Perf Analysis of the Ideal Rocket Motor - Part 1.pdf

7/28/2019 Perf Analysis of the Ideal Rocket Motor - Part 1.pdf

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O ne of the fundamen tal equations in the f ield ofrocketry is the equation for the thrust of an ideal rocketmotor. Th is equation to predict the thrust ofsolid rocket motors an et engine s as a functionressure, ratio of specific heats for the flowthrough the nozzle, nozzle throat area and expansion ratio,and atmospheric pressure.

here:

F = thrustPc = c h a m b e rPe = nozzle exit pressurePw = atmospheric pressurey = ratio of specific heat s for flow through nozzle

on an ideal perfect gas analysis, t heideal rocket motor thrust equation is accurate to within 1-6% for most rocket m otors, and can be accu rate to within

here:= molecular weight of gas

R = universal gas constantTc = combust ion tempera tu reVe = exhau st velocity

Oberth submitted his derivation of the Oberth exhaustvelocity equation as his doctoral thesis at Heidelburg

u m e n t O b e r th

Goddard's experiments proved that a rocket could producethrust in a vacuum. These results showed that the rocketwas a practical mea ns of propulsion in space, ma king space-or's opinion one of the best technical articlesovering the deriv on of the idealmotor thrust equation was by artin Summerfielublished in 1959 in Volume I 2 - Jet PropuSpeed Aerodynamics andtion of the ideal rocketion, specific impulse, characteristic velocity, nozzle exi

pressure as a function of nozzle expansion ratio, non-idealcorrections for nozzle divergence angle,

technical writing have withstlume series was publishender that contract after teublic domain, and the author anpleased to republish this material for a new generation ofrocketeers.

High Power Rocket ry

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2/17

VOLUME XI1HIGH SPEED AERODYNAMICS

AND JET PROPULSION

PROPULSIOJVENGINES

EDITOR: 0 . E. LANCASTERI t 1

PRINCETON, NEW JERSEYPRXNCETON UNIVERSITY PRESS

1959

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3/17

Reproduotion, tramlation ublioation, uae, and din-pond by and for the un ited &ate8 Qovs"pmt anditsoflcers, pte, and emplo eas aoting w i t h he.soopeof,bheir t%olal du$s, for dovernment urn only, m per-rmtted. At the explra tbn of ten eare from the da$s ofpublio~~tion,ll rights in m a d o o n t a i n e d herernhtnroduoed undm aontmt Nonr-09201 ohell be in the5ubEi domain.

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SECTION G

THE LIQUID PROPELLANTMARTIN SUMMERFIELD

G,1. htroduction. The scientific development of the modern rocketengine has been made possible to a great ex tent by the results of researchin th e fields of chemical physics, high temperature com bustion, highintensity heat tranefer, gas dynamics, and heat-resistant materials. Muchof thia work haa taken place in recent years and has not been incorpo-rated heretofore in the standard textbook literature in a form suitable forsystematio atudy. For an up-to-date oomprehensive trea tment of theliquid propellant rooket engine, therefore, it would have been necessaryto introduce in this section all of the new developments mentioned. How-ever, in th e planning of thia seriea, many of these topics were placed inother sections to allow a more logical arrangement of the en tire subjectmatter. Therefore, in thia section, the pertinent results or conclusionsderived from these modern investigations are merely stated in theirsimplest form, and the reader is directed by arose referenoea t o th e othervolumea where the complete treatm ents can be found.In rocketry, as in m y field th at has grown rapidly, the terminologyhas not yet been fully standardized and accepted. Therefore it is appro;priate to start with s, few definitions.Rocket propuleion is a aystem of propulsion t hat depends on forwardthrust created by rearward ejection of a fluid j et through a nozzle mounted

, in the vebicle, with the apeoial oondition tha t the fluid in the jet originateentirely from tanka within the vehiole. It is this speoial condition thatdietinguiehea th e rocket from other classes of jet engines th a t ingest thesurrounding medium (air or water) to form the driving jet. Therefore therooket engine is able to function not only under the usual conditions offlight through the atmosphere (or under water), bu t in the vacuum out-Bide the atmosphere.A rocket propeUaM ia the fluid substance that forms the driving jet,although the term is used most frequently to refer to the driving fluidin it s ohemioal state before oombustion. The te rm is used also to denoteone of the reaotmta in a multicomponent propellant system. A fuel isany propellant that can burn in the presence of oxygen, and the te rm

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5/17

G . TH E LIQUID PROPELLANT ROCKET ENGINEincludes not only hydrocarbons but other substances (e.g. ammonia,powdered aluminum) as well. An 0 ~ i d k W s a propellant that can sup-port the combustion of a fuel, and is applicable to substances th at mayno t contain oxygen (e.g. fluorine) ae well as thom t h at do. A bipropellanteyatem is one th a t consists of two reac tants, uaually a fuel and an oxidizer.A monopropellant is a single aubatance th at can be caused to react in thecombustion chamber to generate hot gas to form the driving jet. Thisterm applies strictly only to a Bingle compound (e.g, ethylene oxide)that undergoes a decomposition reaction, but it has been applied alaoto a propellant mixture tha t is stored in a single propellant tank (e.g. amix ture of m ethyl ni tra te and methyl alcohol).The rocket motm is usually understood to be the p ar t of the enginein which th e propellants are burned and the jet is formed, while the termrocket envine usually refers t o th e entire propulsion system, inoludingthe tanka if they are constructed integrally with the engine. The termt h a t cytinder has been w ed in plaoe of rocket motor in some writings,but the latter term is historically the oldeat and the most widely pre-ferred. In the oonventional solid propellant rocket, the engine and themotor happen to be the same piece of appara tus becausle the propellantis stored in th e oombustion chamber of the rooket motor; th is is not so inthe liquid propellant rocket.G,2. Performance Analyeis of the Ideal Rocket Motor. Theperformance analysis of a rocket motor comprises calcu lations of thethrust F, he effective exhaust velocity c, the adiabatio oombustiontemperature in the ohamber To,he thrust coeffioient Cs, he character-istio velooity c*, and certain ef3eienoiea q . Performance parametersderived from o are the specific impulse I, and the speoific propellantconsumption wan.The thru st equation is the fundamental starting point. In general, th ethrust exerted on a duot of arbitrary shape can be odoulated from themomentum equation written in integrated form appropriate for one-dimensional flow problems. (See II1,B.) Let @ be the rate of maw flowinto th e inlet, ;dc the static pressure at th e inlet, V , the stream velocityat the ialet, and Ad the m a of the inlet, and let the correspondingquantities a t the exit be indioated by the subscript ,. Then,

The stream thrust a t the inlet = (IhrV,+ p,ASThe stream thrust a t the exit = (&,V,+ p,AJ

The tota l force on the external duot surface - F + p,(A, - A,)F + P ~ A , 4 (&V. + p,AJ- WJ~pl-43

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0,2 . PERFORMANCE ANALYSISThe external force on the duct is exprewed here as if the pressureon the external surface were identioal with the ambient pressure p,of th e atmosphere, although in actual flight this is not so. Therefore,for a duct in flight, this equation implies a certain arbitrary separationbetween the thrust F and the aerodynamic drag D. Separation in thismenner is justified by i ts convenience, because the thrus t meaaured in aground tes t of the propulsion system is closely equa l to the th rus t Fthus calculated.In the particultu ow e of a rocket, rik and A' may be set equal to zero.Then,With reasonably well-deaigned exhaust nozzlea, the exit pressure pris nearly or exactly equal to the am bient pressure pa, so th at the secondterm is in th e natu re of a small correction to the thrust. This makes itconvenient to define an ejeective exhaust velocity c such tha t

Clearly, c equ& V, if the nosele is designed properly, t hat is, if th ep. equala p,. For any given values of chamber pressure po and of p,an d h, both F and c reach their maximum values when the exit area A.of the noazle is chosen t o produce a s tat ic preaaure pr a t th e exit exactlyequal to p,. (Thia will be proved later.) The exi t velocity and th e correc-tion term (p, - p,)A./h both vary strongly with A*, bu t in oppositedirections so tha t th e eum is quite insensitive to A,, th a t is, the maximumie very flat. As a result, the effective exhaust velocity meaeured by theratio of F to & can be taken to be the value of (V,)m),,, ven if the ac tua lnoanle used in the test is eomewhat off-design. Herein lies the practicalsignificance of the concept of th e effective exhaust velocity (see Fig. Q,2a.)The specific propellant consumption w,,, defined as the weight rateof coneumption of propellant 3 per unit thrust, is another w f u l indexof rocket engine performance. Let go denote the standard accelerationdue to gravity.

3 w o = f iw,, = - =-F c (2-4)The specific impulse I, (called specific thrust in some writings) isdefined aa the propulsive impulse delivered by th e engine per un it weightof propellant. Fat F 1 cI , p = z = 3 = ' - " -w e goA figure of merit th at is sometimes quoted is th e impulsewe ight ratio,I / W , of a loaded rogket propulsion system or of a loaded rocket vehicle.

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G . THE LIQUID PROPELLANT ROCKET ENGINEIf the firing program calls for oonstant pressure and hence constantspecifio impulse, Iw = IWv (2-6)where v is the propelhnt loadinu fracdion, that is, the ratio of the initialmass of propellant when the rocket is fully loaded to the gross massof the loaded rooket. Obvioudy, P approaohes unity as the structuraleffectivenese is improved. Therefore, I / W measures both the perform-anoe of the engine and th e effectivenees of the struc ture of the rocket.

Th e matter of uni ts of specifio impulse deserves comment. Th edefinition involves the m m ate of flow of propellant expressed in weightunits, so that in the fps system I , 8hould be lb sec/lb. It is commonpractice to denote this ratio &ply ae aec, canceling the lb in numeratorand denominator, disregarding the fsct that one lb refers to a forceand the other to a mass.That this iean error becomes apparent when it isr&ed th a t th e speoifio imp* haa th e dimensions of a velocity.In order to develop several additional performance parameters, it ieneceesary to desoribe in detail the thermodynamic and gas dynamicproo we s in the rocket motor. As a starting point, it i s a pea t hp l i f i c s rtion t o deal with the so-called ideal rocket motor. From a praotioal stand-point, the ideal rocket motor is a useful concept because it leads toaimple theoretical formulas for F, , To, F, nd o*, whioh otherwisewould have t o be presented in tables or in graphs.The ideal rocket motor analysis re& on the following aimpIifications:(1) the propellant gas obeys the perfect gaa law; (2) ita specific hea t i soonstcmt, independent of tem perature; (3) the flow is parallel to the


8/17

G,2 PERFORMANCE ANALYSISaxis of t he motor and uniform in every plane normal to the axis, thuscons tituting a one-dimensional problem; (4) there is no frictional dissipa-tion in the chember or nozzle; (5) there is no heat transfer t o th e motorwalls; (6) the flow velocity in the chamber before the nozzle entrance is0aro; (7) combuation or heat addition i a completed in the chamber atconstant preaaure and does not occur in the nozzle; and (8) the process iasteady in time.The thermodynamic process ia indicated in Fig. G12b, bo th in th ep, V diagram and in the h, e diagram. Combustion a t constant pressure

Fig. G,2b. Ideal thermodynamic prooases in the oombustionahamber and nosale of 8, rooket motor.moves the s tat e point from i to c. The highest temperature occurs a t a.Then, since the frictionleae and adiabatic conditions asaumed for theideal rocke t motor imply an isen tropic expansion in the nozsle (see II1,Bfor proof: also LA). the state aoint moves dona the constant entropyline-from'c to e d&g the exphaion prooesa. ~ & d ic at e d in the mo&eketch of Fig.G,2b, the su bscripts i, c, th , and e employed in th e followinganalyeis refer respectively to fhe &burned atclte &t&e injector, th e all-burned s tate in th e chamber just before expansion, the state a t the throatof the de Lava1 nozzle, and the sta te a t the exit of the nozsle.

The combustion temper&ture (or adiabatic flame tem perature) T.is determined by the heat of oombustion a t constant pressure per unitmaasa. Ah = C ~ ( T . Ti) (2-7)At any station in the nozzle, the entropy, pressure, temperature,


9/17

O . THE LIQUID PROPELLANT ROCKET ENGINEvelocity, and Mach number are given by the following relations (II1,B) :

Since the over-all pressure ratio p d p - is always suffioiently large inrookets to establish sonic flow a t the throat, then

The a p e d o heat ratios of typ ical rocket jet gages range between1.1and 1.3. The 5mt figure oorresponds to mixtures at very high tempera-tures with large concentrations of water vapor and large effective s p e d cheats due to strong dissociation; the latter figure applies to moderatetemperature gasea with moderate oonoentrations of HnO and GO*.With 7 - 1.2, it canbe seen th a t the drop in pressure from the chmnberto the throat is nearly half the ahamber preasure, while the drop intemperature is only about one tent h th e chamber temperature.Th e mass flow throug h the nozzle oan be expressed in t e r m of the flowconditions a t any station. Let A be t he orosa-aeotional area a t any statio n:

. A plot of mam flow per unit area ?h/A against static preasure ratio ,p/pa, shown in Fig. G,2c, exhibits a maximum a t the th roa t of the d e Lava1nozzle, just as expeoted.

By equating th e two expressions, Eq. 2-13 and 2-14, for ph, anexpres-


10/17

0,2 . PERFORMANCE ANALYSIS&on for A /A lh is obtained.

BY nserting A. a d , fo r A and p in Eq. 2-15, the nozzle area ratioe (= A./A*) can be expressed a9 a function of p,/p,. This relation isplotted in Fig. G,2d for several values of y.

Fip. G$o. Vmintion of masa flow per unit are. &/A with preeuure rntio p/p..The maximum exit velocity (V,),),, obtained by eetting p,/pe inEa. 2-10 eqdal to zero, is

It is interesting tha t the maximum exit velooity, obtained by expan-don to eero pressure, is greater than the root-mean-square molecularvelocity in the chamber by the factor [2y/3(y - I)]*, r about 2 fory = 1.2. This result follows directly , of course, from moleoular energyconaiderations.


11/17

G - THE LIQUID PROPELLANT ROC KET ENGINE

17 ( )% [1 -@)$I+p, - pm)A, (2-17)= pad* --_?From this formula, it is dear that the Chrust doea not depend at dlon the oornbuetion temperature To,ut depends mainly on the dimensionsof the nozzle A, and A*, and on the ohamber pressure pa.An important performanoe parameter, the rooket t h a t c o d d e n tCF, oan now be deduoed. The defining equation is

From Eq. 2-17CF can be evaluated:


12/17

G,2 - PERFORMANCE ANALYSISSince p./p. is a function of r according to Eq. 2-15, Cr depends onlyon th e three independent variables 7, ;o,/p,, and e. G r a p h of tbis func-tion are presented in Fig. G12e and 0,2f for 7 1.2 and y - 13,respeotively.There are several features of these ounres tha t d e s m attention.First, each aurve shows a maximum value of CF t n certain area ratiowhich may, for thia purpose, be called ew It can be shown analytically,by differentiating Eq. 2-19 and se tting the derivative equal to zero, thatthe peak value occurs for a value of e such th at p, = pm, hat ie, for a

Area ratioE = kt&Kg, ,%.Variation of rooket thuut ooe&ient with noesle ares ratio andpremwe r d o p&. for y - 1.2. A f h Sutton, Q.P., op, dt,properly expanded no#de. The ares, ratio e~ for proper expansion can bedetermined from the peak of the appropriate m e n Fig.G,2e or G,2f,or, more accurately, from Fig. G,2d. A nozzle having an area ratio leaathan bS is said to be undwexpandad, m d one having an area ratio morethan roS is overeapanded. Clearly, nozsles t ha t em either overexpanded orunderexpanded produce lees thrust than a properly expauded nozzle. Thisaonclusion can be proved in another way, by considering the distributionof pressure on th e inner and outer surfaces of the rooketmotor, s shownin Fig. G,2g. D ownstream of the section indioated as he,he internal

presmm ia lese than the external pressure, 80 tha t this portion of $he ooneacts to produae a force opposed to $he thru st of the rooket motor as awhole. Therefore it iia better to dispense with it and to terminate thenoaale at ew Consequently the higheat thrust is produced with a p ro perly expanded nozzle.


13/17

G . T H E LIQUID PROPELLANT ROCKET ENGINEExam ination of the pressure distribution pictured in Fig. G,2f showstha t Cr must be somewhat greater than unity, except for the unusual caseof low chamber pressure @,/POE 1) and for possibly greatly overex-

panded noz5les. Viewed in the simplest way, the rocket motor ia a prebsurbed vessel with a hole of areeA* in the a ft wall, and so it would be

Area ratioE = klA*Fig. G,Y. Verihtion of rocket thwt coe5oiant with n o~ s le rea mtio andprennure d o / p . for r - 1.3. Aft& &Itton, Q.P., p 08.

Inte rna l pressure Interna l pressuregreater than p, 1 less thon &I

areaFig. a,%. Optimisation of nb d e m a atios.

acted upon by an unbalanced force (pa- pa)Ap. TO his must be addedthe d d i t i o d unbalance due t o th e preseure reduction in th e entrancecone of the nozzle. Then, if pa


14/17

G,2 . PERFORMANCE ANALYSIS

Th e dependence on (p*/pJ can be replaced by dependence on em,through th e relation (Eq. 2-15). The curve (CP), vs. itself reaches anult imate value for inhite e x p d o n . Thus, if (p,/p.) is set equal to

For example, (Cr),,, = 2.246 for y - 1.2.Inspection of Fig. G,2e and G,2f discloses th at , for a prescribednozzle area ratio el the thrust ooefficient increases monotonically aspJp. increases. Thie becomes clear when Eq. 2-19 is written in term s ofthe th rust ooefficient for a given n o d e when it is operating in a vacuum :Thua, for given c and p,, the increaea in thrust with increasing p d p ,stems entire ly from the reduction of the pressure acting on the externalsurfm es of the rocket motor.It is significant th at CF is completely independent of combustiontemperature T.and of molecuhr weight D1Z. Consequently, as a figure ofmerit, it is ineensitive to the effioiency of combuetion, but it is sensitiveto th e quality of the exhaust nozzle. In practice, the t es t engineer oom-pares the measured Cr, computed from pa, A*, and F by means ofEq. 2-18,'with the theoretical CF computed from Eq. 2 1 0 t o determinewhether t he nozzle is function ing properly, and in this way he can looalimto rrome extent the cause of an unexpeoted defeat in speaifio impulse. Theother possible area for loss ia in the combustion process. T o detect com-buation ineffioienoy, the performance parameter c* is ueeful.The characteristic velocity o* is defined se follows:

It follows immediately from Eq. 2-2 and 2-18 th atc - C rc* ( M 3 4

A theoretical expression for c* is ob tainable from E q. 2-14.

From this formula it appears that c* depends mainly on conditionsin th e heombustion chamber, th at is, on &me temperature and combuetion( 449 )


15/17

G THE LIQ'UID PROPELLANT ROCKET ENGINEproduot~oomposition.Consequently, just as Cz lr used as an index of t hequality of th e exhaust nozzle, so c* is used hi praotioe as an index of theeffioiency of combustion. Th e tes t engineer determines a* from measuredvalues of polA*, dl and oomparea it with the theoretioal value (Eq. 2-24).In thieway, a defeot in specifio impulse can be t ramd to a poseible lossin the oombustion process. Although the performance of a rooket motor iaadequately desoribed by the exhaust velocity o, whioh requires only th emeasurement of F an d fi, t is the us ud praotice to meaaurad he sametime fi and A* in order to oompute CF and o* for diagnostic purpwes.

7 8 9 10 11 12 13 14 15mFig. @ah. Variation of ~ o ~ b O i oelooity o* wi thm.(Careful uonaideration of the flow process reveals tha t there exists some& h t orowdependence of CF and c*, that iia, the former ie slightlysensitive to the oombustion prooess and the latter is somewhat a$ectedby the flow conditions in th e nozde, bu t th is is usually ignored.)Curves of o* vs. (Ts/%)* or r 1.2 m d r 2.3 are plotted inFig. G,2h. It is signifioant that cr*, and therefore the speoiflo impulse,depends as muoh on moleoular weight%Is on flame temperature . Thua,it, is just as important for the p roduct gas to have a low mean moleoularweight as a high temperature. Thb point will arise later when actualpropellants are oongidered. It wi l l be pointed out then th a t the optimumfuel-oxidiaer mixture ratio is not neoesrarily the one that produoes thehigheat flame temperature, and that a pa rtioular propellant combinationm y be very hot but no better than another much cooler one from thestandpoint of speoifio impulse.

( 450 )


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G,2 PERFORMANCE ANALYSISThe magnitude of c* is of interest, Because of i ts dependence on(RTo/3n)*,t can be compared directly with the velooity at the nonzlethroa t. Thus, for 7 - 1.2, o* 1 . 5 V ~ .n general, for a properly designed

nonele, Vbb< C*< V a l o tha t c* equak the gas velocity a t some stationin the divergent part of the exhaust noasle.The efficiency of the rocket engine c m be discussed in terms of theconcepts of the ideal rocket motor. Five efficiencies deserve discussionhere: combustion e5loienoy, expasdon or "cycle1' efficiency, nozzleefEoiency, thermal efficiency, and total efficiency. In addition, theat tempts to d&e a so-called propulsive effioiency will be examined.The oombw t i v s&imcy %is defined a9 he ratio of the actua l enthalpyreleased by combuetion to the ideal enthalpy that would be releasedi f the reaction were to go to completion.

The i ded expanaim w cycle e w e n c y qi expresees th e frac tion of theenthalpy available in the combustion chamber that can idedly be con-verted t o kinetic energy in the exhaust jet. Let hi represent the enthalpyof the product gas a t th e injection temperature with reference to OOIC.

From Eq. 810,

The no.& e$ciancy q , is d e h e d aa the ratio of th e actua l kineticenergy in the exhauet jet to th a t whiah could be produoed ideally a t theepecified premure ratio. (Compare with diffuser efticiency in ramjets,Sea. E.)

The t h e d efiemcy qr is th e ratio of actual kinetic energy in th eexhaust jet t o the total enthalpy th at could ideally be produced by theoombustion reaction.

Conversion l o w s in the combustion proceea m d in th e expansionprocess, and the enthalpy discarded in the hot exhsust jet, me all repre-aented in the thermal efficiency.


17/17

G . T H E LIQUID PROPELLANT ROCKET ENGIN8The four efficiencies mentioned so far refer to the rocket motorsimply as a heat engine and do not involve the energy quantities con-nected with flight. Whentlight is conaidered, it is possible .to d e b n

over-dl or a totd ejfeimcy q0 as the ratio of propulsive pow ei(outpu t) tothe ra te of consumption of energy in the rocket motor (input). The outputis aimply V,VI per unit m ws flow of propellant, where V. is the flightspeed. The input il the sum of (Ah+ h,) plue the kinetic energy &V$of t he propellant. The lat ter quantity results from the recognition that aquantity of propellant possesses more total energy when i t ia in motionthan when at rest. -. -

The maximum value of q~ is z/;;;;, a d his value is reaohed when theflight velocity equals the maximum theoretical exhaust velocity. (Thisresult haa led to statement6 in the early literature that a rocket's flightspeed could not axceed the maximum theoretical exhaust velocity.Clearly, this conclusion is not juetified.)There have been several attempts in the peat to define a suitable"propulaive efficienoy" by analogy with the corresponding case of 5propeller-driven airplane, but these attempts have alwayye failed becauaeof th e look of a logical definition of meohanical inpu t. If the analogy is tobe carried through, the so-called propulaive efficiency will have to satisfythree requirements: (1) it should be a ratio of a mechanioal ou tpu t to amechanical input; (2) the produot of the propulsive &oienoy snd thethermal efficiency qe should be equal to the total efficiency 70; and(a) it must under no flight oondition exoeed unity.The definition of propulsive efficiency th at ha8 been most prominentin the litemture isoutput output v,v. z v m v ."''output + oss VmV. + &(V. - v-IP 'fmThe denominator ia suppcwed to represent the sum of the propuleivework a d he absolute h e t i c energy of the jet (r a loas). The preaentauthor objects to this propulaive effi&ency on the grounds that it offerano clear de hi tio n of mechanical input and that, even if this particulardefinition were d lowed, it fails to meet the requirement tha t = q0.1AN conoluaion to th is discuseion of &oieacy, it may he remarkedthat the ooncept of efficiency hasl not been a t al l useful in the field ofrocket engines. Rockets are alwaya compared on the basis of I , or c*values. or on wecific urouellant consumution, and rarely is efficiencv-Editor's n o t e . ~ h e uthor would find the same ditsculty with engine when.ever the kinetio energy of the fuel in included.

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