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THE EFFECT OF RECOMBINATION RATE ON THE FLOW OF A
DISSOCIATING DIATOMIC GAS By
Thomas P. Anderson Gas Dynamics Laboratory Northwestern University
Evanston, Illinois
September 1961
AF~ UM~RV
ARNOLD ENGINEERING
DEVELOPMENT CENTER AIR FORCE SYSTEMS COMMAND
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A EDC-TR-61-12
THE E F F E C T OF RECOMBINATION
RATE ON THE FLOW OF A
DISSOCIATING DIATOMIC GAS
By
Thomas P. Anderson
GAS DYNA/W/CS LABORATORY N o r t h w e s t e r n Unive r s i ty
Evanston, I l l inois
(The r ep roduc ib le copy supplied by the Gas Dynamics L a b o r a t o r y , N o r t h w e s t e r n Unive r s i ty was used in the reproduc t ion of this r epo r t . )
Sep t ember 1961
The r e s e a r c h r e p o r t e d he re in was p e r f o r m e d at N o r t h w e s t e r n Un ive r s i t y under Arnold Cente r sponsor sh ip .
P r o g r a m A r e a 806A, P r o j e c t 8951, T a s k 89104
Cont rac t A F 40(600)-748
AF - A I E : D C Armed # , M
ABSTRACT
The f low of a chemica l ly r e a c t i n g p s through a d iTe rgen t noss l e
Is considered. £ pure diatomic gas i s assumed to d issoc ia te and reoom-
b/he acco rd ing to seven d i e t i n c t recombinat ion r a t e lawe. The e f f e c t of
a o r r e o t i n g L t g h t h i l l ~ 8 " i d e a l d i s s o c i a t i n g gas" f o r v i b r a t i o n a l o o n t r t -
bu t l o r ~ of the molecu la r epec t e s and t r a n s l a t i o n a l c o n t r i b u t i o n s of t h e
atomic spec i e s i s sheen to be s m a l l . The e f f e c t of r e c c = b i n s t i o n r a t e
on the f l o v of o x y ~ n s from t h r o a t c o n d i t i o n s of 2 a t : and ~050 K W i s
shown to he s i g n i f i c a n t f o r r e a c t i o n r a t e s dec r ea s ing wi th tempera ture
and m l a t t T e l y u n t ~ o r t a n t f o r r e a c t i o n r a t e s i n c r e a s i n g wi th t e e ~ e r a -
t u ~ , E l e c t r o n i c ana log computer s o l u t i o n s a re p resen ted f o r a v a r i e t y
of t h r o a t eoapos i t~ons and recombinat ion r a t e s .
t i
Z ,
I I ,
Z Z I .
A*
B ,
O ,
~ ~ C ~ ~
Z l i T J ~ I N C T Z O N . . , , . . . , , . , . . , . , , , . . , .
A , P r S V 2 0 U W o r k
a. s~,t~,~,t ot ~ ~ r o ~
~ D I O ~ U , t l " L O m . . . . . . . . . . . . . . . . . .
A. ~ DSmn~om~ Invi~otd C o n ~ r ~ t ~ n Zquat~o~
B . A u x £ Z t a w Sa_uatlons
1 , T I ~ ~ o b l s t l o n s h l p s
2 . ] ~ u s t ~ m o f 8 t a ~
J . B n t h s l / : ~
h. ] ~ q u L t ~ w £ - . . Oonstant
5. ~ c a b 2 n a t d . ~ Rate
C, R e s o t £ o n S a u s l d . o n
D. Sumary of ~ mmnsSom~sOd S q ~ M
~ O ~ S mJ D~CVSSZ,~ ~ a ~ m . . . . . . . . . .
Jm~Tt~.os.t SoZuCton~
1 . Froun lrlo~
2 , ZsothsrmZ Flow
~ , ~ , 1 1 1 b r ~ m ]~Lov
h. X s c b s r l o - Z s o w l Y ] . o v
5, C m s ~ ~ n s t t 7 l~ew
6, D~sousston
P s p
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27
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2. C o m p u t e r K q u a t £ o n m
3 . R e s u l t . a n d D i a e u u i o n
~ . 0 ~ I O ~ . . • • • • • • • • • • " " " " " " " " " " "
V . R E C O ~ T ~ ~ R F U T ~ ~ F ~ q T I ~ T O I L 9 . . . • • • • •
P a p
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A ~ D I X C - B . o ~ d a r y C , o n d t , t t o n s • • • • • • • • • • • • • • •
£ P P F 2 / D Z X D - &hA, l o g C O m p u t e r P r o g t ~ a . • • • • • • • • • • • •
A P P ~ D I Z I ~ - C o ~ u t e r S o l u t i o n s • • • • • • • • • • • • • • * *
52
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F ~ No.
2 - i
2-2
2-}
3-3
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3-I0
3-11
3 . .~
3-13
I)..I
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D-5
D..6
L~T OF l~olmlm
Zn~%o)" ot Ato-4o Oz~pn
Bn~hs~py of ) ~ o o u l a r
r4ut~br~-un Cous~sn~ for 0~20
Ttwoat, O o u d i t £ ~
Kttoo~ ot V~brst£on~ l;nmeg on Voloo£V
Iosslo ~mtour
Reooubinat~ou RaW lhmot£on
NOI|~L@ ~l.OW P N ~ o r s j ~JJO Z D
]JJosts~J ~ P s N ~ r s j ~ ~ D
Noss2s I~ov Pmltorsp ~JJo ZI I D
Flow ~ W o s ~ l o n , Case Z
nOW COJN~bSitlOIIj C480
now Oowoslt/~nj Case IT~
now Veloo4.ty
FJ.ow P15sUUl~
lel~v ktoa
Bu£o GoUputer CmponenU
Computer S~.bolj
Basio Cmputer C o n f l i ~ s t ~ n s
Conputor Pr~szwt
Computer Prof~sn
~ u p le 8olutton
57pioal F~ror
29
32
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38
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Noss l e n o w PsFanotez~ Case I t
l iossle Flow P a r m e t e r s Cam I B
lJosslo n o w P s z ~ m t o r 8 08so I C
Noss lo F'tow P ~ t e : 8 0 u o I E
Noss~o n o v Pamnetez~ CaN I F
J o s s l o n o w P r a t e r s Case I 0
l ioul,s FJ.ow Pa,.-s~ters Case I I A
l iosu lo Flow Pal~mmter8 Case XI B
llossle F1ov Parsmt~r: CaN I: C
Nossle F1ov Paz~m~ers ~um 11 B
Nossle ~ Pa~met4~ CaN II F
l lossle n o w P a ~ u s t e ~ C u e IX 0
I o s s l o Flow P a z a n e t o : o O u e I I I A
Souses l~ow P a r a ~ t e r s Cam I I I B
Noss~e F~ov Pa.-snotors 08so I3~ 0
l iossle Flow P a r s ~ t ~ r s CaN I l l E
Wusle Mow l ~ a r s ~ e r s Cam I I l F
l l ~ L l s Fl.ov PmNmmtm'~ 0ram 1"£I 0
8o
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82
83
86
87
88
89
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91,
95
96
97
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2-1
C-I
0-2
LIST OF T £ 8 I ~ I
P m t e ~
1hiM.s1 Comlltlmm
17
61
6.1
~LI
I , INTRODUCTI~
Deunds fo r tmpr~ lng the performance of chemical rocknt8 n e c e s s i -
t a t e the use of high erMFa p rope l l an t J with msul t inE higher temperature
md~a~t p z ~ u o t s , The analTst8 of these high energy flow streams becomes
tnomasinSl~ nozw d i f f i c u l t as the tonpemture Is l~ tNd~ sines rea l Ks8
e f f e c t s beoo~ noz~ pronounced and there i s s t ~ n i f l o a n t dee ta t lon from
the e ~ s s t o a l 1deal gas behavior. Therual phononons l ~ l a t i ng to the
v i b r a t i o n a l and olect1.onlc energy l e v e l s as w12 as cbmmloal phenomna
r e l a t i n g to the d i s s o c i a t i o n and the ~ontsa t ion of the d t f f e : e n t species
must be considered. The r a t a of ohan~ of the var ious f l o e p a r a m e k ~
muet be c r i t i c a l l y eummlnod to detezwtno uhotber o1" not equi l ibr ium h u
been N c b e d j and OWeD model.ere olmn8oo Such 88 mtsht be found in the
ezpane£on of a pure gss through 8 b:Vporson4c noss le of ten r e s u l t in non
oqu£11hrlua f l o v oond£tions. Those non equ£11briun f l o w arc p a r t f e u l a r l y
lw?or tan t fo r r e a l reasons , In order to e r m i n e the problem of aero-
dynamic heat iug of : e - e n t r y Tehtcleap the flow condi t ions of the a i r
pass ing through tlw bow shook must be known, Aleop non e q u i l i b r t u a f l o l
mast 13o p n e ~ t e d in t e s t f a c i l i t i e s to simulate these : e - e n t r 7 cond i t ions ,
The t h r u s t of s rocket engine Is s e r i o u s l y a f fec ted bb" the ex t en t to
vhioh the propo~.ant to d i s soc ia ted a t the noss le e x i t and consequently
how e f f i c i e n t the engine has been In oonTert ir~ the chemical ene1.By ~o
k/aer ie oneFiD',
- 1 -
2-
~rs (Reference 1) ~ uas among the first to eezTeet ideal gasas
to compensate for real gas effects. He umd Berthelot's equation of
s t a t e to aoeount fo r the mleou la r 8ise and i n t e r m l e e u l a r forees as
w l l as a Planek t ens to g t w changes in the v i b r a t i o n a l heat eapaeit les.
b then anal~ed a nuabor of one diaenelonal f lows of a dtatoaio gas~
f ind ing t h a t these inpe r feo t gas cor rec t ions beeam s / ~ n i f i s e n t f o r
pera tures eorresponding to the FAov through a shook maw at a Huh
number of 10o Chem~oal roaot iono ms~ not oon~des~d.
Tlw problJm of a obmioal~y m a s t i n g gas f ~ throuzh a n o a ~
has been examined by Edse (Reference 2). In th is 8tudyp the Bases w~e
asS,reed to be in a s t a t e of porfeot thennodynsnic and e~enies l e q u l l i -
brims a t a l l tSJnos and an ef fec t i ve £sentdroFle exponont m detinod tot"
the f e a s t i n g n tx tu r e . E q ~ ~ r ea l gas e f f e c t s haw also been son-
stdered by Kriolmon and Cmkmol'o (Referenoe 3) and Sing (Refelqmse la).
L-iekson has ~ d the flew of a i r and presents a nuaher of sha f t s
f o r the equilAb~Am proper~ims. Eing has 8 tud~d the t h e o r e t i c a l pets
f o r ~ n ~ f o r an e q ~ b r i u s~xture of n o r s s l ~ r o p n during an l s e n -
tropio ex~ion. Waiter (Referenee 5) has included beat additlen to a
r eac t ing f l ov v l t h the hypothesis t h a t the flow i s in t h e n w d y n a a ~
equ i l J .b r£~ a t every po in t .
A l inear~sed t reatment of m a s t i n g nossle f lew i s E lwn by
Pennor (References 6 and 7) , This approxlJnate ans lTs is i s then applJLod
to the tuo extreme cases of very f a s t r s e o t i o n s , near equi l£brluu
RefereMes are l i s ted i n the Bibl touaphYp Appendix A.
f l ay , and v e t olov r e n t i o n s , neer L~rosen l~_ov. H~Lms (Refaranne 8)
~resants a d£acuesion of the various these-leo oonom-ni~ the o~gen
recombination ra ts and also an approximate so lut ion fo r one dimensional
channel f l o v . Dot~Lled meet nuaez'2oal solut ions are presented bY
Br87 (Refe~acoe 9), He considered a L t g h t h t l l "1do81 d4oaociattng
gun and solved several specific cues aa we l l 08 presenting an
approximate method fo r the so lu t ion of the oct of go~mz'~g equat ions.
In a l l except one case he asouaed the reooubtnat ion r a t e ulw eooen t l a l l 7
oons tac t . Later Bray (Reference 10) continued these otudlso audp a l -
though he diooussed the temperature dependence of the recombination
r a t e , oontinueo to use a constant r a t e for most of h is ca lcu la t lono
( t h i s g r e a t l y s l s rp l i t l e s the already complicated ana ly s i s ) . Freeman
(Ro£erenee l l ) has pointed out that, th~s assumption t s doubtful but has
presented only, seu~ oonolusive arguments. In l~Ls l a t e r paper Bray a lso
oon81dm the e f fec t o f nossle contour upon the d e a r i e s ~ou equtl4bl~un
ooupos:Ltion. Re concluded from h~Ls approx~ate c a l c u l a t i o n that there
ere nossle oontmws tha t v t l l have a tendency to keep the coapositLon
close to equ~14bz~um but, in genera l , ably are qu£te long and thez~fore
:Lmpranttoal. AddittLonal ana ly t i ca l attack8 on the nosale f low problem
8re given in References 12-20 4noluAtng a number o f nume~osl oolut~Lona
for par~louler cases .
Wogener (References 21-2)) has conducted a se r i e s o£ i n t e r e s t i n g
exper~ente concerning the f log o f a react ing g u mixture o f n~tl~gan
d i e . d e and ni trogen te t rox lde ca r r i ed in u l t rogen. This p a r t i c u l a r
n txturo changu co lor as the r eac t ion proceeds (ni trogen dlozLde i8
bl~rn and ni t rogen tetroxLde i s co lo r l e s s ) allotting concentrat.ton
smasuremonts to be made by l~ght absorp t ion t s e h n l q u u . S tud ies of both
supersonic expansions and flow through shook waves have bean madsj and
the zwsul te agree m11 with t h e o r e t i c a l pa.ldletlons.
B. S t a t e ~ n t of the Problem,..
This study concerns the e f f e c t of the r a t e of r e ® a b i n a t i o n on the
expansion of a d i a toE te gas. I t wi 11 be assumed t h a t t he va r ious e r m r u
modes are not coupled and t h a t the s t a t e of the gas can be de f ined by
the i d e a l gas law i f con~ensat ion i s provided f o r the Ro leeu la r weijh~
changes as a r e s u l t of the chemical r e a c t i o n . T rans l a t i ona lp r e t a t i o n a ~ B
and v i b r a t i o n a l aodes of energy w i l l be ino luded! h o m v e r , alA the e l e o -
t r o n i o energy l e v e l s w i l l be n e g l e c t e d . An a l g e b r a i c form f o r the recom-
b i n a t i o n r a t e w i l l be assumed and s o l u t i o n s obta ined f o r a number of t i l t -
feawnt r e e e a b i n a t l o n r a t s f u n c t i o n s .
XX. O 0 ~ O EQUATIONS
A. One Dinsnolonal Invincld Conservatle~ Equations
This Invastf4stlon mill consider eu~y one dinonslonal flow along
e
a otrea~ eoerdina~e, r p in a right handedp orthogonal, eurvilinear
eoovdlnate 8ystom. In the abseneo of sources or sinks the steady state
d
c o n t i n u i t y equa t ion i s
- nags d e n o i t y
a - flow awu
V - vslool
Begleoting eleotrcmagnetie effeets and body forees~ the momentum equation
- - "~ - - "-- 0
p - p11880X~
For adlabatle flow the energy equation is
ar
H - e n t h a l ~
I I 11
Although a l l 8yBbols a r e Inc luded in Appendix A. Nomsnelature s t hey m de f ined as t hey a r e In t roduoed in the t e x t .
B, Auxil£ar7 Equations
i . Thermodynm~o Relat ions~tps
Only d~atosttc gasos ~ be conm4dored Lu th£s studyp and Lt ~ l l
be a s e ~ o d t h a t thoro i s no tn toraot ionp or oouplJ~K~ bo~uoon the v a r i -
ous onorsy nodos. Tboroforo~ the p a r t i ~ o n funot£on~ Q~ f o r oaoh d i a t c ~ o
moloeulo may be ~ £ t t o n as the product of ths i nd iv idua l p a r t i t i o n funo-
t i o n s fo r tbo t r a n s l a t i o n a l # v£brat ional# ro ta t iona l~ and olootron:l.o
• lodo8 s Oa"
Q ~ ~ t v a ~ 3
S i L t l a r l y s the p a r t i t i o n func t ion fo r each atom i s
Tho l a s~ form in t h i s oxpross~on se t s t ~ roforonco l e v e l t hn t i s eho~n
fo r the onergy zero f o r the atoxLo spocies a t the seam lowsl as fo r the
molocular s p o t . u ,
I t w i l l be fu r tho r anumod t h a t the compommt8 of the p a r t i t i o n
function8 max be adequately s ~ p s ~ N n ~ d by
1 - t
I s " I kT I
Q ¢lzc - (~
-7-
a - p e r t l o l e mass
k - D o l t m a n u ' s e o n s t a u l
T - a b l o l u t e t empera tu re
h - P l a u o k ' s oons tan t
Y - to ta l volmo
- ~ a m a t 4 r t s t t o f~K luen~ of m l e o u l a r v i b ra t i on
I - moleoular moment o f ~ t A
- r a t a 7 f a o t o r
II - I t a t J J t i o a l m i e h t ( e l e e t a ~ d e p n o m e y ) of the eloet.Yonio
ground s t a t e
g - d i s s o c l a t l o n omrgF
The p a r t l t l o n funo t lono f o r the molsouleo and a t o ~ m t h e r e f o ~
The p a r t i t i o n funo t ion f o r 8 oo:Lleotion of N part, toloo t0 given by
QN N!
The f a o t o r ~ a l in n e o e u a r 7 b e m u s e i n the d e r i v a t i o n of the t e a n e l a -
t i o n a l p a r t i t i o n fvnot$on i t w u assumed t h a t the I n d i v i d u a l p a r t t e l e o
-8-
se re d i s t i n g u i s h a b l e and tho¥ are no t . Noep a l l o~ the ~hez~odymmio
fune t iono may be de f ined in terms of the p a r t £ t i o n fumotion and i r e d e ~ v a -
t i v e o . ( I t i o c o n w n t e n t to choose N oqual to Av&gadroeo nm~oF and uOe a
molar benin f o r these oa leu laUono, )
The Helmholt, s .~rse energy may be u r i t t e n an
T ~ p r e u = and tho entropT a m r e l a t o d to the p a r t i t i o n f m ~ t l o n bF
The on~halpy may be m ~ t t o n u
H
and the Oibbe f rse e n e r ~ ao
A l t h o ~ h oomo of the r e o t r i e t i o n o i n h e r e n t i n the aoaumptiorm
rolatta~g to tbe p a r t i t i o n func t ions e i l l booomo o]:~ioml or ~ l l be
examined in g r e a t e r d e t a i l in the fo l lowing see t io rm m more oxtonel lm d t s -
cuosions of these i t u m may be found in References ~ and 2~.
2, Equatlon of S ta t e
Uotng Ster l£ngeo approximation f o r the f a c t o r i a l o f l a rge numbers
9-
the Belahol t s ~rree energ7 may be wr / t t en
and the psrtLsl pressure exerted by the molecular or atoato opeotes Is
_ R T
V BmnrLtlng in terms of m s dens i ty and adding the p a r t l z l p r e u u r o s of
the ~uo a lx tu r e eoaponen~mj the to~al pressure beeoms
In uhloh Ls defined so the nun fmo~lon of the dlsmetated npeetes
Znape~Lon ot Bquat, Lon (2~ ) show that t t 4,, the eluerJ.oel idea3. Ires
equation ot state utth the noleoul~ might oomotod for the l - , o
£mold.on of the diosooLatod 8peoLeo.
CmbJ~Lng termsj the on ' ,~p7 of eLt l~r opoelos may be m~Ltt4n
The on~34~Loo of the a ~ o and m l o m ~ a r opoc~o m therefore
- i 0 -
,P..T
Rewrit ing in t o m s of mass on, to and adding the cont r ibu t ious of both
eomponentmj the en tha lpy of the m / x t u ~ i s
j, ,E-
Tha d s s r u of approximation in the ea l eu l a t l on of the sn tha lpy m~
be seen by ~ I n l n g P i ~ s 2- I and 2-2. Using the da~a of Bsins
(Reference 26)j the con t r ibu t ions of the various energy modes are shorn
f o r molsoular and atom~o oz~gsn. £1though the o l so t ron le p a r t l t l o n
funot lon has been assumed equal to i t s ground e t a t l e t l o a l vmight and
thus the o).oo~.-'onio onex'lgV has boon nogloatsdp th is r e s u l t s i n only a
ne~1.t~Ibls amount of e r r o r . ~ho e l s o t r ~ l e con t r lbu t lon to the onthalpy
of molsoular oxygen Io more o i ~ f l e a n t j o s p s c l a l l y a t the hlgho~
t smpsmturesI h o ~ r . d l s e o e l a t i ~ deareases the ooncentret ton of the
so l eou l a r opseiso a t h igher temperetureo and oonsequently the e l e e t r o n i e
energy of the moloeuiso may almo bs neglootsd without o~gntf ioont e r r o r .
Almost without oxeeption~ o ther LuvestSgators have used the model of a
d i s s o c i a t i n g gas proposed by L i g h t h i l l (Reference 27).
60
I
m
!
o m
X
,=
o =
n - ILl n
> - n _1 < "1- I - - Z I.d
50
40
30 ~ r
20
I0
EA
0 0 I 2
FIGURE 2-1
ELECTRONIC
,Tj
f
L
3 4 5 0 6_ 7 8 TEMPERATURE °Kx I 3
ENTHALPY OF ATOMIC OXYGEN
9
I
I
6 0
50 I 0
x , v o
4O I.d ._1
U ILl _1 o 30
Ilg W Q.
20 ) - Q. ._1
- r
z I0 W
0 0
ELECTRONIC ~ , ~ CONTRIBUTION
J
f
7-. T
2 :5 4 5 6 7 8 TEMPERATURE °K x I0 -3
9
FIGURE 2-2 ENTHALPY OF MOLECULAR OXYGEN
-1J-
Equation (2-6) correc ts L igh th i l1 ' 8 " ideal dtasootating l~m"
through the fac t that i t allows a var ia t ion in the v ib ra t iona l enea'ey
of the molecular species and the t r a n s l a t i o n a l e n e r ~ of the dtseocia tod
8pe e l se .
h . F, quCl.ttn'iua Coutant
The equi l ibr~ua constant based on p a r t i a l prsseu.-ssj Kp. m y be
m ' I t t e n in t e n l of the Oibbs f ree energy as
= - - - -
R'I"
(The 8 u p o n s ~ p t 0 ind ica tes t h a t the f ree onor~o8 are evaluated a t the
8~endard s t a t e of one atmosphere.) ~n t o m s of the pea ' t i t ion func t ions j
the equi l ibr ium constant f o r the d~ssoolat ion of a d t a t ~ L o gas nay be
written
or~ oonb~:Lng the idea l gas law and the p a r t i t i o n funct ions
This equat ion lwprosont8 the oqutl4brium COherent on~y i f the
o loo t ron io oontz~bution8 to the enor~7 are n o g l £ ~ b l e . Houoverw due
to t ~ high t o ~ o m t u r . ! required fo r 81gni f teant d iesoc ia t ionp th io
assumption t8 ques t ionable , Haneon (Reference 28) ha8 ca lcu la ted the
equllibrCum constant fo r oxygen including two e l e c t r o n i c 1cycle past
the ground s t a t e fo r the aoleeule and four e l s e t r e n i e l eve l s pas t the
ground s t a t e f o r the atom. Ranson's r e s u l t s a ~ eo~oar~ to thJ v e l m a
predieted by the above equat ion in F i g , s 2-J . The omnparimon IB not
too good. Fur ther cons idera t ion and an ana lys i s of t ~ e£fee~ of h l lbe~
elsotz.x~Le states x~veals tha t the elee~.~nAe statas fo r the i tem lllNJ
nueh nOl~ s i g a i f i e a r ~ than the e lee~ronie s t a t e s fo r the noleeulo . This
fol'tUnete oharaete l~ l t io 8 1 1 N & ~oh bet to r 8 p ~ t i o n t~ tbe
e q u L t i b r l - - eon~tant to be rode by negleot~ng the Ttb~at ie=al t e ~ f ~
the aoleeules to offset the eleetronle tarasp He Figure 2-), 1~erefore
the equillbrlm constant nay be eonsldered to be
I 131~~ ~.(r ~ T317" e,/,~ ~. ?-~k Lz-1)
Yn th i s studyp t h i s r ep resen ta t ion of the e q u i l i b r i m oonotant has been
corrected slightly b~ eva1~ating the ooe~flelent to give the e~ N-
se~t at a teaperature near the msxia~ temperature eoneldetwd. Lisht-
hill's "ideal dlssoelating gas" uses a similar approxiaatlon for the
equ l lAbr im cons tan t . Equation (2-7) incorpora tes add i t i ce~ l ~lgor~
s ines i t ino~udee an a d d i t i o n a l square root of teapera ture oorlwetion.
~. Recombination Rate
The recombination of atoms to form diatomie molecules i s genera l l7
considered • termoleeular r eac t ion in whleh three i~dlv~dual pas~tele8
participate in a single kinetic presses, the third particle being neces-
sary to remove the energy of recombination. Humorous theor ies have been
advanned to predic t the valse of the recombination rate f o r t h i s type of
roaction~ and References 8~ 29j and 30 contain discussions of sores of
!
p
I
I0
5
- I0
-15
- 2 0 0 2 3 4 5 6 7 8 g
TEMPERATURE °K x 10 -3
FIGURE 2 - 5 EQUILIBRIUM CONSTANT FOR 0 z ~ 20
-16-
theN theories. Unfos~una~elyw the various appz~aehss do not sjmej
pe~oular17 :eprdlng the tenperat~re dependeneo, Xa photo1 it n8~
be eoaeXuded ~hat the ae~Iva~Lon e ~ required for the meoublnatlon
pl'oeoo8 58 ne~Lt~bla and the =ommbtnation r a t e b u an s l lpbra£e teD-
p e m t m d o p e ~ o
The laok of aKreemon~ ~ o u ~htm po~n~ on i8 evident upon ezaninat ion of
Tablm 2-1 (abetmetod f~on Re£o~eneo 29). Thl8 ~ b l e 1JJr~e ~ho values
the reoombtnat~on r a t e ~or o ~ n ~ t l ~ oct ~oFth In the l l~e~s-
~ ' o . Szpes~umn~ dal~ £or ~ and other high tmupemtm ]~oaotJono
8poAse and quootionable~ oinoo in p n e ~ d ~ dlooooiat4on 1',8~ ~LO
noeou~od and the equt lSbrLm eonetan~ then used to oa leu laM the s, eomt-
bSna~lou m~o. The ~ ae t4va t ton enors~ r o q u l z ~ ~o~ dtseoeta~don 18
~uo~uded in an e z p o n s n t / ~ Vm~p~atus~ dopon~nw ~ h e t t n ~ l ~ s ~ msslm
the e~o t value of n in Squat/nn (2-8), Yni8 study Is of the street of
various z~Inat~on rotes and the ~ens of BqustJ~a (~-8) Is eatlsfae-
O. Reeotlm Beuatloa
Diuoeta t ton of a dlstomle nolooule I F be oouides.od to be t ~
: ~ m l t of a oo111alon betueen a mo~ou~ and another pa~ io le Lf 8uf£1-
etmst e n o r u t o avai lable to break t ~ noleeula: bond. 8 1 m l l ~ y :
~wo atom in the j=eNneo ot • ~ par~lele. The oppooiag reee~tmso
of d548eela~J~a and reeoublns~ou my be n p ~ d ag
b a a s b / n a t ~ o n Rato
m, 6 llS.mols "?. mo ").
q ' 7
z.6 s ~o'- Joo
h.05 z ),0 ).7 ~ 5 ")"~
h.SJ z :w 17 ]~5 ,.o.~
3 z ).o~ ~ 5 o.~
~ . 2 , ~o~+ .~ '~.~
?ab),,e 2-1 Reoombinat, f,on l~to for Ctz~l~S
-17-
.18-
ko
k D - d/ssoa~s~ton rata
i~ - reoonbinst~Lon mrs
The d t H o c t a t l o n m~e J4 4ot2ned ss
(t'be rote ot pzmSuat~on of dtuoc~Latod 8]?oo1441 t.II propol~lon81 to t~o
o ~ o o n t n t L o n of tho d l a t ~ t ~ spe~Los tJ.uos tho tot83, oonoontmtton ot
the n t z t m , ) :he rooonbtnstton ro te JLs de~Lned M
The net ~sn4e of tho a tmde spooLee £s themto=m the sun of ~hoia two
ozpl~lJn/on4, Bzpandlns ~ho Euler tsn doWLvstl~o and l~Otz~Lotln8 i t to
stosdT' Ft.ovp t h i s sun beoouoo
A pe.-t lmLla: ease t o t uhtoh t h e e te no not rata ot preduotlon
o~ o£thoz, ohemlosZ opo~Lo8 Is • ohelLosl equ£Ltbr£uu sta te , to&" oqull:L-
tn~Lm the 8bovo equst~Lon I~eduoos to
or~ 2n ~onm of p s r t l a l pmNures
Irmpactton shoe8 t h a t t h i s r a t i o of p a r t i a l proeSUlW8 t8 equ iva len t to
the equi l ibr ium constant fo r the reaot ion being oongidegwd. Therefore
P.T
Although the above equation has been derived fo r the p a r t i c u l a r ease
of a system in equi l ibr ium (and thus no change in oomposition)j i t wLU
be usod fo r t h i s problem since i t i s gene ra l ly agreed t h a t ~ j ~ p and
Kp are a l l only func t ions o£ temperature and not composition and tho
r e l a t i o n s h i p w i l l hold fo r a l l eases in uhieh a mixture tompoFatum mn
be def inod. Feldumn (Referenvo 31) s t a t e s t h a t t h i s r a t i o may be uood
in non equillbri,u eases ~en the Individual partlele8 haw an equilA-
bri~ statistical distributlon of their enerBy levels. ~ therofe~w
redueeo to the ass~ption that the relaxation time for tbo varlouo
energy modes is very ohort e o ~ to the relaza~len tlao required ~e
attain ehemieal equillbrlem.
The opeciee eontinulty equatlen, or reaetlen equatlen, for the
nozzle flow problem nay now be wr i t t en am
]~. S . m a n of Won D S a o . o ~ l m d ~ i . * t ~
A oct of equationo d u o r i b 2 ~ the flow of • s ~ m t ~ diato~Lo
gao have been dmmloped, To tae i l t te te oubmqmmt ooNmt~ttonop the
-19-
-20-
a r e a . dls tar~ep velooLty, t e e q ~ t u r e e dens i ty , rand pressure otn be non
d:L~ons4onal.£~d w~Lth respoot to e ~ refere~oo aond£~ioas. £ndleated by
the subea~Lpt O. The ~ u l b L n g equat ions are
¢ont~nu$~y ~uat£on
Mce~snt a Equat£on
(,z-~i)
P.,To tt-lz)
V~ = - - l + O ( o
E n e ~ Ilust~Lon
,V cl,~ .+~k,. c i l~t , l , 10c~I" +DK ~. O(~L-Ix~:)l=° t I'-14)
P.,To
0 = ~'---~ (t-i,.) V,To
. ] I , T ~ -- e,L) -l tz-*~)
-21-
Equation of S~a~
Reaction l t u a t t o I
(,t-~e,)
I?.. DAI~ 4 y." r'. k,L ¢- T.'- -~".
( z -~
(,7.-I.o}
C, (,T '~ : 4~. T tl-z,)
III. SOINTIOm8 A~D DISCUSSION OF RESULTS
Ao General, Co, nstdel 'at ioM
~ s reforenos condi t ions ~_th which the dimensionless va r i ab l e s
sre formed are aos t con~snient17 chosen to be those s t the nosslo t h r o a t .
Examination of the con t inu i ty equation shows tha t t h i s oorruponde to the
poin t a t Mhlch
For a wartety of pa r t i cu la r eases th is equation proTtdes an e z p l t o t t
d e f i n i t i o n f o r the v s l m of ~ (which i s a funot lon of the o r t t i e a l
w l o o i t y a t the nossle t h r o a t ) , Hom~mrB in general t h i s oondi t ton
be r e l a t e d to the nozzle geometry through the r~aot ion equat ion and the
reocmblnatlon r a t e funot lon . Phys les l l~ . ~ uan he considered to be an
effeot£Te ra t i o ot speoJ~ie hsatos since i t performs a fonet~on s t a i l a r
to that r s t l o i n c lass ica l sas dynaelos.
AnslysJJ of the equat ions l i s t e d in the prsoeding sec t ion shows
tha t both the con t inu i t7 equation and the energy equation may be i n t o -
grated d l rec t l~ with t lw resul ts
-22-
=~-
1 (s-z)
Stnce the equat ion of s t a t e to an a l p b r a l c equatton and ean be solved
exp]£ottlT~ the problem tha t rematrm t8 the I n t e g r a t i o n of the momsn~m
equat ion and the reac t ion equat ion, A ca re fu l study of the r eac t ion
equat ion Indica te8 t ha t i t ~oald be extremely d i f f i c u l t to I n t e r . a t e
a r~ l . r t l ea l ly~ ovon fo r w r y o:tnplo e s N s . Homver~ oneo tho mmalnder
of ~he problem has been solvedj the motion equation nay be e n Z u t e d
e i t he r n ~ r ~ e a l ~ y or gmphtmLt]~r~ to obtain the nossle con to~ . I t
can be pa~la~Z~ lntegrst~d to
17" c cx
Zna l~ t t ea l so lu t ions to tlw mmntnm equat ion IS7 be obtained f o r s few
eases j bu~ In ~ z m l n ~ s r l e a l techniques mint be employed,
1, l~osen Flew
I f the roooubSns~tou rat~ i s olov enough oo ths t tho eoupoot~ton
of tho f l og doe8 no~ a h a n p appreotablF during tho noszlo expan81onp
the f l o g 4s 8o£d ~o be f rosen. ' l 'b~ ~ be considered to be the ease
for .hteh Dsnkohlsr,s f ~ s t ooef f io tont i s
8Saw t~ to ~ f ~ d ~ a ehsz~oter ls t lo t i ns f o r z~s~ ion divided by
th~ ~ss£dsnee t ~ In ths no~sle.
~ossls th roa t 'must be ouoh t h a t
In t~hts OaN the oondttlons s t the
F/~Lre 3-1 ~ w o th is ~eoult Smphleall~ fo r ths l n l ~ a l eond i t iou
dose~Lbe4 In appsn41z Co
The p u s q ~ equatAon f o r flow 'mloo£1~', Equation (~-.2) m y b~
s p s o l a l i s ~ to the a u g of J~ossn flow.
The Aonsntua equat£on oan be IntI1~ated f o r t h l s o a ~ by oolbln/n8 i t
e i t h ~ n l o o £ ~ - and the equat ion of s t a t e . The ~ u l t
• Y T . I } L .,J ' - '
2, I so the rna l Floe
An ~Lmothorm81 f log oan be mLtn~etnod EhroulJhout the noMlo empma-
8:Lon 4f ouff£etmn~ ono~lF 48 lwloaimd by tho lWoombinat~ton of atoms, In
t h ~ c a n ths oondit lon8 a t the nossle t h roa t must be suoh t ha t
L
This oondi t ion 18 f lmphioal lx swprosontod on Flfluro Jo l .
8pocialtmtnil tho v e l o o l t y to the cane of an i8otholwal flovp tho
mault~ is
1.80
1.70
1.60
1.50
1.40
1 . 3 0 j
1.20 - ~
1.10 •
I i
1.00 L, 0
J FROZEN
J J
EQUILIBRIUM /
J ISOTHERMAL FLOW
0 . 2 0 0 .40 0 . 6 0 0¢' o
0 .80 1.00
FIGURE 3-1 THROAT CONDITIONS
-26-
~ - l'~ k~- , ~ ,
Again tho momentum oquatton nay be in to~ .a tod i f t h i s oquation ~ eombinod
With the equation of o~ato and the p~eoouro may bo oxplweood ao
L ~-~)
3, g q u t l i b r i m Flou
I f tho reac t ion t~no i s very ebort ocmparod to the nu tdenoo timos
OF
DQvv~ / - ~ I,
tho f l o w i n oaid to be an oquil~Lbz'lum flow and the oempoottion ~ •
untquo fumetion of the oqut~br tum eonstmant and the p~onuzw° For
equ i l tb~um flow tbo t h roa t condi t ions nmot 13o ouoh ~hat
t I 1
-27-
The z~aJt~.io~io~ of oqufJ~b~Lum :r~ov does not pz~luoo • ar£Kn~l~ioont slnp14-
££oat/~m fo r the momentum equat /~nj and i t s ~ Z sonnet ha i n t e g r a t e d .
l m r o w oh~1.~8 and tab].os £o1. equ43.ibr£tm plmpert ios have boon pubiS.shed.
4 .O. j ~ o r e o , N 3s to f & O i ~ J . ~ nunorioa~ m~oula t ions .
h. Zsobe~o-~ovo~ F3.ov
k~thoush perhaps not too £ n t o r o s t ~ g ~n a p h ~ l e a ~ N n N ~ an
ans]~p~i0a~ So~utAou m y be obto~nod for the o8o0 of oonstant p15ssu~o
f lm, ' . App3.ioation of the exaLter/on for the noss].e t h ~ a t 7"dLm].de
L
TJmpeotlon of t h i s o q u t l o n show t h a t the oondl t ion for a t h roa t ro -
qu/~es an oxo~hozxLo d ~ s o e 4 a t i o n r e • o r l o n . :Lf such a g u usro u o d .
• unique t lwoat tonpemturo would be roqu~-od s • s p o o ~ o d by the above
equat ion . A o o n v o r p n t - d l v o r p n t noss l s £s /mposslblo for the gaNs
eons£dorod :in Appond/x C; hoverers pure ly o o n v s r p n t or d l v o r p n t m o -
t i o n s nay px~duoo ~ n s t s n t pressure f ~ o l .
Throujh the momentum equat ion i t ~ n he Non ~hat • cons tant
p l ~ S S m n o v n u t oorrospond to a oongtant voloe£t~ ~ o v and the men•n-
tun oquat~Lou s, eduoos to • t .~v~a l Idon t l t~ .
5. C o n s e n t Densi ty
For son• ten t dens l t~ f lov the ~mdl t l ons a t the nosr~e nust be
suoh tha t e i t h e r
-28-
~7o ~ O
01"
~ o
The t ~ s t rost4~Lot4on o o m s p o n d e to the oZus ieaZ ~ of an h ~ o n p ~ o s -
sLblo t l o v wLth no t e a • f l o e . The a l t e r n a t e ros t r4o t~on roquLros t h a t t h e
t h r o a t ooeposLtlon be an ~ q u 4 1 i b r 4 s oompos&t&on. In nsLtber aam ~Lo a
un~quo throat ~loo~t¥ dat~nod.
6. Disouss:Lon
Sons r a t h e r Lntoz~stSJ~ r e s u l t s •an be doduood f r e e a oa r • f e l
s tudy of Fig~wo 3-1. ~ r e p r e s e n t s an e f f e o t i v o r a t i o o f s p e e t f i e
hea t s i n de f tn tn~ the o r l t i e e l v e l o o i t ¥ of 8 r oao t i ng gas n o w a t t~o
t h r o a t of 8 n o s s l e . For the e~ase loa l u s e i t va~_os f ~ n 7/~ f o r •
d~atomle gas to ~/~3 f o r • monet•rile i~m. Hoverer . t h i s f and~ i~ , zsw~o
i s t nozao t , s lneo the v i b r a t i o n a l enorgF has no t been tnoludod. F igure
) -2 shows t h a t the *neZus4on of the v l b m t ~ n a Z t o m 1torero the o f f e c -
t£ve spee~f.e hea t mt~Lo t~ appz~xlnato~y 8 per oen t . S~noo thLs 48
the ra t40 of two t enpe ra tu ro dependent funot£0n8 and vs~Los Zmss than
e l t b e r of them • l o n e . I t oou:Ld be e n t r e / p a r e d t h a t t h e l ~ uould be •
oonaidel~blo oox~root:Lon i n the oalouZLat:Lon of t~o o tbe r f l o v p a n n s t o 1 5
the ~ . twat~onal enoray ~s n o i S e • t e d . The r o d n s n ~ g f e a t u r e £s t h a t a
voz7 high pro•sure vould be roqu i l~d to oxao t l7 appl~aoh t h i 8
• u s a t the f~md t enpors tu ro of h050 K f o r vh~Leh the eul~ee are dravn.
7.00
IN ;>
6.00
5.OO
4.00
3.00
2.OO
1.00
FROZEN FLOW
=o = 0.50
T o " 4 0 5 0 K
J
INCLUDING VIBRATIONAL
/
1.0 0.80
j ~ - - ~ N E G L E C T I Nt.~ v,~,~o~ ~
0.40 0.20 O
f
0.60
TEMPERATURE
FIGURE 3-2
EFFECT OF ON
VIBRATIONAL VELOCITY
ENERGY
29 -
-30-
Howovorm t~o tno lus ion of the v i b r a t i o n a l o n s r g oontinuos to haws an
o f f s e t , of deoma~ng n a ~ t t u d e t as the e o ~ o s i t i o n ohanps to the o ther
luct~ms of a pure monatomlo ~ (oozTespondint to a wsry low pressure) .
~ezqforep exespt fo r almost oompletoly dlssoe~atod f l o w p the v lb r a -
t l o n a l tense should be oonsldeFed. Figure ~-2 shows ~hat t h ~ s t a r esn
exceed ) pea" cent i n the ws loo i ty fo~ O( • " 0,50. Of oomees the e r ro r
wAXl be greater f o r smaller values of K .
The ~ fo r squ~ibr ium flow nate~os t ha t f o r t roson flow a t the
end points ( the 14-4ring oases of the sere and i n f i n i t e pressure) and
drops shup~y in the Intermsdlato n n p , apps'omohtng the mum of I so the r -
z a l flow. This follows f r o s the f e e t t ha t in t h i s region the oomposltion
is psrtieularl~ mnsltiws to pressure . The value of ~, f o r the i s o t h e r -
mal flow of a roas t ing gas i s independent of oompositlon and s lues to
un i ty . In the l i n l t i n ~ u s e of an infinite dissoedat ion euergy, i t i s
sXaot~y un i ty and the flow behaves S ~ L T ~ T to the o l a s s i e a l lootheraal
flow, studied by Rotor and Ca:he1 and reported in Referenes 32, Sines
no ohan~ i n oosposi t ion would be required to mLintain the t e m t o ~ .
The th roa t wslooit~V then oos~uponds to Newton's isothermal speed of
sound.
When o ~ t d o r t ~ the flow of a r eae t i ng gas. i t i s r o l l to z~mm-
her t h a t the oomposition must be appswaohing equillbs~um. The dev ia t ion
from equil£br~um might "inorease throughout the expansion presses , but
t h i s must r e s u l t from the equi l ibr ium po in t shaming sore r ap id ly than
the flow can fo l low. Considering t h i s , I s o t h e n a l flow csn e x i s t only
4~ ~ ) OCe . s ines t h i s w i l l necessar i~y r e s u l t i n recombination.
the r a t e bolnm cont ro l lod b~ tho mcombinatlon ro te funotlon~ F.
c. c o w e r So ut o
I . Nouslo C ~ t o u r
The Ina lus lon of the spoclos aon t lnu l ty equation urlth tho convon-
t i o n a l p s dT~malo o o n s o m t ~ oquatlon8 has addod two addl t lcmal wmrl-
ablesw the ccapositlon and the nossls geometry, Therefore an addit ional
squat ion d o s ~ the nosu~o contour i s l~qutz~l to ob ta in so lu t i ons .
The ~ purpoN of t h i s study i s to azamlns the o f f s e t of the rooom-
binat~on r a t e and thus the c~otw of nossle contour i s a r b i t r a r y . The
n u t eonventont contour
has been ohoNn sad 1o s h o ~ tn Ft~we ) - J , ?h i s p a r t t e u l s r shape has
s s w r a l n o t u o r ~ h y ohsrsoter is t lOSo P r t a s r t l ~ t t s t a p l t t i e 8 the equa-
t i o n s and decreases the ~ o ~ e x l t l e s of the coaputer progrsa, In addi-
t i o n i t i s f ln4d moohenloally £oasiblo s ines the naximm valuo of the
ha l f angle In the oxpansion i s app rox~u to ly 15 d o u s e s . In addt t lonp
i t fo rces the raaot40n equat ion to p rod i s t a soro dsx~Iv&tlvo fo r the
composlt~on a t the throatp u o a t l ~ slmp34~Ing tho s o t t l n g of the i ~ t t l a l
eondlt40ns (me Appondlx C). The naJor d l s a d v s n t a p of tb£s nossle i s
t ha t i t does not aetua1~y haw a t h r o a t , or a tn taua po ln t , but slneo the
teuporaturo nust decroaeo as s roeu l t of the eomputer pz~grem, t h i s
d i f f i c u l t y Is m/nor.
2. Campu~r ~qL~tions
Considorablo care ~ be ~.von to the fora~Tatton of the equations
I
!
FIGURE 3-3 NOZZLE CONTOUR
-)3-
and s o a l f ~ of the ~rtables before t lwy oan be proMmmmd for s o l u t i o n
on an analog eomputer ( d e t a i l s are presented in Appendix D). I~ w i l l
s u t f i m to n o ~ tha t s ~ m s
T - I - lOOv
p " I - lOOt
t T - l - l O O v
- l - l O O t
and the maoh/ns tdLms I s s l o m d down by a £sotor of ~0, £@tsr c o n s i d e r -
ab l s mn/palatlcmm tho c c ~ u t s r equat lons mn be wr i t t en as (M i s the
~ l ~ o e ~ b ot the area)
dT - - I T ~ - d--t I o o o [~-~s~
a t ~o
m
d~. = ~ [ 1 T 2 . =j T TM .
- ~ L ko, ooo * 1>°°°°°° 0< - ZOe d/ dt
• ~eo,, . --j- E . g - e . ] ~.g,. ~o-ooT 2i '
- g D
- - - ~ ~ T - s o , o o o - ~ o o ~o T D o o ~ < ~ a t
dN _ t I I .~ I d__~ x a--~ 1oo M ~ T ~ - 5o, ooo Tboo.~) at
I d~ • t d~< I (S-Ls) S o ~/~ d~. - ~ o o ~ , t ° ° ~ < ~ d -q
- I d G , t CaT" ÷ - - D G {.3-Zo) d t 1 . o o o ~o
_ Sor~ 1 {,s.~.L )
A eoBplets d l s o u u l o n o£ b o ~ d ~ 7 oondl t lons i s ~Lven i n ~ n d J 1
C. Three main eases N r e s tud ied ~Ipendlng upon the I n i t l a l O0~pOl~ti~}Dj
equ4"Jibrium, above equllibr4.um~ and below equilib~Lma. For saoh o f
t hese main oases , seven o u b c a N s o f d i f f e r e n t t e s ~ r a t u a w dependen~ o f
tbe roocubtnatLon fune~ion are ~ . The simetrun of reeoubins t icu
funo~tons sra shmm in Figure ~3-~.
J . RosulW and I)iscussion
~yp~oal so lu t ions from the analog o ~ p u t a r ara sboun in FJLsuras
~3-5s 3-6j and 307, The t l o g v s r a ~ l e s behave 8s oxpeetod and tbe
p r o s s u r a - t e a p e r s t ~ curve i s s l i g h t l y bemd as a r e s u l t of the tno lus ion
of the v i b r a t i o n a l s n o r g as ueLt as the rasomb~at4oa r ean t i on . Tho
value of V 2 (sssentta].l,y the Ho 2 of oZimalelLl gas d~q3amiol) 1"~e8 rap id ly
and thou a p ~ e h o s an asTnptotio ve~Lue ( fo r f rosen f log t h i s must Zie
between L.o (mnJtcato ms) and 6.2 ; (d tonio Sas)).
The oheuloaZ offootaJ in the f l og stream are more in terost tng then
the f l u i d moehantoal v a r i a b l e s . Figures J-Sw ~3-9p and ~-10 show the
chanso in e h e ~ e a l composition as s funot lcu of temperature. Prov~us
4'mvost4gatere, ioO., Roforonose 19 or 20j have Lud4eatod t h a t a t the
l e e pressures s~udiod h e n the anoun~ of ches~oal rosot ion ooeurr ing
dounstl~an of tbe nosr£o th roa t should bo neg l ig ib le . Thus the so lu t ion
ns~ be adoquatel~F rapresentod ~ the f roson f l o g soZution, This eonclu-
8ion i s subs tan t ia ted fl~U th4s study fo r those cases u i th the recombina-
t i o n ro te an inoreasing funo~4on of temperature (n > 0). For these eases
tho rooombination r&te funot iou beeches 8mall quiokly , end thus the
ohesLLoa~ roso t ion d i a l out rap4d~y3 beforo a s ignLf ioant ahsngo i n oom-
positAon can ooour. The o f f s e t of the d4ssocia t ion ensr f~ i s g r s s t l~
d i lu t ed 4f thora ~Ls onl~ a nual l composition chango over a largo tempera-
tram drop, Sinoo the moloeular weight corroc t ion i s a (1 ÷ K ) term.
small ohar3~os in composition ~LZ1 hays nogl tg tb le e f f e c t s on the
5.0
2.0
J
0 . 5 , ~,,~
0 . I
1.0 0.9 0,8 0.7
n=+ l
n-+:
0.6 0.5 0.4 TEMPERATURE
RECOMBINATION RATE FUNCTION
FIGURE :5-4
- 3 6 -
5.0 I
To = 4 0 5 0 K
Po = 2 arm,
(xo = 0.50
n - ' - 2
2.0 f
0.5
0.2
I
0 . 1
1.0 0.5 "0.2 0.1
PRESSURE
|
0.05
NOZZLE
FIGURE 3 - 5
FLOW PARAMETERS, CASE ID
- 3 ? -
5.0
T o= 4 0 5 0 K
po = 2 otto.
=o = 0 .667 n = - 2
2 .0 .
1.0-
0.5
0.2
O.I I
1.0 0.5 0.2 0.1 0 .05
PRESSURE
NOZZLE
FIGURE 3 -6
FLOW PARAMETERS, CASE TrD
3 8 -
5.0 I
T O = 4 0 5 0 K
Po = 2 o t t o .
(Xo= 0.333
2 0 , . = - 2 ! ~ S I
I.O =
I
! | 0.5
0.2
0 . I " i I
1.O 0.5 0.2 | I
0.1 0 . 0 5
PRESSURE
NOZZLE
FIGURE 3-7
FLOW P A R A M E T E R S , CASE TIT D
- : ] 9 -
I 0
0 . 0 3 0
To = 4050 K ! n = - 3 f n = - 2 I Po = 2 o t m
0.025 ~ °¢o= 0.50 I
0.020' I =-I
0.015" ~ n
0 0 0
0 . 0 0 5 " ~
- 0 , 0 0 5 . . . . . . . . . . . . .
-0.010 1.0
I
0.9 0.8 0.7 0.6
TEMPERATURE 0 5
FIGURE : 5 - 8 F L O W C O M P O S I T I O N , C A S E I
- 40 -
tS I
o
0.030
0.025
0 .020
0.015
0.010
0.005
0
I
T o = 4 0 5 0 K n = - 3
Po - 2 otto I ~o = 0 . 6 6 7
- 0 . 0 0 5
-0.010 1.0 0.9 0.8 0.7
TEMPERATURE 0..6 0.5
FIGURE 3 - 9 FLOW COMPOSITION, CASE l"r
I o
0.030
0.025
0.020
0.015
0.010
0.005
0
-0 .005
I To =4050 K Po = 2 otto. O¢o= 0.333
n=+3
n i-l-2
/ J n = + l
-0.010 1.0 0.9 0.8 0.7
TEMPERATURE 0.6 0.5
FIGURE 3 - I 0 FLOW COMPOSITION, CASE TIT
- 4 2 -
no),ooulm, tmisbt.
Susqn~ in i i ~ dollherato)~r •tdu~ing • t the t~-oat v*~b • non
oquL1J.br*m oompoad.~on ham on~ • 8m,1.1~ thoulh prmotmoodp eff•o~ on
tim oompooL~on protL~o•, In par~ th~s L• • rooul t o£ the oh•too s t t~o
noss~o oon~ull's 04~ooj i n ~him osN; t~o ooupog4tion must • t4r~ a t tho
5d~o&t ~ h • so t s r a t e o t ~ in 8 ~ c u s s , Thus the nosslo t h r o s t
n~y be oonsidorsd to be s t • s~ato s t pseudo t rosen ••rip•stOLon,
~ S s t i v s exponents (n < O) far the r e o o u b ~ t t o n rt~e produoe s
d i r t • m i n t o3~ss of so lu te•us f o r ths t~ow eaepositS~n, Inspect ion s t
the mo~Lon oquat~on ~ provide •sue inlrlSht Into th is dlot ino~ion,
Thoro ann bea4oa)J~ ~vo ~orno in t~o roaoLS~n equat4ou~ one Lndtoatin8
t~o dLtuot~on and O n t to vt~oh the f )~v d s v ~ e s ~rou oqullXbr~um
and the o~her~ o o n k d ~ J ~ the roocubS~tt~on rat4 funsa l •n• i n d L u t i n 8
the speed v*th Vhloh the t o s s , I o n v~3~ •sour , For t~o •ass of n ) 0 the
m r 4 s t s~sot lon te rn sona r • l • the oh•r ip in ocapos4tlonD and s , no• £t
oonst4nt j o r noar~y oonst4nt j vs~ue qu*ok)~, For • n q s t l v o ~sluo of n
the ra te s t r u o t £ o n t4rn S~n~ums v~th d e o r o u i n ~ t~npersturo s the •
bsv~n~ • t4ndenc~ to i n o n u o ~he rt~o of ohenp of • • r ip s • i r i sh ,
5~e to rn i n d £ o s t ~ e~on~ of the dov~st£on i ron oqulltbr~um in
the roao~£on equat ion ~s ~n ~onoml • deos~asin~ funot4on v*th do•ross-
t a p o m t u m (although 4~4t~a~)~r 1re value ne~ increase dependtnir on
b L n l t i a l aondit~Lone)= and at ]Low t4~oratores the cl~mp in ooapoe:L-
t~Lon nust spprosoh sero. Houe~er~ fo r the 1or pressures and prossure
rs t los oonsidorod here~ the •topers•ore does not dec=ease s~ffLoLently
f o r ~hls seeondary e t tee~ to l m , i c u t ~ U . Untor~uns~e~y, the eeupu~sr
everloaded f o r tnpemtu~8 ot about 0.60 t o t n s j s ~ t w n ' s , end seas
minor mseaLtag of the problem l u l d be neeosss ty ~o obt~b~ soZu~Aou f o r
grea~er ~aupersture r a t i o s .
The eholee of nosals eon~our 18 of p e ~ i e u l a r s lgn i t i eanee In
de~eraia iag the oeuposit t~n. The LS~ear nossle s~udied here res~r ie~s
~be slope ot the eonposit~on p ro f i l e to sero a t the ~h l~ t t . This
e f t e c ~ v e ~ y smooths out ~ s o o u p o s ~ o u ebsnSe end d e a r e u e s ~he e t t o e t
ot r o o o n b ~ t i o n m~o on ~ o t l o u vs~Lsb~os, I t ~Ls sn~io~pe~md t h a t
othor nossl.os wou~l.d gl.vo oomtido~b~,y dU'fmNml oonpool.ll.on pl'oflloJj
par~lou3arly t o r ~bs eeses in uhieh the eoupostt lon Is not Inlllal.l~y In
oqu4Zt~-~m. Howver~ ~ o oonelus~ons of t h l s sSmdF shouZd not be
at toe~od.
Figures ~-~A~ 3 - ~ j and ~-1~ eoupare ~he veloet~y~ prossure~ and
a r e s fo r the two szWeme ease~ of reoombina~lon ra~e considered. As
~ould be oxpoe~odj the wZoe4~, ~nomasos note mptdl,y in tho ooUFold-
tJ~n ehan~ss as s resu l t of ~ho dt4soetst ton energy being re~urnod to
~he tlou. This sans effee~ aeeoun~s f o r the tact ~hat~ f o r a given
~npem~uze m~io, ~he area i s considerably dLfteren~ f o r the two e x -
t~eus eases. The resu l t ot th i s area lnerosse also causes ~he pressure
~o drop more mp4d3~y f o r ~he case ui~h groa~er reaction (with respoo~
~o taupera~uro), Xn terns ot a given nosslo wi th a t tsnd area rat to~ i t
can be seen ~hat the exi~ pressure is approstnatalF the sane and t~o
ve loc i t y i s grea~er by spprozbuttol¥ 5 per eent f o r the can of neg l ig ib le
reae~lon. Houever~ ~here i s 20 per eent more t h e n s 1 energy rens in ing
4.0
:5.5
3.0
2 . 5
> 2.0
1.5
1.0
0.5
0
J
1.0
T o = 4 0 5 0 K
Po = 2 otm. (xo= 0.50
0.9
/ /
J n - - 5 ~
0 . 8 0 . 7
TEMPERATURE
r
0.6 0.5
J
FIGURE 5-11 FLOW VELOCITY
- k5 -
1.6
1.4
1.2
=- 0.8
I
To = 4050 Po = 2 otm.
(Xo= 0.50
K
0.6
0.4
0.2
0 1.0 0.9 0 . 8 0 .7 0 . 6 0 . 5
TEMPERATURE
FIGURE :5-12 FLOW PRESSURE
- 46 -
1 .8 ~
1.7
1.6
1.5
1,4
1.3
1.2
I.I
1 To = 4050 K Po = 2 o t m .
_ ¢x o 0.50 , , I
n = - 3 i
I
' ~ ~ -
1.0 1.0 0.9 0.8 0.7 0.6
TEMPERATURE 0.5
F I G U R E 3 - 1 3 FLOW AREA
- 4 7 -
the case of t~s t a ro sot&re mg~uro ~ o h ooQld po~sn~gAlly be ~mns-
la~od in to ~ns~Lc o u r w , In ~orms of mob nw~ors tb&s dg~foronoo
flow ~ l o o i t y would be oonsidorably ~ r o pronolnoods stnoo i~ would 9~-
oludo the ~omporaturo s l s o ,
of the e o l u t g ~ ob~tnod in th&8 s~udy ~ h ~ho ~oop~lon
of thoee l~'eeon~d in thg.s 8ootgLon~ 8ro 9J~Og.udod in Appendix IB~ Romultdn.
The oonolwions of th is t m a t t g s t i o u s ly be ausmsrJaed as folloess
1, The sm~log ooapQter has been proven to be a versat i le tool
f o r em~Lning the tlow of a r e a ~ i n g gas. ~ m~or ohanps
An tim prograu umd in ~his stud~ uould be nocossary to
mmmt~ the oomplolm e p e ~ of poss ib le boundary oondlttone,
2, Coffee,Ions to L t s h t h ~ e e widoal d t u o o t a t l ~ gas: are
readt~ disoernabla bu~ not of major t m p o r b ~ ,
The ez~nt of reaction and shape of t~e cmposttion pro f l la
i s pr taar i l7 detomtned by the fora of the reaction rata
law,
~ o t l o u r a t e s t ha t i n e n a m with temperaturo ha~s l i t t l o
o f foo t on the flow eo~os t t~on Cat low pressure) and 8uoh
flows say be a u m e d to be f rosen ,
Roso t~n rat~s ~hat d o o m u~th ~Mporaturo affoe~ ~h8
e o ~ o ~ t ~ n s i g n i f l ~ n t l y and in those oaso8 tbo ohomioal
roao t ion nus~ be eor~f, dorod.
Z n t t t a l l y non oquilibr~Lum eonpos~t~ns r e s u l t In a g rea te r
or lossm" ohanp An couposttion~ bat ~hls effoo~ t s coupled
w r y e loss l~ to tho par~ ieu la r noszle contour considered,
The ef teo~ of t o m of ~hs mccubina t lon ra te on the ~ l w of
a ~'eaotlng gas t8 s~gn~f i~n t and mugt be d o t o ~ d e~aet~y
before d o ~ a ~ d m u ~ m s of p o t e n t i a l appl£eatton8 can be made,
.
.
.
.
- L g -
V, JlK~JO~DATD~I~ YO~ FUTURE INVESTIOaTOR.5
The t u ~ L ~ y u ~ h uhioh the N~ o t 0qus~0nS dosoribin¢ the f!ow
o t s r o s e ~ 8as s u a o s ~ s ns~y sddi tJ~m~ s ~ s of ~ s ~ l o n .
1. The ncss lo oon~our has • detin£~e e t t o ~ on the o ~ p o s l t ~
protL~e 8nd a var4_oty of oon~ours shon3d bo oons~a r sd , Un-
loss considorab3~y t a ro ocupu~n8 oqui~nsnt ~ m s used in
th4t s~udF Is m S ~ b l o p ~ would neoossl~a~e using tho
cons~dersb~y s ~ 3 ~ d mdoZ of a d i s s o ~ s ~ Sss suuos~od
by
20 Tho e ~ b e ~ of bho e~mlF of • d~aoo~t4d gem ~ t,o the
e~ud¥ ot o t t he r • p u r e ~ lon:tsod f lng or oven a psa~tal~y
d~ssoo~a~! and p s r ~ 5 ~ y 4on~od ~_o~ ~s resdL~y f o r s o u b l s ,
Ths gros~os~ d~t f~ou~y in ~h~s extension uou~d bo the 8 v s t ~ -
b l l ~ t ¥ of ooupu~or emponsn~s. Ttn r e s o ~ o n I n ~s nsre3~y •
s ~ p ~ o loop snd a var~o~y o£ dL~feront t o m s or a oo1%oo of
d i f f e r e n t r o a o t ~ n s oen be added u separate c i r c u i t s uhioh
mm be Luoludod o1' oxe~udod a t p r o d o t o ~ d points In
s o l u t i o n . In o thor ~ r d s ~ sovers~ d~tferen~ n schs~sns fo r
~ho pe r t i nen t roaotton8~ eaoh st~nJ~J.un~ fo r var ious pres-
sures and ~enport turosj nsy be oona~dered e i m l t a n e o u s l y .
(Refele~o 33) has ~o~£de~ l tl~L~ probM but ~Lo~g ~,~o
3 ~ o e of h is studLo8 on • d~ssoc:La~ f l o v j and hi8 app rouh
can only be ocue~derod • p r e l ~ study.
-5o-.
3. The s o n s ~ t t v ~ y ot the ooWosl~lon p r o t t l s ~o the reoonbina-
t ~ n ratM s u A o s t s thBt t h i s nay be • usstu~ nothod f o r exper~-
nsn~L~y n s ~ u r / J ~ tho vsluo of the reoombS~stlon r a t e . 8L~o
the oqu~/~a~un cons~snt i s ue3~ knoun and ths ~ o g nay be
estsbLtsbsd f o r • mssonabls period of t i m , the prob~nn of
~ m ' l n g the dgLooooiation r~ te and ix~4~el~.gLng ~ho rooomb4za-
flea eate Is ellsAasted. :he eoapositlon sat be measured ~se:r
FesetLae3r~v m p e ~ p i o a ) A F and ths ).or prsseums c~ns~dsmd In
t h i s s ~ are r s a d l ~ a t t~Lnab~ in the l abo ra to ry .
h . 8peeLfie nessls e o n t o m t o t p e r ~ c u l a r spp%les~cms can be
dos~nsd as an ex t su to , , ot tho emTint s~udy. Pae~leula~
r o l s ~ o n s h / p s bsWwn the v a r ~ u s psraus~ers osn be spec~-
t i e d and the r o s u ~ t l ~ noss l s contour doterainod. This s n s ~ -
s~s n~y u o ~ L~d~oato b e t t e r approaches f o r the m a s e r
of ssooubl~t~on r a t e than tha t suuos~ed above.
5. An 8 n s ~ g ~ s p u t e r a ~ o prov~dos a ~onvont~mt nsans fo r e s -
t ab l i sh ing d~soont/nu~t~os in a t l o v £~eld by suddon~ adding
a s tep r u n , ton (vol taSe) . Therofore i t wou~d be e o n v e n t ~
to usa I t f o r ths study of the e h m l u l r s a e t ~ n s o e e u r r ~ g
behind • s t rong noma~ shook w ~ s .
6. Ths speed of so~nd in a r s ae t / ng a l x t m ~s a tun~tlon of
r ~ o n r a t s and wou~d ~md l ~ t to study ~n a sannsr
s ~ d ~ a r to t h a t app14ed hsl~. ~n a d d i ~ , ths r ~ a t l o n s h ~ p
betlen so~le ~s loe i ty and the eeltleal veloeit~ at the
noss~e th roa t could be ~ .
J j ' l~ l i l ) :~ A
• - 5 2 -
.
.
Is.
.
.
,
.
,
0,
1.
2.
13.
Ilmprs~ A. J . , Jr. "One D~mnaLon~X i~Lmm ot sn 2aperteo~ D~8~on~o
]Mu, it. "Thornod~m£o 9 o s ~ ot 8upersonlo ]~pans~on Boss)~s snd C~ou~t4on ot ~mn~.opl.o BspoNnt t a r 0bm~os)J~ bao4Ani~ 0SNS. = va~o ~ e o ~ d m l Repo~ 57-~8~. 4u~y :L~6.
Rt~dU~n, V. D.j snd W. 8. C~oolmors. mA 8~udy ot Squ£~41br£u h s ~ Oss ~t tooU in lf~q~son:Le ~ r liosslos, Ino~ud£ni~ Cbar~ ot 1~ssmd:rmm~ Propert/m8 for Squ~ULta~Lm A/z." NASA ~M D-2~1,~
1960.
I~Lnlb C. It. "CompS3at~Lou ot ' ~ a e s , m d ~ o Proport~es~ Yransport Propelq~S~ and Yhooret~4mL~ P ~ o l ~ m o o of Oaoeous H~Niroilsn."
Wa£bors S . L . eHos~ &dd£~Lon to • One D~Jnsus~ms~L 8uporsc~o lr~og Znolud~ss ao8~ 0ss r,t~. m ~X)-~I .6~-lS0j September ~960.
l:qmur, 8. 8. Zut4,oduo~:~ou. t~ the $*,uds, ot Chromos). l t u ~ l o n s in FIov 8]ateali. AOAH) olP'spb 7~ But, tervori~he Solentff io Publio~tiono~ London~ 1 ~ 5 ,
Pemmr~ 8, 8, Per~mm~ Pross~ London# ~-9~'T,~----- - --- - --"
h S a s j 8. Y. "Etteo~ ot Os~en ~ o o o ~ a t ~ o u on 0no DSmns~onsl F~ov s t IKsh Huh ]lmbors. • L~CX 11 /~lJ~ January 19~;6.
Bt,~ JC. If. Co "~opsr~uro trou D£ssoela~n Equ~Lbrlun 2n 8 HFper- son~o Xoss~." ARC 19,98)~ Pareh ~7~ 19~;8.
Brays E. N, C. tA~u£o Roeoub£nst~Lon in s HpTorso~o Wind Tunnel ~ss~e." ARC 20~;62~ Novonber 2~ 1~8 .
FrNmSnj N. O. qonsquL~£bs~u Theory ot 8n Zdo~L D£ssoo£atbLng ()so Thmush a Con£oa~L Nossl,. • ARC 2Op~Oj a~us~ 5s 1~8.
Roiohenlboohp Roj and 8, 8. Penner. san Z4botrtt4vo Prooodu15 for tho 8o2u4dLon ot Noss~e F~Lov Prob~eam udL4~h Rmrs£blB Chem£oaZ Rosot4ons." t IT TeohndLos~L Report 28j Ms~V 1959.
LdLs T . T . Ulon S q u t ~ b r i m n o v in Oas Dynamos. w al~flR Tll S9- )89, ~ y 1959.
Drouue~ V, O. ~Jrbo Rooomb£na~Lon Prob~m in a Jet Ezbaue4b Nossle. m Z/t8 Paper ~;9-10~p Jtms 1959.
B2oomp H. H. and H. H. 8tmi4;or. eInv~oLd n o v ~_th Non Squ i l l b r lm Mo~cular DdLsmoo~Lon for Pressure Distr ibut ions Knooun*.erod In i~pereon~o F:Lt~t . , aS]X:-TS-~;9-~2, SevSeuber 1959.
6.
17.
18.
9.
20.
21.
2,
lb~L~ Jo 0 . , A. Q. ~ohem.oodm'j and ,;. ,;. ![1.o£u. "Ctmd.u]. Won,. oqul~tw'J.un F, ffeoW on l~ych'ol[en Roolm~ ~ a m s t Lov Ps.~ammSo" CaL b p o r t AD-1118-,A-Si Imo,vombor 1.%9o
Oessnorj F. ao ~ J r . "Hon lkiull lbrtmt React4Lons in Sol:l~t. Propellant 08s Oenerator SystAms." 1KS i~oprint 10~6-60~ January 1960.
Reynoldoj T. ~.~ and L. V. ]~,,1~vJ.~, "One Dimnelonal Floe ~Lth Chenioal aeaot~on ~n Nosslo ~ x m s i o n s . " PrONnt~d 8~ ~ho IZCh]l ~]mpos:Lum on Thormodyna~oa of Jetb and Rocdml; Ps'opula£on~ Ms,y 1"/- 20, 1 ~ 9 ,
H a ~ J. O,~ and A. L, Ru~eOo "~tud£eo of Che~oa l Non Squi l£ba~m in l~pm~on£o NosLte Ylmm." CAL P.epos~ aD-ll l8.a-6~ liommber Z9~;9.
~bborj D, "~ndannual RepoTt on I m l t i ~ a t i o ~ of Ika ~o.uLlibrlmn Pl~nomna in Rool~i bmml~m. • ~ - 0 0 0 0 - O 9 1 6 8 ~ ~um ~10~ 1 ~ 0 ,
Woge~e~ P. P. "l~aauTement of R~te Coustantha of F u r b s o t 4 o ~ in & Supos.lon£o Nmsslm; u J oulq ~1 of G~me4Ml Ph3~_~_ae- Voltmo 28~ amber b~ Apr~ 1 ~ 8 . '
Weienel"j P. P . , J , E. lqartos and C, ThLele, ~Stud7 o f 8upenon£o
~bo aeaot~on s2"e20h'-2s02 8~ Log Reaetant Conm nm t t ~ n . " J~L Report 20-~9~ June 10, I~U.
23. Vopnere P, P. "Study of ~upo~onio Move v~Lth ChemLoal Rosat,2onss I I , 8UpeTeon£o ltosmZo Floe Tlth a Reaottng Oaa Mix~us~ at l~aotant Conoenta, ation," JPI, Repos"b 20-~70~ Deombe~. 19~ 1 ~ 8 .
2h. Olasstonep S . ~ o r e t ~ o a l CbenLstry. D. Van llostmnd Coups~ Ino.s Bog Zorkm 29~h.
25. Dole, X. Introduction to .S.tat£e,t,i,oal ~ W ~ t . Prentioe- ~lLls Ino . j Nov ][orkp 19r~.
26. lieLno s 8. P. ..Effoot~8 of Che~eal D£ssootation and Paleeular V4- brat~on8 on Steady One DiMnoional Flow. e ~ ~ D-87B August; 19~9.
27. LIEhI~Lllp M. J . a ] ~ e n of a D~osooLat;tng (]as." ARC 18~8J% ~ m n b e r l h j 1 ~ 6 .
28. Han~n, C. F. "£pproxlmat~ns fo r the Thermodynsmlo and ?rlmspo1% Properties of High Temperature A:Lt'." ILLCA 251 ~l~Op Karoh 1 ~ 8 .
~O.
~7.
~8.
• m a , , s . v . ~ ~oun~t£ou of o ~ o a l x ~ e ~ . ~ , - a t l l Book Company, Inc. , ~ev ~ork, 1960.
l~l~man, S. "The Chs~oal ~ £ o s of A~ at l~h ~o~raturoo~ A Problem in l~:~rl;on:Lo £srodynss~oo." AV~O Rasearch Report, 1~ February 19~7.
Cuhel, ~. B., and B. II. Jennf~p. ~B..~. McOrme-P~L~I Book 0ompany, ~o., New ~ork, i~8.
C)s~, E. )1. ¢ . , and J. A. W£~,,an. "a P z ' e ~ Study of lon:l.o /leoembination of Ar&,on in Wind Tunnel Nossles.,' USA,a Beport 1.t~, Februa:7 ).P60.
N a t t ~ u 0 D . L . "lnter~eroms~vlo Mouuremn~ in the Shook Tube of the D~soolat ion Rate of Os~pn." The Physie~ o£ Flu£do~ V o l t a 2~ Numb~ 2~ Mash-April I~9.
VLI£1aM, T . J . 'q'be app).£oatJ.on of ~'~.o1~ Computere t,o Va~Lous Combustion, ~ and Fluid ~ s m ~ o Problems. n WADC Teohntoal ~ te 58-171, Juno 19~8.
LuVaZlej J. B. w and H. Cornel£u. "Teohn£oal Note on the Appl£oatLon of an Analog Computer to the anal~ie of ~q~rlmma~l Kine~to Data." Leos~.~ ~8-527, aura ]5, 1~8.
ICsle s O. A. man Anslol~ Ccmpu~r Study of Iyperson£o ~ £ ~ r Tm,leo- t~ox,Xeo uo~ug Pe~urbattou Te©bniqueo. n Uu£vero£tW of Hiohigsn b p o m , A u l ~ t 1960.
Nm~el, Z. "A Teohn£qm for AnaloE Rawuenta~ian of A~mospher~o b - e n t ~ Ua£nl Pmeturbat£ou ~ b o ~ . . ~n£nz~£tT of M£ohf4~n Rapo~, Aqust 1960.
IPPigiO~[ !1
-56-
A - lblmhol~s f s ~ emm~ (Chapter IT)
a 2- t~pLml d ~ t m ~ o moleoulo
D - dJ~m~4on~oe o o u t ~ , , (d~£nmd by EquabLon 2-15)
S - df~ooofatLon e n o r w
F - OlbIN t s ~ e n o r a ( ~ a p ~ r I I )
F - dLmsnm£on~so moomM~t ,£on m~o (do t i~od by Equst£on 2-20)
Ir - ota~lsU.os) . ~ £ g b t of the o~oL-,on£o ground s,,at, s
0 - d l m n s £ o n l u s equi l ibr ium m n s t a n t (defined bF Equation 2-21)
h - Planok~s oona~au~
I - n o l s o u l a r nonsn~ o t l n o r t £ s
J - d/asnslonlsss 'rJ.brablon fumotion (dotinsd I~ gquat:l~m 2-17)
k - B o l t s a s ~ s oonstant
k D - d lssoolst lon rate
k R - reoonbSns~on rat4
r p . equ£11brlun cons tan t bssod on p a r ~ s l p r e s s u r e s
m - p o ~ l e ~ m o o
n . ~ponou~ f o r moonbina~ton m ~
B - A v o 4 ~ d r o e s n ~ b o r
N . r o o i p r o o ~ of ~ho ~ a ros
p - p ~ o s u r e
Q - psrtt.~l.on funo~£on
r - stres~l.Lno ooordLuate
R - u n £ v e r N 1 IPm o o ~ t ~ a n t
T - a b e o l u t o t o ~ o e r a t u r u
V - v e l o e £ ~ l r
O~ek 5]r'mboIjl t
OC - d l s o o o ~ t l m p ~ . a i W r ( d e f i e d I~, l qua t l on 2..~)
~, - d ~ i ~ n o o n m ~ t (defl.n~d bT S q u a t ~ Z-12)
v~- d tmM£on leoe m ~ a n ~ ( d ~ l d by ~l,~td.on 2-13)
9 - d ~ n e M £ o u l o m o e o u t a n t (dd~d bT ~q~a'~,£em 2 - 1 6 )
:l - o b ~ s ~ z d ~ t t o f r e q u e n a y o f m o l J ~ r v i b r a t i o n
- ~ m m u ~ f a o b o ~
4, - aboM.e speo£os
D - dJ.oeooJAtlou
• - equ£111~£~
e)Jo t - ol~ot~-onlo
M - m ) ~ e u l s r o i )oo too
R - ~ o o n b £ n a t £ o ~
r o b - r o ~ a t £ o n a ~
t r a n s - t m n s l a t l o n a ~
v £ b - v £ b r s t i o n ~
0 - r e f o m n o o condtt~Lon
S U l ~ r O ~ 4 p t ,
0 - s t a n d a r d o~a~o o f one a t m o o p h e r ~
k~:~HDIX O
-.59-
-6O-
Tsble C-1 l i s t s the Taluee of the p l~s las l constants used i n t h i s
study as r o l l as selected propert ies of : t z d is toa lo i~see, Although
o~gen has been usod fo r the spec i f i c solut ions presented berej t~e d l -
mnstonlos8 form of the equations allows seas iPmsrs l isat lon of t h i n
resu l ts to f l t other p, NSo
The choiss of the i n t t ~ l t e m p e r a t : i s d~floult 8rid doponds
upon the compositlon and pressure . Horo p e r t i n e n t to t h i s s tudy than
the ~ n t t l a l temperatu~s Is how f a r the i n i t i a l sompostt~on i s Erda an
oqu l l t b r i tm cc~oostt ion. Using the subscript oe to Indieate ths o q u f l / - t
br4.um composition a t the i n i t i a l pressure and t enpe r s tu re , ~ho o q u l l i -
brtum constant func t lon may be wr i t t en 8s
This oquatlon thus defi~os the I n l t l a l temperature m through the o q u i l l -
b r i m conotantm fo r a ~ p a r t i c u l a r e q u l l l b r l m oospos l t ion and i n i t i a l
p r @ s s U F @ o
Arblt~arlly the value of ~ oe has been chosen to be 0.50. The
t h r o a t pressure has boon chosen to be 28ta , B t h i s low preseure was
chosen oo t h a t the s o l u t i o n s could be a p p l i e d to current l a b o r a t o r y
8ysteus , most of uhlch have g lass CalSLI~ cbezd~rs upstream of the
nos s l e . For o x ~ p n , the correspondins teapera ture i s hOSOK. Thrm
d i f f e r e n t values of the ac tua l initial ooaposi t ion are s tudisdp one
above and one below equtJ.ibr'ltm and the oquilSbrtv~ emaposit lon.
AS discuHod in Chapter I l s there i8 considerable doubt as to
the cor rec t value of the recombination ra te fo r ozTpn . Ezsainat ion
Gas
H
H 2
M
02 i
C1 , i
cz~
I.O1
2.02 | , ,
E/k
o K
26,000
52,000
o K
6,320
I x i0 ~0
2
o.b6
; i
2
¢ g
2
1
L N I~.o 56,5oo
_ N2 28.0 113,O00 3,~00 13.6 2 1
0 I 16.0 29,500 ' 5
19.2 2 ;32.0 59,000 2,270 3
71.0 28,800 813 108 2 1
80.0 lz,~oo
~3
Br
~65 0 l
2 1 22p800
8,950
Br 2 IbO | i
127
Z 2 254 17,900 309 7k2 2 j l
h = 6.62 x 10 -27 erg sec/particle
N = 6.02 x 1023 particles/Ea-aole
R - 8.31 x 107 ergs~K ga-=ole
k - 1.38 x 10 "16 ergs/°K particle
1 - erg sec2/ga ca 2 - I.O1 x 106 erg/ca 3 ata
TABLE C-I Gas Parau.st~rs
-61-
of the recombinat ion r a t e funct ionp Fp shows t h a t the i n i t i a l va lue of
the r e c o a b i n a t i o n r a t e occurs as the product of r o ~ ( T o) and ~hat t h i s
i s the only place t h a t the c h a r a c t e r i s t i c l eng th appears , Thersfomp
any change in the magnitude of the recombinat ion r a t e a t T O can be e x a c t l y
compensated fo r by r e d e f i n i n g the value of r o. The r e s u l t i s t h a t only
the temperature dependence of the recombinat ion r a t e f u n c t i o n need be con-
s ide red° The value o f F has been chosen to he unityp a va lue t h a t O
c o = ~ s ~ s to r o - 1.o7 ~ ~ d ~ ¢T.) - 8.L x 10 ~ ¢ T / 3 5 ~
gin-mole "2 see "1 ( the value p r e d i c t e d by Matthews in Reference ~ ) ,
Seven i n t e g e r va lues of the temperature exponent i n the recombinat ion
rate were considered, -3 ~.n ~- + 3, Table C-2 su,unarises the initial
conditions °
The c r i t i c a l value of the v e l o c i t y , Vo, i s d i f f i c u l t to c a l c u l a t e ,
However, for t h e computer i t i s very simple to determine i t s value on
a t r i a l and e r r o r b a s i s , s ince i t appears only twice as a c o e f f i c i e n t
andp as a r e s u l t of the p a r t i c u l a r no i z l e contour chosen, only one of
these coefficients is initially important, Therefore, ~ is set by
adjusting its value until the derivative of the area is serop the ca-i-
t e r i o n f o r the nozzle t h r o a t ,
PO " 2Mm.
OCoe " 0,$00
T - 60501 Q
F • 1,00 O
0 - o.333 O
" 0.560
O - ~ . 6
Jo " 1.33
CASB u n n n
Z
I I
I l l
%
0.500
0.667
0o333
I I I I nn I
0.667
0,600
0.750
8TJBOJI~Z |
A
B
C
D
B
P
0
n
- )
t2
-1
0
TLBLE C~! I n i t i a l Condi~ione
-63.-
~ Z X D
a.~,,1.~ O-~nuter Pro~
She act ive olmmnt analog omputor j or e lect ron ic dt .~forsnt ia l
m~lTsmr9 oormtaW of a co l lec t ion of high Sain v o l t a p mmplf~iers w
venta~orsp oapaottorsp and oervo~tore arranged in ouch • oonftgu~ation
that the v o l t a p 8 throughout the c i r c u i t behave i d e n t i ~ as the
phy84osZ pal~motel~l in tho d l f£e l~n t ta l equations being 8oZved. The
e l :cuLt i s N t up by ecubinir~ a nunber of i n t e g r a t o r s , susmors, and
nul t lp lXero in a clomd loop ouch t h a t the spot•rictus on the v o l t • p c
oozqmopond to thorn required by the d i l f e r s n t t a l equat ion.
b u 4 c eZenent of the analog eomputor i s a high gain (approxl-
]l~te~, wJ~lO~Oj)CX~) ~p~4~i~ro ~][~ ~p~4411l~r has ~ ~hJrsctl~Jrist~Lc of
/nvorM~g the ai4~n of the v o l t s p being DpZiZied and may be assumed to
have • nagZ~Lg~.bZe current fZow through i t . Figure D-1 shores how t h i s
un l t coupled ~Lth NrlLos J.~out mststanoos 8rid e i ther • £oed1~tck mo le -
tame or ospsoitanoo ~ n be used to add or in te i~s to • v o l t a p . Z n l t t s l
~ud i t t ons a n N t t~ ~Lnpo~lng sn 1nit1•1 vole•go on the u p e o t t o r .
App31ostlon of Etrohof£s lave to the c i r c u i t of the smms~ l~Sulto in
tim output v o l t ~ p being equal to
I f the ampLtftJr MAnp AB 1o l a r p msough tM8 ~ be approximated by
- ~ o
£ o i m l ~ mml~'sts of the i n W ~ t o r m~d a mtmtlar approximation shows
tdmtp In t h ~ CaNe
Ro
R I
e~ e o
et
. . L R=
" - S U M M E R
" C
I
Ri e, e o
. . .L - J - m m
• D
INTEGRATOR
BASIC COMPUTER COMPONENTS
FIGURE D-I
- 66 -
-67-
t f
~° C9., ,u Q
The nes t 8 t~n i f t oan t f ea tu re of these t ~ :moults le t ha t the s m p l t f t e r
oharactmriet io8 do not appear and the gain of the c i r c u i t to s funct ion
on~y of the ex te rna l o l o o t r t o s l component~s.
Another bu4o oonponont used i n analog computers le the 8Ol, VO-
multip3.tor. This un i t oonslet8 of • number o£ o ~ L m ~ r ~ po~ontiomo~om
dz~von by • gozs~motor, (}no of the potonttomotor8 i s oonnoetod to •
l ~ e m n o o 8upp~ and foods I t8 output bask to the Nrvomotoro The
~ t o r pog~t4one i ~ l f so t h a t the input vol tage le Jus t balanood
by ~h~J £oedbsok voltm~o. In o ther gomdep osoh follow up potentlomotor
JJ pos i t ioned go t h a t I t s output i s deomaood in propo~lon to the r e t i e
of the e e r v ~ o t o r dx'i~lLnl; vol tage to the roferenoo Tol tage. Thusj one
func~4on nay bo mult4pliod by a8 manV other funs• lone 88 tha l~ are
fol low up po~onttomo~ows. Mul t ip l i e s • ion by • sons ,ant eoo.Bfiolont tg
• coomplished by feeding the v o l t a p through • p o ~ t i o m o t o r and ~apptng
off the l~qui :od person•see, Tht8 constant ooof f to lon t must the1~fo15
a l l y s be los s than un1~7.
Fl~u:o D-2 shows the Gomputer s~mbols used In t h i s FOpOl~o
F i g s D-3 shows • nunbor of bas le oonf~ura t ion8 used to pes'fos,at s
number of common OpOl~tione. Solving f o r e o ~n the d iv i s i on o t~oui t
eo ~ -- los --~' - tO0 ~.,
I f the amplif~or gain i8 su f f lo ion t lw h~h~ the oooond torn u n be
SUMMER
el , ~ . ~ h ~ , , ~ ~ - o e , - b e = e=
el (0 )
el -o e~ dt
SERVOMULTIPLIER
e, eoO, I00
POTENTIOMETER
.,__Q o < 1
oe,
REFERENCE VOLTAGE TIE POINT
Q ©
COMPUTER SYMBOLS
FIGURE D-2
- 68 -
e,
I00
eo
DIVISION
e, z
- e I eo
SOUARE ROOT
- o e o
e, / ,~ d,
DIFFERENTIATION
BASIC COMPUTER CONFIGURATIONS
FIGURE D-:5
- 6 9 -
-?0-
h e i s t e d and
C o = - ~00 ~--~
S ~ 3 ~ r ~ y . f o r the square ~oot o iz~u i t
~ o : %00
The d:Lfferent~ation ci~ou~t ~8 only an approx~natlon and. as such# ~s
genera l ly not reeonnonded. Solving for the vagus of % .
d e , ~ Co
F~" v ~ s s of a ~ r o a c h l ~ ~'~.tym th l e oesion e p ~ , o x ~ a ~ e t ~
du~vat~ve ~ c lo se ly .
e)Jotronie analog oomputer nay be used ~o solve 8 vat /s ty of
eni~nee~JLni; problem (see Itefe~noe8 3~-38). Zt to best suited fo r
solving systems of ordinary d~f ferent$~ equations desm'iblns inStIA1
oondi t ion p r o ~ ; hovsvor, i t ms7 bo used on non i n t t t s I oondt t ion
prob)Au8 or even sore pe t r i e1 d : L f f e r e r ~ equat ions ,
I nuaber of r e s t o r e n u t bo ca r e fu l l y cono4dered in preparlng a
probhm f o r solut40n on a computer. As evidenced by the p~v~ous d4s-
o~usslon~ the independent vs.-dab, s fo r the ~ u t e r Is time D and a l l of
the dependont var4.ables are vo l t egss . ?h~Ls neee s s l t e t e s scaling of e 33
of the physioal v a t , h i e s i n term8 of volta~SSo Most oonputer ooapo-
nsnts are designed to operate l i n e a r l y f o r v o l t a p s botvoen 0 and * XO0
vo l ts . Therefore a 33 tho m b l o s should be eoalod so t h a t tho i~
absoluto value i s l e s s than 100 w i t s throughout the e n t i r e so lu t i on .
-71-
(The PACE eomputoro used in t h i s study hays an autocratic overload i n d i -
ca tor to stop the so lu t ion i f a t any time any v o l t e ~ exceeds 100 v o l t s . )
In ordoF to ~ s OlTOr8, i t i s ~sIJc'•bk to k ~ p the v o l t a p 8 as
l a rge u possiblep p a r t i c u l a r ~ in sub t rac t ion opera t ions . Scal ing the
t~m fo r the so lut ion presents d i f f e r e n t oonsidemtions. Long t i r os
allow orrors to aoomm~ate u a resu l t of ampl i f i e r d r t f t p s a p • e l i o t
l a a k a p p StOop ~mroa8 shor t t t m u produce orrorsp s~uce tho uapacitozw
have ~ inherent time constants and the Nrvomul t i p l i e ro w i l l t rack
n m t i s f s o t o r t ~ up to a p p r o x , • r o s y 1 spa. (For f a s t e r so lu t ions non-
mechanical eleotronie m u l t i p l i e r s may be ueodj but they are loan depon-
dabls and l s s e aoou,'mto than N r v o m u l t i p l i e r u . )
The form of the equations and the sequence of operat ion i s of
considerable impor~ance in t h e i r so lu t ion . I d e a l l y j the equations should
be cas t in 8 Nlf -oompensat lng formj tha t i s s fop a deormmlng funot lon
tho de r iva t i ve should be equal to the negat ive of the va r i ab le ti,IB8 •
pos i t i ve funot ic~ . This formulat ion has • ~ndon~r to c o z i e r i teml~
in t h a t i f the ver~able i s too l a rge , i t cor rec t s i ~ l f by ad ju s t i ng
~m ~ r i v a t t v e to componmak. E f f e c t i v e l y t h i s approach forces a l l
e r ro r s to o s c i l l a t e around the proper so lu t ion r a t h e r than con t inua l l~
increase am the so lu t ion progFosces. A u r t o u 8 d i f f i c u l t y oeeu~ i f
t~m problem Invelves the d i f fe rence of two values of rmarly equal magni-
tude. In addltlon to the fast that mall error8 am ~agnlfied ~n th~
operatlon~ the oh•raster of the solution may ehan~ Fadloally if the
error is sufficient to ohsnRe the algob~ale sign of t.he result. This
l a t t e r d ~ f t i o ~ 7 may be rmmvad i f the nature of the problem i s such
-72-
tha t the o o m o t eigu f o r t h i s m a l l vol tage i s knmm. In t h i s ease
the r e s u l t m y he foroed to he oor reo t by f eed ing i t through a diode
or by dr~ving a s e r v o a u l t i p l i e r and a l lowing I t to respond in one d i r e c -
t i o n on~y. l n s p e o t l o n of the d lv~elon o l r o u l t shows t h a t the aocuraoy
I s s e r i o u s l y a f f e c t e d I f the d i v i s o r I s a ve ry m a l l number, even though
the q u o t i e n t may be l e e s than I00 v o l t a .
The s e t of e i g h t s4,nultaneous d i f f e r e n t i a l equa t ions d e s o r i b i n g
t h i s problem of a r e a c t i n g nosz le f l o v have been s e t up f o r the computer
s o l u t i o n and are presen ted i n Chapter I l l . I n speo t ion of these equa-
t i o n s shove that they are s e l f nompenseting as mush as poss ib le . The
re&ot4on equa t ion oontatno the d i f f e r e n o e between two n e a r l y equa l
q u a n t l t i e s p the d~ffezwnce be tvsen them be ing p o s i t i v e or n e s a t l v s de-
pending upon the d e v i a t i o n from e q u i l i b r l u a j and a l s o j the a rea funo t lon
s t a r t s a t zero and than i n o r e a s e s . The f i n a l prosren i s shown i n
F igures ~ and D.hB. I n s p e c t i o n shove how these problems were olrcmn-
vented In t h l s ease , The a rea funet lon~ I ~ tiN/dr, d r i v e s a sszwomul-
t l p l i e r ~ a l l o ~ n g only nega t ive valuaep e lnos the s o l u t i o n s t a r t s a t
the n o b l e t h r o a t where t h i s f unc t i on i s l d e n t t m ~ l l y ze ro . The d i f f e ~ -
enos between the two near]~v equal q u a n t i t i e s has not been ad jus t ed m
o t h e r than mhxL~sing the vo l t agssp p r i n a r L l y as a r e s u l t of l e ek of
computer oomponent~. The ohotoe of the r e c i p r o c a l of the area~ N s as a
oomputer v a r i a b l e was made to formulate a s e l f oompensatin8 equa t ion as
v i i i as g i~ ,ng i t a mud~um value a t the beginning of the s o l u t i o n ,
The oholoe of p/T as a v a r i a b l e m made to reduce the r a t e of e h a n p
of p ressu re and glve nero seourete m o u l t s . The ehoioe of d ? / d t - -T 2
!
L~
I
¢ ~ " ~ . _ _ _ _ ~ , ~ n<o '=0
~__ d_..V.V" tOO I:-"T dt~VL/ I0o Va at
• ,~ ~ 0 _ _ ~ / ' 3
Z I"=
d t , ~ ~5oG
do~ i d~
d ~ * dt d~.
dV L
a~
i I
I dK -too ~.o¢ dt
- 5 , o o o P t d
~ T[.Ioo , ~% a t
, 5 E-r v =
,_ ~'~"L N d t ~4~. * .~ ,., a t dK
~at ~ dt
0
- I ~ - - ~ L T
~ V L dt
FIGURE D - 4 A COMPUTER PROGRAM
I
I
i +0( *-~EX
o----@ * So
-~* , ~ _ 6_.~ i D d ~
~t . ~ 50 d t
_ L _ t.
I0,000
I +__ T ~
I~O00
d=(
d r .
I 1-
IOO d t
I |00 ,-Or.. Z z.
I , 1 0 o r ± d'~z ~oo.t K d t
i ocT ~ I00,00o
i@
ioo+ I dq~
T{ iOo .i i ) dt ~ ' ~ I00
I00, 0 0 0 I 00 ~t.
dt
do~
+ 5 ~ T ~ - ~' T I o o 3 o o o i ,ooo
-~oc dt
-zo d3 V T ~
_ _ _ _ _ d J i d 3
-Z0J (J,=)
i T L + 1 , 0 0 0
• ~oJ , ~ - ~ t J + , } -4J U,,) *2OJU, , I
FIGURE D-4B COMPUTER PROGRAM
f o r the independent var iab le funct ion was made to m . ~ l a l ~ the number
of computer components nooouary fo r the problem so lu t i on , NOVOl~lO- I I
l o n , . t~s problem requtawd the fo~1o~ng eo~oonents
8 - l n t e g r a t o ~
2~ - Sum~ng J~mpl~Lfte~
8 - Se~vomult tp l~rs (18-Nparato m u l t i p l i e r c i r c u i t s )
27 - Pc•on•tome•ere
2 - 8~ tches
F l g u l w D-~ i s a reproduction of • sample so lu t ion to the oct of
d i f f e l ~ n t i a l equat ions . I t i s the output of • s ix chem~ll Sanborn
Recorder, I~spi to the f a s t t h a t the p r o ~ lure boon opttmtMd ~o
minJxtse e r ro r and the actual components e ros i s to l5 and ospsottonsp al, o
aocuna~o to 0,1 par sent and onoloNd in • t h e z ~ m t t c a l l y con t ro l led
ovonp e r ro r s u ~ •coumu~t4 end mus~ bo ohocimd, Although • complete
el~Or ana lys i s 18 out of the quostLon fo r • probleu ~his oonplezj •
VOlT good check on ~ •sour•oF £8 afforded t h r o u ~ the f a c t t ha t the
equation of s t a t e n~y be intoK~atod. Zn o ther wordap the ez~or i8 •
dilwot func t ion of hog close ~ 8 fo l log tng ~dontt ty £8 s a t i s f i e d
= t
Y ~ u r o D-6 shams hey c lose ly the so lut ions obta4nod In t h i s study sstLsFF
t h i s ros t r io t4ono The curve presented roprooen~s the average of • nunbor
of so lu t ions° Ccaparison ~l th Figure D-~ 8hou tha t the accuracy i s
gel1 ~ t h i n the l ~ u t t s u~th which the eonputer output can be road. Tbo
sharp increase in e r r o r a t the l o m r temperatures ~8 caused by the mmll
4- 0.6
d= dt
0
- 0 . 4
+ 2.5
&CX
0
- 2 . 5
1.0
N
0
I.O
.E T
0
I0
V 2
!.0 0
i.0
T
0
60 SECONDS
FIGURE D-5 SAMPLE SOLUTION
- 76 -
1.10
1.00
0.90,
0.80,
0.70.
0.60,
0.50" 1.00 0.80
\
0.60 0.40 TEMPERATURE
0.20 0
FIGURE D-6 TYPICAL ERROR
- 7 7 -
-78-
values of most of the palwmo~rra and the ro su l t i ng l n a o ~ Of nsadins
the da~a. lSmmverw ainoe tho me, or ~n~ereat 1o oentered upon ~ o t n l ~
por t ion of tho | o l u ~ n p t h i s prooonto no major problmmo
• t
API"SNDIX lz
Oomput4r 8o.lut, J.one
-79-
5 .0
T o = 4 0 5 0 K
Po = 2 a t m .
¢x o 0 . 5 0 ~ n = + 3 ~ =
2.0 , ~ I , , ,
1 1 . 0 '
0 .2
0 . 1 ' 1.0 0 .5 0 .2 O. I 0 . 0 5
P R E S S U R E
NOZZLE FLOW PARAMETERS, CASE I A
FIGURE E-I
- 80 -
5.0
2.0
I
To = 4 0 5 0 K
Po = 2 otto.
=o = 0.50 n --:3
0.5
0.2
0.1 1.0 0.5
NOZZLE FLOW
| 1
0.2 0.1 0.05 P R E S S U R E
PARAMETERS, CASE TB
FIGURE E - 2
- 8 1 -
5.0 T O = 4050 K Po = 2atm. uz _ . . . ~ ~ ao= 0.50 n = + 2 J
2.0 t
I,O ~
0"51 ~ ~ ~
,
0.2
0.1 1.0 0.5 0.2
PRESSURE
O. I 0.05
NOZZLE FLOW PARAMETERS, CASE IC
FIGURE E- 3
- 8 2 -
5.0
2.0
1.0
0.5
0.2
0.1
To = 4050 K Po = 2 a t m .
=o = 0.50 n - + l J
1.0 0.5 0.2 0.1 PRESSURE
0.05
NOZZLE FLOW P A R A M E T E R S , CASE I E
F IGURE E - 4
- 8 3 -
5.0
2.0
I
T o - 4 0 5 0 K
Po = 2 atm. m o 0.50 . - - I ~
"Jl
V 2
0.2
0.11 1.0 0.5 0.2 0.1 0.05
PRESSURE
NOZZLE FLOW PARAMETERS, CASE TF
FIGURE E- 5
- 8 4 -
5.0
2.0
I
To = 4050 K Po = 2 atm v2 ~ ao= 0.50
n =0 J l
S f
0.5
0.2
0 . 1 L
1.0 0.5 0.2 0.1 0 .05 PRESSURE
N O Z Z L E FLOW P A R A M E T E R S , CASE IG
FIGURE E - 6
- 8 5 -
5.0
2.0
1.0
0.5
0,2
0.1" 1.0
T o = 4050 K Po = 2 atm.
a o = 0 . 6 6 7
n = + 3
I ,---a..
Lv
0.5 0 .2
PRESSURE
! I
0.1 0 . 0 5
N O Z Z L E FLOW P A R A M E T E R S , CASE TrA
F IGURE E - 7
- 8 6 -
5.0
2.0
T o = 4 0 5 0 K
Po = 2 otto.
o¢ o = 0 . 6 6 7
n - - ' 3 ~ f
j f
0.5
0.1" 1.0 0 .5 0 .2 O. I 0 . 0 5
P R E S S U R E
0.2
NOZZLE FLOW PARAMETERS, CASE 11"B
FIGURE E- 8
- 8 7 -
5.0
2.0
T o = 4 0 5 0 K
Po = 2 arm, i = o¢ o - 0 , 6 6 7
0.5
0 .2
0.1 1.0
| |
0.5 0.2 0.1 0 . 0 5
P R E S S U R E
NOZZLE FLOW PARAMETERS,. CASE Tr C
FIGURE E- 9
- 8 8 -
5.0
2.0
T o = 4 0 5 0 K
Po =2 atm. = 0.667 (X o
n = + 1 ~
J 0.5" "
0.2
0.1 1.0 0.5 0.2
PRESSURE O. 0.05
NOZZLE FLOW PARAMETERS, CASE Tr E
FIGURE E - I 0
- 89 -
5.0
2.0
I T O = 4 0 5 0 K
Po - 2 otto.
. __ ~
f
S 0.5
0.2
0.1 1.0 0.5
I
0.2 0,1 0 ;05 P R E S S U R E
NOZZLE FLOW PARAMETERS, CASE TrF
FIGURE E-I I
- 9 0 -
5.0
T o = 4050 K Pa = 2otto.
= 0.667 ~ ~ - O( o
n = 0
~o~ - - ~ /A
0.5 " ~ r ~ '
0.2
0,11, 1.0 0.5 0.2 O.
PRESSURE
|
0.05
NOZZLE FLOW PARAMETERS, CASE TrG
FIGURE E-12 - 9 1 -
5.0
2 ,0
1
= 4 0 5 0 K
Po = 2o t to . ~,2 ~
ot o 0 .333
n = + 3 1 J
S , o
r
0.2
O.I • I I
I.O 0 .5 0 .2
P R E S S U R E
0.1 0 . 0 5
NOZZLE FLOW PARAMETERS, CASE TITA
FIGURE E-13
- 9 2 -
5.0
2.0 . = - 3 ! ~
S ,.o
0 . 5
0 .2
0.1 1.0 0 . 5
i |
0 .2 O. I O. 0 5 P R E S S U R E
I T o = 4050 K
Po = 2 otto
¢x o = 0 . 3 3 3 ,,,
NOZZLE FLOW PARAMETERS, CASE "rrr B
FIGURE E-14
- 9 3 -
5.0
2.0
I T o = 4050 K Po " 2 otto. (x o • 0.333 n - - + Z ~
0.5
0.2
0 . 1 u 1.0 -0.5 0.2
P R E S S U R E
NOZZLE
0.1 0.05
FLOW PARAMETERS, CASE TIT C
FIGURE E-15
- 9 4 -
5.0
T o = 4 0 5 0 K
Po = 2otto. v 2 ~
=o 0.333
2.0.i n = +t J
1.0
0.5" ~
0.2
O.i 1.0 0.5 0.2 0.1
PRESSURE 0.05
NOZZLE FLOW PARAMETERS, CASE Trr E
FIGURE E-16
- 9 S -
5.0
2,0
1.0
0.5
0.2
0.1
I To : 4050 K Po : 2 otto.
(x o 0.333
n =--I l J
f
1.0 O. 5 0.2 O.
PRESSURE
0 .05
NOZZLE FLOW P A R A M E T E R S , CASE I E F
F I G U R E E - 1 7
- 9 6 -
5,0
2.0
0.5
I T O = 4 0 5 0 K
Po = 2 arm.
a o 0 . 3 3 3
° ° ~
S
0.2
0.1 1.0 0 .5 0.2 O. I 0 . 0 5
P R E S S U R E
NOZZLE FLOW PARAMETERS, CASE ]]I G
FIGURE E-18
- 9 7 -
