- Email: [email protected]
On generalized scaling procedures for liquid-fuel rocket engines
On Generalized Scaling Procedures for Liquid-fuel Rocket Engines* S. S. PENNER and A. E. Fu~s The scaling procedures o/ Penner and Tsien, of Crocco and o/ Barrkre have been generalized by using the assumption that the mean drop size is proportional to the product of powers of the Weber number and the Reynolds number, together with the hypothesis that the total conversion time varies as a power (usually the second) of the drop diameter. The results obtained for the steady aerothermochemistry and for unstable motor operation (low-frequency and high-[requency oscillations) are shown to reduce to previously published rules when suitable simplifying assumptions are made.
IN RECENT publications on similarity analysis for the scaling of liquid-fuel rocket engines 1-~, procedures have been developed both for scaling the steady aerothermochemistry and for maintaining similar conditions with respect to low-frequency and high-frequency motor oscillations. An important practical problem concerned with rocket motor development is the result of occurrence of uncontrolled high-frequency motor oscillations. The value of similarity analysis must ultimately be assessed in terms of our ability to design large engines with predictable performance on the basis of small-engine test experience. In this connection, the most urgent experimental problem is that of determining minimum conditions required for effective practical scaling. We may consider qualitatively three separate components which are changed on scaling, any combination of which may be of dominant importance in producing uncontrolled motor oscillations. Thus highfrequency oscillatory combustion involves, in general, the following separate phenomena: (a) a driving mechanism for producing the oscillations: (b) a coupling process which allows sustained oscillations; (c) damping mechanisms for preventing or minimizing motor oscillations. A study of published scaling procedures shows clearly that it is impossible, except under very special conditions and for particular (and generally invalid) assumptions relating to the factors determining the chemical conversion time, to maintain exact similarity with respect to all phases of motor operation. For this reason, a preliminary experimental programme on the development of rational scaling procedures should be designed to show which physicochemical processes are of dominant importance for a particular motor design. If a strong driving mechanism is generally sufficient to induce unstable motor operation, we may be reduced to a study of microscopic phenomena involving the details of injector performance and combustion reactions. On * T h e w o r k d e s c r i b e d in this p a p e r w a s s u p p o r t e d b y the U . S . A i r F o r c e Office for A d v a n c e d S t u d y . u n d e r C o n t r a c t A F 18(603)-107 with the A m e r i c a n M a c h i n e a n d F o u n d r y C o . a n d J e t P r o p u l s i o n R e s e a r c h , Inc.
229
S. S. PENNER AND A. E. FUHS
the other hand, if we can prevent disastrous motor oscillation merely by preventing strong coupling between the driving and sustaining mechanisms, then motor oscillations can presumably be avoided by retaining an invariant ratio of chemical conversion time to wave propagation time. Finally, it is conceivable that motor oscillation can be prevented by maintaining adequate damping, irrespective of the nature of the driving and sustaining processes. In this case a detailed analysis of the steady aerothermochemistry, of frictional losses, and of damping associated with oscillating gas flow through the nozzle may be required. It is the purpose of theoretical studies on similitude in engine scaling to provide useful guides for experimental work. Meaningful answers to the questions raised in the preceding paragraphs can only be obtained empirically. To facilitate this programme, a concise and more general approach to the scaling problem than has yet been used is presented in the following section. The results contain the rules of S. S. PENNER and H. S. TSt~N, of L. CROCCO, and of M. BARREREas special cases. GENERALIZED SCALING RULES
For a given propellant system with fixed injector temperature the important similarity groups for stable motor operation are the Reynolds (R,,) and first Damk6hler (D,) similarity parameters; the Mach number (M) may become important for high-velocity processes involving oscillations. Here R~ = p v L / , 1 . . . . [1] D, = (L/v)/r
. . . . [21
M = ( o r ~/~,p)~
....
and
[3]
where ,o denotes density, v linear flow velocity, L a characteristic chamber dimension, ~j viscosity coefficient, T chemical conversion time, 3, ratio of specific heats, and p chamber pressure, Except for the orifice diameter, we shall impose the conditions of geometric similarity with respect to all chamber dimensions in order to make all Reynolds numbers constant. For fixed propellants and injection temperature, the coefficients 71 and 7 are fixed whereas p becomes effectively equivalent to p. Surface reactions are neglected throughout the following discussion although they may be important in selected motor applications. We denote by n~ the ratio i , / i ~ , , where H and M o denote, respectively, the large motor (H for H a u p t a u s f i i h r u n g ) and prototype ( M o for M o d e l l = model). Here i may represent any of the independent variables. We expect, in general, that the ratio of mean droplet diameter D to orifice diameter h is a function of the Weber (We) and Reynolds numbers based on the orifice diameter*. In particular, we assume that D / h , ~ (W~) -~ (Re) -~ ( h A p / o - ) - , (pvh) t~ *For a d e t a i l e d discussion of this p r o b l e m , as well as for some e x p e r i m e n t a l correlations, SHAVER a n d H . L . B o v E Y , J. Rex. nat. B u r . . S t a n d . 52 (1954) 141.
9213
. . . . [4] see M .
R.
ON G E N E R A L I Z E D SCALING P R O C E D U R E S
FOR L I Q U I D - F U E L ROCKET ENGINES
where &p is injector pressure drop, o- interfacial tension, and ~ and /3 represent parameters which must be determined empirically. Finally, we introduce the hypothesis that the conversion time r is determined through D and p, viz. 7 ~ D ~ p - '"' . . . . [5] with ~; and m' denoting parameters which must also be obtained experimentally. Equation 5 is known to hold when m ' = 0 . 2 , fi=2, for the burning of stationary fuel droplets in an oxidizing medium '~. A value ~=2 has been observed also in the burning of monopropellants and premixed bipropellant droplets in an inert atmosphere '~. Finally, the observed dependence of r on chamber pressure and injection pressure drop for a nitric acid-furfuryl alcohol rocket engine, which follows from equations 4 and 5 with fixed values of h and (3 =0. is in accord with the values: ,~= 2. m' 0-3, and ~.--0.8". In the following discussion we shall neglect the Mach number as a similarity parameter, thereby restricting our analysis to the development of scaling procedures for stably operating rocket engines. This is of practical importance in engine development programmes+. Similarity
o[ steady
aerothermochemistry
The .conditions for maintaining constant Reynolds and first Damk6hler numbers are p.v.L.--p~1,,v~,,Lt~,, and L . / v . r . = L , ~ . , / v , ~ o r , t ~ o whence it follows that n,. = 1/n,,n,.
....
[6]
and n,./n~. = n .
.
.
.
.
[7]
Combining equations 6 and 7 we find the universal relation n~ = 1 / n~n,;-'
....
[8]
which has been discussed for chemical reactors, particularly .... for the c a s e r/t~ = l .
From equations 4 and 5 we find the following alternate relation for n~: l l r ~ II~ ~ Ilh g~(1 ~ - f l ) F l p - ' ; ~ g + m ' )
=n~tt~, ~~
H~l~
;~n,,-:~+''~n,,
~;~ t ~,,3)
.-,~,:~+,;3~
. . . . [9]
where we have assumed that Ap ~ v '2, n~,, =n,, =
. . . . [10]
Eliminating n~ between equations 8 and 9 we find F/v I -(,V3 + m
')/1r2- "-',~(~+
~ ) ~ 1j - , % tT/ ;~(~+ t~- 1 )
*It might be m o r e logical to r e p r e s e n t r as the sum of times associated with e v a p o r a t i o n , diffusion. cheirl!cal reaction, etc. In the abscence of a d e q u a t e d a t a . however, we shall use the functional f o r m of e q u a t i o n 5. t i n l a b o r a t o r y tests which are designed t o test scaling rules r a t h e r t h a n to develop engines, it m a y be i m p o r t a n t t o scale also a n u n s t a b l y o p e r a t i n g c o m b u s t i o n c h a m b e r . In this case the M a c h n u m b e r should iaot be neglected as a n i m p o r t a n t similarity group.
S.
S.
P E N N E R AND A. E. F U H S
and after using equation 6 to eliminate np nv 1-2~+'~'" = m-~m,, 5~+~-' ~nL~-(~ +m'~
. . . . [11]
By the use of equations 6 and 11 we find also. that np - 1 + 2 ~ - ' '
(1)
n,-~'n~, ~(~+~-~) n,~2-~2"+~)
=
. . . . [12]
Relation between geometric scale factor nL anti thrust scale [actor
n--The continuity equation for the rocket chamber is ( p v d O , / (pvd~)Mo = n
. . . . [13]
where d identifies the chamber diameter and n is the thrust scale factor. Hence
npnvn~7 = n
and, using equation 6, . . . . [14]
n=nL
i.e. the geometric and thrust scale factors are identical. The relatively small effect of n~ on thrust, for pressures exceeding the normal operating pressure, is neglected. (2) Scale [actor [or the orifice hole d i a m e t e r - - T h e continuity equation for flow through the orifice holes is (vh~)n/(vh")Mo=n=nL
..
[15]
since the number of orifice holes remains unchanged if geometric similitude is enforced. Hence n~n,, ~ = nL . . . . [16] We may combine equations 11 and 16 in order to obtain the appropriate scaling rule for the orifice diameter• The result is 3:t nh =nor 2(l+m')-a(l
2m'--5(2a--~) +3
~-fl)nL2(1-rm')-3(l+3 a-fl)
....
[17]
(3) Scale [actor [or the linear flow v e l o c i t y - - T h e scale factor for the linear flow velocity may be obtained from equations 16 and 17. The result is --23~
2(1 - m ' ) - 3 ( 1 - - ~ + f l )
n~ = n,2" +")-~tl + 3"-~)nL2" +'')~t1+3"-~) (4)
. . . . [18]
Scale [actor [or the c h a m b e r p r e s s u r e - - F r o m equations 6 and 16 it
is apparent that n v -_ n h 2 Int. 2
Hence, using equation 17, we find 25=
23(~t-}- 1)--4
n v = n , 2 ( l + m ' ) - 8 ( l + 3 ~-fl) nt, 2¢l +m')-a
....
[19]
(5) Scale [actor [or the nozzle throat d i a m e t e r - - F r o m the continuity equation for flow through the nozzle throat of diameter r we obtain ( p f ) n = n = nL (pf)uo
. . . . [20]
ON
GENERALIZED
SCALING
PROCEDURES
FOR
LIQUID-FUEL
ROCKET
ENGINES
since the linear flow velocity at the nozzle throat is sonic irrespective of the motor size. Hence npn,. 2 =nl.
....
[21]
....
[22]
and, using equation 19, -~
n,=n
(:~+ m') + ½~;(/3-:,~- 3)
2. ~,,'~ ~(,+3~ ~)n~Z(,
~m')-~(l+3=
/3)
(6) S c a l e / a c t o r / o r the/eed system--From the continuity equation for the feed system in which the linear flow velocity is v, and the cross sectional area A1 we obtain ( v i A j ) , / (vIA:)Mo = n = n ,
whence n,t =n,/nal Similarity
o/ regenerative
....
[23]
cooling system:
Since the Reynolds and Prandtl numbers are fixed, the Nusselt heat transfer number also remains the same in the model and in the large engine. Hence the total heat Q which must be transferred to the interior motor walls, for chambers operating under similar conditions of temperature, pressure and internal aerothermochemistry, must be directly proportional to the propellant consumption rate; this, in turn, varies linearly with the geometric scale factor, as has been shown in equation 15. Thus nQ = Q , / Q ~ o = n , .
. . . . [24]
Correspondingly, the heat transfer per unit area q must obey the scaling requirement n,~=qH/q,~1o= 1/n~, . . . . [25] For motors operating with maintenance of complete similarity, the interior and exterior wall temperatures are fixed. Therefore, since the heat transfer per unit area varies as 1/nt,, the temperature gradient must be changed by the same factor, i.e. the wall thickness W must be scaled in geometric proportions: nw-WMWMo=nl. . . . . [26] The heat transfer problem has now been reduced to a determination of the requirements in the cooling passages for transferring heat per unit area in accordance with the scaling condition expressed by equation 25. The Reynolds number in the cooling coils remains fixed if the cooling coils of diameter C are also scaled in geometric proportions, viz. nc - C ~ / C,~o = nL
....
[27]
In the special case of a regeneratively cooled motor, the cooling passages correspond to the feeding system. For fixed propellants, constant Prandtl, Reynolds, and Ntrsselt numbers are maintained if / v i A : q , , / [v1A?L,o = n, lnA1 ~ = 1
or n,, I = 1/n,.
....
[28]
S.
S.
PENNER
AND
A.
E.
FUHS
a relation which is evidently consistent with equation 23 if geometric similarity is maintained for the cooling coils. In practice, the scaling requirements for the wall thickness and cooling coil dimensions may be undesirable because of excessive weight, or they may be inconsistent with design restrictions such as those imposed by similarity analysis for maintenance of low-frequency stability. The thickness of the outer wall of the cooling coils may be changed at will without affecting any of the other design restrictions obtained from similitude considerations. Similarity o[ sweat cooling
If a fraction ~' of the total propellant flow is used for sweat cooling, this fraction must be maintained at a fixed value as the engine is scaled. Details concerning the scaling of pore size in motor walls are easily worked out but will not be considered in the present discussion. Low-/requency stability
L. CROCCO: has discussed scaling procedures for a rigid and rigidly supported feeding system neglecting the compressibility of the propellants. Low-frequency stability depends on several parameters including the ratio of the chemical conversion t i m e r to the residence time ~,., the relative pressure drop A p / p (Ap=_ pressure drop across injection orifices, p=steady operating pressure in the combustion chamber) and an inertia parameter which equals the ratio of the kinetic energy of the propellants in the feeding system to the work done on the propellants by the pressure drops during the residence time. The ratio of the chemical conversion time r to the residence time r,. is maintained invariant by enforcing the condition that the first Damk6hler similarity group must remain fixed. This requirement has been introduced in the discussion relating to the steady aerothermochemistry and is given explicitly in equation 7. In view of equation 18, the result may be stated in the form n. ~ n,, = n~ 2" +m'~-~¢~+3 ~-~ nl2. +,.'~ ~. +3 ~-~>
. . . . [29]
Next we consider the implications of the low-frequency stability requirement that (Ap/p) n ( A p ) . [Mo (Ap/p)Mo - (Ap).~,o p,,
-
1
....
[30]
Since /Xp ~ v =, equation 30 is equivalent to the relation (v'-'/p)./(v'-'/p).,,o=n.~/n,= 1
By using equations 18 and 19 it may then be shown that equation 30 reduces to the condition -~ /,1 2(1 ~m J - ~ l ~ 3 ~ - B )
1('
m ' ) - 2,~(/3+ :' >
n t 2¢1 ~m)-~¢1+3,-/~)
_
1
....
[31]
ON
GENERALIZED
SCALING
PROCEDURES
FOR
LIQUID-FUEL
ROCKET
ENGINES
For n , = 1, n,,@l, equation 31 corresponds to either of the relations 2 ( 2 - m') = 8 (/? + 2) "~ l 2(1 +m') - ~(1 +3 ~--/~)= ~
or
. . . . [3 la]
The inertia parameter .
.
.
.
must also remain invariant to maintain low-frequency stability'-'. Here K is a constant, 11 repres6nts the equivalent length of the feeding system with cross sectional area A,, F is the total propellant injection rate, and the other symbols have their usual meaning. The condition J,,/J,,,.- 1
. . . . [331
then becomes (11/ AI),, = n,,n: / n, .
.
.
.
.
[34]
(1!/AI)~,,, in view of equation 30 and of equations 13 and 14. Using equations 19 and 29 we find the result l£z
,~q34 ~÷/3) ~ "-'fro' - 3)
(It~A,)1, _ n, m-,,,'i-.~(~ ~ 3 '-/~;n,. 5~. ,,r}-2(i :: ? ;:~r (l!/ A,):u,,
. . . . [35]
For scaling of the feeding system and cooling coils with maintenance of geometric similarity, the left-hand side of equation 35 evidently reduces to (ll/ A l ) z / ( l l / Aj).u,, = 1/nj.
Crocco ~ has indicated the need for extending the analysis presented in this section in order to make allowance for compressibility of the propellants, non-rigid feeding systems and/or non-rigid mountings, systems involving pumps for pressurization, etc. Details concerning studies of this type are best worked out in connection with motor development programmes. High-jrequency
stability
High-frequency instabilities can presumably be controlled, for operating conditions otherwise similar, if the ratio of the chemical time 7 is fixed with respect to the wave propagation times 7w. But the latter are linear functions of the chamber dimensions for sound waves as well as for shock waves provided the sound velocity remains invariant. Hence, for fixed temperature and 7, the similarity requirement for high-frequency instability may be written in the form it: = n ~ . = n , .
.
.
.
.
[36]
it is apparent that the condition expressed by equation 36 is not usually consistent with the similarity requirement for the steady aerothermochemistry or for low-frequency stability, both of which lead to the result given in equation 29.
S.
S.
PENNER
AND
A.
E.
FUHS
Equation 36 expresses only the condition for similitude of the driving and coupling forces for oscillations. The forces producing oscillations must be balanced against the damping forces, of which the principal sources must be frictional losses. These, in turn, should be similar as long as the Reynolds number is constant. Damping associated with the oscillating gas flow through the nozzle is not similar unless the nozzles are scaled with maintenance of complete geometric similarity. However, for a well designed prototype it appears unlikely that damping through nozzle oscillations plays a dominant role. SUMMARY
OF
RESULTS
AND
DISCUSSION
In T a b l e 1 the similitude requirements for different aspects of the scaling problem have been summarized. The. Penner-Tsien (P.-T.) relations correspond to the special case in which m ' = ~ = f l = 0 , n , = n ~ = l , and ~=2; they can be derived from the generalized results formally by letting m' approach infinity. The scaling procedure for the P.-T. rule is summarized in T a b l e 2. Crocco's second rule corresponds to the special condition g=0; the results thus obtained are listed in T a b l e 3. Barr~re's scaling rule is based on an empirical expression for 7 (derived from motor tests with red fuming nitric acid and furfuryl alcohol) which is inconsistent with the assumptions expressed by equations 4 and 5. Thus Barr&e starts from the relation r ~ Ap-,"p-,,,"
. . . . [37]
which is in agreement with equations 4 and 5 if ~a=r', f i = O , n , = 1, and nh = 1 in the relation for D only. This last requirement is probably to be attributed to the fact that motor tests were carried out without variations in orifice diameter. The statement n,,= 1 in the expression for D is inconsistent with Barr6re's scaling rule for n,, derived from other considerations. It will be interesting to see if motor tests with variable injection orifices are really correlated better by equations 4 and 5 than by equation 37. The scaling procedure of Barr6re can be obtained by repeating the analysis described in detail in the text with equation 37 substituted for equations 4 and 5. The results derived from this study are summarized in T a b l e 4. Reference to T a b l e s 1 to 4 shows that the scaling requirements for the steady aerothermochemistry are not generally consistent either with those for low-frequency stability or with those for high-frequency stability*. Thus it is impossible to scale with maintenance of complete similarity with respect to all the important variables. From the practical point of view, the most important discrepancy is that between the usually contradictory requirements for n~ in the analysis for the steady aerothermochemistry and for high-frequency stability. As was stated in an earlier pape r3 and mentioned again in the introduction to the present survey, we are as yet *The only exception is 0rovided by Crocco's second rule in the unlikely case that m ' = 1; see T a b l e 3. 9q~
ON G E N E R A L I Z E D
SCALING
PROCEDURES
FOR
LIQUID-FUEL
ROCKET
ENGINES
unable to assess the significance of this discrepancy. For this reason a preliminary test programme on engine scaling should involve an investigation of high-frequency stability of properly performing engines in which only the scaling rule n~ = m~ is accounted for. The steady aerothermochemistry and high-frequency stability requirements are satisfied simultaneously if 2~
2('.'m'-2,~+t;fl)
n72(1 F m ' ) - t ~ ( l - 3 z fl) n l . 2 ( l ~ m ' ) - ~ ( l q - 3 a - / 3 ) __nL or 2~ n
.'2(m'-l)+~(l + B ~)
2~1 ÷ m ' ) - , ' ; ( l + 3 ~ - / z b n l . 2 ( l + m ' ) - , ' ; ( l + 3 ~ h') = 1
....
[38]
According to equation 38, variation of m can, in principle, be used to maintain similarity for the steady aerothermochemistry and high-frequency stability. Whether or not this programme is practically feasible remains to be seen and cannot, in any event, be decided until reliable empirical information is available, not only on the validity of the functional relations given in equations 4 and 5, but also on the absolute values of the parameters 3, ~, fi and m'. It is interesting to note that equation 38 is satisfied exactly when n , = 1 if either 2(l-re') 2(l+m')
=~(1+,~-2) ~S(1+32-fl)
] }1
. . . . [39]
or
2(1 + m ' ) - ~ (1 +3 2-/?)
~}] 2(1 - m ' ) - ~ ( l + / 3 - ~) ~ oo !
. . . . [40]
The latter relation would never be expected to apply*. For m ' = 0 , S--2, ~=/?=0, the first expression given in equation 39 is satisfied; however, in this case all the exponents for n,. appearing in the scaling rules for nh, nv, n,, n,. and n, become indeterminate and, furthermore, the second condition given in equation 39 is not satisfied. In fact, the rule m ' = 0 , ~=2, 2 = / ~ = 0 for n , = 1 corresponds simply to the P.-T. procedure. According to Barr6re's findings for a bipropellant motor utilizing red fuming nitric acid and furfuryl alcohol, m'=0.3, ~=2, 3~ = 1.6,/3=0. Then 2(l-m')=14 and 3 (1 +/3 - 2)=0-4 Therefore the conditions given in equation 39 are far from satisfied. Rather than pursue the interpretation of the generalized results for other speculative cases, we conclude the present discussion with the observation that extensive tests yielding a correct functional form for r are the first *The statement given in equation 40 should not be confused with the lormal procedure used to derive the P.-T. results from the generalized expressions by setting m ' - c ~ .
237
2(l+m')-~-(l
~
(;.s.
2 ( l + m ')-g,(l-c3
=
~3=
(
z,~( = + , ) . , )
##l.~2(It, m')-3(l-.3~-fl) 1
~-B) it L
(l+m')-;
fl)]
(1 ÷ 3 ~ /?)]
b(I-13=
((.~+,,,')+'~ ~n-~ ~ ~
i3)]nL\2(l+m')
~
/3] HL 2 (l-} m ' ) - ~ 0 - - 3 ~-/~)
)
(ll)
(10)
(9)
(8) l/Hi.
n(,=nl,
n,,.=nj
,1,:
,,.=l/n,.
B Regenerative cooling
D High-frequency stability
( 1~ \ = ns .' c.s.
(IJA,),t
~ , ,
(7) Repeat
(13)
(14)
n~=nl.
~_nn(2(l_fm,)_~-if3~_/3)jlll\~,~--+-m:)-~)]
(12) 11=(2(T+mr)--g(i+3~/3')lit\2-(f-+~-~r-)-~rl~-3~L--g)]=l ~ ' ( ~ ~. . . . "~ ~ ~/J+'-'> '~
C Low-Jrequency stability
nv=
)
(11)
) n~t=nL-a
) n~----nlg
(10)
(9)
(8)
no=n1.
nw=-nt.
na=l/n L
n.s=l/n r
B Regenerative cooling
) n"=nL ½
1
n.=nL -1
t
) nh=nl.
) n~n L
A Steady aerothermochemistry
nt-'=l
hi'-'= l
(7) n:=nt"-
(13)
(12)
C Low-frequency stability
(14)
n~=nt,
High-frequency stability
D
Ible 2. The P.-T. scaling rules for liquid@tel rocket engines (m'=~=fl= 0, n~=nv=l, 3=2); these rules are obtained from the general results most simply by setting n~= 1, m'=zx:.
he subscript G.S. identifies results appropriate for scaling with geometric similarity
nr~n~
-~
.,( 1)
)* nvl--- -hA!
n,.=n~
2(I-Fm')-3(1+3=
no~n ~
?~(1+3 ~-fl)
26:
2 ( I #m')
2(l+m')~(l~3~g))nLt2(l+m')--~(l+3=--n) ]"~ ~ ( ~'"' -~ ~'-'~ ~
n l =~ n j
n,,=nJ-
li~n L
A Steady aerothermochemistry
Table 1. Generalized scaling rules for liquid-fuel rocket engines deduced ]rom similitude considerations
nL
(l--m')/(l+m')
,~'/(1 +m')
u
m'/(
n 4/
n L
L
I
+ ,. ')
1
.
.
.
(11)
(10)
(9)
(8)
= l/nl,
=nt
nc=nt.
n,
n i = 1,In I
n
,13)
(12)
C
(1,/A I)n / ~ .o \ ~ ' ,. - 1 =
J~7) , ' - ' 6 : ..... '~t+'"'
nt= !'-'-"")/"' { '"')=1
Low-/reqtwncy stability
D
High-[requency stability
I--2
r'
'
ii =nl
",
av~nL
.
nr
n4t \
]
"(t
%~r::i-;;,')
'.~ ~m" t)
"'( ])
2
)
(:{+,,,,-,,,~
m'
\l+t
,
+
m'--r'
np_nL(-'2,,_ 1 ~ )
n v ~ n 1.
n h --nl,
n~-n L
.,~ (11)
(10)
(9)
n,~n t
n ----n~
t =l/n1.
/R) n,. ----I/hi.
B
Regenerative cooling
A
Steady aerothermochemiatry
(12)
t,,= (¢
nt /~ s.
~]/7";2"-';::1=1
(I¢/A )M, \
lit.
"
Low-/requency stability
C
~/41
n,=tt~
High-Jrcq tem'y .stability
D
(7) n~=n, =,,,'.~1+,,'~ (14) n~=n L able 4. Barrbre's scaling rtde [or liquid-[uel rocket e'l!;ines (e~=2. ~=r'. 13=0, n~= I, nt = l in the expression lot" D/h); the effect ol setting n.,=l in the expre.~sion [or D / h can be accounted for by repeating the entire analyMs
nr ~nT,
nr=n
1/2+l/(l+m')
n p = n I - g / ( 1 + m ')
nv~.l~ L
nh=n L
n =
B
Regenerative cooling
A
Steady aerothermochemistry
]able 3. Crocco's second ,waling rule/or liquid-[uel rocket engines (~;=0, n== 1)
S. S. PENNER AND A. E. FUHS
requirement of an intelligent scaling programme. I m p o r t a n t steps in this direction have already been made by Crocco and Barr6re but additional quantitative experimental work, for a variety of propellant systems, is urgently needed.
Daniel and Florence Guggenheim Jet Propulsion Center, California Institute of Technology, Pasadena, Cali[ornia (Received October 1956) REFERENCES
1 Modelli nella Tecnica Vol. 1 pp. 652-669, Accademia Nazionale dei Lincei, Rome, 1956; See also Teeh. Rep. No. 13 Contract DA 04-495-Ord-446. California Institute of Technology, Pasadena, 1955 2 CROCCO, L. Considerations on the Problem of Scaling Rocket Motors, Selected Combustion Problems 11 p. 457. AGARD, Butterworths Scientific Publications: London, 1956 :l PENNER,S. S. Jet Propulsion 27 (1957) 156 BaRR[RE, M. Similarity of Liquid-fuel Rocket Combustion Chambers Office Nationale d'Etudes et de Recherches A6ronautiques: Paris, 1956 :' See, for example, PENNER, S. S. Chemical Reactions in Flow Systems Chapter 4 Butterworths Scientific Publications: London, 1955 BARR/~REM. and MOtJTET, H. Rech. adro. 50 (1955) 31 7 BARR/~RE, M. Preliminary experimental results obtained at the Office Nationale d'Etudes et de Recherches A6ronautiques, Paris; personal communication DAMK6HLER, G. Z. Elektrochem. 42 (1936) 846 PENNER, S. S. Combustion Researches and Reviews 1955 Chapter 12 AGARD, Butterworths Scientific Publications: London, 1956 1 FENNER, S. S. Models in Aerothermochemistry in
240
IN RECENT publications on similarity analysis for the scaling of liquid-fuel rocket engines 1-~, procedures have been developed both for scaling the steady aerothermochemistry and for maintaining similar conditions with respect to low-frequency and high-frequency motor oscillations. An important practical problem concerned with rocket motor development is the result of occurrence of uncontrolled high-frequency motor oscillations. The value of similarity analysis must ultimately be assessed in terms of our ability to design large engines with predictable performance on the basis of small-engine test experience. In this connection, the most urgent experimental problem is that of determining minimum conditions required for effective practical scaling. We may consider qualitatively three separate components which are changed on scaling, any combination of which may be of dominant importance in producing uncontrolled motor oscillations. Thus highfrequency oscillatory combustion involves, in general, the following separate phenomena: (a) a driving mechanism for producing the oscillations: (b) a coupling process which allows sustained oscillations; (c) damping mechanisms for preventing or minimizing motor oscillations. A study of published scaling procedures shows clearly that it is impossible, except under very special conditions and for particular (and generally invalid) assumptions relating to the factors determining the chemical conversion time, to maintain exact similarity with respect to all phases of motor operation. For this reason, a preliminary experimental programme on the development of rational scaling procedures should be designed to show which physicochemical processes are of dominant importance for a particular motor design. If a strong driving mechanism is generally sufficient to induce unstable motor operation, we may be reduced to a study of microscopic phenomena involving the details of injector performance and combustion reactions. On * T h e w o r k d e s c r i b e d in this p a p e r w a s s u p p o r t e d b y the U . S . A i r F o r c e Office for A d v a n c e d S t u d y . u n d e r C o n t r a c t A F 18(603)-107 with the A m e r i c a n M a c h i n e a n d F o u n d r y C o . a n d J e t P r o p u l s i o n R e s e a r c h , Inc.
229
S. S. PENNER AND A. E. FUHS
the other hand, if we can prevent disastrous motor oscillation merely by preventing strong coupling between the driving and sustaining mechanisms, then motor oscillations can presumably be avoided by retaining an invariant ratio of chemical conversion time to wave propagation time. Finally, it is conceivable that motor oscillation can be prevented by maintaining adequate damping, irrespective of the nature of the driving and sustaining processes. In this case a detailed analysis of the steady aerothermochemistry, of frictional losses, and of damping associated with oscillating gas flow through the nozzle may be required. It is the purpose of theoretical studies on similitude in engine scaling to provide useful guides for experimental work. Meaningful answers to the questions raised in the preceding paragraphs can only be obtained empirically. To facilitate this programme, a concise and more general approach to the scaling problem than has yet been used is presented in the following section. The results contain the rules of S. S. PENNER and H. S. TSt~N, of L. CROCCO, and of M. BARREREas special cases. GENERALIZED SCALING RULES
For a given propellant system with fixed injector temperature the important similarity groups for stable motor operation are the Reynolds (R,,) and first Damk6hler (D,) similarity parameters; the Mach number (M) may become important for high-velocity processes involving oscillations. Here R~ = p v L / , 1 . . . . [1] D, = (L/v)/r
. . . . [21
M = ( o r ~/~,p)~
....
and
[3]
where ,o denotes density, v linear flow velocity, L a characteristic chamber dimension, ~j viscosity coefficient, T chemical conversion time, 3, ratio of specific heats, and p chamber pressure, Except for the orifice diameter, we shall impose the conditions of geometric similarity with respect to all chamber dimensions in order to make all Reynolds numbers constant. For fixed propellants and injection temperature, the coefficients 71 and 7 are fixed whereas p becomes effectively equivalent to p. Surface reactions are neglected throughout the following discussion although they may be important in selected motor applications. We denote by n~ the ratio i , / i ~ , , where H and M o denote, respectively, the large motor (H for H a u p t a u s f i i h r u n g ) and prototype ( M o for M o d e l l = model). Here i may represent any of the independent variables. We expect, in general, that the ratio of mean droplet diameter D to orifice diameter h is a function of the Weber (We) and Reynolds numbers based on the orifice diameter*. In particular, we assume that D / h , ~ (W~) -~ (Re) -~ ( h A p / o - ) - , (pvh) t~ *For a d e t a i l e d discussion of this p r o b l e m , as well as for some e x p e r i m e n t a l correlations, SHAVER a n d H . L . B o v E Y , J. Rex. nat. B u r . . S t a n d . 52 (1954) 141.
9213
. . . . [4] see M .
R.
ON G E N E R A L I Z E D SCALING P R O C E D U R E S
FOR L I Q U I D - F U E L ROCKET ENGINES
where &p is injector pressure drop, o- interfacial tension, and ~ and /3 represent parameters which must be determined empirically. Finally, we introduce the hypothesis that the conversion time r is determined through D and p, viz. 7 ~ D ~ p - '"' . . . . [5] with ~; and m' denoting parameters which must also be obtained experimentally. Equation 5 is known to hold when m ' = 0 . 2 , fi=2, for the burning of stationary fuel droplets in an oxidizing medium '~. A value ~=2 has been observed also in the burning of monopropellants and premixed bipropellant droplets in an inert atmosphere '~. Finally, the observed dependence of r on chamber pressure and injection pressure drop for a nitric acid-furfuryl alcohol rocket engine, which follows from equations 4 and 5 with fixed values of h and (3 =0. is in accord with the values: ,~= 2. m' 0-3, and ~.--0.8". In the following discussion we shall neglect the Mach number as a similarity parameter, thereby restricting our analysis to the development of scaling procedures for stably operating rocket engines. This is of practical importance in engine development programmes+. Similarity
o[ steady
aerothermochemistry
The .conditions for maintaining constant Reynolds and first Damk6hler numbers are p.v.L.--p~1,,v~,,Lt~,, and L . / v . r . = L , ~ . , / v , ~ o r , t ~ o whence it follows that n,. = 1/n,,n,.
....
[6]
and n,./n~. = n .
.
.
.
.
[7]
Combining equations 6 and 7 we find the universal relation n~ = 1 / n~n,;-'
....
[8]
which has been discussed for chemical reactors, particularly .... for the c a s e r/t~ = l .
From equations 4 and 5 we find the following alternate relation for n~: l l r ~ II~ ~ Ilh g~(1 ~ - f l ) F l p - ' ; ~ g + m ' )
=n~tt~, ~~
H~l~
;~n,,-:~+''~n,,
~;~ t ~,,3)
.-,~,:~+,;3~
. . . . [9]
where we have assumed that Ap ~ v '2, n~,, =n,, =
. . . . [10]
Eliminating n~ between equations 8 and 9 we find F/v I -(,V3 + m
')/1r2- "-',~(~+
~ ) ~ 1j - , % tT/ ;~(~+ t~- 1 )
*It might be m o r e logical to r e p r e s e n t r as the sum of times associated with e v a p o r a t i o n , diffusion. cheirl!cal reaction, etc. In the abscence of a d e q u a t e d a t a . however, we shall use the functional f o r m of e q u a t i o n 5. t i n l a b o r a t o r y tests which are designed t o test scaling rules r a t h e r t h a n to develop engines, it m a y be i m p o r t a n t t o scale also a n u n s t a b l y o p e r a t i n g c o m b u s t i o n c h a m b e r . In this case the M a c h n u m b e r should iaot be neglected as a n i m p o r t a n t similarity group.
S.
S.
P E N N E R AND A. E. F U H S
and after using equation 6 to eliminate np nv 1-2~+'~'" = m-~m,, 5~+~-' ~nL~-(~ +m'~
. . . . [11]
By the use of equations 6 and 11 we find also. that np - 1 + 2 ~ - ' '
(1)
n,-~'n~, ~(~+~-~) n,~2-~2"+~)
=
. . . . [12]
Relation between geometric scale factor nL anti thrust scale [actor
n--The continuity equation for the rocket chamber is ( p v d O , / (pvd~)Mo = n
. . . . [13]
where d identifies the chamber diameter and n is the thrust scale factor. Hence
npnvn~7 = n
and, using equation 6, . . . . [14]
n=nL
i.e. the geometric and thrust scale factors are identical. The relatively small effect of n~ on thrust, for pressures exceeding the normal operating pressure, is neglected. (2) Scale [actor [or the orifice hole d i a m e t e r - - T h e continuity equation for flow through the orifice holes is (vh~)n/(vh")Mo=n=nL
..
[15]
since the number of orifice holes remains unchanged if geometric similitude is enforced. Hence n~n,, ~ = nL . . . . [16] We may combine equations 11 and 16 in order to obtain the appropriate scaling rule for the orifice diameter• The result is 3:t nh =nor 2(l+m')-a(l
2m'--5(2a--~) +3
~-fl)nL2(1-rm')-3(l+3 a-fl)
....
[17]
(3) Scale [actor [or the linear flow v e l o c i t y - - T h e scale factor for the linear flow velocity may be obtained from equations 16 and 17. The result is --23~
2(1 - m ' ) - 3 ( 1 - - ~ + f l )
n~ = n,2" +")-~tl + 3"-~)nL2" +'')~t1+3"-~) (4)
. . . . [18]
Scale [actor [or the c h a m b e r p r e s s u r e - - F r o m equations 6 and 16 it
is apparent that n v -_ n h 2 Int. 2
Hence, using equation 17, we find 25=
23(~t-}- 1)--4
n v = n , 2 ( l + m ' ) - 8 ( l + 3 ~-fl) nt, 2¢l +m')-a
....
[19]
(5) Scale [actor [or the nozzle throat d i a m e t e r - - F r o m the continuity equation for flow through the nozzle throat of diameter r we obtain ( p f ) n = n = nL (pf)uo
. . . . [20]
ON
GENERALIZED
SCALING
PROCEDURES
FOR
LIQUID-FUEL
ROCKET
ENGINES
since the linear flow velocity at the nozzle throat is sonic irrespective of the motor size. Hence npn,. 2 =nl.
....
[21]
....
[22]
and, using equation 19, -~
n,=n
(:~+ m') + ½~;(/3-:,~- 3)
2. ~,,'~ ~(,+3~ ~)n~Z(,
~m')-~(l+3=
/3)
(6) S c a l e / a c t o r / o r the/eed system--From the continuity equation for the feed system in which the linear flow velocity is v, and the cross sectional area A1 we obtain ( v i A j ) , / (vIA:)Mo = n = n ,
whence n,t =n,/nal Similarity
o/ regenerative
....
[23]
cooling system:
Since the Reynolds and Prandtl numbers are fixed, the Nusselt heat transfer number also remains the same in the model and in the large engine. Hence the total heat Q which must be transferred to the interior motor walls, for chambers operating under similar conditions of temperature, pressure and internal aerothermochemistry, must be directly proportional to the propellant consumption rate; this, in turn, varies linearly with the geometric scale factor, as has been shown in equation 15. Thus nQ = Q , / Q ~ o = n , .
. . . . [24]
Correspondingly, the heat transfer per unit area q must obey the scaling requirement n,~=qH/q,~1o= 1/n~, . . . . [25] For motors operating with maintenance of complete similarity, the interior and exterior wall temperatures are fixed. Therefore, since the heat transfer per unit area varies as 1/nt,, the temperature gradient must be changed by the same factor, i.e. the wall thickness W must be scaled in geometric proportions: nw-WMWMo=nl. . . . . [26] The heat transfer problem has now been reduced to a determination of the requirements in the cooling passages for transferring heat per unit area in accordance with the scaling condition expressed by equation 25. The Reynolds number in the cooling coils remains fixed if the cooling coils of diameter C are also scaled in geometric proportions, viz. nc - C ~ / C,~o = nL
....
[27]
In the special case of a regeneratively cooled motor, the cooling passages correspond to the feeding system. For fixed propellants, constant Prandtl, Reynolds, and Ntrsselt numbers are maintained if / v i A : q , , / [v1A?L,o = n, lnA1 ~ = 1
or n,, I = 1/n,.
....
[28]
S.
S.
PENNER
AND
A.
E.
FUHS
a relation which is evidently consistent with equation 23 if geometric similarity is maintained for the cooling coils. In practice, the scaling requirements for the wall thickness and cooling coil dimensions may be undesirable because of excessive weight, or they may be inconsistent with design restrictions such as those imposed by similarity analysis for maintenance of low-frequency stability. The thickness of the outer wall of the cooling coils may be changed at will without affecting any of the other design restrictions obtained from similitude considerations. Similarity o[ sweat cooling
If a fraction ~' of the total propellant flow is used for sweat cooling, this fraction must be maintained at a fixed value as the engine is scaled. Details concerning the scaling of pore size in motor walls are easily worked out but will not be considered in the present discussion. Low-/requency stability
L. CROCCO: has discussed scaling procedures for a rigid and rigidly supported feeding system neglecting the compressibility of the propellants. Low-frequency stability depends on several parameters including the ratio of the chemical conversion t i m e r to the residence time ~,., the relative pressure drop A p / p (Ap=_ pressure drop across injection orifices, p=steady operating pressure in the combustion chamber) and an inertia parameter which equals the ratio of the kinetic energy of the propellants in the feeding system to the work done on the propellants by the pressure drops during the residence time. The ratio of the chemical conversion time r to the residence time r,. is maintained invariant by enforcing the condition that the first Damk6hler similarity group must remain fixed. This requirement has been introduced in the discussion relating to the steady aerothermochemistry and is given explicitly in equation 7. In view of equation 18, the result may be stated in the form n. ~ n,, = n~ 2" +m'~-~¢~+3 ~-~ nl2. +,.'~ ~. +3 ~-~>
. . . . [29]
Next we consider the implications of the low-frequency stability requirement that (Ap/p) n ( A p ) . [Mo (Ap/p)Mo - (Ap).~,o p,,
-
1
....
[30]
Since /Xp ~ v =, equation 30 is equivalent to the relation (v'-'/p)./(v'-'/p).,,o=n.~/n,= 1
By using equations 18 and 19 it may then be shown that equation 30 reduces to the condition -~ /,1 2(1 ~m J - ~ l ~ 3 ~ - B )
1('
m ' ) - 2,~(/3+ :' >
n t 2¢1 ~m)-~¢1+3,-/~)
_
1
....
[31]
ON
GENERALIZED
SCALING
PROCEDURES
FOR
LIQUID-FUEL
ROCKET
ENGINES
For n , = 1, n,,@l, equation 31 corresponds to either of the relations 2 ( 2 - m') = 8 (/? + 2) "~ l 2(1 +m') - ~(1 +3 ~--/~)= ~
or
. . . . [3 la]
The inertia parameter .
.
.
.
must also remain invariant to maintain low-frequency stability'-'. Here K is a constant, 11 repres6nts the equivalent length of the feeding system with cross sectional area A,, F is the total propellant injection rate, and the other symbols have their usual meaning. The condition J,,/J,,,.- 1
. . . . [331
then becomes (11/ AI),, = n,,n: / n, .
.
.
.
.
[34]
(1!/AI)~,,, in view of equation 30 and of equations 13 and 14. Using equations 19 and 29 we find the result l£z
,~q34 ~÷/3) ~ "-'fro' - 3)
(It~A,)1, _ n, m-,,,'i-.~(~ ~ 3 '-/~;n,. 5~. ,,r}-2(i :: ? ;:~r (l!/ A,):u,,
. . . . [35]
For scaling of the feeding system and cooling coils with maintenance of geometric similarity, the left-hand side of equation 35 evidently reduces to (ll/ A l ) z / ( l l / Aj).u,, = 1/nj.
Crocco ~ has indicated the need for extending the analysis presented in this section in order to make allowance for compressibility of the propellants, non-rigid feeding systems and/or non-rigid mountings, systems involving pumps for pressurization, etc. Details concerning studies of this type are best worked out in connection with motor development programmes. High-jrequency
stability
High-frequency instabilities can presumably be controlled, for operating conditions otherwise similar, if the ratio of the chemical time 7 is fixed with respect to the wave propagation times 7w. But the latter are linear functions of the chamber dimensions for sound waves as well as for shock waves provided the sound velocity remains invariant. Hence, for fixed temperature and 7, the similarity requirement for high-frequency instability may be written in the form it: = n ~ . = n , .
.
.
.
.
[36]
it is apparent that the condition expressed by equation 36 is not usually consistent with the similarity requirement for the steady aerothermochemistry or for low-frequency stability, both of which lead to the result given in equation 29.
S.
S.
PENNER
AND
A.
E.
FUHS
Equation 36 expresses only the condition for similitude of the driving and coupling forces for oscillations. The forces producing oscillations must be balanced against the damping forces, of which the principal sources must be frictional losses. These, in turn, should be similar as long as the Reynolds number is constant. Damping associated with the oscillating gas flow through the nozzle is not similar unless the nozzles are scaled with maintenance of complete geometric similarity. However, for a well designed prototype it appears unlikely that damping through nozzle oscillations plays a dominant role. SUMMARY
OF
RESULTS
AND
DISCUSSION
In T a b l e 1 the similitude requirements for different aspects of the scaling problem have been summarized. The. Penner-Tsien (P.-T.) relations correspond to the special case in which m ' = ~ = f l = 0 , n , = n ~ = l , and ~=2; they can be derived from the generalized results formally by letting m' approach infinity. The scaling procedure for the P.-T. rule is summarized in T a b l e 2. Crocco's second rule corresponds to the special condition g=0; the results thus obtained are listed in T a b l e 3. Barr~re's scaling rule is based on an empirical expression for 7 (derived from motor tests with red fuming nitric acid and furfuryl alcohol) which is inconsistent with the assumptions expressed by equations 4 and 5. Thus Barr&e starts from the relation r ~ Ap-,"p-,,,"
. . . . [37]
which is in agreement with equations 4 and 5 if ~a=r', f i = O , n , = 1, and nh = 1 in the relation for D only. This last requirement is probably to be attributed to the fact that motor tests were carried out without variations in orifice diameter. The statement n,,= 1 in the expression for D is inconsistent with Barr6re's scaling rule for n,, derived from other considerations. It will be interesting to see if motor tests with variable injection orifices are really correlated better by equations 4 and 5 than by equation 37. The scaling procedure of Barr6re can be obtained by repeating the analysis described in detail in the text with equation 37 substituted for equations 4 and 5. The results derived from this study are summarized in T a b l e 4. Reference to T a b l e s 1 to 4 shows that the scaling requirements for the steady aerothermochemistry are not generally consistent either with those for low-frequency stability or with those for high-frequency stability*. Thus it is impossible to scale with maintenance of complete similarity with respect to all the important variables. From the practical point of view, the most important discrepancy is that between the usually contradictory requirements for n~ in the analysis for the steady aerothermochemistry and for high-frequency stability. As was stated in an earlier pape r3 and mentioned again in the introduction to the present survey, we are as yet *The only exception is 0rovided by Crocco's second rule in the unlikely case that m ' = 1; see T a b l e 3. 9q~
ON G E N E R A L I Z E D
SCALING
PROCEDURES
FOR
LIQUID-FUEL
ROCKET
ENGINES
unable to assess the significance of this discrepancy. For this reason a preliminary test programme on engine scaling should involve an investigation of high-frequency stability of properly performing engines in which only the scaling rule n~ = m~ is accounted for. The steady aerothermochemistry and high-frequency stability requirements are satisfied simultaneously if 2~
2('.'m'-2,~+t;fl)
n72(1 F m ' ) - t ~ ( l - 3 z fl) n l . 2 ( l ~ m ' ) - ~ ( l q - 3 a - / 3 ) __nL or 2~ n
.'2(m'-l)+~(l + B ~)
2~1 ÷ m ' ) - , ' ; ( l + 3 ~ - / z b n l . 2 ( l + m ' ) - , ' ; ( l + 3 ~ h') = 1
....
[38]
According to equation 38, variation of m can, in principle, be used to maintain similarity for the steady aerothermochemistry and high-frequency stability. Whether or not this programme is practically feasible remains to be seen and cannot, in any event, be decided until reliable empirical information is available, not only on the validity of the functional relations given in equations 4 and 5, but also on the absolute values of the parameters 3, ~, fi and m'. It is interesting to note that equation 38 is satisfied exactly when n , = 1 if either 2(l-re') 2(l+m')
=~(1+,~-2) ~S(1+32-fl)
] }1
. . . . [39]
or
2(1 + m ' ) - ~ (1 +3 2-/?)
~}] 2(1 - m ' ) - ~ ( l + / 3 - ~) ~ oo !
. . . . [40]
The latter relation would never be expected to apply*. For m ' = 0 , S--2, ~=/?=0, the first expression given in equation 39 is satisfied; however, in this case all the exponents for n,. appearing in the scaling rules for nh, nv, n,, n,. and n, become indeterminate and, furthermore, the second condition given in equation 39 is not satisfied. In fact, the rule m ' = 0 , ~=2, 2 = / ~ = 0 for n , = 1 corresponds simply to the P.-T. procedure. According to Barr6re's findings for a bipropellant motor utilizing red fuming nitric acid and furfuryl alcohol, m'=0.3, ~=2, 3~ = 1.6,/3=0. Then 2(l-m')=14 and 3 (1 +/3 - 2)=0-4 Therefore the conditions given in equation 39 are far from satisfied. Rather than pursue the interpretation of the generalized results for other speculative cases, we conclude the present discussion with the observation that extensive tests yielding a correct functional form for r are the first *The statement given in equation 40 should not be confused with the lormal procedure used to derive the P.-T. results from the generalized expressions by setting m ' - c ~ .
237
2(l+m')-~-(l
~
(;.s.
2 ( l + m ')-g,(l-c3
=
~3=
(
z,~( = + , ) . , )
##l.~2(It, m')-3(l-.3~-fl) 1
~-B) it L
(l+m')-;
fl)]
(1 ÷ 3 ~ /?)]
b(I-13=
((.~+,,,')+'~ ~n-~ ~ ~
i3)]nL\2(l+m')
~
/3] HL 2 (l-} m ' ) - ~ 0 - - 3 ~-/~)
)
(ll)
(10)
(9)
(8) l/Hi.
n(,=nl,
n,,.=nj
,1,:
,,.=l/n,.
B Regenerative cooling
D High-frequency stability
( 1~ \ = ns .' c.s.
(IJA,),t
~ , ,
(7) Repeat
(13)
(14)
n~=nl.
~_nn(2(l_fm,)_~-if3~_/3)jlll\~,~--+-m:)-~)]
(12) 11=(2(T+mr)--g(i+3~/3')lit\2-(f-+~-~r-)-~rl~-3~L--g)]=l ~ ' ( ~ ~. . . . "~ ~ ~/J+'-'> '~
C Low-Jrequency stability
nv=
)
(11)
) n~t=nL-a
) n~----nlg
(10)
(9)
(8)
no=n1.
nw=-nt.
na=l/n L
n.s=l/n r
B Regenerative cooling
) n"=nL ½
1
n.=nL -1
t
) nh=nl.
) n~n L
A Steady aerothermochemistry
nt-'=l
hi'-'= l
(7) n:=nt"-
(13)
(12)
C Low-frequency stability
(14)
n~=nt,
High-frequency stability
D
Ible 2. The P.-T. scaling rules for liquid@tel rocket engines (m'=~=fl= 0, n~=nv=l, 3=2); these rules are obtained from the general results most simply by setting n~= 1, m'=zx:.
he subscript G.S. identifies results appropriate for scaling with geometric similarity
nr~n~
-~
.,( 1)
)* nvl--- -hA!
n,.=n~
2(I-Fm')-3(1+3=
no~n ~
?~(1+3 ~-fl)
26:
2 ( I #m')
2(l+m')~(l~3~g))nLt2(l+m')--~(l+3=--n) ]"~ ~ ( ~'"' -~ ~'-'~ ~
n l =~ n j
n,,=nJ-
li~n L
A Steady aerothermochemistry
Table 1. Generalized scaling rules for liquid-fuel rocket engines deduced ]rom similitude considerations
nL
(l--m')/(l+m')
,~'/(1 +m')
u
m'/(
n 4/
n L
L
I
+ ,. ')
1
.
.
.
(11)
(10)
(9)
(8)
= l/nl,
=nt
nc=nt.
n,
n i = 1,In I
n
,13)
(12)
C
(1,/A I)n / ~ .o \ ~ ' ,. - 1 =
J~7) , ' - ' 6 : ..... '~t+'"'
nt= !'-'-"")/"' { '"')=1
Low-/reqtwncy stability
D
High-[requency stability
I--2
r'
'
ii =nl
",
av~nL
.
nr
n4t \
]
"(t
%~r::i-;;,')
'.~ ~m" t)
"'( ])
2
)
(:{+,,,,-,,,~
m'
\l+t
,
+
m'--r'
np_nL(-'2,,_ 1 ~ )
n v ~ n 1.
n h --nl,
n~-n L
.,~ (11)
(10)
(9)
n,~n t
n ----n~
t =l/n1.
/R) n,. ----I/hi.
B
Regenerative cooling
A
Steady aerothermochemiatry
(12)
t,,= (¢
nt /~ s.
~]/7";2"-';::1=1
(I¢/A )M, \
lit.
"
Low-/requency stability
C
~/41
n,=tt~
High-Jrcq tem'y .stability
D
(7) n~=n, =,,,'.~1+,,'~ (14) n~=n L able 4. Barrbre's scaling rtde [or liquid-[uel rocket e'l!;ines (e~=2. ~=r'. 13=0, n~= I, nt = l in the expression lot" D/h); the effect ol setting n.,=l in the expre.~sion [or D / h can be accounted for by repeating the entire analyMs
nr ~nT,
nr=n
1/2+l/(l+m')
n p = n I - g / ( 1 + m ')
nv~.l~ L
nh=n L
n =
B
Regenerative cooling
A
Steady aerothermochemistry
]able 3. Crocco's second ,waling rule/or liquid-[uel rocket engines (~;=0, n== 1)
S. S. PENNER AND A. E. FUHS
requirement of an intelligent scaling programme. I m p o r t a n t steps in this direction have already been made by Crocco and Barr6re but additional quantitative experimental work, for a variety of propellant systems, is urgently needed.
Daniel and Florence Guggenheim Jet Propulsion Center, California Institute of Technology, Pasadena, Cali[ornia (Received October 1956) REFERENCES
1 Modelli nella Tecnica Vol. 1 pp. 652-669, Accademia Nazionale dei Lincei, Rome, 1956; See also Teeh. Rep. No. 13 Contract DA 04-495-Ord-446. California Institute of Technology, Pasadena, 1955 2 CROCCO, L. Considerations on the Problem of Scaling Rocket Motors, Selected Combustion Problems 11 p. 457. AGARD, Butterworths Scientific Publications: London, 1956 :l PENNER,S. S. Jet Propulsion 27 (1957) 156 BaRR[RE, M. Similarity of Liquid-fuel Rocket Combustion Chambers Office Nationale d'Etudes et de Recherches A6ronautiques: Paris, 1956 :' See, for example, PENNER, S. S. Chemical Reactions in Flow Systems Chapter 4 Butterworths Scientific Publications: London, 1955 BARR/~REM. and MOtJTET, H. Rech. adro. 50 (1955) 31 7 BARR/~RE, M. Preliminary experimental results obtained at the Office Nationale d'Etudes et de Recherches A6ronautiques, Paris; personal communication DAMK6HLER, G. Z. Elektrochem. 42 (1936) 846 PENNER, S. S. Combustion Researches and Reviews 1955 Chapter 12 AGARD, Butterworths Scientific Publications: London, 1956 1 FENNER, S. S. Models in Aerothermochemistry in
240