THE EFFICf OF DEPARTURE FROM IDEALiff OF A MULTIPLY IONIZED ,i

MONATOMIC GAS ON THE PERFORP..ANCE OF ROCKET ENGINES

by

John Noble Perkins "'

B. s. in Aeronautical Engineering

M. S. in Aeronautical Engineering

Thesi.s siibmitted to the Graduate Faculty the

Virginia Polytechnic Institute

in candidacy for degree of

DOCTOR OF PHILOSOPHY

AEROSPACE ENGINEERING

Apd.1 1963

Blacksburg, Virginia

I -

I

3.

... z ....

INTRODUCTIO!t • • • • • • • • • • • • • • • • • • • • • .. • • • • • • • • • • • • • • • • • • • • • • • • • 5

EQUILIBRIUM TF..ERMODYli.M-UCS •...•....•..•.....•..••.....•.... 2.1

2 .. S

2.6

2.7

1

2

................................ . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .

.............................................. ..............................

........................................ o • • o • • • •. • • • 1' • • • " • • • fi • 41 Ii • • II • • • • • ,i, • • ilJ

taw of .................................... •••••••••••••••••••••••••••••••••••••••

EQUATION .............. • ............................. .

.............................. for Particle

11

11

13

15

16

17

20

21

µ ................. • • • • • • • • • • • • • • • • • • • • • • 28

.3

METHOD OF

4.1

4.3

5.

5.2

µ = r •••..•••.••.•

••• Ii •••••••••••••••••••••••••••••••••••••

........................... 34

••••••••••••••••••••••••••••••• 37

...................... "' . CALCULA~IONS •••••••••••••••••••••••••••••••••••••

ti •••••••••••••••••••

•••••••••••••••••••••••••••••

Ps.ge

6.. llStlLTS .AND CONCLUSIONS ••• ,. ........... • ...... • ........... • •• • •• ,. • 46

1. smtlAll.t ••••••.••••••.•.•••.•••••.••.• --. ............... • .. _............... 49

8. PllNCtPAla: SDmoLS •• •. ~ •••• • •• ·• ........... • •••• •· •• ~ • •• • •.•. ,. ., • • • • • 50

9. ACKNOWLEDG!fBNTS ·• • , a ........ • •· • ~ • ~ •• • •••. •: •.•••.•• • • ·• • • • • • .. • • • • • • • 53

10. ~CBS •• • .-•.•.• ot1i •••• , .... • •- •. , ~ ••••• -•••• • ..... • • ••••• • ••••• , • •. 54

1.1. 171.TA ••••·••••••·•·••••-•••••~••-•··~··••••••·••••••••••••••••··•••••• 57

AitPDIJ)lX A •••••••. -. •• • • •-•• -• ,, • ........ • •••. ., •• • •• • ....... • •• ·• •••••. • • 58

A.1 Collision lntegra.1s for lnela.stic Collisions ••••••••• 58

A.2 &$action 'tem$ for µ • t:· •............................. 62

A.3 Derivation of Reaction Velocities•••••••••••••••••••• 63

AP'PBNDlX · B • ••• • ............ •· ............ • • , ·• •••• • ••••••.••• • • • • • • • • • 65

B.1 Equilibrium flow Equations••••••••••••••••••••••••••• 65

B.2 NonequiU.brium Flow Equations ••••• •• •• "........ •• • • • • • • 67

B.3 Reaction Velocities for Argon ......................... 68

,Pase

Figure l tllusttation of Convergence of Starting Procedure • • • .. 71

Figure 2

Figure 3

31igui-e 4

Figure 5

Figure 6

Figure 1

li'igute 8

1!igure 9

Figure lO

Figure 11

Figure 12

Figu'te l.3

Figure 14

Figure l5

Figure 16

Ionization fraction vs. Area Ratio

Ionization ft,;1.et;ion vs. Are.a Ratio

Ion:i.zat:i.on Fra.etiO'i.1 V$.. Area Ratio

l:onization Fraction ve. Area Ratio

Ionization lh:act:lon vs • .i;\t'ee latio

' ......... ' ... •· ....... .

.....................

.....................

.................... Temperature vs. At'e·a Ratio ................ , .............. .

'.l:eniperature vi. Area. Ratio • • ............................. ..

Teraperature v-s.. .Area Ratio ............................ .

temperature vs. At-ea Ratio•••••••••••••••••••••••••··

Tempe:1:ature vs. Ate:a '.a.atiQ ................................. .

Velocity vs. Area Ratio ................................. .

Velocity vs. Area Ratio •••••••.• , •••• • ••••••••••••••••

Velocity vs. Are$ Ratio••••••••••••••••••••••••••••••

Velocity vs. Area l:t.atio ................................... .

Velocity vs. Area Ratio •••••••••••••t•••~··••••••••••

72

73

7l~

7S

76

77

78

79

80

81

82

83

Bl~

85

86

Performance calculations rockets :ln flight show that the

velocity of the vehicle at: burnout increases linearly with the specific

impulse and decreases 1d.th an increase the inert mass. the specific

impulse, or thrust per unit rate. of flow, is determined chiefly

by thermodynamic properties of propellant gases, the operating

cu)l"lditions tdthin the motor., and geometry of the nozzle. Since the

specific impulse the parameter linking the exterior: ballistics of

vehicle and the interior design of the power plant, it is imperative

that the specific impulse developed by the rocket engine be known to a

high degree accuracy.

1:he assumption tht.tt the working subetance (propellant products)

obeys ;he perfect laws provides a &imple and rapid method for

estimating the specifi~ impulse a given tocket engine, and bas been

treated extensi:vely the literature (see, for example, refs. 1 .. 3),

While this method does not, general, yi$ld sufficiently accurate

results for practical applications; it does point out the important

fact th.at the specific impulee increases id.th an increase in the chamber

temperature and decrease$ with the :molecular weight of the combustion

product~. High $pecifie impulse thu$ results from operating at high

temperatures and/or low weight propellant gases, Since the

choice of molecula:r weight of the propellant gases is obviously limited.,

the only alternative, obtaining high specific impuhei to heat the

working subttanee to extremely

concept of tr$ating tbe propellant produ<!ts a* perfec:t gas.es 1s no

longer applicable. That is, the eltcitat.ion of dissociation and ioni•

dete:tmirling specifie impulse is no longer a simple one ..

ln there ia only a the effect::i dissociation

ot1 speeif :Le impulse and evren is known about the effects of ioni•

3 10 eeconds 1 or more, ate to oe(:Omle a reality, it will be 11ecessa:ry that

problem ionization

theory for

for the case

its ~ssociated eff~cts on rocket performance

molecular dissociation and recombination

molecular·dissoclation 's theory

adequate; however, once gas becomes ioni~ed, tbe con•

as an ideal gas is not $0 apparentj, However, Bray' s

diverge from equilibrium. was also found that emi.lt'i'Y radiated as a

from adiabatic flow.

thesis to study tbe equilibrium,

flow of an :c•times ionized gas in the nozzle

engine including

radiation. the investig:ation aasumes an ionized gas which is

in thermal equilibrium. (1.e •. , a one temperature f1u(d) and wbieh is

eleetrically neutral. Also, dis.sociati.011 of a mole~ular gas is

essentially complete before 1onizat.ion becomes ~ppa:eeiable, only mon•

atomic gases will considered.

the equilibrium flow eol.ution is used both as a refe1.;ence for

measuiing departure$ from equilib,:ium as initial conditioni for the

noneqt.tiU.brium calculations. The equiU.briwn ca.1culations are based on

a system of algebraic equations consisting of th~ isent:ropie oondition.,

law of mass atttion, an equation of etate, and the equations for the

conservation of raas& and ~nergy • 1:hie system of equations is nonlinear.,

and iterative methods solution. lquilibrium flow

calculations an upper limit speei£1c impulse since a.11 of

the energy invested in ionii.ation. ,iegained,.

The (i.e. fixed eompQsition) flow solution gives the lower

limit for the specific since none of energy invested :ti regained.

these ealcula.tiona are relatively simple since gat behaves as a

'rbe nonequilibri'U'nl solution accounts fot the fact that the internal

structure the gas lag behind that required for thermodynamic

equilibrium., and, as a result,. may be considered as the "eorreetH

solution. The ®n$quilibrium calculations differ :from equtlibi:ium or

frozen calculations in that the composition of the gas is governed by

a different:l.al equation of fi~st•order which will be referred to as the

rate equation. Thus, the calcul~tion the nonequ:Uibrium flow involves

the numerical integration ,of a system of first .. order 1 nonlinear differ­

ential equations.

The derivation of the equations governing the equilibrium, frozen

and nonequilibrium flow ts based on the assumption that ionization is

due to atom•atom, atom-ion,. and atom•eleetton collisions. Since the

methods of cbemical kinetics fail iii. the ptesenee Qf Coulomb interactions,

recourS1e is made to the methods of kinetic theoey. The kinetic theory

.approach con,iats of stating the fundamental kinetic equations for the

ga.s c01.1sidered (Maxwell•Bolumann equations) .which are to be solved at

least a,pproximately. 'lhe solution of the Boltzmann equations gives not:

only the macroscopic equations of mot.ion, but also the equations of

transport for the single particle components, i.e. the rate equations.

The kinetic theory represents,. therefore, a uniform start.ing point for

nonequilibrium flow problems ... lt: tnU$t be pointed out that since quantum

statistics include Boltmnann statistics as a limit:i:ng case, it would

have been more approprLate to use quantum statistic$. 1-k)wever, their

use results in increasing the complexity of the al;eady coinplicated

expressions of the thermodynamic properties and therefore, sueb a

refinement not seem to be warranted at this stage.

The the,:modynamie variables are derived etarting from the appropriate

express.ion for the Helmholtz free energy. ln this expression two approxi•

mationa are employed. The :first assumes that the rth ionization is

complete before the r + 1 ionization starts; thus, three species exist

in the plasma at a given time. ?he second assumes that the eleetronie

partition function., which is represe11ted by e.n infinite series., can be

approximated over any temperature range by a constant. fot the singly

ionized case this constant may be taken a& the degenlt!racy of the ground

state.

rhe de.rivation of the rate equation req.uires an investigation of

the rate$ at which ioniiaation and recombination take place. Ionic

recombination is a t:eh.tively complicated process since the electrons

and ions can recombine in several ways7~ Also, the presence of Coulomb

interactions excludes the uee

kinetics 1 which assumes that

the methods of classical chemical

rate constants are independent of the

concentrations 8• Examination of the Helmholtz f1:et energy taking into

account the free energy reeulting frol'n Coulomb inte,:ac.tions showa that

the equilibrium constant is a function of both temperature and con•

centration. This implies that the reaction rate constant$ .are functions

of temperature and concentration if interaction taken into account.

ln addition to the fact that Coulomb interaction, preelude the

use of cb.uieal chemical kinetics, their presence also gives rise to

difficulties in calculating the Helmholtz free energy and certain

collision integralslt As a result of the long range Coulomb forces

there ,....,. .......... ;;;;> an energy of interaction between the eha:rged particles.

This energy of interaetion is a function only of the distanc:e between

the particles and, thus, ma.y be tt:ea~ed as a potential energy whose

total derivative defines the forces. However, the presence of this

•10•

potential causes certain integrals in the expressions for the free

energy and the scattering cross sections to diverge unless some method

the potential is etnployed. ln the present work, the

Debye ... ffuckle approximation :ts used for cut:•off distance. ?he

choice of the J)ebye•Huckle theory due to the fact that it is the

only self-consistent theory availab1e 9• Because a number of assumptions

t1ere involved in the formulation of the theory" the coriditions under

'Which valid are not well generally

believed that the Debye.,..Huekle theory is valid.as long as the Coulomb

en,~rg:y does not exceed the thermal energy 10 ..

Sample calculations are presented for an inviecid :flot1 in a

Coulomb interactions result in an increa.se the

impulse and a decrease energy, enthalpy, etc~ ;

and an ap1pa1~e11,~ decrease in the ionization potential.. However, the

specific impulse at'(! insignificant

from a practical viewpoint,

more

· to

2.1. Helmqoltz.rree Eners,:.

as the .sum of two parts: (1)

derivation of

One of the

a·gas

F. Given the free

Helmholtz free ~ergy uaa:y 'be expressed

action, FP, and (2) free en~arg;y reaulti'll8 from Coulomb interaction.,

F 1 • That is cou.

(2.1)

the an

(2.2)

where, under the assumption each proee,s of ionization is

essentially complete na,rn?~ the ne~t begins, subscriptµ tal(QS on

the values r + 1, and e. llebye•Huc.kel apprC>ximation leads to 11

263 . ·. 1/2 :, coul. • ~ ... 3 ( k:l:v ) (

µ

2 3/2 N z ) µ µ , (2.3)

for the free energy due :l:nte;1raction. ln Et.\~ (2. 3) the 1µ. are given by

=r (2.4)

Substitution of Eqs. (2 .. 2) and (2.l) i11to iq. (2.1) gives for the free

energy of including Coulomb interactions

F • ... N - k'l V.

Note the Nµ are not :Lndep$ndent of each other; electrical

neutrality requires

M.+N. 1 .. =N • r . t+'·

AU of

(2.S)

(2.6)

(2. 7)

2.,2 Partitionrunctions. :ror all cases considered herein, the total

partition fQnction may ba written as

, (2.8)

where Bi

respectively. Thus, for the atom., ion, and el.eett:on,

(2.9)

(2 .. 10)

and

V • (2.U.)

Equations (2.9) and (2.10) assume that the reference energy level is

that the neutral atom, that it the average value of er E

over temperature rang~ !n which r~ti1nes ionized atoms exist,

Note that ain.ee the Htemperatute boundariesn of the ioniting process

Ar to may be computet:121:1 it iB necessary to compute only the

first few terms tlle infinite series representing e • . rl

Using the above expr.es$10ns for partition functions in Eqs.

obtained as follows ..

defined as

, (2.12)

or,

ki' p -- v

Lett:b1g 11 = N/V be

density the

end

3 k .. ·. ·.. . ·. .f!I. (.· :It p ~ - tll (r + a ) • ~ · · ..... ) m ·· ;,r 3 ld.'

•

atomic nuclei

given by

'

(2.13)

noting

(2.14)

a.re

(2.15)

the (r + 1) .. t~es 1onU:ed

written as

(2.18)

unit mass

or

l Tr + · · · (1 "'" a"" .•. >• ·;-.· .· + t+C\,: • 4

where Tr • 11/k is a "ebaraetei:'i$tic:r temperature of the r•times

ion1eed atom. Note that Tr = 0 for r - 11 by definition.

i = u + p/p • (2.21)

Substitution Eqs. (2,18) (2.20) Eq. (2.21) yields

de.fined as

(2.23)

where r f :;: ,,_ • pV • (2.24)

Using Eq$. (2.S), (2.20), and (2 •. 24) i:n lq. (2.2.3)

k s=­m.

where

(t

composition

atraint.s

Eq. (2.5)

a > r. · i • loa ( .Q , ~· S!. ··1,. + r L2 Q.1n 3..J

l•a ( ....... r) ·<'.'\·

(2.25)

(2.26)

tb~ ionizing.monatomic A$ is well known, eq,uil:tbrium

by lqs. (2,6) and (2.7). Thus, using Eqs. (2.6) and (2.7),

be ,,:d .• tten as

0r J 0fr+ll • log ( u·r·· ) + (N • N~.) log ( · < c ) ..... • N • N r

oe 2 3 . n 1/2 lr 2 ( r. N • "'r. · ) + 3 ~ ( 3 ··3 . . l (r • l) Nr. ,,~. it T. V ,

(2.27)

r 2(r + alt) . . l. I ...... · .. ·. . . a , L (t" • 1 + <-,;.-) ..J (2. 21)

or, substituting the partition functions Eqs. (2119)•(2.11)

2(r+a:r) . . a

(r~l+2 etr> J .

(2.28)

the condition equilibd.uni; is, the lali .of

mass action.

( SU?) ;Iii. ·. d . • ' I) s

(2.29)

emptoying Eq. (2.18)

l . 0 ) } +-.·(l·- . p 2 (2.30)

Taking total derivative of (2.iS) and setting the i-esult

to aero gives (dT/dp)$ of (d a1/dp)

8, which when sub•

stituted into (2.30) gives

8r o . 3 (r • '\-) log ], °'..

1 . . + 2 log (ET) + log .. ( .. 1 • . . )· -

Ot+l,Q · O!t.' t' · . T at p

( .. ) .. m

(r + (% ) ..____ .. r 3(: + a )(3 + a) a ·]. do · a -. · · r. . ·. ··. · ... · .. (l + .......__~ ........... - ) ( -. r ) J· ,

r • l, + 2 ar 6 + cr r .. 1 + 2 dp s

(2. 31)

where, for equilibrium, (d CY.rr;ldp)8

!s fou.ud from Eqjl (2. to be

! 0 + t1) ~ ~ ......... l

6(:t· + ar) a l6(r + or) . 2 -.· .. 2+·. .•... 2(5 (r .... 1 + 2 a,;.) (t • 1 + 2 ~)

1£ .. ·....L.· .. l ... ·T· • ... '···.·.?.... J'"'.' r'/' ... \.· do 1 {. . 't . ·. • ,: ,.-r • VJ (d/\ =p .c6t,>< t+\ r)•

1 . 3(r + a~) --1 •·1

[ 2 + (r .. 1 + 2 a,/ J O' J ' (2.32)

lqs. (2.18) through (2.25) shows that the effect

Coulonib interact.ions,. wu.J1.cu are represented by the quantity a 1 is to

decrease the pressure, enf!rgy, enthalpy, entropy of the gas.

Equation (2. 28) shm-ts that thete an apparent decrease in the

ionization potential.

Equations (2 ... 18) u, i, etc .. as functions

p, 'l, a.nd C\1

• In the case of equilibrium, ~ is

given by law of mass a.etion, lq •. (2.28). l.n tha presence of non•

equiU .. brittn1 an equation for '\:' whi~h the rate equation, is

obtained for the assumed ior.diation mechanism frcnn kinetic theory

considerations. This problem. the next section.

It1 ode~ to oJ,tain 4s eQmplete a picture tlf poss:tble of the flow

of tile iord.itng 34$ tkreugh the XlO~z1e of a r~t engine., it is

ntc.,,uaar;y ~t the ut11u1t to which the flw depart$ &om equilibrium

ha ~t!Uted. As wall Nntio®d previouflY, the -rate equation for the

it•f.times ionis.:tng gas will he detemin-ed from • Jyatem of equations of

the Bol.tanana ~pe~ ln add:i.tion. to col.U.aio:ns which t:esult in no

change ill the nwnber of partielest the followi:pg co1.1f.tioris, which

· at:e aa,umed to b.rtng about the ionii•tton, wi.11 b6 eonside~ed;

0 1 r ,

13·1 ,;

(3,.1)

To .deisortbe the tolU .. sion eq,u~ti~s., a eetni*'quantu:ra•mechanical de• .12 . . . · ·

sei-iption $.$ used ·• l'o; the t.rans1atiori.a1 tt10tion the c1asa:tc-.1

deJaeription ls. t.'etained; but for the 1nte.f1Ul.1 motion, the atCtlls and

ions are eontidered aa betug able. to exist in difftin:-ent iatEJmal ene"tgy

states" !he inten.ta1. energy is deuot$d by et.ti' where., a, beftn'e,

µ stands for an abbreviation for i = N, L (atoms).,

1 = N', l,i (ions). N., N' are the total quantum numbers and L, L 1 are

and

has

£or rotation.

energy eri·

advantage

course, e . ii 0. A .(v ., e . ) e1 r r ·r~ r ... tin1e.s ;,ll., .... ~ ... .,.. ... ,...,...... a.tom with velocity "r

tha state of a part:ic le

total

e,tistence of total inverse collisions

one of main assumptions in derivation the claseical

3.1. J~e Bo1t~mann Bguat;lons. With the abbreviation for the velocity

diin:ribution function

of. 1 of . c,£ . of . = ~ + v . ~ + f (;r, t) • ~--"~ = ( ·~ ) . ,

µ µ oVµ coll.

df where ( d~& ) . . is an abbreviation for the collision term.

coll ..

rate at which the velocity distribution

(3.2)

(3.3)

It

is being altered by encounters and may be exrn:essad as the sum of

collision terms,.

of . (~) =

ot coll.

of elastic

( '<,~1. ) . + · eoll • .,j

'£hat is, the eoU.ieion term may be divided into encounters tn which

the number paiticlei remains conE1.tant (defined here to be elastic

collisions), those in which the number of paiticles changea

(;i:nel~i;tic col.lis.tons).

an ela$tic eneountet, oonsidtat' the eollieion

Jlµ(vµ, ~µi) + Av(v,,., EVJ)~Aµ(Vµu ~\.i'k) + t\,(Vv" '\,p•> • (3.5)

the e~:ptession atµ, .elastic

< ot > dv di dt eoll .. .,v µ

(3.6)

$ignifies the net increase, during

A (v , e. ,.) (without restriction to µ µ. vi

in the number of particles

velocity of the latter

particles). Thi$ net inerease the difference bett1een the numbers

of particles A , within df., which during µ

Qv1ing to encountera the particles A • 'V

enter and leave the set,

nuniber Qf pattieles to the set during tim.$ d.t, owing to

the pa:rticul•r group of encounters particles 4v<vv' E11

j), i$

equal to

f . i f .~ O'(V. E . 4 • V E: ·J .... ,v ... • '.µ";. f • Vv. ·I E I ) dv dv " d:t t df dt " µ · VJ ·· µ, IJ.•' \I 11 ~ • "" ' vp ·. V µ • \I '

(3. 7)

to

fµk,i :i\,p• a(\1 €µk,H V11, 6vp'/Vµ Eµi' VV Elli dvV d~, dvV, dr dt •

(3.8)

Since, number particles

Eqs. (3 .. 7) {3.8)

(3.9)

Defining peculiar velocity of~ particle of speciesµ as the

average velocity v0

, that

...... . ..... ~ :\ .. ~ ...:a,. V (v , r, t, == v. • v , µ µ . µ 0

the velocity distribution functions may be written 14

where

u,1ng lqs. (3.11)

requires

then,

. m . .. .. ,

"'._.. . ·. ~. V . + 'i . j k! 2 µ tJ.l. [ 1 (" u .. 2 . ) l

exp ,.....,. ______ ,.,._........,....._~-

e ._ 1 exp(--~l kT

f 1 .. (v_ > • µ' ,µ <v > • µ

(3.10)

I (3.11)

(3.12)

(3.13)

q(v .. e .·..t.' µ µ ....

Usins (3.14)

(3.14)

Eq. (3. 8) ""'""'""'·"b '\-11th Eqi. (3. 6) and (3. 7), the

probabilities fQr be obtained by a

generfl~liiation of the above s.ri~~tui £0-r t.al.af.ttie encounte,:t. As an

~ampl.e of an inelastic. coll~ef.011,

By analogy with Eq. (3 .. 9), the condition for

be local ,quilibrium

(3.16)

~eaetion (3.16) to

(3.17)

Subst:t.tutins Bq.s .• (3.11) into lq. (3.17)

m · '" 2 .. . ·· )· . ( .. e 2 ) ( y(r+t)• + £(t+1)p• + 1(~1) + · .. 2 "e• • '

(3.18)

~r:~ ne • g ( 2gr:;·o·) ,:3/Z eltP • ( T~! .. Tr) .. (;~;;~) a] '

i:r r(~l)e· cr~(ttl)e = A er ,:,: . ,

where

rr c e

i-(r+l)e

I:lavi11.g established the t:eJ~a1;~or.ls between the t,!'~~'t'\a'i

probabil.itie,, £'1111 expression of 1 .

:( ?,~ · ) . . • •. . . , · may h~ written as coll.

( d.· .. f· Y .. 1 )··. at · · •· • eoll.

th~ collision

J (et) µ.i (a)•

(3.19)

(:l.20)

(3.21)

(3.22)

(3.23)

with the following abbreviation, 14 fot the elastic collision term.a;

(a) (tr) (rr) (re) J sJ.,. +J ... +Ji ,

t.i (a)' r1. (rr) • (r(r+l)e)• r (re)' (3.24)

(a) ((r+l)(:r:+l)) (r(tt-1)) 3

(rtl)t (a,),= 3

(r+1)1 ((r+l)(r+l))'+ 3

(r+-1)i (r(~l))' +

(b+l)e) J

(r+l)'i. ( (r+l)e)• ,. (3.2S)

(a) (ee) (te) ((r+l)e) J . $:=J ... +J ... +J ' a (a)' e (et) • e (re) ' · e (rt+l)e) *

(3.26)

(a) (rr) (re) (rb+l)) J ..... 1.. s 2 J + J ·. + J.. +

4'. (fl)' rt (i:(t+t)e) • J>i ((r+l)ee.) • d. ((:r+1) (t+l)e)'

(r(r+l)a)

(rit')'

(a) (r(r+l)) (r(t*l)e)

J(t+l)i (f;i)' = J(t+l)i ((r+l){r+l)e)'+ J(r+l)i ( rr)•

(h+l)ee) ((t+l)(r+l)e) J 1.~1) • · · + 2 4(.....t.l·) ." ,

\P' i ( (r+l)e). I..T. l. (r(r+1)) I

+

(a) (re) (r(r+l)e) ( (r+l) (t+l)e) J = J . + ·J . + J • e O)• - e ((r+l)ee)' e (ri-)' e (r(r+l))'

(3.29)

(3.30)

(t) (3.31)

etc. wm;r'.e ,-·; means the summ.a.tion L, (i)

taken indices except i

(of co,urse, the subscripts r., r,,t,-11 e are not .... ....,,..11,p.1.·,1.n: .. .i.,;;:.u to be indices

following properties:

(3.32)

collision integrals.

3.2.

Let fµ-1 some p,:operty of the patti.cte Aµ i fJ ,. (v , r, t" G .i) ; ..... . ', . . . . . . µ1· µ· . µ.

µ = r, t+l, Eh ''the mean value of properties iµi is d$fined by

(3.33)

r (3.34) ,J

•\1.iµ are tl1e densities of the property iµi iu f•spac,a. The

eqvations transport for mean val~e (t\,. follQW Sqs. (3. 3),

if thetse equatto.n.s are multiplied 1\.t.:tdvtJ., integrated Qver vµ, and

summed over tc

r ·. .· of i ..... J t .. 1 ( ·~ ) . .. dv.. • .

" f!'¥' · ···· coll. .f""

(3.35)

'l!he e~lUations of trmlsport fo:t the gas are thu$ obtained by subst.ttuting

f .. the quantitieEl µ1 .

2

equat:i.on

+e. 4 +1 , µ. µ. (3.36)

elf • ( ~. ~ .. 1

) . dv , 0 coll. JJ.

(3.31)

or

on. d ~. ·. . . + - • n (v + V ) ;;:· R , at ot µ · o µ µ

(3~38)

V • V • V µ µ. 0 (3.39)

is the diffusion velocity of tlle species µ 1 and

r of .· ( I ~-sM\ ) . . dV

d ot coU •. · µ (3.40)

of produetion of ps:.rticle$ of species µ. Sin.ee only

the b~e1astic colUsions, lqs. (3.1..)., eontd.bute to 1\,., Eq,. (3.40)

may be written

a • µ

of . inelastic ( ~.i >. . dV , at . 1·1. L fJ. co .. •JI.\.

(3.41)

(3.42)

The ecf!ations (3.37) are not independent $ince conserva.tion of mass

requires that

µ

•30·

tn R = 0 , µ µ

( r . .,.1.·.·) n ·. + rn ... · .. l. "" n.. = 0 • r ff e

(3.43)

(3.44)

l.l. 111.a· Rate Bquation for the Species µ = t. Since the f ·i· are . . µ

as.sumed to be Maxwellian1 V ~ 0 and lfis. (3.38) may be -written µ

(3.45)

(3.46)

(3.47)

Substituting Eq. (3~46) into lq. (3.45) gives,

D(m n ) .·.. m. ti Dp MMr .. ~~=mR

Dt p Dt µµ • (3.48)

Da .......!. = •. :Ot .. (3.49)

I ij

(3.50)

(3.51)

the "co111tant" o.f the law of mast action defined

lq. · (2. 28). the quanUtie,s urt' ut·(,utl)' ure. are the ret.t.etion

velocitiet and fit'e. g:tven by (sae Eq. (A.24))

whete la;.t-1 • lvr ~. f,r\ , tte • ., Ql)d;p the Q't •• the co1U.ston croes

-.ctions def1n$1 by

ICn:J ~ "'J '\,(=l)• .tfr; dV(r+l)' a;., , (3,SS)

rb'+l)

I '"(JM.t> I ~<i-+1> • f ct ... ·1 ·. . .•.. ,, <i.i+1> • 8<.+1). ,. "• •. ·~ . .. (t.+.)b+l)e. (3.56)

tbe crosa•sections are known for the. colU .. sions in

question, Sqs. (3 .• 52) .... (3.54) be intagt'ated to yield the reaction

velociti••· Once the ,teaetion veloc;it:les have been detet'nlined tbe

rate equation, Eq. (3.-49)., compl$tely specified. ln t;he following

seetton ineocyoi-ation of this rate equation with the quaei•one•

dimensional flow equations is along with the met.hods of

solution for equiU .. brium. and f:ctnien flow case$.

This section sets down the method of analys:h, for the quasi.""'one•

dimensional equi1ib:d.um, frozen, and nonequi1ibrium flow of the real

r--time$ io1;d.2ed gas tbr<>ugh '1 ccnvel:$ing•diverging noza1e. for each

ease conside:red, the neceeaaey system of equ.ath'>ns written down by

a suit~b1$ combination the one•dimena::lonal £low ~quat.ions with the

equations of 1ection(a) 3 and/or 2. The present investigaU.on thus

eon,ists

order.

t.h;ee separate part• which wilt be treated in the following

these flow

properties necu~s-sa~ not only to gain an understal.lding 0, depart•

u1:es equiU,brium, but to obtain ixd.i:tal conditions for the non~

equilibrium calculations.

'?he ~quilibrium calculations are l>ased on 4 system of nonlinear

algebraic eq,u~ttons c,onsisting of the one•dimensiona.l expressions for

coufJervation of mass and energy;

(4.l)

(4.2)

togetlle't with the thermodynanie t'elationships of lqs. (2.18), (2.22),

~d (1.2S)l

(4.3)

,., ... r ( ; ) • j + ! l log (DT) +

and the law mass act.ion, 2tg. (2.28)t

T .. •i 2(r+a) ( . £ti ... t ) - .. . . . ··,:

(t"• 1 + 2etr)

(4.6)

where quantitie.s at a sonic throat are denoted

:s0

are the total entbalpy and entropy, ltespeetively, which are const$:lt

for equilibrium flow.

p* v* A*, has been determined. the flow is isentropic ,.

speed sound at point.

g .·.. ··.·· 3 ( .... r,o..,_ ) + _ 2sr+. . l. ·. o a ,

where., (dar/dp)8

is

a. simultaneous Qv.i:,.u.t..11-1.1,.~

is then calculated

are

(ET)+

41 (4.3)

t:emperatute (T >

downstream of the th11oa.t)

simultaneously for a

The velocity, which

from Eq. (2 ... 31)

(4.6)

da ~1 < or:{ > s J'

(4.7)

p* are speeif:i.ed,

...,. ....... _.;-o r? and p*.

(4.4)

Eq. (4!0 S). 1111.ally, the.

the noazle

for upstream of the throat., and

solving Eqs. (4.S) and (4.,6)

$pecifie impulse,

fol.101-,s fr.om Eqs. (4.2) and (4. 4), and iq. (4.1) then gives the area A 1 •

Stagnation conditions a.re obtained from a siwltaneous solution of

Eqs. (4.4); (4.S), and (4.6).

4.2. l'ro~eil Flgw £9.taagionp. ,;b,e f~ozen f-t.ow calculations provide the

lower limit to the ~pec.ific impulse. the. composit,icm is astumed t:o

remain conftettf!, m<i thus, the f:r11n:ep. flow eal.eulations a.re identic.$l

to the pet:fect; ~u ea.1culatiQnS. !hat is, for a specified value of

11 pi& ealeulated directly from iq. (4.5),. l?he remain11,1g state an(l

flow properties folJ.ci;1 from Jqi;,. (4.1).,(4 •. 4)* Note that the ca.lculations

a~e 4a:td.ed out ouly for tbs:~ portion of the noia:ale whtch :i.$ dow&tream

of the tllJo~t., and at'e lla•~ on the equtU.brium tnaa!ll £low rate.

4.3. Hpnawail1'briµm !1• lifat,ione. Jf the ~eaetiOll t~a for the

eo1l1$1onl given by lq$... (3. l) ·l:W~ of th$ ,_. otdei as the chal!'4etertetic

tren$it time th~ugh the flow belng considered, tlle f1owwi11 depart

from e4ui11brtuin .. · As a result, th$ f \ow, while ad:btbat:tc, is no longer

isenttopic. '.thus, the eq,uatl<ms whieh de1;1cribe the nonequilib1:'ium fl.ow

through a nozie:le of ;lowly va.~tng c:toss•sectional a3i&a 4\l'.\lthe eon ..

Sliltv~tion equ$tions (4 .. 1) and (4.,2) and the equation of eonsetva..t:ion of

vdv+ldp•O I) . (4.8)

together with the t:be,:inodynamic relatienthips o.f lfis .. (4,.3) Ci\nd (4.4)

Bald the r~te eq_uat;.1® (l<b (3.49)) which., fci- •t~ady 1 one•.d:tmensiou1

fl.ow, •1 'b<! writ.ten in the fem

do: ..... p . r J=[. v dxl • (; >t, (1 • o:t) urr + o:r u-r(r:t-1) + (t: .. 1 + aJt) ut-e .. {r .. ar)

• Kt' ar (i-•1 + a,:,> J , (4~9)

where xis the ~ia1 distanc.• n,aqured from the throat.

speeified, so

the area A' Eq. (4.1).

c:!1".l!:l:nn.tll ~ in Eq. (4. 9) may 'be CQ~reb.ted with

a no~zle with tlle hypE:?rbolie

(4 .• 10)

(4.l1)

b

at large values

(4.12)

(4.13)

·[ 2 ! . . . . . . ' .. •·· . ' 'fl-... ' ·' . . ... ... J.., . ·. . · · • i - {~)c, + 3 .• · .. ·· 2 2 .··

(. .a... "" ... >· 1. 1 + .! ... )· } J!d_ {· ... ~. 1 "· ···· .... · ·., · , ik~ml ,S, £1:tt il . · [ 1 +· . t' V ""t· '\ .a . 3 V•. ·. 4' T . .. . . .. . . . . .. . . ..... · . ··. . .. _ 2

('t + ~2 ('· + _a. \:.· '1.:· j ! ,4 if! ,.

(i.J V

i . I .. f 3Et • a > 1- .. # .. !'+ , i r )·.. . . . . L l .· .. ~ .... ; ~ . . i. ·. ·.tr, =. l!ffl .. ·. . . .; ·~. ) .. · i ' · · · , · · "' r ;;.· · , + ~ "' • a "" "' ,,..r,+.l • ·~~ ""'

(4.14)

p ...... ~~~--------------~~-- ......... ....._-------------

2.. : . . k ··[ 1 ... · . . . ·. . . . 4 . . ] L112 (l+g ) 1''~·2 (;). 2 f~) + c:t,.::rt+-1 + (l~t) T'*l.• 3 (r+o:r)Er J

(4.15)

(4.13) (4.14) are .,...~ .. _._u~..,U. for ar and 'r

by nnt.1tJ.er:tclU, 11cu;;eigr,at:1kon

Eq. (4.15)

readily possibl~

difficulties

constant M

not occur at the tbl!'oat, must be

a ltunge-Kutta pr<>cetu.11:e employing

are then

must be emphasized that although this

principle, in actual pract;ie~ ce.:rtain

the solution that the

ftom a. tedious iterative

ealculatton pro¢edure 16, *?his ..... N'ol• ................ ...

that flow may be tre.ated. as a.n

equilibt'ium flow unt;il a point downstream of th.e throat reached ..

However, the :rate equation (4.14), the

above procedure gives riie to a second difficulty11 Since Kr is the

equi.libriUlT! cous~t, then ·i=. 0 for equilibrium flow. '?his conditio-u

has been to lead problems when the integration is

the equilibrium flow eonditions1 5, 16; '?his difficulty

however,

be satisfied at

tllia

s.ection in connection with

a:rgon.

conditions

(4.16)

end each integration step. A

procedure given in the next

As an, illustra.~ion of. the proposed ~naly:$if, t~ flow o~ At'gon

th.Tough a ~oi~le Q~ sl.ow'lyvaeying c~ss•$erzt:toual _.ea for the three.

cases of; equU.tbi-:tum, frl;)a@,. 4nd m;,~qutU.b~i.Utn ts con,i4e1:ed.. Since

a a~h of tbfl pteacmt day 11te~atui~ ievealed that no eoiJ;tstort

c~oss s~e;ion. data was ~vatlab1e for :more tba.u singl~~stage ionization,

all numed.cal ,;~u.tationa •e. for tha c.alffi or r • l. ~ five

txam.p 1e.s considel:'ed ¢orr~pond t<> the fo1l<>Wf.ng tempera.~,;es and

pre,,~ure$ . at the ndntnun eros$._,·•eotd.onal 4lZ.8a of the nozale:

Case. t <>(K) p (a.tm~)

1 1!000 O.\ 2 13000 0.1 3 14000 o ... i 4 14000 1,0 5 14000 10.c

tt :ts tho~t lbat thue e!($1.iplt$ give ,:epyese~•ttve ~su1.t$ for ne vwt~Uons of thE) effects of Cou10tllb i~taetiona wtt:b temperatuie and

pl'e&sur.e tor the ea~ of r • 1.

s.1. l9iillt?•,:Lmn ,.tn1J IJ;pzt1 Jo,\uti9u. tb.e equations ot Section 4.1

(Jpetd.ali:t•d for t • )*) h.ave been fll•J.ved by ite1:,-ative methods,. using

.. ~a•~•. U!h• corresponding •ttdeat.•• JOiutions (er l',9 0) have. also been

found .. ' 9\e value cf the ioutli.ltion fra,c:.t1.on va,:ied f~ appt®tf.tnat$1y

O,.S to o, and ·tae .. tmwn. value <>fa w-.s i<>u® tq b~ apptOl(imately

O .. tl2.$.. fte mt:tld:iaum n.~i-ie~l di.f.ftt'ellCe !~ the t-eal aud ideal sq1utiop;s

w~ fQuttd to be a 12 .. pf!.'.f.e(!nt vm;tat.i.~ tn the temperatutE¥. 'J:hl$ led to

f:he oonc1u;ion t.hat, for singfy,•ioniaed ~e•, tbe Pl"th!lCe Qfflo\J.lom.b

intff&etiona ean have 4 n<.J~tc.•a1>1e ,it,et on the t;:Jter:mad:vn•e,

pro~i:tf.ee if equllil.bd,\.tm flow 1, acltteved«

s.2. ~~bill!! !olP,tion. $olution; of the dif.'fer$ntial e1u-.tiqns

(4.13) and (4 .. 14) (epeeiali.z~ foi t • t•) ~re been fo\lnd numerically

t>y a foutth•ord¢r R~~kutta method, which included •trapolatic>n to

ae,:o 1ntetv$1 size as~ eonEW:tton fS¢tor, using an l»M 7890 digital

computei fen: th~ rea1 and idea:i eas$&.

!fhe coU.iston Ct:0$$ Sftct!o.ns Uled for the, atom•atoa.l,. atom-ion, ant\

atom-el~tron co1U.,ion, a.re (s1u~ AppentU.x ll)

(S.1)

(5.2)

2 ~ m le I ~

Q. = 6 0 .. · 10.., 22 I e le • T jl le · • · x · · l 2k exc . . ' (5.3)

respectively. Thus,. the reaction velocities become (see Appendix I)

3 /2 T .·· " •i' ··. 12T .. s 'k!l r .... exe .. 2 . 1 exc u12 = S.5 :& 10 ·• . 112 ( - ) + 8 j e , (5.5)

(nm) L T

and

• (5.6)

reaction velocity u18 contributes

to the rate equation. Thus, the expression for the rate. equation,

assum.ing :: 1.0 om2 and b su¢.h that e • 1.so,. is

(5.7)

.. 44 ...

As ll1eU.t1o~ bti(tfly in the plreceeding sec.tion, t:h~ equU.ibri.um

s.olut1on$ are uted to st~t; tba iid;ag;aeion. the sttartins pttQCed11re

that is u$ed to join. tb~ two solutions is as f1;>ltow,,, Ufing

eq,uil:liu:1• vatu•• cort:$$vonding to .~· point Jqs~ dowstl'eatn oi the

th1t'oat, au ~tt~t :L$ 'Ul.ade to eta.it. •ne intesx-at.tcm.. At th~ and of i:be

fltst i1tt$g1:"at:t.on $tep 1 the eo.tlditi.onf (4 .• 16) a,:,e ~becked. lf either

c,f tn,.,, C()$liti0tis it vto1a.ted1 th• :tnteg;ation :bi atoppe<J •m a

poi:nt aligbtl.; futthe~ downstreem·it: tried. It bas tlteri fomld that

onee the conditions (lh16) ~e simultaneouJly s.aitt1tfied 1 t:;Jm integration

will. proeaed d()tffl the ll()Zzle with no t$ndency :fttr tnatability to

dev&lop ..

tt: :ls to be poitd;ed cut th.at thil atQ.rtlng point ie 1 to some

extent, d$pen~ntupon 1:he number of significant ziguies eari-ied in

the comput$t:ions. Wherefore., the st:a.i:tiug pt"oetdtn:~ described above

does not a;:tve uni4ue re.$t11ts i~ the vicinity of the ste,rting point ..

Howavai, itwl:is f<1und that if the integration wa.i:t at:atted at a point

slightly downttream of the. :bd.tial starting point, .th('!· two solutions

rapidly l!!on.versed (see figure t). For the cases considered., the

aveJ:ase range of t over which t.he soltJ.tion. could 'be· st.a.rted, and sive

results independent of the stat'ting poi11t, w,1.s found to be approximately

0~06.

thiing the method tf solution described jbove fot' the non-­

eq,uilibriU111 fl.ow, th~ ~i~ numerical di.f f$renees between the teal

•d :tde•l solt.ttions wete found to be. less than 3 pereent for th.e cases

next

... 45 ...

actual

view ..

are :tnsignifi•

uJ.:~cin.u::i.1.on of ·these

'1le tesults of the analysis and catcuiations 4re p~$1;1$n~d

gr-.phi..e31ly. figu~e 1 tbows the coll'V'$t.'8G~e of the intes~ation

solution uai:ng dUfetent st~ting points fl)r a typ1c:it1 ease.

figures 2•6 show plots of a vs. A. Jigures 7.-U. give thf vafiation

of T with A, figur.e.s 12 .. 10 show plots of v vs. A,. which, since

1,, • i , itepte,sent tb• vt.riatton of $peeifie impu1ae. Xn all plots

e~lus1:ve of Figute 11 t~sults !ncludt~ the •ffeets of iatera.ction

at$ plott;:ed - ,atoU.d tines., Ca,e, whet:~ only solid lines 4ppeat

indic4te that th$ i:esults wt.th and without Coul.otid) tntei;-aetione are

ee•entia.lly td,nttea1~

!the wtesu1ts indicat• thit f<;;'t tile c:on<l:lt.1Qns und~i eons.iderat:t.on,

the speeJ.ftc 1mpub$ :l$ •1me4t ~omplettly unaffected by the. presence

of Coulomb 1nt.el:"•ct1()n6. th• ealculat.ions cattt.ed out indicate that.

the -.~:!mum eor'ttct1on ts l$s$ tbau 1 p-.rceJ*;t. Alaot since the

•f:fect of negleettag Coulomb interacti.Olltt is .to unde.te$tin.late the

.apeeific impul~, ide4l gas c•l¢u1at:tons raay be consideted to give

conservative. tesults when used to pr4dict t'Oe~t engine. performance ...

Note, h~r, that itgur~ U. shows. that 11eglecttng tnte:r:actions

for the ~quiU.b'rium flow results. in unde'testimati~ the t$1UPeJ:"at.ure

by !lppro~:bnately 12 per.cent at 1*n area i:atio of one hundred. While

this hae little or no ed:fect ot1 t;he performance of rocket engiues, it

would be of importance :ln -.11•tns data obtained from a hypersonic

arc jet facility. Figure 11 also that interaction ef:Eects are

preseu,t in nozzle flow after the deg~ee of ionization has become

negligible. That is, the two solutions do not converge as a approaches

zero. Too, it can be seen from Figure 11 that for A> 16, the equili•

br:t1trn is less t.he nonequilibrium temperature. The

e::q,lanation of: thio lies in the :fact. that since a appro,;:i,:::hes zero

there a:ce 110 J:eac tions place the equilibrium flow and, as a

result, the temperatcrte g:a:.adient is ve1:y large. The effect of th:i.s

large :i.n temper,:ature on the veloc is to mako it cipproacb. its ,.

value of ,J i 0 fa.ster.

tl1at tl1e reac tidn ty usl:'ld in the anslysis is

correct;, the wonequil:tbrium results represent the actual conditions in

the nm:~zle., and consequently the equilibr:i.um and. solutions

merely deJ:itte bm:m.daries c:>£ problem. 1'iote, however, that: for all

down.stream 0£ throat. This conclusion comes

from the of ' . t:n-e pi:ocedure for starting the integration.

Finally, the analysis ind3.catcr; that the or<lel'.' magnitude

of the Coulomb corrections. For example; at A = 100 1 Figure 11 g:i.ves

the percent correction in T too .. 5

:;; O. 20 ::t 10 to be

a.ppro:dmate:l.y 12 percent., whereas Figure lO gives the percent correction

"" .? c.orrespond:tng to a = 0. 25 x 10 '"' to. less than 1 Also, as

noted a.bove, Coulomb interactions can influence the flow of an

ionb:.ing

concluded

parameter rs

deviations

lt muse

hold only

of r n.,,.,~l!);\"'A'l"

of more

.,.,. ........ __ a has

a is

not sufficient

an ideal gas.

emph~si~ed

ease

one,

However,

small. Thus, while can be

0 smallU, t:he

actual

above results eonelusions

:ta to that for values

Coulomb inlt'ieX'.8.Ct1<)ns v:rill become

demonstration w.ust . wait ........ , ..........

multi•st:~ge ionieation crios:3•!U!!c~t:1.on data is

7. -

Vsit.ll the hi,ye-Bueki• 4Jprn,cimat1® 1 the e·ifct~ta of t:oul.omb

iti.'t'erae.tl®s on .th$· tM1u:tU.brtum" froeen, $id ~tiectuU.ib'd.um flow of

~ 1-i•ed a•1 bavt b~ uwtsttgated. ·fhe gas :ia as~d eo k

®na•te1, el.ae~~!eally neutral, and- in tbetrltd equi.U.\»rjum (i.e., •

one t!~etaturt flu.id) a but th• eompo$iti® of th~ 1• i, •td.tr~wy,

thai :ts1, tllUlUple ionuat!o~ of •:r 4egtee 1,. 4l1ov4d.

·~fil thet"mO~~ ~iil!t.b1es •~ derived iJtai-tt~ ft:a the

app1:0prlilte. exp,:esston fo:r the hlmbo:J.:tt• fr~e. eAet"BY~ Using Boltzmenn

st•tttstJ.cs ~ uawm.na that t,.,. velocity di$1!:d,bution fu®t;t-one •~

given ·'by t'ne:lr -.111an vdueJ, the tat• of twi•tion ts t.ied,ved

:for ab:ml•atQll1 atom-c1on, en4 atom~\\l'1et:ro..ti colliatona.

'?he ~$1:.tlttng ~--stone .a,;e tben employed il\ soJ.vina t~ <,tua$i•

one, .... d~t~ flow !n • con.v•si~•divfJgtq noot.e fo~ ti. equtU.ltrtum ,

fl'o•e.n, and not1tqutU.brl'Ull\ <;... lumertcal tx•1es# using &rgOIJ, as:

ti. w~1:kf.ng subs~e, fte dis~.ussed and the reeulttt pt't•t11ted St'aJ>-_:l.ea11yl'

·':ftlf\l ~suJ.tf of tbe$e 4a1.cu14ti~n$ indicate thai, fqt tiing14\ :l.otdeation.,

tbe effect of Cow.Otm) :i.nt.tu:act:Ltms cm the peitf~f.lt of to~t •~ines

iJ1 negligibt..J but that datf.t ttbtatiied fl'Ol'l.l byper$0n1e ~ jet; wiadw

tu:IU.1els e• be ti;ntftcantly inf1uell'Ced br the pt<,uuim.~e <>f tee bltet'•

t1.Ctions ..

µ

n

h

A µ

N µ

V

N

m

particle deneity,

Planck's constant

Helmholtz

particle

energy

µ

particles A µ

a pa~ticle A µ

number of atomic nuclei

1; 2; 3; •• 1 = a.toms;

rnase . of atom

ionized atoms;

g,:ound state degenercy for e1eet·~onic partition function

of a particle A µ

of

degree of :loni~ation

dimensionless 1uantity defined by ltq" (2.17)

u

i

s

D, E

a

a, f3

-V µ

e µi -r t

fµi

~

rµ

a£ .. (~)

dt colt.

(a) J;.

µ1;. o>•

(a)

Ji µ (a)'

rr at(t+l)e

specific energy

specific entropy

constants defined 'by lq. (2.26)

speed of sound

coordinate system

internal e1nn:gy af a p~rticle Aµ

a fixed Cartesian

position vector in a fixed Cartesian coordinate system

time

fµ (vµ, Eµi' $, t) distributiQn function for particle$ '\

With internal ene:rgy eµ:t

collision integrals particles A µ

abbreviation a $Unl of collision integrals belonging

to the Boltm.nann equation particles A where the µ number pai:ticle$ at encounters changes due to t:<eactions

as be.fore, but where tb.e nwnber of

eneountere

transition probability

Ar + Ar_... Ar • + Ar+1' +

v 0 ~

''t.J. Iµ

E .. µl.

l\,. t

fJ

K

u rµ

M

ielat:lon totil forward and total t'e4rwa-i-d

transition probabilities, defined by lq. (3.22)

tJ 1(v, a • :, r, t) some property of the particle A µ .. µ. ···µ1 µ

masa avetage 0£

peculiar velocity of particle

ionization energy

energy of

a particle A µ

e. particle A µ

constant

partic1El Ar relative to the

coU.ision cross-section def:t.ned by lqs. (3.S5)•(3.57)

total electronic

length along centerline throat

a particle

4 particle A µ

nozzle measured from the

<!imensiot1les$ distance along noz~le centerline defined by Eq. (4.11)

specific impulse

is desire the his appreciation

to Drt! A. Ras&an, Mechanical Engineetbtg Department at the

Notth Carolina State College., for nun1erou1 suggestion$ tnd patient

ass:tst:ar1ce in the successfu1 of proje.e.t~

'to Dt. J1l J. lades., J,:., of Aerospac.e Sn.gineering Depart•

mant at Virginia J?olytechnie Institute, wish~, to express

appreciation hi.s and consultation$.

l. f. J~ Malina ~ C1.Wast•£llt&sa 1of .Ra,ag !!qtOtE Jlei~s II,~ !J.f1 $'l

~ of ttrfesti !l41ii, Soutnal of tbe huk1ia Xaat.,. V<tl.. llO,

Ito. 4, 1!140.

~. u. s. &eifut • I!!&@ ISR!melou, John W:Ucay Ind SoM> tnc.,

New to,:kji l9S9.

3. G, ,. Sut~Oll <jl,· ltett -~11.0l'l(JJ.eaast, John Wil.$31' ,md Sou,

In¢., tiew Yo:rk, 1956.

4. a. 1. o. lray • naut5,9,r1. ,,. pisg9t9yp. ,11911,~;- in ,a

Dz2iaU2D&s lu.lkh A.a ... c. 19, 1,983, Mwch ltJa.

S .. M, I:• U$bthill • ~I gf a tis109iaJiy .111.t Part l, •

'SJ"&hJ:&• Vmt J. fluid Mech., fol. !, l'art 11 ... January l9S7.

6. It. N. t. Bway amt J~ A. w1t11on • & :,rs~ §tu!&% o.f lon&'r

~'91£~ ti .AtSQn. \1l li!!i1,lmle1 MialQi, U.S.A.A. le,t.

Uo. 14~ fell.\ 1960,

1. :,. w. sou<t 'I! !lmbn . 13i1,, t siws1 11;og1a. AJ.atn, lhysical aevtew, Vol... ios,. 19S7..

s. s.. s.. 1$nner ... ~-am~1ton t~ ea !b.tv 2S .. SJ!tekf!l: le,stw@ &a 11ft S11t1.1.1, ACA1$o&l."4Ph *· 7, lutte~~th Sct.euc1.f$.a lubl:tcattons•

1'>ndon, l.9SS.

9. R •. Fol11ex-$Ud E. A. Guggenbeim .- Statistical 'l?hE\rmodynmpic$,

Cambridge at the Unive.rsity Pre,s, New York, 1956.

10.. a. Fowl.er • Statisti~al. Mechanies, Cambridge ,t the Utd.vei.-&:tty

Press, New York, 19.55.

11. L. D. Landau and I. Lifshitz • Statiatical siot, Addison•

Wesley Publishing Company, Inc., Reading, Mass., 1958.

12. e. S. Wang Chang

Polz~tomic . Gas~s, Eng. Res. Inst., Univ. Michigan, ItepOtt CM 681,

July 1951.

13. S. Chapman and t. G. flf)wling "" ~e. ~tbet®ltical 'l'beog of. >ton•

Unive,:sity iress, New York, 1960.

14. G. Ludwig and M. Heil • :BQundaQr.-Lax;e,: .. 'tbeog. with l)isso.eiation

a:nd loni~a.tton., Advane.e$ ii,; .AppU.ed MechlUlice, Vol. 61 Academic

Prass, )iew York, 1960~

15. G. Emanuel and w. G. Vincenti • Methoc.i for .Calculation of the One•

Dimensicrnal litoneguilibrium Flow pf .. a generail Oas Mixture Tb:t'ough

a,H:z.2ersonicNozz1e, AEOO•iDl•62•Ul, Atnold lngineet'ing Devel.op•

Center, June 1962.

16. J. G~ Hall

H 1'02;;1.e , Co:rne11 Aero. Lah • ., !ept .• No.

AD•lUS•A,.6, Cont. No. Gt 18(603)-..141, 1959.

17. H. Petscbek s. Bryon• AJ.!J!?;Oach.to Jiqu:tU.brium lonima.tio:n §~bind

Strong 6-hock wa11s in Arson, Physics; Vol. 1957.

H. ~ier•Leiu,;d.tz, z. Jpys1,k., Vol. 9S, p. 499,. 19.35.

19 .. A. von Engel• ]pni2;;ed Gas~s, Oxfot"d at the Clarendon Press,

Phenomena, Qgford at the Clarendon Press,: tondon, 19S2.

21. R. w. 'tt'Uitt and J. N. :Petkins • fhermodz;nand.cs .of t:ne Ideal

r .. t.imes :toniZE Monatamic .. Gas I Developmet1t$

Plenum Press, New Yo1:k, 1961.

Mechanics, Vol. 1,

22. J. B. l(nc,ff • ?he Bo~t!!ann B9uation and Non.eguilibrium Pheno•ena

lnstitute

NQrway.

Theoretical Astrophys:tca, Oslo University,. BU.ndern.,

1:be author was born

attended schools

'!he £ollotving fall

tmbse.quently g:rl:lduated

l.l.

award Bachelor 's uej~t'e(e

school at the

on. Api;il. 7, 1934. He

completing high school in 1952.

.i..4;.i;:;.,1,.,1,;4g. Polytechnic lns.titute, and was

1956, his Bachelor of

In July, following the

he eniolled 111 the engineeii,ng graduate

In September of 19S7 he

accepted cl position as on the Aeronautical Bngineeiing Staff

a.t the Vit'ginia Polytechnic Institute, ln 1958 the e.utbor received his

Master Science degree in Aeroniuti~al Engineering, In January, l9S9

he enrolled in the engineering graduate school at the Virginia Poly•

technie At the pres~nt he bolds the position of

Assistant Professor in Aeiroi~:oit.ce !ngineeting and is completiug the

requix-ement;s the degree Doetor Philosophy Aerospace Engi'""

neering at

APPENDIX A

are revtrU.ten in the form

(A.l•b)

The colU.sion integrals, (of .• lat) . 1. 1 .. , are obtained from the eolU.sion µ1. co • equation& (A.1) by considering each react.ion foll!' each value ofµ. lot

) rr ' O'

(A.2) ;(l:+l)e

Note that lq. _(A.2) mut be m.ultiplied by 2 since each eolU.sion

results in loss of two particles

proceeding to the left gives

(A.3)

Similarly, collisions (A.l .. b) and (A.1 .... c) give fo,: µ = r

(r(t+l)) J,. • r 1 ((r+l)(r+l)e)*

(i)

(A.4)

and

dv. dv~ 1 , av • di , , e ·:;;T e· · e (A.5)

(t(r+l)e)I .-, r·. J. ·.. « =. (i\

(r+l)i .(· rr), , ,,1 . (i)

.) a ·r(t'+l)e

(A.6)

V 1

(A. 7)

J . . ((r+l)b+l)e) • \·,

(r+l)i (r(..+1))' li'J

,,) (S (r(r+1). dv P (r+l)(l:+l)e r+l

(A.8)

and ((r+l)ee)

J(r+l)i b:e)'

=

(i)

f . t t'J

..

(A.9)

E!q,, (A.2), Bq. (A.8) must be ,multiplied by 2 to

account for the loss of two

collision. lri:nally, Eq$. (A., l) give µ•e

(r(1*1)e) J •

e (r-,:)' -rr

• ~P*' er. • dv\. dv~1 dv, cf , , ... r(7N-1)e "' ,l;.T . r r

(A.l.0)

((r+l)(t+1)e) J • e (r(t"+l))t i j

(A.U.)

(re) J = e((t+l)ee)'

(A.12)

((r+l)ee) J =

e (1:e)'

(A.13)

wher~,

that the summation ove:t all indices, Eq. (A.12)

is equal to negative of iq. (A.13). 'I!herefore, it is necessary to

y"""'""'""¥l>~;;.. only lq,. (A.10') through (A.12) for the case of µ. = e.

A

J of .. inelastic < r:1) uv \ ot coll., k r

(A.14)

ri [ A (rt) .... (r(r+l)e) A (r(t+l))

lr = ~ j 2 1r1 ·· + 11:1 ·(· ) , + '~1((.-..t..1··) (·..;J;.1) )' +

(r(r+l)e), · • J:'r . . «-T '",.. e

(A.15)

sun1ned over all values of i

(rt) 3 . (r(r+l)e) rt '

(A, 16) "" J ri (r(r+l)e)' (rr)'

therefore

J .. (rr) + J (r(r+l)) + J¥.1. (re) J av ...... ri · · ri ~ ~

(t(r+l)e)• ((t'tl)(~l)e)' (h+l)ee)'

gives

dV = r

2

r "' (rr) nr . m ·. 3 /2 L Jt:i = ·z; ( 2,rl,i'f )

(r(r+l)e)'

(A.17)

(A.19)

Bq. (A.18) yields

(A.20)

(A .. 21)

in Eq. (A.20) may be t-alated to ea.eh other through the conservation of

[ r- lil • ...Lj !! (• . 2 ·. ·. •. 2)· . ~ ... 2

. 2kt .' 2 v. ' + '~1• ·· + ,, V • + L · J!' ~r 6 e

J (A.22)

JA (rr) , Jr1

(r(,r+l)e) • ... n J r (A.23)

where

~ -L -<·v• ·. 2 ..1... v· 2). ~.· "" 2· f ,.;m ... ·.. "" -:r T ~. t' . . r

rr _.... --., r1 dV l

r(r+l)e 5 f (A.24)

reaction, velocity

u are obtained re

APPENDIX B

:B. l. E.s,uilibrium Flow Equation::}. The equilibrium flow e<.Juations for

the special case of r=l are obtained fron1 section 2. These are:

Equation o~ State k (J

p l\'lO ; pT ( l + a) (l .. 3 ) , (B.1)

Spe.cific Enthalpy

i !t(l a) i 5 a 'X2 4 l = + - + - .. -cr J m I 2 1 + a T 3 ' ""

(B.2)

Specific Entropy

l·J·· rs P ai.3r 'i £1 ;::;; -m'" L (1 + a) 1,i -2. .. log ( - ) ... - J + -.. i log (DT) + o; log (ET)J! .. m 3 2 1

L

Law of Mass Action

2 2g2 T.3/2 r· T ,1 w.. l a ;O ) ...£ + (B. l}) = E ( exp ... .. IJ I 1 .. a gl,o (p/ro) L T a J '

and

(B.5)

wttb

1 i (.. 6 + cr ).· .'2 .· ( 3 + 4a ·)· ~l -;t·· 3 r· 2et· (JI· (ll.6)

the equations of conservation

03.1)

(B.8)

fcn:m the system Qf nonlinea,: elgebr.aic equations used to solve the

r=-l are

Ir- 2T ·_ .. · .. _ .... M2 - - (1 + a) cr + _ ·•··· . ·· :~

1 L 3 P .- . _ . .· (k/m) p3 (1 + t ) 2 t . . .

! (1 + a:) (1 + !16... ) ~ .

J . . -' [ f (1 + a)(3 + cr) J "

fl + a) ( l ... + 1 ... --a) } ~ + • .· · · 2 · 3 . d~

~ T

(B.9)

(B.10)

where

M

(J =:= .,_,.,.._,...,.......,._""""""'!.-. .......-. -·~···--· -· """". --· ..... """'· .--.!"""' .. --"'""*"'· .-. --· ·•· """"·-· ~ ................ """·~·--·. -1 /2 I

2 10

.. 2( ! ) [ f T(l-.()$) + (l .. a:) Tton .,. ! (lffl)cr ]}

(B.11)

B.3. leact:i.on Velocitif!1S foll'. Ar&t?J!• Since electton•atom collisions

ao1:,eaLr to be the dominant mechanism of ioniaation. 1 the reaetio.n

will discussed first.

tbe at:Qms which

ev·entually ionize, Petschek

mation fot'

is electron energy

~1a = 1.0 Jt 10 · (w • 11.S) , e

ev and the constant 11.S

(B.11)

the energy,

ev, the first excited argon. Equation (B.11) represents

a straight line fit to data obtained. by Maier•t.eibnit~ 18•

E<1uat:ton (B.11) mtpressad in ·0&. (Te,m = 1.34 ~ 105 °K)

(B.12)

-where c1e is the initial velocity of the atom relative the electton. ..... -lo.

terms of c18 and G, the mass•eente.r

. 13 SJ.nee

a. <.,r .... ,.~1> ........ ......._e~ ....... ·.· .. = l • o<v;,, ve)

, .. ' .. ·· 3 1/2 t ~ 106 I 2<kT) l ( · ~e + 2) e .. L ,t J .. '

(B.13)

(B.14)

(B.15)

11 in,'fo¥ the a$sumptton that eeeentially al.l excited atoms eventually

to twice the energy

an a:tom etrikes 191J nterefore, assuming

and using cross data given in

r,~ferences l!:1 and 20 fo-r Q11 and Q12., respectively, the following

approximations are obtained fot' atom.iiiatom and atom•ion cross

(B.16)

(B.17)

Ud~g !qi. (B.16) and (lh 17) t;.he expre11ions fQr u11 integrated to give !qst (S.4) (5.5).

-71-

0.4

Case 2

(~ i = • 22901, a:i = • 31911)

a:

T* • 14000 OK

p* • 1.0 atm~ er ,;. 0

Case 1 <~1 • .16372, 0:1 = .32455)

Equilibrium

o":'------------------~------------------,r 1 10

A

Figure 1. Illustration of Convergence of Starting Procedure

-72-

0. 8 ~------------------~------------------------------------,----------------------------------------------------------,

0.7-----

* 12000 °K T = 0.6 p* = 0.1 atm

- a :; 0

a = 0

0.5

a

0.4

---Frozen

Nonequilibrium -

o.__ ________________ ___."-------------------1 10

A

Figure 2. Ionization Fraction vs. Area Ratio

100

-73-

0--~------

T* = 13000 01(

p* = 0.1 atm

- cr,'O

cr • O

Frozen

Nonequilibrium

Equilibrium

o'-----------------.....iL.,,-----------------"'!!"!!!! 1 1 100

A

Figure 3. Ionization Fract.ioll vs. Area-Ratio

-74-

a

Frozen

Equilibrium

T* • 14000 <>K

p* • 0.1 atm

cr ., 0

cr • 0

o"'r---------------~~--------------~ 1 10 A

Figure 4. Ionization traction vs. Area Ratio

-75-

0:

Frozen

0.3

T* • 14000 OK

p* • 1.0 atm

a =I, O

(1 = 0

Nonequilibrium

Equilibrium

o'-!!-----------------__.~-----------------1 10 100

A

Figure 5. Ionization Fraction vs. Area Ratio

-76-0.8

0.7

T* = 14,000 OK

p* ... 10.0 atm 0.6

-a ,, 0

--a = 0

0.5

a

0.4

0.3

0.2

-- Frozen ~------ - '-- - -- -- -- -- -- -- ----0.1 ~

~'..;;::'! _<Equilibrium ~

~--= Nonequilibrium --.:::: --==-.;; 0

1 ll 100

A

Figure 6. Ionization Fraction vs. Area Ratio

-77-16--------------------------...-------------------------------,

T* = 12000 OK

12 p* = 0.1 atm

-{: ., 0

= 0

10

Equilibrium

it 8 (")

I 0 ,-1

>4 E-t

6

Nonequilibrium

Frozen

01----------------~~--------------~ 1 10 100

A

Figure 7. Temperature vs. Area Ratio

M I 0 ,-4

>4

E-4

Equilibrium

Nonequilibrium

T* = 13000 °K

p* • 0.1 atm

-{a=,O a = O

61-------+----....,..,..,---------+--------------------I

Frozen

4r----~~---~----------===----..;;;;;;;;::::::::::=:J

0,...---------------~----------------~ 100

A

Figure 8. Temperature vs. Area Ratio

M I

Equilibrium T* • 14000 °K

p* • 0.1 atm

_. { (J. 0

(J - 0

sa~--+~~-----------~r----------------. ,cl

rt Nonequilibrium

Frozen

o'----------------f::---------------:;;~ 10 100

A

Figure 9. Temperature vs. Area Ratio

~ 0

M

Equilibrium

Nonequilibrium

-80-

T* = 14000 °K

* p • 1.0 atm

er .;. O

er = 0

'o 8 ~---\..--------:!io.----------+------------------....:::,,::lll

,-l

Frozen

0 ~1----------------.....,.1 ..... 0 _________________ 10-0

A

Figure 10. Temperature vs. Area Ratio

-81-

T* = 14000 °K

p* = 10.0 atm

(J ., 0

er = O

Note: Frozen and Nonequilibrium

12 curves show no effects of er

CV")

~8~--~-------~~--~~------+-------------------1 .....

Equilibrium

Frozen Nonequilibrium

o.__ ________________ ...,,.j _________________ _

1 10 A

Figure 11. Temperature vs. Area Ratio

100

-82-8-----------------------------------

7t-------------------t------------------l

6t-------------------t-----------------Equilibrim

u51---------------------+------::i...-~ Nonequilibrita IJ • ...... e

T* • 12000 OK

p* = 0.1 atm

{ (J .. 0

a• 0

Frozen

2-----------------------------------1

lt---~--------------+-----------------....,

o"T----------------..,..---------------..... A

Figure 12. Velocity vs. Area Ratio

an •

-83-a---------------....----------------1

71-----------------+-----------------t

94~---~~-----------~------------------1 H

>

T'* • 13000 °K

p* • 0.1 abl

-{a•O a• 0

2.------------------+------------------t

lt-----"'lc---------------+-----------------1

o.._ _______________ .._ ______________ __.

1 10 A

ll'ipre 13. Velocity vs. Area Ratio

100

(.) G) CD -fl

Ll"I I 0 ,-1

>< >

-84-8-----------------,------------------,

Frozen

5

4

T* • 14000 OK 3 p* = 0.1 abn

{ (1 ~ 0 -(1 a 0

2

ll------3-.,._ ____________ __,1---------------------a

o'-------------------1.---------------- .. 1 10 A

Figure 14. Velocity vs. Area Ratio

100

-85-8-----------------,-----------------,

71--------------'---+---------------.

61-------------------;-- Equilibrium

Nonequilibrium ~5~------------~~~~~j==~~~~~~~--------, (I.I -5

It\ I 0 ~

Frozen

~41-----~L------------+-----------------, >

T*"" 14000 °K 3 p* = 1.0 atm

CJ :I, 0

er = 0

2

o.__---------------:-,!-----------------:~ 10 100 A

Figure 15. Velocity vs. Area Ratio

(.) <1l Cll -s (.)

in

-86-s --------------------..---------------------

7 I-------·------------------------- ----------------

6 !---------------- --- - ----- -----,f------

Equilibrium

Nonequilibrium

Frozen

6 4 1----------------i"~------------+------- -------------1 ~

T* = 14000 OK

p~' = 10.0 atm

er -:# 0

a = 0

l ,____ __ ,.. __________________ ------ -------+-----------------

o',;-------------------=-':::------------------,,-,,-,! 10 100

A

Figure 16. Velocity vs. Area Ratio

Usi11g the Debye•liuckle approximation, the effects of Coulomb

interactions Oll the equi1Un:ium.., frozen, and nonequilibrium flow of

an ioni~ed gas have been investigated. The gas i$ assumed to be

monatomic, electrically neutra.:t.,·and in thermal equilibrium (i.e., a

one temperature fluid); 'but tbe composition of the gal$ is a'tbitrary,

that :ts, multiple ionization of any degree 1s allowed.

The thermodynamic variables are derived starting from the appropri•

ate expression for the Helmholtz free energy. · U$i.ng Boltzmann

statistics and asstun.ing that the velocity distribution functions are

given by their Maxwellian va.lues, the rate of ionization is derived

for atom•atom, atom. ... ton, and atom .. electron collisions.

The resulting tmpressions are then employed in solving the

qua.$1 .. one .. dimen,eional flow in a eo'1werging ... diverging noz21le for the

equilibrium, frozen, and nonequil:i.brium eases. Numerical examples,

using argon as the working substance, are discussed and the results

presented graphically. 'the results of these calculations indicate

that, for single ionization, the effect of Coulomb inti$1taetions on the

performance of rocket engines is negligible; but that data obtained

from hypersonic arc jet wind-tunnels can be significantly influenced

by the presence of the interections ..