A simplified combustion model, which is motivated by available performance studies on the diverging rocket reactor, has been used as basis for an engine performance

evaluation. Comparison with conventional rocket configurations shows that an upper performance limit for the diverging reactor is comparable with performance

estimates for engines using an adiabatic work cycle. Development of the diverging reactor for engine applications may, however, offer some advantages for very hot, high-energy, propellant systems

Eckel, E. F.

Gill, G.

Jacobs, T. A.

Penner, S. S.

English

Caltech Authors - Main

Approximate Theoretical Performance Evaluation for a Diverging 
Rocket 
By 
'I'. A. Jacobs!, S. S. Penner!, G. Gill2 and E. F. Eckel2 
(With 4 Figures) 
(Received February 22, 1960) 
Abstract - Zusammenfassung - Resume 
Approximate Theoretical Performance Evaluation for a Diverging Rocket. A sim-
plified combustion model, which is motivated by available performance studies 
on the diverging rocket reactor, has been used as basis for an engine performance 
evaluation. Comparison with conventional rocket configurations shows that an 
upper performance limit for the diverging reactor is comparable with performance 
estimates for engines using an adiabatic work cycle. Development of the diverging 
reactor for engine applications may, however, offer some advantages for very hot, 
high-energy, propellant systems. 
Niiherungsweise theoretische Ermittlung der Wirkungsweise eines Raketen-
motors mit durchwegs divergierender Diise. Es wurde ein vereinfachtes Modell der 
Yerbrennung, welches aus verfiigbaren Untersuchungen iiber das Verhalten eines 
Raketenmotors mit durchwegs divergierender Diise erhalten wurde, als Basis fiir die 
theoretische Ermittlung des Betriebsverhaltens eines solchen Triebwerkes verwendet. 
Ein Vergleich mit iiblichen Raketenmotoren ergibt, daB ahnlich wie bei den iiblichen 
adiabatischen Triebwerken eine obere Grenze fiir den Betrieb besteht. Die Ent-
wicklung dieses Triebwerkes diirfte vor allem auf dem Gebiete hoher Temperaturen 
hei Verwendung hochenergetischer Treibstoffe liegen. 
Evaluation theorique approchee des performances d'un moteur-fusee It chambre 
divergente. Un modele simplifie pour la combustion, justifie par les etudes de 
performance connues, a ete utilise pour l'evaluation des chambres divergentes. La 
comparaison avec les configurations conventionnelles place la limite superieure de 
performances au niveau des moteurs utilisant un cycle acliabatique. Le developpement 
de la chambre divergente pour les moteurs-fusee peut neanmoins etre avantageux 
,wec les ergols a enthalpie specifique et temperature de combustion tres elevees. 
I. Introduction 
In previous publications we have discussed application of the diverging reactor 
for the determination of overall kinetic parameters under the conditions actually 
existing in rocket combustion chambers [lJ, [2J. In the present discussion we 
1 Consultant, Rocketdyne, :'\orth American Aviation Corp., Canoga Park, 
California; Division of Engineering, California Institute of Technology, Pasadena, 
California, U.S.A. 
2 Rocketdyne, North American Aviation Corporation, Canoga Park, California, 
U.S.A. 
152 T. A. JACOBS, S. S. PENNER, G. GILL and E. F. ECKEL: 
shall outline an 
rocket chamberl . 
approximate engine performance evaluation for a diverging 
As basis for this evaluation we choose a simplified combustion 
Isentropic rlsentroPic 
subsonic floW7 flow 1_~3 
~,2 
o 
(2) 
supersoni c 
model that we have used previously 
for the computation of various 
dimensionless groups [1]. 
A schematic diagram of the en-
gine and combustion front is shown 
in Fig. 1. For the sake of simplicity, 
we assume that all of the heat re-
lease occurs at the plane I, 2 in 
Fig. 1, that the gases behave as ideal 
gases with a constant ratio y for the 
specific heat at constant pressure to 
choking and completion of the specific heat at constant volume, 
chemical reactions occur at the plane 1,2. and that isentropic, one-dimensional, 
expansion occurs in the regions up-I:ig. 1. Schematic diagram of a diverging 
stream and downstream from the rocket engine 
reaction front. Weare led to the 
model implicit in these approxima-
tions through the experimentally observed fact that the sonic plane occurs 
very close to the injector [1J and that the effective reaction rates are very fast 
(with low overall activation energy) near the injector end whereas they 
are relatively slow farther downstream (with large effective activation energy). 
Since the limited available experimental data suggest that most of the reactions 
are completed very close to the injector end, we may consider the present 
analysis with the characteristic length L in Fig. 1 equal to zero, or at least very 
small, to provide a reasonable upper limit for the performance which can be 
achieved with a diverging reactor. 
II. Outline of 'fheoretical Considerations 
1. Pressure Ratio Across the Reaction Front 
The performance of a diverging rocket engine is a sensitive function of the 
stagnation pressure ratio Ps2/Psl across the reaction plane. Fortunately it turns 
out that reasonable values of the heat release lead to values of Ps2/psl which 
are practically constant at about 0.8. We shall now prove the validity of this 
last statement by utilizing appropriate, simplified, versions of the conservation 
equations. 
The overall conservation of mass equation at the plane 1, 2 in Fig. 1 is 
(11 VI = fl2 ~'2 
where (! and V identify, respectively, the density and linear flow velocity. :VIul-
tiplying and dividing by Vy-R;t and replacing (! by P/Rg T we find that 
(} l' ~~. -It Vy-R-;-f VY~~T = (t~)li2 pM 
where Rg = R/I;jJ' is the specific gas constant, R denotes the molar gas constant, 
W is the molecular weight of the gas mixture, T stands for the local (random 
1 With appropriate changes in wording, the present analysis applies also to a 
throatless motor. 
Performance Evaluation for a Diverging Rocket 
translational) temperature, and M identifies the local MACH number. 
the continuity equation becomes 
YI WI Y2 W 2 ( )~2 ()~2 T--;- PIMI = ~ P2 M2' 
Similarly, the equation for conservation of momentum IS 
PI + Ih VI2 = P2 + 1?2 v22 
where 
whence 
153 
Hence 
(I) 
(2) 
In terms of an integral over stagnation temperatures and the total heat release 
per unit mass q, the energy equation reduces to 
Ts2 
q = Icp dTs (3) 
Tsl 
where cp is the specific heat at constant pressure. 
The stagnation temperature Ts and the translational temperature Tare 
related through the expression 
or 
v2 
Ts= T+--
2 cp 
Ts = T (I + 2 :p2 T~ ~~~) = T (1 + ~ ~ M2) = T (I + 
smce Rg = cp - c" and Y = cplcv. Hence Eq. (I) becomes 
-I M2) 
2 
(Y~:It2pIMI(1 +YI ~ 1 MI2t2 = (y~:gt2P2M2(1 +~~ I M22t2 
\\"e may now eliminate the pressure between the continuity and momentum 
equations by dividing this last relation by Eq. (2). We then find that 
(
T W)1/2 U2(Y2, M2 = 1) = U1(Yl> MI) T:: W: (4) 
where 
( )
1/"2 
112 '1 1 Y - 1 M2 y' "" T -~,-
U = ----T+-
Y
-c;;!-;c2,---- , (5) 
The function U has been tabulated in [3J for various values of Y and M. 
For specified values of the heat release and cp, the change in stagnation 
temperature and, therefore, the stagnation temperature ratio, are defined by 
Eq. (3). Hence, Eq. (4) may be used to determine M1 for known values of Yl> Y2' 
WI and W 2 . The stagnation pressure loss may then be evaluated by using the 
expression 
(6) 
154 T. A. JACOBS, S. S. PENNER, G. GILL and E. F. ECKEL: 
since M2 = 1. The pressure ratios (PdP2) and (PI/PSI) are conveniently obtained 
by using existing tabulations for isentropic expansions [4 J. For these calculations 
it is actually not necessary to utilize constant values of Wand y. 
o. 
0.8 
10 20 30 
Fig. 2. The stagnation pressure ratio Ps2/Psl as a function of the parameter (Ts2/TsI) (W1/W 2) 
for a diverging rocket (Yl = Y2 = 1.30) 
The stagnation pressure ratio Ps2/PSI has been calculated as a function of the 
parameter (TdTsI) (WI/W2) from Eq. (6) for YI = Y2 = 1.3 by using Eq. (4) 
for the determination of MI. The results are plotted in Fig. 2. Reference to 
Fig. 2 shows that ps2/Psl == 0.8 for (TdTsI) (WI /W2) greater than about 10; 
additional data may be obtained from [6]. 
2. Definitions of the Thrust Coefficient for Diverging and for Conventional 
Rocket Engines 
The thrust coefficient CF,con for a conventional rocket engine (see Fig. 3) IS 
defined by the expression 
o 
The chemical re - I 
actions go to comi 
pletion in the com-j 
bustion chamber I 
Isentropic expansion without 
composition change 
* 
(2) 
* I 
3 
Fig. 3. Schematic diagram of a conventional 
rocket engine 
F 
CF,con= P-A- (7) 
sO * 
where F is the thrust on the engine, 
pso identifies the chamber pressure, 
and A * is the cross-sectional area 
at the nozzle throat. For properly 
expanded nozzles with P3 equal to 
the cxternal pressure, 
F = (e3 V3 A 3) V3 = A3 P3 Y2 M3 2 (8) 
if Y2 denotes the (constant) heat 
capacity ratio for the expanding 
gases. \Ve note that in a conven-
tional rocket 
Pso = Ps3 
if we neglect dissipative terms during expansion and chemical reactions during 
nozzle flow. Thus 
(9) 
Performance Evaluation for a Diverging Rocket 15.3 
For the diverging reactor we define a thrust coefficient by the analogous 
expression 
(10) 
But 
~~ = ~32 = (:::) (~31) = (:::) (~j 
whence 
CF div = Y2 M3
2(A3/A1,2) . 
, (PS 2) (PS 0) 
psI P3 
(11) 
It is interesting to observe that Eqs. (9) and (11) differ formally only through the 
occurrence of the term (Ps2/Psl) in the denominator of Eq. (11) since A* and A I ,2 
are analogous quantities. This pressure ratio, in turn, has been related to the 
heat release through (Ts2/TsI) (W1/W2) in the preceding Section II 1. 
3. Performance Evaluations lor the Diverging Reactor 
It is clearly possible to make a number of different performance evaluations 
for the diverging rocket. The performance estimate will then depend somewhat 
2 
& 
4.4 20018 
4.2 18016 
40 
38 
3.6 
140 
./ 
./' 
200 
----
_ Diverging engine 
- - - Conventional eng i ne 
300 400 
Fig. 4. The quantities lY13 and ".).3/A* ([or a conventional rocket) and lYla, PS3/Pa and A3/A"2 
(for a diverging rocket) as a fnnction of the pressure ratio Pso/Pa f(1's2/1'sl) (W1IW2) = lO 
for the diverging rocket, )12 = 1.30J 
on the predetermined design restrictions. We shall now consider two instructive 
examples. We restrict our discussion to the representative case with Y2 = 1.30 
and (Ts2/Tsl) (W1/W2) > 10 for which Fig. 2 shows that PdPsI::--: 0.80. The 
numerical work is facilitated by constructing a universal plot in which Ps3/P3' 
A3/A*, A 3/A I ,2, and M3 are related to the pressure ratio (d. Fig. 4). For conven-
H,6 T. A. JACOBS, S. S. PENNER, G. GILL and E. F. ECKEL: 
tional rockets this calculation involves only the use of tabulated isentropic 
functions which first yield M 3 through the expression 
~= 1+L_M2 P ( 
_ 1 )}'!(?-1) 
P3 2 3 
and then A 3/A* in terms of M3 [4]. The curve PsS!P3 = PsO/P3 for the conventional 
engine is not plotted in Fig. 4. For the diverging rocket it is convenient to compute 
first the ratio Ps3/P3 and then to evaluate M3 and A3/A1,2 through use of the 
isentropic expansion relations. 
III. Comparison of Engines with :I<'ixed 'l'hrust and Pressure Uatio PSO/P3 
Consider a conventional and a diverging engine both of which deliver the 
thrust F = 25,000 lbs and which are designed with pressure ratios PSO/P3 = 200. 
Also assume that P3 = 1 psia and (Tsz/Ts1 ) (W1/W2) > 10. 
For the conventional engine we find from Fig. 4 that M3 =~ 4.0 and 
A3/A* = 15.9. Hence, using Eq. (9), CF,con = 1.63; it now follows from Eq. (7) 
that 
. 2.5 X 104 " . 0 A. =-----.-.. = 76.3 Ill" 
'" 1.64 X 200 
and 
A3 = 15.9 A* = 1211 in 2. 
Similarly, for the diverging rocket, M3 ~= 3.87, A3/A 1,2 = 13.8, Ps3/P3 = 162, 
CF,div = 1.64 [from Eq. (10)J, 
') 5 X 104 Al 2 = -""-- .. - .. = 94.8 in2 
, 1.63 X 162 ' 
and 
A3 = 13.8 A 1,2 = 1308 in 2. 
Using customary design estimates, the weight ratio of the diverging nozzle 
sections for the two engines [5J is given by the relation 
m~iv = (:4;; fsLv = 1.07. 
1n~OIl (A3 F) 
A* Ps3 cony 
Hence it follows that the diverging rocket will be relatively lighter provided the 
chamber and converging sections of the conventional engine have more than about 
7% of the weight of the diverging section and the effective weight of the diverging 
reactor between the injector and reaction planes is negligibly small. We expect 
in practice that the combined weights of the motor and converging sections of 
a conventional engine will constitute about 10% of the total weight. Hence it 
follows that optimal design of the diverging reactor will be roughly comparable 
with the usually achieved design of a conventional rocket for equivalent engine 
thrust and pressure ratio Pso/Pa. 
IV. Comparison of Engines with Identical Diverging Sections but Different Pressure 
Uatios PsO/P3 
Consider two engines which develop equivalent thrust but with 
A3 = A3 = I5.H. 
A* A 1,2 
Performance Evaluation for a Diverging Rocket li,7 
Then, from Fig. 4, Pso = 244 psia for the diverging reactor whereas Pso = 200 psia 
for the conventional engine. The corresponding weight ratio for the pumping 
equipment [5J is 
Therefore, if the chamber and converging section of the conventional engine 
weigh about 14% more than the pumping equipment, then the diverging 
rocket will be the lighter engine if we neglect the influence of the slightly 
higher injector end pressure on the weight of the diverging reactor and, 
furthermore, assume again that the chemical reactions are completed very close 
to the injector plane. 
Comparison of the preceding two examples suggests that the use of relatively 
large pressure ratios PsO/P3 for the diverging rocket is impractical since the 
pumping equipment makes a major contribution to the mass of the engine. 
V. Conclusions 
Examination of the performance data specified in the preceding paragraph 
shows that diverging and conventional rocket engines should yield comparable 
facilities after optimum development of the relatively untried diverging rocket. 
I t is difficult to assess the practical utility of the new device since it possesses 
both an obvious advantage and an obvious disadvantage. The advantage lies in 
the possibility of utilizing very hot, high-energy, propellant systems under 
conditions in which excessive temperatures and heat losses are not encountered 
because the chemical energy may be transformed directly into translational 
energy and because the diverging rocket may be relatively easier to cool l . On the 
other hand, in order to approximate our assumed combustion model (compare 
Fig. 1), it is apparent that the diverging reactor must give very efficient 
combustion near the injector plane, i.e., severe heat transfer and erosion should 
occur when the practical performance of the device approximates optimum 
design. 
In conclusion it is appropriate to speculate on the possible existence of high-
frequency instabilities in diverging reactors for which neither experimental nor 
theoretical data are available. The nature of the design actually appears to be 
rather favorable for the suppression of high-frequency instabilities: transverse 
modes should be damped out because of the rapidly changing cross-sectional area 
of a device in which strong composition inhomogeneities must accompany 
chemical changes; furthermore, it is probably impossible to sustain longitudinal 
oscillations because of extreme damping which must accompany diverging, 
supersonic, flow with chemical reactions. On the basis of the preceding considera-
tions, it appears highly desirable to initiate a large-scale engine program for a 
high-energy liquid propellant mixture (e.g., ClF3-N2H4). It is unlikely that small-
scale engine tests will provide significant information either concerning the 
practical difficulties inherent in proper injector design or concerning the nature 
of combustion instabilities. 
1 A practical diverging rocket design might well involve the use of ablating material 
near the injector end. The performance of the engine should not suffer significantly 
because of enlargement of cross-sectional area near the injector associated with 
ablation. 
158 T. "\. JACOBS et al. Performance Evaluation for a Diverging Rocket 
References 
1. G. GILL, E. F. ECKEL, F. A. \VILLIAMS and S. S. PENNER, Seventh International 
Combustion Symposium, pp. 712~ 724. London: Butterworths Scientific Publica-
tions, 1959. 
2. F. A. WILLIAMS, S. S. PENNER, G. GILL and E. F. ECKEL, Combustion and Flame 
3, 355~371 (1959). 
a. T. A. JACOBS, Tables of Mach Number in a Constant-Area Channel. Rocketdyne, 
DDR 711~35, Canoga Park, Calif., 1957. 
4. C. J. WANG, J. B. PETERSON and R. ANDERSON, Gas Flow Tables. Space 
Technology Laboratories, GM-TR-I54, Inglewood, Calif., 1957. 
5. R. W. BUSSARD and R. D. DELAUER, Nuclear Rocket Propulsion, pp. 45~ 85. 
New York, N.Y.: McGraw-Hill Book Co., 1958. 
6. G. P. SUTTON, Rocket Propulsion Elements, 2nd Ed., p. 82 et seq. New York, N.Y.: 
J. Wiley and Sons, 1956. 


Approximate theoretical performance evaluation for a diverging rocket

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