Report No. 123

L

THE COLLEGE OF AERONAUTICS

CRANFIELD

THE TWO-DIMENSIONAL FLOW OF AN IDEAL DISSOCIATING GAS

by

J. W. CLEAVER

.::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

-by -

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 for the Diploma of the College of Aeronautics,

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 to the expansion of an Ideal Dissociat ing Gas 13

Conclusions 14

Acknov/ledgement 15

References 13

Ajrpendix A 17

Appendix B 19

Figures

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

u Velocity in strearawise direct ion

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. eto) Points in flov; field

W Conditions at the wall

a atomic species

m molecular species

- 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,

- 2 -

The cha rac te r i s t i c density P^, see Ref ,11, i s s t r i c t l y a function of temperatixre, but for the temperature r:aige 3000 K < T < 7000^C tlie var ia t ion 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 in the assi:iiiption that P^^ i s a constant i s the requirement tha t tlie v ihra t iona l degrees of freedom of "tlie molecxxLe are alv-'-aijrs ju s t half excited even a t low temperatures. This enable tlie in te rna l energy to be wr i t ten in 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 Ligl i thi l l points out, t h i s idea l associat ing gas y.dll only be representat ive over the teniperatxare range 3000 A! <T< 7000 Iv. Outside these l imi t s ionisa t ion and v ibra t iona l exci ta t ion Yrill becoiiie inroortant and t h i s has not been allowed for . However in the examples presented below, since only small expansion angles a-re considered, the L igh th i l l approximations are adequate,

3 , The Rate Equa.tion

By assuming tha t the molecular and atanic s t a tes of the assembly are independent, the equation describing tlie reaction 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)

v/hero brackets represent tiie concentration of the species in mol,/co,

-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 constant ,

= K (8)

As the flovTs discussed l a t e r are of an expansive natxire i t i s convenient to express the ra te equation in terms of Ic, ra ther than kj, ,

Thus i f the concentration c i s evaluated a t loca 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 noticed that r = m /p^ kp(l + c) has tlie dimensions

of time and may be taken as a cha rac te 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 in tlie following approximate form,

1 dc' 1 / ^ °e 1 ( = „ ^ J = const , (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^^ -2 6 -1 N ti

k. -

Recent i n v e s t i g a t i o n s Ref , 1 3 , give k^ = 0,8 x 10^^ a t a tenrperEtture of

3,500 K v/hich i s i n c lo se agreement \ , l t h lief , 2 .

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

By mailing the s i r ip l i fy ing assxjmption t h a t v i s c o s i t y , h e a t condxxjtion and d i f fu s ion may be n e g l e c t e d , the genera l eqxiations gov3rning the sxipersonic flov/ af a r e a c t i n g gas a re reduced to a hyperbo l ic 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 the 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 ob ta ined ,

To ob ta in a nxomerical s o l u t i o n i t v/as foxmd advantageox:is t o woi-k i n the n a t u r a l c o - o r d i n a t e system. I n t h i s cxarvi l inear system s i s measxired a long a s t r eaml ine i n the 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 he i n c l i n a t i o n of the veloci t j ' - u t o some a r b i t r s i r i l y chosen datxjm. I t shoxxld be n o t i c e d however, t i ia t a l though t h i s c o - o r d i n a t e system i s convenient i t has the disadvantage 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 the 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 or g l o b a l contdnxoity equa t ion may the;i be v-^i t ten as

9p 3u ee at + Ts + ^ ^

U 9p 9u 9® n fj,A\

and the cont inxi i ty of the atomic s p e c i e s i s

• u | 2 - f = 0 (12) t

where K i s the r a t e of mass p roduc t ion of the atomic spec 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 the tliermod;7namics of the mixture the energji" eqxjations may be used t o express the ontrapy change along a s t r eaml ine i n the form

^ - ^ + ^ ( ' ^ o - ' ^ ) ' ' = 0 (15) OS P T a m' \ ^/

•; / ' • • •

where S is the specific entropy and M is the chemical potential,

- 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 . (17)

J i j OS i j On 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 is 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) is specified such that in a linear combination L = X ^ L j al 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 ^ is a

parameter associated f/ith the cliaracteristic curve a = -^^ / 9s .

ag / ^ The condition that al 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 ,

. -é -

9n 9s /MJ -1

(19)

Ts^iere M„ i s the Mach nxmiber based on the f rozen 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 co inc iden t v/itli the 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 th tlie a i d of equa t ions ( l 8 ) two more equa t ions v / i l l J

r e s u l t . These combined v/ith equa t ions (17) y i e l d s e v e r a l de ter r rdnanta l equa t ions which g ives r e l a t i o n s l i i p s betv/een the v a r i a b l e s u . a long 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 the f ol lovdng

/ 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

ds ~ ~ ^ 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 rov ides a convenient groxiping of the thermodynamic v a r i a b l e s invo lved and was f i r s t i n t roduced 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 ( equa t ions 12 , 1 3 , 15) a re ob ta ined §nd a l l h o l d a long the s tareamlines,

Thus p rov ided the k i n e t i c s of the r e a c t i n g gas a re lmw.n and provided sxaitable i n i t i a l d a t a are given then eqx.iations (12) (13) (15) and (19) and (20) enable a l l the xanknown de;Tendent v a r i a b l e s to be c a l c u l a t e d a t a l l p o i n t s i n the flov/ f i e l d ,

_ 7 -

^•1 • In tegra t ion of the clic'-rac_ ter is_tlc .egi^tioris

The equations are non-l inear and are not reducible in the sense of Coxarant and Friedrich Ref, 4» and therefore in order to obtain a solution a numerical step-by-step 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 in terms of f i n i t e differences,

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

provided P , P are sxxfficiently close togetlier the point p^ a t which

the two cha rac te r i s t i c l ines throxigh P and P in t e r sec t way be foxmd by

l e t t i n g the tangents to the frozen Mach l ines a t P and P represent

the cha rac te r i s t i c l ines in a f i r s t arrproximatiai. This v/il l of coxirse intaroduce an e r ror 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 rva l chosen for pipa , This has not been analysed ana ly t i ca l ly , but t r i a l and error revealed tliat for the cases discxissed in t h i s repor t , i t i s desirable thatP^ï 'g shoxild not be greater than 0,01 of a dimensionless xmit. Here length has been non-dimensiotialised by u T .

• o o

After non-dimensionalising the var iables v/ith free stream condit ions,

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 , are defined by

(23)

(24)

(25)

(26)

- 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) express ions 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 U2 0 o

; 2

p 2

iC'

- 1

" (P l - Pa )

K C

P u 2 2

(27)

and

K^ U2

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

(P2P3 •

M fz

2 2_^

P u 2 2

(28)

where p p (say) i s the d i s t a n c e betv/oen the 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 at 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

found. Once P. ( 1 , 2 )

has been l o c a t e d and t h e valxxes of the dependent

v a r i a b l e s a re l-niown a t t h a t p o i n t then the fo l lowing r e lei t i ons ho ld a long the s t r e a m l i n e ,

- 9 -

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

P h P/ >, A h / N - p A p , V = O , (51) 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 = hj - h / >, V 1 j 2 ; \ i » 2 ^

and u/ \ e tc define valxjes of tlie dependent variables a t the point P/ v,,

In order that t h i s approximation sha l l be useful i t i s necessary'' t o locate the point P/ \ f a i r l y accurately. I t v/as foxmd that a t l e a s t

^ 1 , 2 ; one i t e r a t i o n v/as needed in order tha t the values of tlie dependent var iables a t P/ \ v/ere located to the same order of accuracy as the increments Ac, Au, Ah andAp. At th i s stage p , 6 , u , c , h are knov/n a t P

and nov/ the otlier variables may be calcxilated.

The ca lor ic eqxiation of s ta te for an ideal dissociat ing gas may be v/ritten in terms of f i n i t e differences to give -

Ah AT Ac

and the thermal equation of s t a te becomes

AP Ap Ac^ ^ + ^ (33)

p p 1 1 1 1

To e^alxiate the slope of the cha rac t e r i s t i c l ine 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 local pressxire and tenrperatxire, then o at; the point P may be obtained from the lav/ of mass ac t ion , i . e .

- 10 -

1 - C2

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 are known a t P , I n a lilce manner the v/hole flov/ f i e l d may be c a l c u l a t e d by 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 , see F i g . 1 fo 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 the flov/ i s cons ide red t o be f roaen or i n equi l ibr ixmi,

( i ) Frozen Flow :

For f rozen flovT the c h a r a c t e r i s t i c chemical t i n e T becomes i n f i n i t e and thus a l l the terms invo lv ing the chemical r e a c t i o n s d i s appea r , Equat ion (20) becomes

lit, - 1 rt ± A6 = 0 {•53) p u

and tlius i t i s seen t h a t the flea? w i l l occur i s e n t r o p i c a l l y . This 'type of flca7 occxirs a t the apex of a Prandtl-Jvieyer expansion 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;

For the o t h e r extreme v/hen r = 0 the flow i s i n eqxiilibrixim and f o r t h i s i d e a l case i t i s foxmd t h a t the d i s tu rbance f r o n t p ropaga tes vnth a .

e The equa t ions of motion governing the equi l ib r i imi flow are :

- 11 -

u dp du 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 tha t the equilibrixim concentration c (S,p) has been evalxiated a t

loca 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 in te res t ing to note that for a react ing gas , althougli the order of the eqxiations i s reduced when T = 0 the system s t i l l remains hyperbolic,

The nximerical solut ion i s analogoxis to that of section (5 .2 . ) and as before i t i s convenient to v/rite the re la t ions holding along the cha rac te r i s t i c s in terms of f i n i t e differences, 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 =" ° • ^ )

- 12 -

The the rmal and c a l o r i c eqxiation of stante becoross

Pe = ^ (^ + % ) ^ ^e ' (50)

• \ = ( 4 + c^ )RT^ + 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 ^ ( ^ ^ % ) • (52)

thus g iv ing the tempera ture and d e n s i t y from eqxiations ( 5 0 , 5 1 ) .

To evalxiate the 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 speed must be used and may be ob ta ined from the 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 has been tabxilated f o r vary ing temperatxire and pressxi re .

For the smal l expansion ang les v/hich have been used t o i l l x i s t r a t e the c h a r a c t e r i s t i c method the equil ibr ixim condi t icms appear t o provide an asyniptot ic va lue f o r the dependent v a r i a b l e s behind the c o m e r . Fur t l i e r examples cons ide r ing l a r g e r expansion angles and o t h e r i n i t i a l c o n d i t i o n s a re r e q u i r e d hov/ever t o see the t r ue s i g n i f i c a n c e of those ' asyniptot ic ' v a l u e s .

The frozen and eqxii l ibrium cases ( o r expansive flows have a l s o been discxissed by Heims, Ref. 9 . I n the l a t t e r tlie c o n d i t i o n s behind the c o m e r a re foxm.d by cons ide r i ng Prandt l - Ideyer flow i n v/hich allowance has been made f o r the chemical composi t ion of the gas by i n t e g r a t i n g , s t e p - b y - s t e p , a long 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 .

- 13 -

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

For a high temperature i-eacting gas the flov/ roimd a comer v/ i l l no longer occur i sent ropica l ly and narked diffci-ences fx-om the cla.ssical Prandtl-Meyer flow v/i l l a r i s e . Thus in order to invest igate the flow f i e l d for such an expansion the theory and method given in sect ion (4) has been applied to three specif ic exanples. The three cases for 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 sections are l ike ly to be of more importance in the laboratory, the abc3ve ccjnditions were chosc;in to correspond to those conditions l i ke ly to be encoxmtered in shock tubes. The conditions of the wal l are generally those of i n t e r e s t and. so the v/all temperature, pressure , concentration and Macli numbei' have been p lo t t ed against the non<~dir.-iensional length nieasured from the comer of the wal l . Further to obtain a more general v isua l pictxire of the overal l flov/ f i e ld behind the expansion isobcT p lo 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 in the region very close to the comer there -vidll be a lag in the ine r t degrees of freedom. Generally, the iner t degrees of freedom v/i l l include vibrat ion as well as dissociat ion but in t h i s report hov/evcr only the r e l a ra t ion effects due to dissociat ion ace considered. The vibi 'at ional degree of freedom i s taken to be always excited to jus t half i t s c l a s s i c a l value. For the cases discussed here the degree of d issoc ia t icn i s f a i r l y high (c-'O.S) and the energy stored in the v ibra t iona l modes of the remaining molecules v/ i l l be small conTpa.rcd v/ith that of d issocia t ion. Thus, the assxjmption that the v ibra t iona l mode i s instantaneously adjusted to be in local eqxiilibrixim v/ith the t r ans l a t iona l and ro ta t iona l modes should not affect the resxilts appreciably. At the ajjex of the fan then, the i sent ropic •bemperatx:i;ire drop w i l l caxise the atoms to be no longer in chemical equilibrixjim v/ith the molecules. Thus a t rans i t ion region must ex i s t 'in v/hich the system has time to approach a nev/ e-quilibrium valixe, In th is region the atoms recombine and supply the flov/ with some of i t s d issocia t ive energy. This process continues xmtil a balance i s achieved betv/een the concentrations of the a.toms and molecxiles. This re laxat ion to a nev/ equilibrium le ' /el i s an i r revers ib le one and i s accompanied by an increase in entrojy .

The zone over v/hich th i s re laxat ion occurs i s very dependent on tLe free streaiii conditions ahead of the comer and tlie expansion angle. As the

- 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 approx­imately 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.

- 15 -

Of the cases examined for expansive flov/s of an ideal 'oxygen-like' d issociat ing gas roxmd a sharp comer i t v/as foxmd that the re laxat ion zones v/ere considerably influenced by the free stream condit ions, and for the small expansion angles consideixïd the flow wi l l qxiickly be adjusted to nev/ equilibrixim condit ions.

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 th 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. & Fr iedr ich , K.O,

Clarke, J , F ,

Feldman, S,

Feldman, S,

Fowler, R,H, & Gxiggenheim, E.A,

Heims, S,P,

10, Isenherg, J , S ,

Character is t ics of the eqxiations of moticm of a react ing gas. J n l . Fluid Mech. Vol.4 p . I l l Ju ly , 1958,

Chemical relaxat ion in a i r , oxygen and ni trogen, I ,A.S, Prepr int 802.

Wave-Propagation and the method of charac te r i s t i c s in react ing gas mixtxires with application to hypersonic flov/. Brov/n University Report W.A.D.C. TN-57-213 May, 1957.

Supersonic flow and shock waves, Interscience Publ ishers , 1948.

The flov/ of chemically react ing gas mixtxires. C. of A. Report 117, November 1958.

The chemical k ine t ics 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 Fluids , Vol, I . No.6 1958.

S t a t i s t i c a l tliermodynamics, Cambridge P res s , 1949.

Prandtl-üvleyer eDqjansion of chemically react ing gases in local chemical and thermodynamic equilibrixim, N,A.G,A. TN,4250 March 1958

The method of charac te r i s t i cs in compressihle flow, P t , I . Tech. Report No. P-TR-1173A-ND

- 16

RefoTCnces (Continued)

1 1 . L i g h t h i l l , M.J,

12 . Logan, J.G-,

1 3 . Matthexvs, D,L,

14 . Penner , 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 equil ibrixim flow. J n l , F l u i d Mech, Vo l ,2 p , I 1957

Relax:ation phenomena i n hypersonic aerodynamics, 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 the shock tube of the d i s s o c i a t i n g r a t e of o^qygen. The Phys i c s of Flxiids Vol .2 Ko,2 1959,

Chemical r e a c t i o n s i n f l a v sys tems , A,G,A.R.D. p u b l i c a t i o n 1955.

Hydrodynaitdcs of a r e a c t i n g and re lax ing f l u i d , J n l , App, Phys , Vol .28 N0.4 1957,

16 . C l a r k e , J , F , To be pub l i shed ,

- 17 -

The spee^ds of soimd for an idcs l j i i ssoGiat ing gas

(^) l!?^° jj:.- -Qljnd Speed

To v/rite the frozen speed of soxmd in a forro v/hich i s sui table for nximerical computetion i t i s convenient to 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 in te rna l energy •e ' of the gas to be functions of the pressxire, tempera txire and concentration, together v/ith the fac t tha t the r ea l t i on h = e + £ holds , 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 specif ic heat a t constant pressxire, and C,^

i s the frozen specific heat a t constant volxime. For an idea l dissociat ing gas equation (A3) i s readi ly shc3v/n to be

BZ 9p\

9 p ^ (A 4) S,c

- 18 -

( i i ) I jqxii l ibriumSound Speed

The e q u i l i b r i u m speed of soxmd can , 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 equa t ion ( A 2 ) becomes xmvri.eldy and thus the b e s t metliod i s t o eva lua t e a by cons ide r i ng

a r a t i o of the soxmd speeds .

Consider ing the d e n s i t y as a fxmction of pressxire , 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 equ i l ib r i imi , and v/hose l o c a l entroT:)y va lues a re cons ide red c o n s t a n t , t h i s l a s t equa t ion 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)

Ra ther than eva lua t e i^A ^ t [-^ ) ^ i n d e t a i l , r e fe rence i s m.ade t o \ ' ^ V p , S \ 9 P / S

Clarke (1958) \7here thSs has been e v a l u a t e d fo r the 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 has been evalxiated f o r vary ing tempera txire and pressxiare, and the resx i l t s p r e s e n t e d t h e r e i n are used i n t l i i s r e p o r t v/hen the flow f i e l d has been evalxiated f o r eqxiilibrixmi c o n d i t i o n s .

- 19

APEENDIX E

The thermodynamic v a r i a b l e er

I n the g e n e r a l a n a l y s i s of cheiidcally r e a c t i n g flcrws Ref, 5 , the

v a r i a b l e c p rov ides a convenient groxiping of the thermodynamic v a r i a b l e s i nvo lved , and i s def ined a s

1- ( ˣ, ^^ \ 9 i

p , o

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

P , P J

As p o i n t e d out i n Ref, 5 , t h i s i s n o t tlie most convenient 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 as

9£ 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

4ii .

>.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 • = lotm.

>. C = O 76

\

\ \

\ \ \

\ \ \

FIG. 2. ISOBARS FOR 5° EXPANSION OF AN IDEAL DISSOCIATING GAS

0 - 7 9

C...

0 - 7 8

0 - 7 7

0-76

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_

2'07S

2 -OSO

M, 'w

2'OCK)

I - 9 5 0

l ' 925

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- IO O-IS I

Uoto

0-OS O-IO 0 - I 5 I

FIG. 4 . CONDITIONS AT WALL FOB 5 EXPANSION.

o'SO

0 - 7 8

O-76

0 ' 7 4

0 - 7 2

0 - 7 0

EQUILIBRIUM VALUE ^

^

FREE

+0 =•'! C o = 0

k L V

STREAM CONDITIONS Ootm. TO = 3 7 5 O ' ' K •83 M f o » 1-83

0 - 0 5 o- io 0-I5

l-OO

Too

O-96

0 - 9 2

O-88

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

O C

o

«4

w

83

0 - 8 2

0 - 8 I

o-SO

\ ^

^ • - - — _ _

EC3UILIBRIUM VALUE ' / ^

0-05 O-IO 0-I5 l_ u„t„

2-07S

2 ' 0 5 0

2 - 0 0 0

•9SO

•925

\

/ ^

ECXJILIBRIUM VM.UE —

FREE STREAM CONDITIONS

If a - ''lO atm

T , =37SO "K

Co = 0-83 M f o = l ' 8 3

^

O-OS O-IO 0-15 I

Uo t .

FIG. 6. CONDITIONS AT WALL FOR 5 EXPANSION.

EQUILIBRIUM VALUE-

y^ ^

\~~^^ ,

^ . - - " ^

—•

0 - 0 5

0 -05 O-IO O- 15

0-80

0-I5 _g_ 0'20

Uoto

EQUILIBRIUM VALUE

, » ^

- . - . ^

^

FREE STREAM \MLU • , a' / lOotm 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

Co=0-83 M^„=l -83

1

^ ^

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

UTto

FIG. 7. CONDITIONS AT WALL FOR ICf EXPANSION FIG. 8 . CONDITIONS AT WALL FOR lO EXPANSION