G THE LIQUID PROPELLANT ROCKET ENGINE

Preaaure drop in the combustor due to bat rekase. The analyeis of the ideal rocket motor b baaed on the postulate that combustion takes place at constant pressure. In praotice, the release of heat in the gas stream in the combustor is accompanied by a deorewe of both static and stagnation pressures. The decrease is dependent on the Mach number at the hot end of the combustion chamber, and beoomes larger aa the ratio of combustor cross-sectional area A. to throat area A& approaches unity. The loss in stagnation prwure represents an undesirable inorease in entropy, a decrease in the effeotive pressure ratio for expansion, and henoe a loss in speoifio impulse. The following analysis deals with these effects.

As before, the subscripts i, c, th, and e refer to states a t the injector end of the combustor, the hot end of the combustor, the nozale throat, and the nozzle exit. Except for the nonzero velocity in the oombustor, the oonditions of the analysis are the same as for the ideal rocket motor.

Since the expansion in the nozsle ia isentropic (heat release completed in the chamber) the Mach number at c ie obtainable from Eq. 210:

The integrated momentum equation, for frictionlw flow in a con- stant area duct, provides an expression for the dearwe in statio pressure in the chamber. It ia wumed that Mi c! 0.

The simplest way to treat the performance for the owe of nonzero M, is first to calculate the equivalent plenum chamber pressure and temperature oorresponding to the oonditiow a t c, and then to oaloulate the flow proparoiee in the nozzle on the assumption that the gw flow originatee from this plenum chamber. The temperature T,O and pres- sure p: of the equivalent plenum ohamber are really 'the stagnation temperature and pressure at station a. (See UI,+4 for disouaaion of atag- nation quantities.)

Ah cp(T: - TI) (3-18)

Eq. 3-18 is simply a statament of the con~ervation of energy, with the kinetio energy term d;Vi absorbed in the stagnation enthalpy c,Ti.

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C,3 DEPARTURES PROM IDEAL PERFORMANCE

Therefore, Ti is the same as T, in the ideal rocket motor of Eq. 2-7, that is, the so-called adiabatia flame temperature, or simply the chamber temperature. The static temperature T. is lees than the adiabatio flame temperature, by aa much a9 10 or 15 per cent for a throatlees motor, but this is of no consequenoe for the e x h ~ w t velocity. Only the stagnation temperature matters, and this depends only on the thermoohernioal heat of reaction Ah.

Fig. 0,3d oompares the conditions of the equivalent plenum chamber with thorn of the aotua ohamber by prmenting the ratio8 p&,, p:/p,,

Fig. 0,Sd. Compuiaon of conditions in equivalent plenum ohsmber with those in aotual combustion ohamber.

M., and T,/Tf aa funotiona of A./A,L. This figure ie useful as a guide in carrying out the following computational prooedure recommended for the performanoe analysis of a rocket motor.

In general, the particular propellant oombhation, the feed pressure, the nornle lest area, the external pressure, and the ohamber dimeneiana are speoibd in advance. The problem ie to oompute the effective exhawt velocity c, taking into aooount the effect of the pressure drop in the oombuation ohamber.

1. From the specified AJAla and 7, oompute Ha from Eq. 3-14. 2. With fhis M. and the specified pi, oalculate pa and then p,O from Eq, 3-16

and 8-16. 8. From the apeoified e of the node, and the deotive presme ratio

G . THE LIQUID PROPELLANT ROCKET ENGINE

p p determine C$ for the equivalent plenum chamber, from Fig. G,2d and (32%.

4. The thrust is then oaloulated as follows:

5. The oharaoteristic velocity c* depends on the stagnation temperature, which ia exactly equal to the adiabatio flame temperature. Therefore, (c*)Q is known, either by thermoohemioal calculation or by experiments with plenum-type oombustors.

6. The mass flow is then oaloulated in terms of (c*)O:

7. Finally, the deotive exhauet velooity is calculated:

c - Cg(c*)O (3-21)

In rooket testing, the chamber preasure is frequently measured at the forward end of the chamber, through a pressure tap drilled into the injeotor face. Clearly, this is pi and not p:. Unlaas the measured p1 is converted by Eq. 3-16 and 3-16 to p:, the reported value of c* wil l be too high by as much as 26 per cent (Pig. G,8b) for a throatleaa motor, and the reported value of CF (experimental)/C~ (theoretioal) will be low by a comparable error. There effeots ore signifioant for a rocked motor with a throat diameter equal to 4 or more of the chamber diameter. If the combustor is not oylindrical, the preceding analysis must be modi- fied. In partioular, Eq, 8-15 holds only for oylindrical combustors; for other shapes, it is necessary to know the axial diitribution of the heat release in order to integrate the momentum equation.

It may be noted that the Bame theory can be used to determine the axial distribution of heat release, i.e. the over-all kinetics of combustion. Thus, in a cylindrioal ohamber, the variation of M with axial position x is obtainable from the pressure distribution by means of Eq. 3-16. Then, from the equation of continuity (Eq. 3-22), the point-by-point static temperature oan be computed, and from thia, the stagmiion temperature and the heat release.

G,1. Theoretioal Sped50 Impulse Calculations. The calculation of spec%o impulse in Ohe theory of the ideal rooket motor in Art. 2 is deficient in two very important reapeots: the oombustion gas mixture is chemically reactive, and the specifio heats vary strongly with tempera- ture. Both these factorrr were ignored there. It is the purpose of this article to develop the complete theory, with both effeots present. Inek

G,8 . BIBLIOGRAPHY

with respect to its oprating parameters. The most important of these is the combustion pressure-the higher the pressure, the lower the propellant consumption rate for (L given tbruet; but at the same time, the greater the weight of the pressurization system. For the com- pressed gas type, the optimum pressure usually works out to lie be- tween 260 and 400 lb/in.'; for the turbopump system, the optimum is usually near 1000 lb/in.n, but a lower pressure is always employed to ease the cooling problem. It is clear that the turbopump type i s the most suitable one for long range, large dse rockets with long firing durations.

3, simplicity and ruggedness: The great simplicity of the compressed gas type makes it the moat appropriate system for such applicatione as emall anti-aircraft guided mid=, which have to be ready for imtan- taneous firing with high reliability with a minimum of pre6ring inspec- tion and adjustment. 'l'he same consideration applies to detachable JATO units.

3. Low cost: The consideration of cost ie important mainly for civilian applications such as the take-off of heavy cargo plrtnes, and perhaps rocket travel in the future, For military purposes, cost ia s ignihnt mainly as anindication of manufacturing difficulty, and, therefore, for mimiles or JATO8 required in very large numbers, the.gas-preasuriaed type is preferred.

6.8. Bibliography. Arlick 1.

Diationery of guided missile terms. Compiled by Ordnnnae Dept., U.S. Army. Anli-Air& J., 1849.

Dloaaaxy of guided miwile terms. Prepnred by Cmtltscr on (ha'ded M~&~.ss, Readaroh md Devdop. Board, U.B. Depf. of Defaud, UM 61/8 and OM 61/8, sept. 1848.

Layton, 3. P., nnd Youngquint, R. Proposed A.S.A. letter ~ymbols for rocket propulsion. JetPropuleion 86,684 (1861).

A r t i l d. Malina, F. J. Ghnr&okristos of the roaket motor unit baed on the theory of

perfeat gaoes. J. FrmkUn lmt. 880, 488 (1840). Wort , H. S., m d Crum, J. Thru6t aoeBaient nnd expansion ratio tablee.

RawWoddddgs Corp., Feb. 1860. Wert, H. S., Millrr, 1M. M., and Summerfield, M. The phyaian of roaketo,

Part I. Am. J . Phw. 16, 1 (1847). h p i r o , A. H. Dytrclmics and T h d m c o d ~ ' c 0 of Comprdssiblo Fluid F k ,

Vol, I , Chn 4. Ronald PMM, 1868. Sutton, a. P. gookal ~ropukion Bllmum~lr, 2nd ed., Chap. 8. Wiley, 1968.

ArlicD 8. &n&, a. Rooket psrformnnoe with h e ~ t added to inlet propellante by

regenerative nnd external m e w . J d Propcel. 86,712 (1866). Beakwith, I. E., nnd Moore, J. A. An aaourats md rapid method for the design

of auperaonio nosdw. NACA TmA. Noh IS#, Feb. 1866.

( 617 )

G THE LIQUID PROPELLANT ROCKET ENGINE

Dillon, P., and Line, L. E., Jr. Heat t r a d e r between aolid particleu and gas in a rook& nos&. Jet Prqztl. 88, 1091 (1966).

Durham, B. P. T M ohareoteriitics of underexpanded noaalaa. Jet Propul. 66, 696 (1055).

Qflbert, M., Davie, L., and Altmm, D. Veloaity lag of pwtioles in linearly aooelerated oombustion gases. J . Am. Rocket Soc. $6, 28-30 (1966).

Qhsman, I. Impulse expremiom for rocket systams eon-g a nolid phase. JBt Propul. 87,542 (1967).

Kogan, k Boundmy layer correction in superaonio norzla scaling. 3. Awonazct. Bd. W, 64 (1968).

Puokett, A. E. Super~onio nor& design. J . Applied M&. 18, AZ6bAd70 (1946). Beifert, H. S., and Attmlur, D, A comparison of ediabatio and iaothermal ex-

pdanion procwss in rocket noadea. J . Am. Rocket Boc. $8, 160-162 (1962). Bhapiro, A. E. The Dynam'w and Thsrmodpam4c.a qf Cowrssmble Fldd F b ,

Vol. I, Chap. 7 and 16. Ronald fiw, 1983. Bummerfield, M., Foater, C. R., and Swan, W. C. Flow separation in overex-

panded supeb80aio exhaut nosden. Jef Prqul. 84,819 (1954). Button, Q. P. Roake$Pr@ulalm, E h m t a , 2nd ed., Chap. 8. Wiley, 1956.

drbick 1. Battelle Manorial kmt. Phymod properties m d thermodynamic functions of

fuek, o x i d h s , and products of oombustion. Project Rand Rgta. R-187, 188, 186. Columbu, Ohio, 1949.

Ohm% of theoretical periormmoe of several rocket propellant oombinationa. ROblosrol$m, Dh. North Antw. Auintim, Inc., 1956.

Donegan, A. J., m d Wber , M. Solution of thermochemioal propellaat oalcuh- tiona on a high speed digiCal computer. Jet Propl , 88, 164-172 (1966).

Fickett, W., and Oowan, R. D. Valuen of thermodgnamio funotions to ~ ~ , C I K ~ K for aaverd aubstanoes. Lon Alamim Sd. Lab. Rwt. 1787, Bept. 1964.

Hottel, E. C., Wiluame, G. C., and BatterUd, 0. N. Thrrnrodwmic Chatts jor C o m b w h Prwedtae. Wiley, 1949.

Huff, V. N., and Cs lvd , C. 8. Oharts for computation of equilibrium comporG tion of chemical maotiona in the 0-II-0-N afrptem from W)OO•‹K to 600O0K. YACA Teoh. Note 1668, July 1948.

Huff, V. N., and ndrdon, 8. Tables of thermodynamic ffuntionn for analyei. of airorafb propulsion syatems. NACA Tech. No& 8181, Aug. 1960. (Imluda. light metals and halogem, as well an O, H, 0, and N oompounda.)

Huff, V. N., and Momell, V. E. Qenerd method for wmputation of equilibrium oompdtion m d temperature of chemical reaotions. NACA T d . Note dilS, June 1950.

Krieger, F. J. Chemiod kinatics and rocket nosale design. J . Am. Rocket BOG. $1, 179-186 (1961).

Lewia, B., and von Elbe, G. Combusdim, Flame8 and E2plosioc~ ofUasse, Chap. 13. Aoademio Prw, 1951.

Netl. Bur. h d a r d s . Beleoted d u e s of properties of hydrocarbons. Cirwtar 461, aov't. fiintiner OBO~, 1947.

NatL Bur. Standardu. T a b h of selected values of ohemioal thermodynamio properties, 8 vol. Cirwlar 600, Oov't. Printing OfBce, 1862.

P m e r , 8. 8. Thennodynamios and chemical kinatics of one-dimerwional non- Viscous %ow through a h v a l nozzle. J . Chem. Phya. 18, July 1961.

Pock4 Data Book for Rockt Ev@e. Bell Airoraft Oorp., Bufido, N.Y., 1964. (Pro ertiss of f u l , midisen, and propellant oombimtiaud.)

Button, 8. P., Rocket Propla'm Elemsnta, 2nd ed., Ohnp. 4. Wiley, 1956. Viohnimky, R., M e , B., m d M~sroldst, J. Combustion temperaturea and ~a.3

oompoaition. 3. Am. Rocket Boc. 86, 105 (1955).

Xrf(els 6. Baker, D. I. Mixture ratio and temprat& BUrVe of n m m w t ~ r ~ x y p n rocket

motor combuation ohmbers. Jef Prowl. WBb, 215 (1956).

Additional Material on Nozzle Flow Separation:

Performance Analysis of the Ideal Rocket Motor, Section G,3, Page 458, Equation (3-5); gives the nozzle static pressure (wall pressure) for flow separation for severely overexpanded conical nozzles as a function of atmospheric pressure, for determining the flow separation point for severely overexpanded conical nozzles.

The additional material which follows from NASA SP-8120, Liquid Rocket Engine Nozzles, provides the equivalent information for bell nozzles; the nozzle static pressure (wall pressure) for flow separation for severely overexpanded bell nozzles (contoured nozzles, parabolic contour nozzles) as a function of atmospheric pressure and chamber pressure, for determining the flow separation point for severely overexpanded bell nozzles.

rex

Nozzles designed for vacuum operation have large expansion area ratios in order to achieve high specific impulse. It is desirable to ground test engines in the course of the development program. During ground testing, most altitude engines are overexpanded, often to the extent that the exhaust gas separates from the nozzle wall. This flow separation can result in serious problems. For example, a nonoptimum (parabolic) contour was selected for the nozzle of the 5-2 engine in order to raise the exit wall pressure. The high-exit-pressure nozzle was supposed to run unseparated at an area ratio of 27 with a chamber pressure of 700 psi. A wall-pressure minimum that occurred between area ratio of 14 and the nozzle exit produced an unstable condition that caused unsteady asymmetric separation, especially during the startup. The large loads that occurred caused various thrust-chamber structural failures. A short bolt-on diffuser was developed to eliminate separation during mainstage operation of the J-2 (ref. 23). Restraining arms were attached from the test stand to the nozzle skirt to absorb the separation loads at startup.

.

Separation of flow occurs when the gas in the boundary layer is unable to negotiate the rise to ambient pressure at the end of the nozzle. The exact atmospheric pressure at which flow will separate from the wall of a nozzle cannot be predicted accurately. Various rules of thumb to predict separation have been suggested; however, general agreement on one of these methods has , v t been reached. An early rule stated that a danger of separation existed when the ratio of exit pressure to ambient pressure was equal to 0.4. Later methods based on fitting of experimental results accounted for the increase in overexpansion that can be obtained with increasing Mach number. A fit of experimental data for short contoured nozzles over a broad range of nozzle area ratios (ref. 24) indicates that separation will occur when

where

Pw, = exhaust-gas static pressure on the wall at separation

Pam, = ambient pressure

PC = chamber pressure = exhaust-gas total pressure

, The method of reference 25 was the basis for a separation-prediction criterion that includes the effects of gas properties and nozzle shape on separation. A recent and fairly complete treatment of flow separation in nozzles is presented in reference 26. The results of the various prediction methods have shown agreement with experimental data in many cases, but in general predictions are used only as a guide.

-81 20 Overexpanded ozzle References:

Anon.: 5-2 Bimonthly Progress Report, November-December 1965. R-6300-3, Rocketdyne Div., North American Aviation, Inc., January 1966.

Schilling, M. T.: Flow Separation in a Rocket Nozzle. M. S. Thesis, University of Buffalo, June 1962.

Crocco, L.; and Probstein, R.:, The Peak Pressure Rise Across an Oblique Shock Emerging from a Turbulent Boundary Layer Over a Plane Surface. Princeton University (princeton, NJ), March 1964.

Schmucker, R. H.: Status of Flow Separation Prediction in Liquid Propellant Rocket Nozzles. NASA TM X-64890, November 1974.