is comparable with the time it takes for a particle to pass through the flow, then areas of the flow field will arise in i*ich

non-equilibrium states are encoxmtered. The object of this work is to investigate these non-equilibrixjm regions or relaxation zones, •v;4iich arise v»rhen a high temperature reacting gas is expanded roxmd a sharp comer. To do this, plane flov"/ has been considered.

The problem has been sin-iplified by neglecting viscosity, heat conduction and diffusion, and by assxming that the translational, rotational and

vibrational degrees of freedom are always adjusted to be in thermal equilibrium with each oth.er. The equations of motion governing the flow of an ideal ' oxygen lilce' dissociating gas have been f ormxilated and treated in a manner similar to that suggested by Ghu (1957). Due to tlie entropy production associated with the chemical reactions sinalytical solutions are not possible and thus a nimierical step-by-step solution similar to that encountered in the familiar method of characteristics has to be employed,

2, The Ideal Dissociating Gas

The dissociating gas is considered to be composed of symmetrical diatomic molecules and is ' ideal' in the sense that for a specified

temperatxjre range the thermodynamics can be simplified gx:ea.±ly by noting

that a groxjp of temperature dependent terms arising in the lav/ of mass action may conveniently be talcen as a constant, Lighthill (Ref.1l) first noticed this and foxmd that, for a diatomic gas T/hich is in chemical equilibrixam, the law of mass action niay be ivritten as

"where o is the concentration by mass of the atomic species, D is the dissociation energy per imit mass, R the gas constant for the molecxilar species, the suffix 'e' iinplying that a state of equilibrium exists,

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2

-The c h a r a c t e r i s t i c density P ^ , see R e f , 1 1 , i s s t r i c t l y a function of

temperatixre, but f o r the temperature r:aige 3000 K < T < 7000^C tlie

v a r i a t i o n of P_ i s small for oxygen and thus may be considered to have

a constant value of 150 gn/co,

I n p l i c i t i n the assi:iiiption t h a t P^^ i s a constant i s the requirement t h a t

tlie v i h r a t i o n a l degrees of freedom of "tlie molecxxLe are alv-'-aijrs j u s t half

e x c i t e d even a t low temperatures. This enable tlie i n t e r n a l energy t o be

w r i t t e n i n the following sitnple form,

e = 3RT + Dc " (2)

and thxis the enthalpy i s

h = ( 4 + c) RT + Dc. (3)

Finally, the equation of state may be deduced by considering the partially dissociated gas to be an assembly of tvra independent perfect gases and is

p = p(l + c)RT (4)

As L i g l i t h i l l p o i n t s out, t h i s i d e a l a s s o c i a t i n g gas y.dll only be

r e p r e s e n t a t i v e over the teniperatxare range 3000 A! <T< 7000 Iv. Outside

these l i m i t s i o n i s a t i o n and v i b r a t i o n a l e x c i t a t i o n Yrill becoiiie inroortant

and t h i s has not been allowed f o r . However i n the examples presented

below, since only small expansion angles a-re considered, the L i g h t h i l l

approximations are adequate,

3 , The Rate Equa.tion

By assuming t h a t the molecular and a t a n i c s t a t e s of the assembly

are independent, the equation describing tlie r e a c t i o n may be T,xitten

formally a s ,

A + X ^ 2A + X (5)

Trdiere X represents a third "body, which a'ji collision v/ith a raolecule or an atom may act as an heat sink or soiurce. For a pure diatomic gas this third body may be an atom or a molecule. Ic^ and ICp are the specific

reaction rate coefficients for dissociation and recombination; both being

dependent on local tenperatiire and concentration in general, 1!hey tiay be

obtained from statistical considerations, or from experiiiient,

From Ref ,14 the net rate of production of the atcanic species may be written as,

4 ^ = 2kj^(A,)(x) - 2 1^(A)'(X) (6)

(8)

-3

-It is more convenient to express the concentration in mass fractions,

Noting that (A) = ^ , (Ag) = " ^ ^ and that (X) = ^^-^-^

then equation (6) becomes

ft = F (^ (^ -

^'^ - >^

( i j

(^ ^

°)=^ (V)

dc

At eqxxilibrium " ^ = 0 and therefore

e\ / e

k^ \ m y i 1 - c_ j ""e

K , being the equilibrium c o n s t a n t ,

= K (8)

As the flovTs discussed l a t e r are of an expansive natxire i t i s

convenient t o express the r a t e equation i n terms of Ic, r a t h e r than kj,^,

Thus i f the concentration c i s evaluated a t l o c a l p and T, eaxiation (7)

e ir- 7 ^ \ ' >

becomes

dc

dt

p^ kf^ (1 + °)

m 1 - C \ 2 2 = » — — = - \ C - C Jt 2 1 e

1 - c /

e -^

( 9 )

I t may be n o t i c e d t h a t r = m^ / p ^ kp(l + c) has tlie dimensions

of time and may be taken as a c h a r a c t e r i s t i c chemical time. By considering

a small deviation from, equilibrixra xmder conditions of almost constant

pressxare and ternperatxire then equation (9) niay be wTitten i n tlie following

approximate form,

1 dc' 1 / ^ °e

1 ( = „ ^ J = c o n s t , (10)

e \ 1-G2/

e \ 1-c2

e

Here c' is a small deviation from the initial eqxailibrium conditions e,

Thus it is seen that T can be talcen as proportional to the time which it

takes for a small deviation of the concentration to retiarn to its initial equilibrixmi value and may thus be used as a measure of the rate at which the chemical reaction proceeds tovvards equilibrixjm,

Yftien r ^ 0 then c -» o and equilibrium, flow exists and as

r -» 00 c will remain constant and the flwY is called frozen,

The recombination rate coefficients are difficxüt to determine and Lshed results suggest that it lies v/ithin the

for 03(ygen (iCp. measxared in mole" . cm , sec" . ) .

published results suggest that it lies v/ithin the range JIQ^^ < %. <-io^^

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k.

-Recent i n v e s t i g a t i o n s Ref , 1 3 , g i v e k^ = 0 , 8 x 10^^ a t a tenrperEtture of 3,500 K v/hich i s i n c l o s e agreement \ , l t h lief , 2 .

h-» The Two DjjTgjisionaJ^FJLCTV; ^ f _ a n _IdeaJ- D i s s o c i a t i n g Gas

By mailing t h e s i r i p l i f y i n g assxjmption t h a t v i s c o s i t y , h e a t condxxjtion and d i f f u s i o n may be n e g l e c t e d , t h e g e n e r a l eqxiations gov3rning t h e

sxipersonic flov/ af a r e a c t i n g gas a r e reduced t o a h y p e r b o l i c sj/stem of d i f f e r e n t i a l e q u a t i o n s . By u t i l i s i n g t h e method of c h a r a c t e r i s t i c s Ref . 4 , a nxjmerical s o l u t i o n may thxis be o b t a i n e d ,

To o b t a i n a nxomerical s o l u t i o n i t v/as foxmd advantageox:is t o woi-k i n t h e n a t u r a l c o - o r d i n a t e s y s t e m . I n t h i s c x a r v i l i n e a r system s i s measxired a l o n g a s t r e a m l i n e i n t h e d i r e c t i o n of tlie velocil^'' v e c t o r u and n

i s measured noiroal t o i t . 6 i s t h e i n c l i n a t i o n of t h e v e l o c i t j ' - u t o some a r b i t r s i r i l y c h o s e n datxjm. I t shoxxld be n o t i c e d however, t i i a t a l t h o u g h t h i s c o - o r d i n a t e system i s c o n v e n i e n t i t h a s t h e d i s a d v a n t a g e of n o t a u t o m a t i c a l l y r e v e a l i n g a l l t h e c h a r a c t e r i s t i c d i r e c t i o n s i n t h e subseqxjent a n a l y s i s .

The o v e r a l l o r g l o b a l contdnxoity e q u a t i o n may the;i be v - ^ i t t e n a s 9p 3u ee

at + Ts + ^ ^

U 9p 9u 9® n fj,A\ and t h e c o n t i n x i i t y of t h e atomic s p e c i e s i s

• u | 2 - f = 0

(12)

t

where K i s t h e r a t e of mass p r o d u c t i o n of t h e a t o m i c s p e c i e s c p e r xjnit volxame.

The momentum eqiKitions a-re

^ ^ + -p St = °

^^^^

and

2 ae 1 „ , .

u^-^^

. 1 ^ = 0

(14)

By c o n s i d e r i n g t h e tliermod;7namics of t h e m i x t u r e t h e energji" eqxjations

may b e u s e d t o e x p r e s s t h e ontrapy change a l o n g a s t r e a m l i n e i n t h e form

^ - ^ + ^ ( ' ^ o - ' ^ ) ' ' = 0 (15)

OS P T a m' \ ^/

•; / ' • • •

(10)

5

-Finally, the equation of state is specified by expressing the densitj?-as a function of piressxjre, entropy and concentration

dg = p^ dS + p dc + — , dp, (16)

"P ^S ^ • ^c

4

Tiiiere p~ = (-^] , and p = (^" j and the subscripts indicating ^ ^ ^ p , c ° \ ° /'SJP

which variables are held constant dxaring differentiation.

Y/'riting the equation of state xrith the density as tlie dependent variable enables the results obtained in Ref,4 to be included in the follaving analysis,

Equations (11 - 16) constitute six quasi-linear partial differential equations in six unknown dependent variables u, p , P, ®c, S, and in the t\70 independent variables s and n of tlie form

^ 9ui , 9xii . . J y r^-,\ L . = a . . ^--" + b . . Tï— + c . = 1 , 0 = 1 , . . , . , 6 . ( 1 7 )

J i j OS i j O n J " ^ , . . . . , . \ 1/

In this equation the cartesian tensor notation has been employed and tlius to obtain the values of the coefficients and dependent variables in the six equations (17), the subscripts ( i , j ) must be taJcen over a l l possihle combinations of i , j , each of v/hicli may talce any of the values 1 , 2 , . . . . , 6 ,

As the e-quations (l 7) stand the dependent variables appoejr differientiated in the directions ' s ' or 'n' . To obtain a solution, Ref.4, i t i s

necessary to have the dependent variables differentiated in directions along wliich the normal derivatives are not necessarily continuous,

These directions are called characteristic directions and the cxnrve vdiich they describe is called the characteristic curve. To find these directions a system of multipliers "^ . dependent on (s,n) i s specified such that in a linear combination L = X ^ L j a l l the dependent variables vd. appecj? differentiated in the same direction given by _dn _ pf^ ( \

J CIS

If the cxarve specified by "5- = « is given by s ( ? ) , n(5) v/here ^ i s a parameter associated f/ith the cliaracteristic curve a = -^^ / 9s .

ag / ^

The condition that a l l the variables in L = X . Lj are differentiated in the same direction a is thus given hy

If the six equations gi-rcn by equation (17) are compatible, i.e if a characteristic direction exists, tlion the \ . . may be eliminated to yield a characteristic deterrrdnantal relation x^hicli defines tlie directions of the tangents to the characteristics curves at the point (s,n) for specific values of the dependent variable, i , e ,

(11)

. é

-9n

9s

/MJ

-1

(19)

Ts^iere M„ i s t h e Mach nxmiber b a s e d on t h e f r o z e n speed of sound, A t h i r d c h a r a c t e r i s t i c d i r e c t i o n e x i s t s and i s c o i n c i d e n t v/itli t h e s t r e a m l i n e ,

= ^j L. = 0

J

The dependent variables u. will also satisfy the equation L at the point (s,n). Thxis by eliminating first i and then

a s 9n

from L = '^j L . =- 0 w i t h tlie a i d of e q u a t i o n s ( l 8 ) two more e q u a t i o n s v / i l l

J

r e s u l t . These combined v/ith e q u a t i o n s (17) y i e l d s e v e r a l d e t e r r r d n a n t a l e q u a t i o n s which g i v e s r e l a t i o n s l i i p s betv/een t h e v a r i a b l e s u . a l o n g

c h a r a c t e r i s t i c d i r e c t i o n s . One r e l a t i o n s h i p of importance wliich resxxLts i s t h e f o l l o v d n g /

u:

- 1 u dp ± d M, K C ~ U d ^ ± = 0 (20)

where d^i are increments of length along characteristic lines corresponding to the characteristic directions dn _ + J

d s ~ ~ ^

v/.?

and

1.

PT 9P p , c L a "^m 9S p , P - 1 (21)

i s a v a r i a b l e which p r o v i d e s a c o n v e n i e n t groxiping of t h e thermodynamic v a r i a b l e s i n v o l v e d and was f i r s t i n t r o d u c e d by Kirlw/ood and Wood Ref , 1 5 , Tliree o t h e r c h a r a c t e r i s t i c r e l a t i o n s ( e q u a t i o n s 1 2 , 1 3 , 15) a r e o b t a i n e d

§nd a l l h o l d a l o n g t h e s t a r e a m l i n e s ,

Thus p r o v i d e d t h e k i n e t i c s of t h e r e a c t i n g gas a r e lmw.n and p r o v i d e d sxaitable i n i t i a l d a t a a r e g i v e n t h e n eqx.iations ( 1 2 ) ( 1 3 ) ( 1 5 ) and (19) and (20) e n a b l e a l l t h e xanknown de;Tendent v a r i a b l e s t o be c a l c u l a t e d a t a l l p o i n t s i n t h e flov/ f i e l d ,

(12)

_ 7

-^•1 • I n t e g r a t i o n of the clic'-rac_^ter^is_tlc .egi^tioris

The equations are n o n - l i n e a r and are not reducible i n the sense of

Coxarant and F r i e d r i c h Ref, 4» and therefore i n order to obtain a s o l u t i o n

a numerical s t e p - b y - s t e p method must be employed. To f a c i l i t a t e nximerlcal

coiiiputation i t i s convenient to express a l l the eqxiD>.tions i n terms of

f i n i t e d i f f e r e n c e s ,

Assxaming t h a t conditions are knovm a t ti70 points p and p ^ , then

provided P , P are sxxfficiently close togetlier the p o i n t p^ a t which

the two c h a r a c t e r i s t i c l i n e s throxigh P and P i n t e r s e c t way be foxmd by

l e t t i n g the tangents to the frozen Mach l i n e s a t P and P r e p r e s e n t

the c h a r a c t e r i s t i c l i n e s i n a f i r s t arrproximatiai. This v/ill of coxirse

intaroduce an e r r o r v/hich can be reduced by i t e r a t i o n ; the convergence of

t h i s i t e r a t i o n depending on the size of the i n t e r v a l chosen for pipa ,

This has not been analysed a n a l y t i c a l l y , but t r i a l and e r r o r revealed

tliat f o r the cases discxissed i n t h i s r e p o r t , i t i s desirable thatP^ï'g

shoxild not be g r e a t e r than 0,01 of a dimensionless xmit. Here length

has been non-dimensiotialised by u T .

•^ o o

After non-dimensionalising the v a r i a b l e s v/ith free stream c o n d i t i o n s ,

i , e , p = " ^ e t c , eqxiations (19) nxay be v/ritten

Pa ^O

1 / ^ ^

and y ^ H-^jy^^^ ^ p + A6 + ; r - ^ — i - A^ + = 0

p u ^ p u2 ^^ 1 M„ V p ^ y

X) o 1 1 ^ f \ 1 1 /

, . . . (21)

AG + 1 - f - ^ 2 - ) A.^ - = r; 1 ^ 2 2 (22)

where subscripts d ) (2) represent the valxjes of the variables at the points P andP ,, >'•

Ap

1

Ap

2 Ae 1

AS

2

The f i n i t e differences

= P - P

3 1

= Ap + p - p

1 1 2

t= 6 - 0

3 1

= AS + 6 - 6

1 1 2

^ P , ,

.^v^.

Ae 1

e t c ,

a r e

defined by

(23)

(24)

(25)

(26)

(13)

8

-This of course, is only a first order approximation since It has been assximed tliat the coefficients in the differential equations (21. 22) are constant along the characteristic lines joining the points (P^> Pj) §nd (P , P' ) , their values being token at the points 1 and 2 jcespectively. To obtain a better approximation an iteration process must be used, i,e, values obtained at the point 3 aj'e used to obta^in a mean value for the coefficients and the slope of the characteristic lines, and the procedxire

is repeated. Once again trial and error revealed that only one iteration

need be x.Tsed v/hen examination is carried out along the characteristic lines dn + 1

ds

fi^-

1

Prom eqxaations (21 - 2é) e x p r e s s i o n s can be foxmd f o r tlie change i n p and Ö from p o i n t s P t o P^ , i . e . P u^ o o

he -1 Jit -1"

Ap = ( e e )

-P U 2 0 o ; 2 p 2

iC'

- 1 " ( P l - Pa ) K C P u 2 2 (27) and

K^

U 2 1

JC^"

u ' ( 6 _ 6 ) . \ , 2 '

Ae

K^

(B^P^) M •fi 1 K. (J P U

K-i

1' u" (P1 - ï ^ )

K^'-1 ( P 2 P 3 • M fz 2 2_^ P u 2 2 (28)

where p p ( s a y ) i s the d i s t a n c e betv/oen t h e r p o i n t and s p o i n t . X^ s

Thus the flow inclination and pressxire are knov/n at P . Before tlie other

variables can be foxmd the relations holding along the streai::lines must be used, Enov/ing the flov/ inclination at P^ a first approxiiDation may be obtained by projecting the tangent to the streamline a^t P3 to intersect

tlie line '^~é at P/ s {ss.y), Then by assximing that a linear variation

of the variables exists over the segment P^Pa conditions at E/ s may be

f o u n d . Once P.

( 1 , 2 ) h a s b e e n l o c a t e d and t h e valxxes of t h e d e p e n d e n t

v a r i a b l e s a r e l-niown a t t h a t p o i n t t h e n t h e f o l l o w i n g r e lei t i ons h o l d a l o n g t h e s t r e a m l i n e ,

(14)

9

-" o ^ ' ' ' -" ( . i -" ( „ a ) ' ^ „ O ' ^ o '^PCa) = ° ('5)

P h P/ >, A h / N - p A p , V = O , ( 5 1 )

o o ( 1 , 2 ) ( 1 , 2 ) ^o ^ ( 1 , 2 ) ' ^-^ '

v/here Au/ v = u - U/ N ; Ap/ > = P - P/ \ J Ac/ % = c - C/ s

( 1 , 2 ) 3 ( 1 , 2 ) ^ 1 , 2 ) 3 ( 1 , 2 ) ( 1 , 2 ) 3 ( 1 , 2 ) ,

A h / s = h j - h / >, V 1 j 2 ; \ i » 2 ^

and u/ \ e t c define valxjes of tlie dependent v a r i a b l e s a t the p o i n t P/ v,,

I n order t h a t t h i s approximation s h a l l be useful i t i s necessary'' t o

l o c a t e the p o i n t P/ \ f a i r l y a c c u r a t e l y . I t v/as foxmd t h a t a t l e a s t

^ 1 , 2 ;

one i t e r a t i o n v/as needed in order t h a t the values of tlie dependent

v a r i a b l e s a t P/ \ v/ere located to the same order of accuracy as the

increments Ac, Au, Ah andAp. At t h i s stage p , 6 , u , c , h are knov/n a t P

and nov/ the otlier v a r i a b l e s may be calcxilated.

The c a l o r i c eqxiation of s t a t e for an i d e a l d i s s o c i a t i n g gas may be

v/ritten i n terms of f i n i t e differences t o give

-A h -AT -Ac

and the thermal equation of s t a t e becomes

AP Ap Ac^ ^ + ^ ( 3 3 )

p p 1 1

1 1

To e^alxiate the slope of the c h a r a c t e r i s t i c l i n e a t P^ tlie frozen Mach

nxmiber i s needed and may be evalxiated (Appendix A) from the follov/ing

f i n i t e difference equation

Aa2 AT ( 5 + 2c )

^1

I f the equilibrium composition i s evaluated a t l o c a l pressxire and tenrperatxire,

then o at; the p o i n t P may be obtained from the lav/ of mass a c t i o n , i . e .

(15)

10 -1 - C 2

J-a

_

pTr"'+ c 7

3 3 exp

[ - Ï J

05)

The r e a c t i o n r a t e i s dc d t v;±Lere K P (1 - C^) 2 - c; ^ 3 % )

(36)

o K ^ p ; (1 . c ; (37)

The thermodynamic variables O"may be written. Appendix B,

c ^ (1 + c ^ ) ( i + ^ ) - ( 4 - H c ;

(38)

(1 + c ^

( 4 + c^) f o r an i d e a l d i s s o c i a t i n g g a s .

Thxis a l l tJie flow and thermodynamic v a r i a b l e s a r e known a t P , I n a lilce manner t h e v/hole flov/ f i e l d may be c a l c u l a t e d b y c o n s t r u c t i n g a c h a r a c t e r i s t i c n e t , s e e F i g . 1 f o r example.

T\70 r e l a t i v e l y siinple s o l u t i o n s a r i s e vAien t h e flov/ i s c o n s i d e r e d t o be f r o a e n o r i n e q u i l i b r i x m i , ( i ) Frozen Flow : F o r f r o z e n flovT t h e c h a r a c t e r i s t i c c h e m i c a l t i n e T becomes i n f i n i t e and t h u s a l l t h e t e r m s i n v o l v i n g t h e c h e m i c a l r e a c t i o n s d i s a p p e a r , E q u a t i o n (20) becomes

lit, -

1

rt

± A6 = 0 {•53) p u

and tlius i t i s seen t h a t t h e flea? w i l l o c c u r i s e n t r o p i c a l l y . T h i s 'type of flca7 occxirs a t t h e apex of a Prandtl-Jvieyer e x p a n s i o n i n a r e a c t i v e f l u i d and i t s s o l u t i o n i s e a s i l y c o n s t r u c t e d ,

( i i ) Eqxiilibrixim Flow;

F o r t h e o t h e r extreme v/hen r = 0 t h e flow i s i n eqxiilibrixim and f o r t h i s i d e a l c a s e i t i s foxmd t h a t t h e d i s t u r b a n c e f r o n t p r o p a g a t e s v n t h a .

e The e q u a t i o n s of motion g o v e r n i n g t h e e q u i l i b r i i m i flow a r e :

(16)

1 1 -u dp d-u ae e e e e _ / i j \

p a s a s e an ^^ '

e

(42)

v43)

(4(1.)

''e

-1

^e

9u

u e

«Sa

^ ^ = 0

9n

0

as

dP = — dp + (-3=2 ) dc + (—^\ dS (45)

e Pe^e e / p , c ^

Note t h a t the equilibrixim concentration c (S,p) has been evalxiated a t

l o c a l pressure and entropy and i s only the same as c (p,T) v/hen equilibrixma

e exists. ' •

Once again equations (4'1 - 45) constitute a quasi-linear system of differential equations and may be solved in much the same maimer as before. It is now foxmd however that the characteristic directions are defined hy

K^

where M i s based on the equilibrium speed of soxmd. As pointed out by

Chu Ref^ 3 i t i s i n t e r e s t i n g to note t h a t f o r a r e a c t i n g g a s , althougli

the order of the eqxiations i s reduced when T = 0 the system s t i l l remains

hyperbolic,

The nximerical s o l u t i o n i s analogoxis to t h a t of s e c t i o n ( 5 . 2 . ) and

as before i t i s convenient to v/rite the r e l a t i o n s holding along the

c h a r a c t e r i s t i c s i n terms of f i n i t e d i f f e r e n c e s , i . e .

/ M ^

- 1

^ . . j _ „ Ap ± Ae = 0 (47)

p XT e e ^ ' e e P U Au - Ap = O (2j£) e e e -^e '

Pe ^\

+ ^Pe =" ° • ^^^)

(17)

12

-The t h e r m a l and c a l o r i c eqxiation of stante becoross

Pe = ^ (^ + % ) ^ ^e ' (50) • \ = ( 4 + c ^ ) R T ^ + G ^ D . (51)

At this stage p , h , u are known and in order to evalxiate c , P , and e e e e e T the equilibrixim concentration is first foxmd from

p ^ ( h - c D )

D ^ e e '

^ "JXTTT ^^ l - Th-r^D)'

^D ^ ( ^ ^ % ) •

( 5 2 )

t h u s g i v i n g the t e m p e r a t u r e and d e n s i t y from eqxiations ( 5 0 , 5 1 ) .

To evalxiate t h e c h a r a c t e r i s t i c s d i r e c t i o n s the equilirbixjm soxmd s p e e d must b e u s e d and may b e o b t a i n e d from t h e r a t i o i n Ref. 5

2

( 1 - o J [ d + Vi)(1 +=e' -(4+c^)]^

10^ 1 . i

-a / 3 T„ * e (^ ^ " ^ 4 ) ^ 0 (1 - c^ )+ 8 + 2 c ^ ' e e e •Viiiere i t h a s been t a b x i l a t e d f o r v a r y i n g temperatxire and p r e s s x i r e .

For t h e s m a l l e x p a n s i o n a n g l e s v/hich have b e e n u s e d t o i l l x i s t r a t e t h e c h a r a c t e r i s t i c method t h e e q u i l i b r i x i m c o n d i t i c m s a p p e a r t o p r o v i d e an a s y n i p t o t i c v a l u e f o r t h e dependent v a r i a b l e s b e h i n d t h e c o m e r . F u r t l i e r examples c o n s i d e r i n g l a r g e r e x p a n s i o n a n g l e s and o t h e r i n i t i a l c o n d i t i o n s a r e r e q u i r e d hov/ever t o see t h e t r u e s i g n i f i c a n c e of t h o s e ' a s y n i p t o t i c ' v a l u e s .

The f r o z e n and e q x i i l i b r i u m c a s e s ( o r expansive flows have a l s o b e e n discxissed by Heims, Ref. 9 . I n t h e l a t t e r tlie c o n d i t i o n s b e h i n d t h e c o m e r a r e foxm.d b y c o n s i d e r i n g P r a n d t l - I d e y e r flow i n v/hich allowance h a s b e e n made f o r t h e c h e m i c a l c o m p o s i t i o n of t h e g a s by i n t e g r a t i n g , s t e p - b y - s t e p , a l o n g i s e n t r o p e s v d t h t h e a i d of a M o l l i e r c h a r t .

(18)

13

-5 . ^^^.ApplA'^.at.iPn.-Q^. jj^g.^.^'70.Il^Ijpnaj^gnal The_qr^ ,"^o „.'^hg-^€;_:?gyns_ion _of

an I d e a l Dissociating Gaa

For a high temperature i-eacting gas the flov/ roimd a c o m e r v / i l l

no longer occur i s e n t r o p i c a l l y and narked diffci-ences fx-om the cla.ssical

Prandtl-Meyer flow v / i l l a r i s e . Thus i n order t o i n v e s t i g a t e the flow

f i e l d f o r such an expansion the theory and method given i n s e c t i o n (4)

has been applied t o three s p e c i f i c exanples. The three cases f o r v/hich the

flov/ f i e l d has been examined ca'e

( i ) p = 1 a-tm., T = 4250 K., o = 0,78 and expansion angle = 5 .

( i i ) p = 0 . 1 atm., T = 3750 K,, c = 0.83 and expansion angle = 5 .

( i i i ) p = 0,1 aim», T = 3750 K„, c =; 0.83 and expansior angle = 10 ,

a l l a t a Mach No, of 1.83,

As the flov/ roxjnd •»edgo s e c t i o n s are l i k e l y t o be of more importance in

the l a b o r a t o r y , the abc3ve ccjnditions were chosc;in to correspond t o those

conditions l i k e l y t o be encoxmtered i n shock tubes. The conditions of the

w a l l are generally those of i n t e r e s t and. so the v/all temperature, p r e s s u r e ,

concentration and Macli numbei' have been p l o t t e d against the non<~dir.-iensional

length nieasured from the c o m e r of the w a l l . Further t o obtain a more

general v i s u a l pictxire of the o v e r a l l flov/ f i e l d behind the expansion isobcT

p l o t s have been made of cases ( i ) and ( i i i ) .

The t r a n s i t time through the apex of the expansion fan i s vanishingly

small so i n the region very close to the c o m e r there -vidll be a lag i n

the i n e r t degrees of freedom. Generally, the i n e r t degrees of freedom

v/ill include v i b r a t i o n as well as d i s s o c i a t i o n but i n t h i s r e p o r t hov/evcr

only the r e l a r a t i o n e f f e c t s due to d i s s o c i a t i o n ace considered. The

v i b i ' a t i o n a l degree of freedom i s taken to be always excited to j u s t h a l f

i t s c l a s s i c a l v a l u e . For the cases discussed here the degree of d i s s o c i a t i c n

i s f a i r l y high (c-'O.S) and the energy s t o r e d i n the v i b r a t i o n a l modes of

the remaining molecules v / i l l be small conTpa.rcd v/ith t h a t of d i s s o c i a t i o n .

Thus, the assxjmption t h a t the v i b r a t i o n a l mode i s instantaneously adjusted

to be i n l o c a l eqxiilibrixim v/ith the t r a n s l a t i o n a l and r o t a t i o n a l modes

should not a f f e c t the resxilts appreciably. At the ajjex of the fan then,

the i s e n t r o p i c •bemperatx:i;ire drop w i l l caxise the atoms t o be no longer i n

chemical equilibrixjim v/ith the molecules. Thus a t r a n s i t i o n region must

e x i s t 'in v/hich the system has time t o approach a nev/ e-quilibrium valixe,

In t h i s region the atoms recombine and supply the flov/ with some of

i t s d i s s o c i a t i v e energy. This process continues xmtil a balance i s

achieved betv/een the concentrations of the a.toms and molecxiles. This

r e l a x a t i o n t o a nev/ equilibrium le'/el i s an i r r e v e r s i b l e one and i s

accompanied by an increase i n e n t r o j y .

The zone over v/hich t h i s r e l a x a t i o n occurs i s very dependent on tLe

free streaiii conditions ahead of the c o m e r and tlie expansion angle. As the

(19)

14

-method of solution for such flox-/s is necessarily a lengthy iDrocess only a limited investigation of these effects has been obtained. Of the cases solved, the most pronoxmced variation occxirred with the reduction of free stream pressxire. By reducing the pressure from one atmosphere to one tenth of an atmosphere for the same ejcpansion angle the relaxation zone at the wall increased from approximately 0,026 cms to approximately 1«735 cms, This is because the recombination of the atoms requires a three body collision, Thxis, as the density becomes smaller -öie probabilitrj'- of sxjch collisions is less, and so tlie tims to reach a nev/ equillbrixun level v/ill be Icmger, Variation of the expansion angle from 5 to 10 in oases (ii, iii) showed that the relaxation zone at the v/all increased by approx-imately 0.9 cms.

A fxirther influence on the relaxation zone arises from the recombination rate ccjefficient. Owing to the lack of data concerning the recombination rate coefficient for oxygen, in the cases discussed an arbitrary value of

17 2 2

10 cc /mol sec v/as used. Recent evidence however Ref .13 would suggest that a value of -k^ = 1015 cc /mol sec would be a better estimate. It

woxild appear that due to this eirbitrary choice of k^ the length of the

relaxation zones are xmderestimated by a factor of one hxindred in this report. Also ICj has been assumed constant throughout. Over the temperature range

which these examples cc3ver Ref, 13 suggests that kp v/oxild only change from

0,78 , 10 cc /mol sec to 0,69 x 10 oo /mol sec and so the assxJEiption k- = const, throughout the expansion appears to be satisfactozy.

In Ref, 7, Feldman reports the existence of a recombination shock

out in the flow field due to the relaxation. In the cases discussed, the

relaxation along the wall caused a steepening of the Mach lines at the

v/all but dve to the relaxation effects (x:cxirring thrcxigti the expansion,

av/ay from the comer, it is xmlikely that the Mach lines will coalesce even at great distances from the v/all. To investigate such recombination shocks it suggests that much larger expansion angles must be considered,

In a recent paper by Clarke, Ref,l6, the linearised flow of an Id.eal Dissociating gas is discxissed. For the small expansion angles considered tlie variation of the non-dimensional dependent vjxriables in the rela:cation zone are in good agreement vdth the non-linear solutions presented in figures 3 - 8 ,

6, G one Ixis i ons^ _

The method presented in this report provides a means of investigating the effect of dissociation on the flov/ field of a high temperatxire reacting gas. The Kietliod is successful in so far that results can be obtained, but the number of computing hours required for each solution makes it Impracticable for extensive investigations to be made v/ithout the aid of a digital computer or the further possihility of a nximerical graphical procedure being developed.

(20)

15

-Of the cases examined for expansive flov/s of an i d e a l 'oxygen-like'

d i s s o c i a t i n g gas roxmd a sharp c o m e r i t v/as foxmd t h a t the r e l a x a t i o n

zones v/ere considerably influenced by the free stream c o n d i t i o n s , and f o r

the small expansion angles consideixïd the flow w i l l qxiickly be adjusted

t o nev/ equilibrixim c o n d i t i o n s .

7.

The author wishes to express h i s appreciation to Dr. J . F Clarke,

for h i s supervision and encoxiragement dxuring the coxurse of t h i s work.

8 , References

1 . Broer, L . J , P .

2 . Camae, M,

3 . Chu, B.T.

4.

5.

6.

7.

8.

9.

Coxirant, R. &

F r i e d r i c h , K.O,

Clarke, J , F ,

Feldman, S,

Feldman, S,

Fowler, R,H, &

Gxiggenheim, E.A,

Heims, S,P,

10, Isenherg, J , S ,

C h a r a c t e r i s t i c s of the eqxiations of moticm

of a r e a c t i n g gas.

J n l . Fluid Mech. Vol.4 p . I l l J u l y , 1958,

Chemical r e l a x a t i o n i n a i r , oxygen and

n i t r o g e n ,

I , A . S , P r e p r i n t 802.

Wave-Propagation and the method of

c h a r a c t e r i s t i c s i n r e a c t i n g gas mixtxires

with a p p l i c a t i o n to hypersonic flov/.

Brov/n University Report W.A.D.C. TN-57-213

May, 1957.

Supersonic flow and shock waves,

I n t e r s c i e n c e P u b l i s h e r s , 1948.

The flov/ of chemically r e a c t i n g gas mixtxires.

C. of A. Report 117, November 1958.

The chemical k i n e t i c s of a i r a t high

temperatxircs.

Heat»-transfer and Flxu.d Mechanics Institut-e 1957,

On the existence of recombination shocks

The Physics of F l u i d s , Vol, I . No.6 1958.

S t a t i s t i c a l tliermodynamics,

Cambridge P r e s s , 1949.

Prandtl-üvleyer eDqjansion of chemically

r e a c t i n g gases i n l o c a l chemical and

thermodynamic equilibrixim,

N,A.G,A. TN,4250 March 1958

The method of c h a r a c t e r i s t i c s i n compressihle

flow,

(21)

- 16

RefoTCnces (Continued) 1 1 . L i g h t h i l l , M . J , 1 2 . Logan, J.G-, 1 3 . Matthexvs, D,L, 1 4 . P e n n e r , S . S . 1 5 . Wood, W.W. & ICirkwood, J , & . Dynojnd-cs of a d i s s o c i a t i n g gaa f o r e q u i l i b r i x i m f l o w . J n l , F l u i d Mech, V o l , 2 p , I 1957 Relax:ation phenomena i n h y p e r s o n i c a e r o d y n a m i c s , I . A . S . P r e p r i n t 728 I n t e r f e r o m e t e r measxirement i n t h e shock tube of t h e d i s s o c i a t i n g r a t e of o^qygen. The P h y s i c s of Flxiids V o l . 2 Ko,2 1959, Chemical r e a c t i o n s i n f l a v s y s t e m s , A,G,A.R.D. p u b l i c a t i o n 1 9 5 5 . Hydrodynaitdcs of a r e a c t i n g and r e l a x i n g f l u i d , J n l , App, P h y s , V o l . 2 8 N 0 . 4 1 9 5 7 , 1 6 . C l a r k e , J , F , To be p u b l i s h e d ,

(22)

17

-The spee^ds of soimd for an i d c s l j i i s s o G i a t i n g gas

(^) l!?^°^jj:.-^-Qljnd Speed

To v/rite the frozen speed of soxmd i n a forro v/hich i s s u i t a b l e for

nximerical computetion i t i s convenient t o consider the following energy

r e l a t i o n , i . e ,

T dS = du

- f '^'e.-'J

dc (A1)

Teücing the enthalpy and i n t e r n a l energy • e ' of the gas to be functions of

the pressxire, tempera txire and concentration, together v/ith the f a c t t h a t

the r e a l t i o n h = e + £ h o l d s , then the following expression may be obtained,

dp

For dissociating gas, the pressxire does not appear explicitly in the energy relation, tlius if the above expression is evaluated for constant local values of entropy and concentration it is reduced to,

S,c

P

P vf

(A3)

p,c

yjhere C „ i s the frozen s p e c i f i c heat a t constant pressxire, and C,^

i s the frozen specific heat a t constant volxime. For an i d e a l d i s s o c i a t i n g

gas equation (A3) i s r e a d i l y shc3v/n to be

BZ 9 p ^ 9 p \

(A 4)

(23)

18

-( i i ) Ijqx iilibr iu m S oun d Speed

The e q u i l i b r i u m speed of soxmd c a n , i n p r i n c i p l e , be e v a l u a t e d i n a lilce manner b u t xmfortxmately tlie r i g h t hand s i d e of e q u a t i o n ( A 2 ) becomes xmvri.eldy and t h u s the b e s t metliod i s t o e v a l u a t e a b y c o n s i d e r i n g a r a t i o of t h e soxmd s p e e d s .

C o n s i d e r i n g the d e n s i t y a s a fxmction of p r e s s x i r e , entrcjpy and c o n c e n t r a t i o n , we have

d p = ^ d p H - ( ^ ) dS + f e ) ^ dc (A5)

4.

\ 9 s / \

°/p,S

f p , c ^»

Nov/ f o r a gas which i s i n e q u i l i b r i i m i , and v/hose l o c a l entroT:)y v a l u e s a r e c o n s i d e r e d c o n s t a n t , t h i s l a s t e q u a t i o n may be v / r i t t e n

l^

= i . = 1-

+ / 9 £

c e 9py~ ~ 2 g V 9 c y - o • ^^2,0^ Sg a^ \ / p , S p S 'ac (A6) R a t h e r t h a n e v a l u a t e i^A ^ t [-^ ) ^ i n d e t a i l , r e f e r e n c e i s m.ade t o \ ' ^ V p , S \ 9 P / S C l a r k e (1958) \7here t h S s h a s b e e n e v a l u a t e d f o r t h e i d e a l d i s s o c i a t i n g g a s , and i s , , 2 , 1 - f ~ 2 ; ( i + c) - ( 4 + c ^ ) ^ (A7) 1 + ^ J Ogd - c^^) + 8 + 2

This h a s been evalxiated f o r v a r y i n g tempera txire and pressxiare, and t h e r e s x i l t s p r e s e n t e d t h e r e i n a r e u s e d i n t l i i s r e p o r t v/hen t h e flow f i e l d h a s b e e n evalxiated f o r eqxiilibrixmi c o n d i t i o n s .

(24)

- 19 APEENDIX E The thermodynamic v a r i a b l e er I n t h e g e n e r a l a n a l y s i s of c h e i i d c a l l y r e a c t i n g flcrws Ref, 5 , t h e v a r i a b l e c p r o v i d e s a c o n v e n i e n t groxiping of t h e thermodynamic v a r i a b l e s i n v o l v e d , and i s d e f i n e d a s

1- (

Ë£,

^^

\ 9 i

p , o ^ - /J + T ( ^ a m \o n P , P J

As p o i n t e d o u t i n Ref, 5 , t h i s i s n o t tlie most c o n v e n i e n t form f o r a v a i u a t i o n and i t may be w r i t t e n a s

ah\ 1

^ % T * "

a i ; ^ ^'-^

p , o 9P

ao

p , T

For tlie i d e a l d i s s o c i a t i n g gas

9p 9h p , c p , c P T / p , T = ( 4 + c)R (^ p , T P 1+c thus cr = _ , (1 + c) (1 + - ^ ) - ( 4 + c) and ^a '^m RT T

r

^°Êe

m ''D ( 1 ^ ^ ^ c^

(25)
(26)

>.c^ o ^o O' \ : ^ \ \ O O I C3F A D1MENSK5NLESS UNIT SCALE I 1 OOOrr Cms. WHEN KpstO*^

M O L E " * Cm.f S E C " ' \ \ \ \ \ . \ \

N .

\ > < \ \ \ . l \ . r=0-735 \ ^ ^ < : < ' : 0-760 \ 0-745 \ \ ;=0-7S5 \ > - = 0-74 \ ,^ \

- ^NÏ.'

O-765 INITIAL CONDITIONS. M , „ . l - 8 3 To = 42SO "K • = l o t m . >. C = O 7 6 \ \ \ \ \ \ \ \ \

(27)

0 - 7 9 C... 0 - 7 8 0 - 7 7 0 - 7 6 0-7S ^ ^ "

1

~ EQUILIBRIUM VALUE '

FREE STREAM CONDTriONS 1 *o = 1 Ot m. To'42 S O ' K M f o * ' - 8 3 C o = 0 - 7 8 | '^ O ' O S O • lO O-IS i_ 2 ' 0 7 S 2 -OSO M, 'w 2'OCK) I - 9 5 0 l ' 9 2 5

k

EQUILIBRIUM VALUE'

'7

0-05 o-io 0-I5 ^

FIG. 3. CONDITIONS AT WALL FOB 5 EXPANSION

o - 7 0 EQUILIBRIUM VALUE. ^ - ^ V

.V

o - 0 5 O - I O O-IS I Uoto 0-OS O-IO 0 - I 5 I

(28)

o'SO 0 - 7 8 O - 7 6 0 ' 7 4 0 - 7 2 0 - 7 0 EQUILIBRIUM VALUE ^ ^ FREE +0 =•'! C o = 0

k

L V

STREAM CONDITIONS O o t m . TO = 3 7 5 O ' ' K • 8 3 M f o » 1-83 0 - 0 5 o - i o 0 - I 5 l-OO Too O - 9 6 0 - 9 2 O - 8 8

FIG, 5. CONDITIONS AT WALL FOR 5° EXPANSION.

O C o «4 w 83 0 - 8 2 0 - 8 I o-SO \ ^ ^ - - _ _ EC3UILIBRIUM VALUE ' / ^ 0 - 0 5 O-IO 0-I5 l_ u„t„ 2 - 0 7 S 2 ' 0 5 0 2 - 0 0 0 •9SO • 9 2 5 \ / ^ ECXJILIBRIUM VM.UE —

FREE STREAM CONDITIONS

If a - ''lO atm T , = 3 7 S O "K Co = 0-83 M f o = l ' 8 3 ^ O-OS O - I O 0 - 1 5 I Uo t . FIG. 6. CONDITIONS AT WALL FOR 5 EXPANSION.

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EQUILIBRIUM

VALUE-y^

^ \~~^^ , ^ . - - " ^ —• 0 - 0 5 0 - 0 5 O-IO O- 15 0-80 0-I5 _g_ 0'20 Uoto EQUILIBRIUM VALUE , » ^ - . - . ^ ^

FREE STREAM \MLU • , a ' / l O o t m T , = C , = 0 - 8 3 M^^=|. ES 83 2 - 3 2 - 2 2 - 1 I 0'20 uTto 2 - 0 EOUIUBRUM

VMLUE-FREE STREAM CONDITIONS >^ ='/lO atm T^3750 " K C o = 0 - 8 3 M ^ „ = l - 8 3

1

^ ^ 0-05 O-IO 0-I5 i 0-20 UTto

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Celem kwerendy archiwalnej było zapoznanie się z niemieckimi osiemnasto­ wiecznymi publikacjami i rękopisami z zakresu teorii nauki. Materiały te wskazują na

Under time- varying wind directions, it shows time delays in wake direction as inflow changes propagate through the farm with the wind speed, although the dynamics still differ from

Franz Ruppert o twórcy wspomnianego syte- mu wspomina niewiele, bodaj tylko raz, i to w formie polemicznej, z którą nie- koniecznie też trzeba się zgadzać.. W bibliografii

Таким образом, он ищет счастье в том, что уже отошло в область воспоминаний, но при этом изображает минувшее как незаконченный процесс, несмотря на