Nozzle design paper.pdf

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by Charles E. Rogers

Introductionhis article is part of a continuing series inHigh Power Rocketry on the solid propellantrocket motor. In this series solid propellantselection and characterization, internal ballisticsand grain design, and solid rocket motor perform-

ance analysis and prediction will be covered inextensive detail. Previous installme nts of this serieswere published in the following issues of HighPower Rocketry magazine; Performance AnaZysis ofthe Ideal Rocket Motor (Part "On of th e serie s), in t heJan 1997 issue (Ref. 1) ; Parts 1 and 2, SolidPropellant Selection and Characterization, in theFebruary (A) 2001 and February (B) 2001 issues(Ref. 2); and Part 3, Solid Propellant Grain Designand Internal Ballistics, in the October/November2002 issue (Ref. 3).In wha t was in retrospect Part "0" of this series,Performance AnaZysis of the Ideal Rocket Motor, th ederivations of the equations for the theoretical idealperformance of a rocket nozzle and rocket motorwere presented. In Parts 1 and 2 of this series, SolidPropellant Selection and Characterization, themethods and equations for the theoretical specificimpulse for solid propellants based on theoreticalideal performance were presented. In this install-ment of the series departures from ideal perform-ance will be presented. Losses from the ideal thr ustcoefficient to the actual thrust coefficient for noz-zles on solid rocket motors, liquid rocket engines,and hybrid rocket motors will be covered. Methodsfor quantifjring the losses from the theoretical spe-cific impulse to the delivered specific impulse forsolid rocket motors and hybrid rocket motors will becovered. Losses from ideal performance will be pre-sented for conical nozzles an d bell nozzles, straight-cut throats and rounded throats.Of particular intere st is tha t for the first time forhigh power and experimentallamateur rocket motorsthe thrust coefficient losses from straight-cutthroats on conical nozzles will be quantified, bothfrom historical professional solid rocket motor da ta

and recent experimental static firing data. Wbetter understand ing of the thru st coefficient lfrom straight-cut throats, it will be shown hoptimize the design of straight-cut thro ats to mize performance. As will be shown, simple dmodifications have the potential to increasthrust, total impulse and specific impulse ofhigh power solid rocket motors, and proalmost all experimentallamateur solid rocket mby 3.5% to 8%, a significant across-the-increase in performance for two entire classrocket motors.While this article is part of a series on solidet motors, an d the specific results presented asolid rocket motors, the methods and e quationwill be presented for determining the thrust ccient for conical nozzles, bell nozzles, and routhroa ts ar e applicable to solid rocket motors, rocket engines, or hybrid rocket motors. Whioverall design optimization lessons-learnestraight-cut throats are probably applicablexperimentallamateur liquid rocket enginesspecific straight-cut thro at thru st coefficient rare applicable only to solid rocket motors an d hrocket motors. The methods presented for deteing the los ses from the theoretical specific imto th e delivered specific impulse are only applto solid rocket motors and hybrid rocket motorferent techniques are used for liquid rocket enThe present author would like to give stha nk s to G ary Rosenfield of AeroTech, Inc. anRocket Motor Compo nents, Inc., for providing sed AeroTech nozzles drawings from the Resource Library Compact Disk (CD) (ReAnthony Cesaroni and Mike Dennett from CeTechnology Incorporated (CTI) for providing thrust and chamber pressure test data inclpropellant geometry data and detailed nozzle ings, Derek Deville from Environmental Aeences Corporation (EAC) for providing motor and cham ber pressure test da ta including propgeometry da ta and detailed nozzle drawin gs, aTrong Bui from the NASA Dryden Flight ReCenter for performing corrections to chambersure time history data using a MATLAB comprogram.

Copyright 02004 by Charles E. Rogers. All rights reserved. Published with permission.24 High Power Ro

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Straight-Cut ThroatRounded Throat Entrance

ConvergenceDivergence Half Angle = 15 deg Half Angle = 45 deg

Figure I- traight-Cut Throat Conical Nozzle with Rounded Throat EntranceConical Nozzle with a Straight-Cut Throat

The typical nozzle design used on high powerrocket motors and experimentallamateur rocketmotors is a conical nozzle with a straight-cut thro at,a n example of which is sho wn in Figure 1. The con-ical nozzle typically has a convergence half an gle of30-4S0 (for reduced convergent section length), arounded entrance to the throat, a straight (straight-cut) throat, an d a divergence half angle of 1 5 O . Theoptimal divergence half angle of 1 5 O results in agood compromise of minimizing divergence losseswithou t excessive nozzle length and weight. Thelength-to-diameter ratio of the straight-cut throatsection, shown in Figure 1, will turn out to be animportant parameter affecting the nozzle perform-ance. Another example of a conical nozzle with a

straight-cut throat is shown in Figure 2; in thisthe entrance to the throat is not rounded. Mexperimental/amateur rocketeers use this typdesign with a conical convergent section, a stracut throat, and a conical divergent section, wirounding of the corners between the sectioreduce the machining required for the nozzle.While some professional solid rocket motorsused straight throats (sounding rocket mwhich will be presented in this article), routhroats are known to be more efficient, and high performance large professional solid rmotors use rounded throats. A typical conical ndesign with a rounded throat is shown in FiguWhy do most, if not all high power and exmental lamateur rocket motors use str(straight-cut) throats? Primarily for ease of m

I IFigure 2- traight-Cut Throat Conical Nozzle with Sharp Throat Entrance.

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--AXIS

% THROATFigure 3- ypical Design for a Conical Nozzle with a Rounded Throat.

facture, and ease of using a single nozzle "blank" formultiple motors with different throat a reas. A singlemold can be used to make injection-molded pheno-lic nozzle "blanks ," or with autom ated machininggraphite nozzle "blanks" can be machined in largenumbers. Then each nozzle throat can be drilled toa specific throa t area to tailor the nozzle for differ-ent rocket motors. These throats are called "straight-cut" throats from the action of the vertical drillingwith a drill bit used to open up the throat area.Many high power and experimentallamateur rock-eteers use equations for thrust coefficient and spe-cific impulse either not taking into account lossesfrom ideal performance, or usin g typical losses fromideal performance representative of roundedthroats. Clearly straight-cut th roats w ill cause addi-tional losses in thrust coefficient and specificimpulse. One area that will be focused on in thisinstallment of The Solid Rocket Motor series will beto quantify these ad ditional losses, and to see if thedesign of straight-cut throats can be optimized toreduce thrus t coefficient and specific impulse loss es.

CF = thrus t coefficient, dimensionlessF = thrust, N (lb)p, = chamber pressure, Pa (lb/in2)The specific impulse of the rocket motor ithrust of the motor divided by the propellantrate.

In SI Units:

Where:F = thrust, Ngo = acceleration due to gravity a t sea 9.8066 m/s2 (32.174:ft/sec2)I, = specific impulse , N-seclkg (sec)Thrust Coefficient and Specific Impulseli.1 = propellant mass flow rate, kglsecTwo parameters will be of interest in calculatingthe losse s for, and optimizing the design of straight- In English Units:cut throats; the thrust coefficient and specificimpulse. The nozzle thr us t coefficient is defined asthe thrust divided by the product of the chamberpressure and the nozzle throat area. Where:

-F = C ~ A t hC (2) I ~ P -Where:=

Ath = nozzle th roa t are a, m2 (in2)

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thrust, lbspecific impulse , lbf-secllbm (sec )propellant flow rate, lblsec

As will be seen, the thrus t coefficient and spimpulse are interrelated. If a nozzle produHigh Power Ro


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higher thrust by having a higher thrust coefficient and the thrust coefficient.for a particular propellant flow rate and chamberpressure, the specific impulse of the rocket motor Isp = (C* cF) gowill also be increased.Eqs. (4)-(7) provide insight into h ow com bulosses in the motor upstream of the nozzle tCharacteristic Velocity (c*) and t hru st coefficient losses in the nozzle, low

Another parameter of interest for understandinglosses in specific impulse from combustion lossesupstream of the nozzle throat is the characteristicvelocity (c*). The chara cteristic velocity is defined byEq. (4). PC Ath* -------m (4)Where:

c* = characteristic velocity, m/sec (ftlsec)In Ref. 1 a theoretical expression for characteris-tic velocity is derived showing that the characteris-tic velocity is a function of the combustion condi-tions and th e ratio of specific heats.

Where:-M = average molecular weight ofcombustion gases, kg/mol (lblmole)R = universal gas c onstan t, 8314.3 Jlmole- OK (154 5 ft-lblmole - OR)Tc = adiabatic equilibrium flametem per atu re, OK (OR)y = ratio of specific heats, dim ensionlessWith the characteristic velocity based on howmuch chamber pressure is generated for a giventhroa t area by a given m ass flow from combustion(Eq. (4) ),and addit ionall y by inspection of Eq. (5), itbecomes clear that the characteristic velocitydepends primarily on the combustion conditions,and therefore is a relative measu re of the efficiencyof combustion.Combining Eq. (4 ), the definition of the character-istic velocity (c*), with Eqs. (1) and ( 2 ) , the defini-

tion of the th ru st coefficient (C,), resu lts in Eq. (6) ,the thr ust of the rocket motor as a function of ma ssflow, characteristic velocity, and thr ust coefficient.

Eq. (7 ), from Ref. 5, gives th e rocket mo tor specificimpulse as a function of the characteristic velocity

thrust and specific impulse of the motor. Suthere are losses upstream of the nozzle throat dinefficient comb ustion. From Eq. (5 ), with lossthe fundamental combustion process, sucreduction in the ad iabatic equilibrium flame teature (Tc , the combustion temperature), we wexpect to s ee a reduction in the characteristic ity. From Eq. (4 ), the los s in characteristic vewould result in a reduction in chamber pressua given mas s flow through the motor. A reductchamber pressure will reduce the thrust, thlower characteristic velocity will result in a redthrust (Eq. (6)).A reduced thrust , for the sameflow (propellant flow), results in a reduced spimpulse. Thus the lower characteristic velocitresult in a reduced specific impulse (Eq. (7) ), would expect because th e reduction in charactevelocity was caused by inefficient combuwhich would of course lower the specific impuIf efficient combustion is occurring in the mthe motor will have a high characteristic velThe motor will be producing a high chamber sure for a given propellant mass flow (Eq. (4)what if the nozzle is inefficiently expandinproducts of combustion, which creates the thThen there will be a reduction in the nozzle tcoefficient. A loss in thrust coefficient caudirect loss in th rus t (Eq. (6)) . For a given p ropmass flow, creating a given chamber pressunozzle thrust coefficient losses reduce the thrust, there will be a loss in specific impulse(7))-As will be seen, losses in specific impulse ddepartures from ideal performance will be bdown into losses in the fundamental combuprocess upstream of the nozzle throa t (charactevelocity, c* losses) and losses in the expansithe combustion products through the nozzle (tcoefficient losses) .

The Ideal Thrust Coefficient

To determine the thr ust coefficient for the nof a rocket motor th e ideal th rus t coefficient iculated first. Corrections are then made to thethrust coefficient to take into account depafrom the ideal performance assumptions usderive the ideal thrust coefficient equation.The derivation of the ideal thr ust coefficient October 2004


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tion was presented in, what was in retrospect Part"0" of this series, the article Peformance Analysis ofthe Ideal Rocket Motor published in the January1997 issue of High Power Rocketry (Ref. 1). Notethat there are sm all differences in the no menclatureused in the equ ations in Ref. 1 and the equ ations inthis article. The nomenclature u sed in th is article isconsistent with the nom enclature used in the rest ofThe Solid Rocket Motor series of articles.Note that the d erivation of the ideal thrus t coeffi-cient equation presented in Ref. 1 is based on a co n-trol volume drawn around a liquid rocket enginethrust chamber and nozzle. The assumption is m adein the derivation that the stream thrust from inject-ing the propellants into the combustion chamber iszero, due to the velocity of the propellant flow intothe chamber being much lower than the nozzleexhaust velocity. This same assumption can bemade for a hybrid rocket motor. This assumption isexactly correct for a solid rocket motor, as no propel-lant is injected at all into the motor, the propellantis cast into the motor case. With the stream thrustinto the com bustion chamber being zero, the streamthrust at the nozzle exit remains, which with thepressure differential acting on the exit area of themotor results in the classic equation for the thrustof a rocket (Eq. (8)).

Where:Ae = exit area, m2 (in2)m = mass flow rate of propellant, kglsec(slugs/sec)p, = nozzle exit pressure, Pa (lbIin2)p, = atmospheric pressure, Pa (IbIin2)Ve = nozzle e xha ust velocity, mlsec (ftlsec)The ideal thrust coefficient equation is derivedfrom Eq. (8), based on a n ideal perfect ga s analysisfor the flow through the rocket motor nozzle. Thederivation is p resented in Ref. 1 , resulting in theideal thrust coefficient equation (Eq. (9)).

Where:COF = ideal thr ust coefficient, dimension& = nozzle expansion area ratio,dimensionlessIn the deriva tion in Ref. 1 of the ideal thru st ficient equation (Eq. (9 )), he characteristic vel(c*)equation (Eq. (4 )), and the nozzle expanarea ratio equation as a function of the nozzle

pressure to chamber pressure ratio (wh ich wsubsequently presented as Eq. ( I I) ), i t is assutha t the ratio of specific hea ts is con stan t, anda single value for the ratio of specific hea ts is in Eqs. (4), (9), and (11). When several va luethe ratio of specific hea ts are available (typicalthe cham ber, throat, and nozzle exit), since thestant ratio of specific heats assumed in the detion in Ref. 1 is for the expansion from the mchamber to the nozzle exit, the average of the cber and nozzle exit ratio of specific hea ts vshould be used. In particular the present aprefers to use the average of the chamber valuethe nozzle exit value, with the nozzle exit based on eq uilibrium flow.Many stu dents at engineering un iversities, neers in indus try performing general rocket caltions, and high power and experimental/amrocketeers use Eq. (9),with a divergence correadded, as the equation for the thrust coefficienThe present author prefers to more accurdescribe the thrust coefficient from Eq. (9) aideal thrust coefficient, OF, .e,, the ideal tcoefficient with no departu res from ideal perance included, consistent with the nomencluse d in NASA S P-8076 (Ref. 6, portion s rep rintRef. 3) , NASA S P-8039 (Ref. 7), and NASA SP(Ref. 8, portions reprinted in Ref. 2).Based on Eq. (9) the ideal thrust coefficienfunction of the chamber pressure to atmosppressure ratio and the nozzle expansion area for two typical ratios of specific hea ts, is plottFigures 4 an d 5. Note in Figures 4 an d 5 that fnozzles the ideal thrust coefficient increases aatmospheric pressure decreases, and that aatmospheric pressure the maximum ideal tcoefficient for a given chamber pressure is prodwhen the no zzle is ideally expanded, i.e. the nexit pressure equals the atmospheric pressure.The nozzle exit pressure to ch amber pressure

( pe lpc ) tha t is required for Eq. (9 ) is a fun ctithe nozzle expansion area ratio and the ratio ocific hea ts for the g as flow through the nozzle

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Area ratio E = A,/&Figure 4 - ariation of ldeal Thrust Coefficient with Nozzle ExpansionArea Ratio and Pressure Ratio pc.p, for y = I 2.

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4 6 8 1 0 20 40 60 80 100Area ratio E = A, /Ath

Figure 5 - ariation of ldeal Thrust Coefficient with Nozzle ExArea Ratio and Pressure Ratio p,/ p, for y I 3.


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Based on Eq. (11) the nozzle expansion area ratioas a function of the chamber pressure to exit pres-sure ratio (the inverse of p, lpc ), for several ratiosof specific heats, is plotted in Figure 6. Note inFigure 6 that a s the expansion a rea ratio of the noz-zle is increased the chamber pressure to exit pres-sure ratio increases, mean ing tha t for a given cham-ber pressure the nozzle exit pressure will decrease.If the desired nozzle exit pressure to chamberpressure ratio is known, and the nozzle expansionarea ratio for that exit pressure to chamber pressureratio needs to be determined, the exit pressure tochamber pressure ratio can simply be entered intoEq. (11) to determine the nozzle expansion arearatio. The more common situation is th at the nozzleexpansion area ratio is known, and it is the nozzleexit pressure to chamber pressure ratio th at needs tobe determined. Data for the cham ber pressure to exitpressure ratio (the inverse of the exit pressure tocham ber pressure ratio) can be read off Figure 6, butfor computer simulations a numerical solution of

Eq. (11) is required. Given the nozzle expaarea ratio, Eq. (11) must be solved using a n itenumerical method to determine the exit presschamber pressure ratio. The present author rmends the Newton-Raphson numerical mealthough the half-interval method or other siterative methods can be used. Or for simplifieculations usin g a calculator, the d ata can be rethe plot in Figure 6.The fact that the maximum ideal thrust ccient for a given chamber pressure is achievedthe nozzle is ideally expanded is discussed in Rand can be derived theoretically by sub stitutin(11) into Eq. (9 ), and then taking the derivatthe re sulting equation with respect to exit preThe derivative is equal to zero when the exitsure is equal to the atmospheric pressure, icase indicating a maximum value of the fun(rather than a minimum value), the maximumof the ideal thrust coefficient.Note that while PF Eq. (9)) is called the "

Figure 6 - ariation of Nozzle Expansion Area Ratio withChamber Pressure to Exit Pressure Ratio (p, /p&30 High Power Ro


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thr ust coefficient, it is called "ideal" because it isbased on an ideal performance analysis, with nolosses from ideal performance. The truly "optimum"or "ideal" nozzle is an ideally expanded nozzle (noz-zle exit pressure equals atmospheric pressure, p, =p,), because this will produce the maxim um thr us tcoefficient at the atmospheric pressure for which thenozzle is ideally expanded. This poses a n interestingquestion; are underexpansion and overexpansionlosses considered to be departures from ideal per-formance? The answer is no. Based on the idealrocket motor performance analysis from Ref. 1, Eq.(9) gives the ideal perform ance of the nozzle giventhe chamber pressure, nozzle area ratio and atmos-pheric pressur e, including the effect of nozzle under-expansion or overexpansion. While there are sever-al corrections to the ideal performance from Eq. (9)which will be covered in detail in this article, thereare no corrections required for underexpansion oroverexpansion, because the effect of underexpan-sion or overexpansion is already included in Eq. (9).Note th at after the derivation of Eqs. (8 ), (9), and(11) (presented in Ref. 1) is complete, with theresults plotted in F igures 4-6, the only ga s propertyrequired for the flow through the nozzle is the ratioof specific heats. The exit pressure to chamber pres-sure ratio for the nozzle (a function of the nozzleexpansion area ratio) and the ideal thrust coeffi-cient are both independent of the combustion tem-perature and the molecular weight of the gasesmaking up the nozzle flow. For most solid propel-lants, liquid rocket engine fuel and oxidizer combi-nations, and hybrid rocket motor fuel and oxidizercombinations the ratio of specific heats for the flowthrough the nozzle is typically between 1.2 and 1 .3,the values used for Figures 4 and 5, and which areincluded with additional ratio of specific heat valuesin Figure 6. (Althou gh for some propellants th e ratioof specific heats can be as low as 1-13 .)Again it's important to note t hat based on the con-trol volume analysis used to derive the equation forthe thrus t of a rocket (Eq. (8) ),and the ideal perfectgas analysis used to derive the ideal thrust coeffi-cient equation and associated equations (Eqs. (9)-(1 ) ), tha t Eqs. (8)- (11) are valid for solid rocketmotors, liquid rocket engines, and hybrid rocketmotors. It's also important to note tha t Eqs. (8)-(1 ) ,and the dat a presented in Figures 4-6, are valid forboth conical nozzles and bell nozzles. There arehowever impor tant differences in the divergence cor-rection to ideal performance for conical nozzles andbell nozzles, which will be presented later in thisarticle. There are no divergence losses an d no throa tlosses included in Eqs. (8)-(11) and Figures 4-6, theequations and da ta presented as sum e no divergencelosses an d a "perfect" throa t with no losses.Despite being based o n a n ideal perfect gas analy-

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sis, the ideal thrust coefficient equation (Ecan be accurate to within 1% o 5% for solid motors, liquid rocket engines and hybrid motors with rounded throats , and can be accuwithin 1% to 3% for nozzles with rounded twhen a non-ideal correction for nozzle diveangle is included. As will be seen though, fozles with straight-cut throa ts the ideal thrust cient equation can be in error up to 9.5% dhigh, unaccounted for losses.To reduce the errors in the thrust coefficiendicted by the ideal thrust coefficient equatieffects of departures from ideal performance wconsidered, and quantified for both rounded tand straight-cut throats, conical nozzles annozzles.

Theoretical Specific ImpulseThe counterpart to the ideal thrust coefficthe theoretical specific impulse. From NASA SP(Ref. 8, reprinted in Ref. 2) an d othe r referenctheoretical specific impulse is determined usi

(12).

Where:Pg, = theoretical specific impulse,N-seclkg (sec) , lbf-secllbm (sec)The theoretical specific impulse is calcassu min g ideal expansion (p, = p,), with no

gence losses or other losses associated with tures from ideal performance.For the theore tical specific impulse there areoptions for calculating chamber and exha ust csition and performance; frozen flow, equiliflow, and chemical kinetics. Frozen flow prthe lowest performance, equilibrium flow theest performance, with the chemical kinetics-performance lying between frozen an d equiliEquilibrium flow is also known as shifting erium, because while chemical reactions are assto occur instantaneously under equilibrium tions, the equilibrium values are continuchanging as the flow pressures and tempervary through the combustion chamber and nWhile including chemical kinetics provides thaccurate calculation of theoretical performancvery numerically intensive. With the much scalculations for equilibrium flow versus incchemical kinetics, and with equilibrium flow p


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ing an upper bound for the theoretical performanceof the propellant, it is custom ary in the solid rocketmotor industry to calculate the theoretical specificimpulse,P, , based on equilibrium flow.The JANNAF Thermochemical Equilibrium Code,the NASA Lewis Code, and the Air Force ChemicalEquilibrium Specific Impulse Code (AFCESIC, alsoknown as the United States Air Force [USAF] ISPCode) are three of the compu ter programs in use inthe solid rocket motor industry for calculation ofP,. The PROPEP program based on the NavalWeapons Center PEP program, d ocum ented in Ref. 9,is widely used in the model rocket and high powerrocket industries, and by experimentallamateurrocketeers.Many high power and experimental/amateur rock-eteers run programs such as PROPER and assumetha t the theoretical specific impulse predicted by theprogram for their propellant will be the specificimpulse delivered by their rocket motors using thepropellant. This will not be the case, with lossesfrom the theoretical performance the actual specificimpulse of the rocket motors will be less. It's impor-tant to quantify these losses from the theoreticalperformance, which will be covered in a later sec-tion.

Actual Thrust Coefficient andDelivered Specific ImpulseIt's important to note that beyond the ideal thrustcoefficient and the theoretical specific impulse,what we're really interested in is predictions for theactual thrust coefficient and the delivered specificimpulse for the nozzle and motor.C , = actual thr us t coefficient, including allnozzle losses, dimensionlessIspd = delivered specific impulse,N-seclkg (sec) lbf-secllbm (sec)The actual thrust coefficient (C, ,& is also refer-red to as the "measured" thrus t coefficient, the actu -al thru st coefficient which would be measured basedon test d ata. The delivered specific impulse (Ispd)sthe actual (delivered) specific impulse of the motor,based on the motor total impulse and propellantweight. It is the actual specific impulse the motor

would deliver when fired on a thrust stand.While the ideal thrust coefficient and theoreticalspecific impulse are theoretical values, the actualthr us t coefficient and the delivered specific impulserepresent the actual performance of the nozzle andmotor, including all departures from ideal perform-ance and all real-world losses.

The procedure used is to star t with the theore(ideal) prediction for the thrus t coefficient (the thrust coefficient,0,)nd the theoretical (iprediction for the specific impulse (the theorespecific impulse,P,), and make corrections forideal performance to arrive at predictions foactual thrust coefficient and the delivered speimpulse.

Nozzle Divergence Correction Factor,Efficiency Factor, and the Momentumand Pressure Differential ComponentsThrust

One of the primary corrections to the ideal tcoefficient is the effect of nozzle divergenceconical nozzles, the classic equation for the dgence correc tion factor (Eq. (13)) is used NASA SP-8076, also Refs. 1, 5, 10, 11, and mother references).1h = - - ( I + cos a )

Where:a = nozzle divergence half a ngle, degh = nozzle divergence correction factordimensionless

The geometry definition for the nozzle diverghalf angle (a)or conical nozzles was presentFigure 3.For bell nozzles, the divergence correction fis determined using Eq. (14) (fro m NASA SP-[pages 23-24 of Ref. 31, original reference Ref.

Where:8, = nozzle exit plane lip angle, deg

The geometry definitions for bell nozzles fonozzle divergence half angle (a ) and the nozzlplan e lip angle (8,) will be presente d in FigureReturning to the equ ation for the thrus t of a et (Eq. (8)),we can see tha t the thrus t of a rocmade up of a momentum component ( m V,) apressure differential component, (p, - p,) A,

Momentum PressureComponent DifferentialComponent

As noted previously, many students at engiHigh Power Roc


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ing universities, engineers in industry performinggeneral rocket calculations, calculations done byhigh power and experimental/amateur rocketeers,and in many of the references and course materialused at engineering universities, the nozzle diver-gence correction factor (A) is simply applied to theideal thrust coefficient to determine the actualthru st coefficient.

Where:Fa = actual thru st, N (Ib)Fi = ideal thrust, N (Ib)

This ha s the effect of app lying the divergence correc-tion factor to both the momentum component andthe pressure differential component.

The divergence correction factor should clearly beapplied to the mom entum component, but shou ld itbe applied to the pressure differential componentalso? Additionally, we can group the remainingthru st coefficient losses from th e ideal thru st coeffi-cient into a CF efficiency factor term (qF).Shouldthis CF efficiency factor be applied to the m omentumcomponent only, or to both the m omentum compo-nent and the pressure differential component?Note that the divergence correction factor equa-tion for conical nozzles (Eq. (13 )), and the diver-gence correction factor equ ation for bell nozzles (Eq.(14 )), are valid for when the divergence correctionfactor is applied only to the momentum componentof Eq. (8) (which will be su bsequ ently described a sthe NASA SP Method), or when the divergence cor-rection factor is applied to both the m omentum com-ponent and the pressure differential component ofEq. (8) (which will be sub sequ ently described a s theStandard Method).Depending on how the divergence correction fac-tor and the CF efficiency factor are applied, there arethree different methods for correcting the idealthrust coefficient to the final actual thrust coeffi-cient for the rocket m otor nozzle.

Sutton Performance CorrectionFactor MethodIn Sutton, Rocket Propulsion Elements (Ref. 5 ) amethod u sing performance correction factors is pre-

sented for correcting the ideal thrust and idealthrust coefficient to the actual thrust and actualthrust coefficient. On pages 90-91 of Ref. 5, anempirical thr us t correction factor is defined w hich isequal to the ratio of the actual thrust to the idealthrust .October 2004

SF = thru st correction factor, dimensioThe thrust correction factor can also be adirectly to the thrust coefficient. From Ref. 5:

Note that from Ref. 5 the th rus t coefficient(17) is the ideal thrust coefficient. The pauthor will now substitute in nomenclature tent with the other thrust coefficient equatiosented in this article for the final equa tion, E

Note th at from Ref. 5 the thrust correction faequal to the product of the velocity correctionand the discharge correction factor.SF = rv rdWhere:

Sd = discharge correction factor,dimensionlessSv = velocity correction factor, dimensFrom Ref. 5; the velocity correction faapproximately equal to the ratio of the actual ic impulse to the theoretical (ideal) specific imand thus can be used to correct the theoreticcific impulse to the actual specific impulse.nomenclature cons istent with the rest of this

The discharge correction factor is defined ratio of the actual mass flow rate through thzle to the ideal ma ss flow rate.

Where:ma = actual mass flow rate through nkglsec (lbmlsec)


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m i = ideal mass flow rate through nozzle,kglsec (Ibmlsec)Interestingly, from Ref. 5, the value of the dis-charge correction factor is usually larger tha n 1 (1.0to 1.15 ); i.e. the actua l flow through th e nozzle islarger than the theoretical (ideal) flow, due to (1) themolecular weight of the gasses increases slightlyflowing through th e nozzle, increasing the g as den-sity, (2) heat transfer into the nozzle walls, decreas-

ing the temperature of the flow in the nozzle,increasing its density, (3) changes in the specificheat ratio and other gas properties compared to theideal analysis, and (4) incomplete combustion,which increases the density of the exhaust gases.While the discharge correction factor corrects themass flow through the nozzle, and the velocity cor-rection factor corrects the exhaust velocity (butbased on a energy conversion efficiency, see Ref. 5),by multiplying the ideal th rust coefficient by thethrust correction factor (the product of the velocitycorrection factor and the discharge correction fac-tor), the net effect of the Sutton PerformanceCorrection Method is tha t both the momentum com-ponent and the pressure differential componentfrom Eq. (8) are multiplied by the thrust correctionfactor (rF). hus the Sutton Performance CorrectionMethod fits into the method category where both themomentum com ponent and the pressure differentialcomponent from Eq. (8) are multiplied by the samecorrection factor(s), which captures the divergencecorrection and the correction for other thru st coeffi-cient losses, although in a different form using asingle thrust correction factor (6,) as described bythe equations above.The NASA SP Method

In the NASA SP Method presented in NASA SP-807 6 (Ref. 6, reprinted in Ref. 3) and NASA SP-8039(Ref. 7), the divergence correction factor a nd a noz-zle CF efficiency factor a re applied to only themomentum component in the rocket thrust equation(Eq. (8)), resulting in the following equations forthrust coefficient and thrust.

Where:LdF ,vat = ideal thrust coefficient (Eq. (9 )),wp, = 0 (vacuum ), dimensionlessTF = C efficiency factor, dimensionlessFrom Section 2.1.1.2.1 of NASA SP-8076 (p18-19 of Ref. 3), the deliverab le motor efficiencyis defined such that

Where:q = deliverable motor efficiency,dimensionlessPYd = theoretical delivered specific impulN-seclkg (sec ), lbf-secllbm (sec)Note from the glossary section of NASA SP-(pages 56-5 7 of Ref. 3), the delivered speimpulse (I, ) and the theoretical delivered speimpulse ( 4 d ) are both based on a particular cber pressure, atmospheric pressure, nozzle exsion area ratio an d nozzle divergence half anglIspd measured (delivered) propellant specifimpulse, N-seclkg (lbf-secllbm ) (ratio o/o p rope llant weigh t, cor respondin

to p, , E, p,, and a [nozzle half angleof the motor [ref. 121)Pqd= theoretical delivered propellant specifimpulse , N-seclkg (lbf-secllbm)(theoretical lSporresponding toparticular va lues of6 E,p, and a[ref. 121)

Where:I,,, = total impulse, N-sec (lb-sec),(tota l integral of thrust-time)

-p, = average chamber pressure, Pa (lbliTherefore, while divergence losses a re includthe theoretical delivered specific impulse andelivered specific impulse, the additional tcoefficient losses captured by the CF efficiency f

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(qF)are not included in the theoretical delivered spe-cific impulse. Thus, as noted in NASA SP-8076(pages 18-19 of Ref. 3), the deliverable mo tor effi-ciency (qp) is appro ximately eq ual to th e p roduct ofthe c* efficiency fa ctor (Q) and the CF efficiency fac-tor (qF). It is "approximately equal to" due to thecomplexities of the NASA SP Method thrust coeffi-cient and th rust equations, Eqs. (23) and (24).

qe = c* efficiency factor, dimen sionlessThe present author proposes, based on equationsfrom Section 2.1.1.2 .1 a nd Section 2.1.2.2 of NASASP-8076 (pages 18-19 and pages 23-24 of Ref. 3),the definitions of theoretical specific impulse anddelivered specific impulse, and Eqs. (25) and (26)above, that the delivered specific impulse can bedetermined from the theoretical specific impulseusing Eq. (27).

The ratio of the actual thrust coefficient to theideal thrust coefficient with ideal expansion is usedin Eq. (27) because with the complex equation forthe a ctual th rust coefficient (Eq. (23)), there is nosimple way to directly include the nozzle losses fromideal perfo rmance (A, qF ) in Eq. (2 7).

The Standard MethodIn many of the references and much of the coursematerial used for undergraduate and graduate engi-neering courses, engineers in industry performinggeneral rocket calculations, and calculations per-formed by high power and experimentallamateurrocketeers, to determine the actual th rust coefficientthe ideal thrust coefficient is simply multiplied bythe nozzle divergence correction factor (A).

The present author proposes what will be calledthe Standa rd Method, where the product of the idealthrus t coefficient and the nozzle divergence correc-tion factor is simply multiplied by the C' efficiencyfactor (qF).

This ha s the effect of applying the divergence cor-October 2004

rection factor and the CF efficiency factor to bmomentum component and the pressure diffecomponent in the equation for the thrus t of et (Eq. (8)).

= A q F CmVe + ( P e - P c o ) ~As was done in the NASA SP Method, baNASA SP-8076 the deliverable motor efficienis defined such that

As in the NASA SP Metho d, whil e divergences are included in the theoretical delivered simpulse (pqd)nd the delivered specific i(Ivd),he additional thrust coefficient lossetured by the CF efficiency factor are not incluthe theoretical delivered specific impulse.A difference in the Standard Method relathe NASA SP Method is that rather than haviproduct of the c* efficiency factor (qe) and efficiency factor (qF ) being ap proxima tely eqthe deliverable motor efficiency (q& , the p rothe c* efficiency factor and the CF efficiencyare exactly equal to the deliverable motor effi

Including underexpansion or overexpeffects, the equation for the delivered simpulse from the theoretical specific imremains the same as used in the NASA SP M

If ideal expansion (pe = p,) is ass um ed forthen

Eq. (30 ), the de livered specific impulse e qthen reduces to:

Assuming ideal expansion (p, = p,).Approximate (nearly exact) for close to expansion.

The Standard Method equations, includingtions for the divergence correction factor (L)ical nozzles and for bell nozzles, are preseFigure 7.


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The Standard Method for Actual Thrust Coefficientand Delivered Specific Impulse

Ispd = % 7 ] 0 I : p

For Conical Nozzle:

1h =- 1 + co s a)2 Eq. (13) RfI- NOZZLLn I[-

For Bell Nozzle:

. [ 2 + co s ( a @ex)Figure 7 - he Standard Method for Actual Thrust Coefficientand Delivered Specific Impulse.

Recommended Method-The Standard MethodThe present author recommends that theStandard Method be used for correcting the ideal

thrust coefficient and the theoretical specificimpulse to the actual thrust coefficient and thedelivered specific impulse. The rationale for recom-mending the Standard Method will be presented inthis section.First, from a contrary view, Section 2.1.3.2.2 ofNASA SP-8039 recom mend s tha t the divergenc e cor-

rection should only be applied to the momencomponent of the rocket motor thrust (the NASMethod). Similar arguments can be made throat losses captured using the C efficiency faespecially losses from straight-cut thro ats, will a larger effect on the mom entum component thathe pressure differential component of the equafor rocket thrust, and thus applying the C' efficy factor to the m omentum component only mamore a ccurate (th e NASA SP Metho d).The only differences between the StanMethod and the NASA SP Method arise

36 High Power Roc


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Dth = 0.75inpc = 500 psiap , = 14.69 psiaBased on basic equation for rocket thrus t (Eq. (8)).No divergence correction or C efficiency factor corre ction.Nozzle Half Angle (a)= 0 deg?L = l . OqF = 1 . 0y = 1 . 2

E = 3.0 E = 4.0 E = 5.0 E = 5.31 E = 6.0(idealexpansion)4- in) 1.299 1 1.5 1.6771 1.7283 1.8371- - -Pe PC 0.06562 0.04344 0.03 188 0.02937 0.02484Thrust 323.78 329.04 330.77 330.85 330.49(total) (lb)Pressure 24.02 12.42 2.76DifferentialComponent (Ib)-Percent of 7.4% 3.8% 0.8%Total ThrustMomentum 299.76 3 16.62 328.01 330.8 5 336.5 1Component (lb)-Percent of 92.6% 96.2 99.2% 100% 101.8%Total ThrustDe = nozzle exit diameterDth = nozzle throat diameter

Table I- ercentage of the Total Thrust for a Typical Large High Power Rocket from the Momentum Component and the Pressure Differential Component oRocket Thrust Equation (Eq. (8)).whether to apply the divergence correction factorand the CF efficiency factor to either both themomentum com ponent and the pressure differentialcomponent in the thrust equation, or to the momen-tum component only. If the motor nozzle is ideallyexpanded (p e = p,), then there is no pressure dif-ferential component, and as can be seen in Eqs.(23) -(31 ) the NASA SP Method equ ations reduce tothe Standard Method equations; i.e., for idealexpansion, the two methods produce identicalresults.

With the Standard Method and the NASA SPMethod identical for ideal expansion, for nozzlestha t are nearly, or close to ideally expanded, the twomethods will be very close. Table 1 shows the per-centage of total thr ust from the momentum compo-nent and the pressure differential component fromthe rocket thrus t e quation (Eq. (8)) for a typicalOctober 2004

large high power rocket motor nozzle. Tableconstructed using the basic equation for thrus t (Eq. (8)), with no nozzle divergence divergence half angle a = 0 deg), and wdivergence correction factor h = 1 O, and theciency factor qF = 1 O.Note in Table 1 that when the nozzle is expanded, the pressure differential componencourse zero. Also note in Table 1 that for expratios greater than the expansion ratio foexpansion the pressure differential componnegative, it subtracts from the total thru st. Tpercentage of the total thru st tha t is the momcomponent goes over loo%, ince it is makingthe negative thrust from the pressure diffeterm.As can be seen from Table 1, when the noclose to ideally expanded, or even somew


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from ideal expansion, the majority of the thrustfrom the nozzle is from the m omentum component.With the portion of the thrust from the pressure dif-ferential com ponent being relatively small, the errorfrom applying or not app lying the dive rgence correc-tion factor and the C efficiency factor corrections tothe pressure differential component will be small.Thus for nozzles th at are close, or relatively close toideal expansion, there is little difference betweenthe two methods.In Sec tion 2.1.3.2 .2 of NASA SP-8039 (Ref. 7), an don pages 3 59-36 1 of Mechanics an d Thermodynam-ics ofPropulsion by Hill and Peterson (Ref. 1 ), der-ivations based on a point-source flow assumptionare presented to determine where to apply the diver-gence correction factor (1). he point-source flowgeometry used in the derivations presented in NASASP-8039 and Mechanics and Thermodynamics ofPropulsion is presented in Figure 8. From Mechanicsan d Thermodynam ics ofPropuZsion the equation forthe thrust of the nozzle in Figure 8 is Eq. (32).

Where:u, = nozzle exhaust velocity, mlsec (ftlsec)Note from Eq. (32) that when the conventionalnozzle exit area (A,) is used th e con ical nozzle diver-gence correction facto r (see Eq. (13 )) is applied on lyto the momentum component of the nozzle thrust.The spherical exit area of the nozzle (A',), whichresults from the point-source flow assumptionshown in Figure 8, can be determined from the con-ventional nozzle exit area using Eq. (33).

Where:A', = nozzle spherical exit area, m2 (in2)Note that Eq. (33) can be used to substitute area

A', in place of ar ea A, in Eq. ( 32 ), resulting in(34), the nozzle thrust equ ation based on the spical nozz le exit are a A', .F = 1+ cos a2 [mue+ P~-P,)A:I

Note from Eq. (34) that when the nozzle thequation is based on the spherical nozzle exit A', , the nozzle divergence correction factoapplied to both the momentum component andpressure differential component of the nothrust.Clearly for conical nozzles with typ ical diverghalf an gles there is very little difference betweenconventional nozzle exit are a A, an d the spheexit area A',. Thus there will be very little differbetween applying the nozzle divergence correcfactor to both the momentum component andpressure differential compon ent of th rus t (usingspherical exit ar ea A',) or applying the nozzle dgence correction factor to the momentum conent of thru st only (using the conventional exitAe).While us ing a different technique in applyingrection factors, fundamentally the SuPerformance Correction Method multiplies the thrus t coefficient by a single correction factorthrust correction factor ( cF) . Thus the SuPerformance Correction Method share s the primtrait with the present author-proposed StanMethod of not applying a correction factor tomomentum component only, but applying a cotion factor to both the momentum componentthe pressure differential com ponent by simply tiplying the ideal thrust coefficient by a correfactor.In the opinion of the present author, the coused in the NASA SP Method and the StanMethod of breaking down the losses into thcoefficient losses and c* efficiency losses is a powerful concept and much more useable for power and experimentallamateur rocketeers, the concept used in the Sutton Perform

Control surfaceJ

Figure 8 - pherically Symmetric Nozzle Flow Based OnPoint-Source Flow Assumption.High Power Roc


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Correction Method of a velocity correction factor an da discharge correction factor. The present authorprefers that the thrust coefficient corrections beapplied to both the momentum component and thepressure differential component (the StandardMethod), rather th an just to the momentum compo-nen t (th e NASA SP Method).The present author proposes that the dischargecorrection factor from the Sutton PerformanceCorrection Method will be very difficult for highpower and experimentallamateur rocketeers toquantifjr from nozzle test data. While the Suttonthrust correction factor could be used directly, thedivergence correction factor is buried within it. Thepresent author prefers to separate out the diver-gence correction factor, since it is e asy to calculateand has accepted values, leaving the "all otherth ru st coefficient losse s" in the CF efficiency factorterm.Finally, the present auth or recommends th at theStandard Method be used primarily because of thewidespread u se by engineering stu dents , engineersin industry performing general rocket calculations,and high power and experimentallamateur rocke-teers, of the technique of simply multiplying theideal thrust coefficient (PF)y the nozzle diver-gence correction factor (h) to determine the actualthrust coefficient (CF ,,,, ).

To the present author's knowledge at the time of

the writing of this article, with the exceptsome internal-use computer programs by theent author, every solid rocket motor, hybrid motor, and liquid rocket engine computer prsoftware, spreadsheet, performance charts, epredicting performance and calculating thruschamber pressure used by model, high poweexperimental/am ateur rocketeers use s Eq. (3of these computer programs, software pacspreadsheets, performance charts, etc., can bly updated to the S tandard Method proposed present author by simply multiplying thethr ust coefficient and the divergence correctitor wi th the CF efficiency factor (qF ), i.e., byEq. (28).

Rep resenta tive valu es for the CF efficiency(qF) for straight-cut throats and rounded twill be presented later in this article.Additionally, the PROPEP program th at is used by high power rocketeers and experimamateur rocketeers, the JANNAF ThermochEquilibrium Code, the NASA Lewis Code, aUSAF ISP Code all produce results for the theospecific im pulse, P, . Using Eq. (31 ) the despecific impulse can be determined frotheoretical specific impulse from the abovgram s based on representative or calculated for the divergence correction factor (A), the Cciency factor (qF),and the c* efficiency factor

October 2004


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Providing high power rocketeersand experimentallamateur rocke-teers for the first time with amethod for determining the deliv-ered specific impulse from the the-oretical specific impulse obtainedfrom PROPEP-type prog rams .

Specific Impulse Losses-Characteristic VelocityLosses, Thrust CoefficientLosses, and the Variationof the c* Efficiency Factorwith Motor Size

The Standard Method equationused to determine the deliveredspecific impulse from the theoret-ical specific impulse (Eq. (31)),illustrates some important con-cepts in terms of departures fromideal performance. Specificimpulse losses from ideal per-formance are du e to (1) fluid flowlosses including two-phase flowin which particles fail to achievekinetic and thermal equilibrium,(2) combustion inefficiency, (3)heat losses to the motor hard-ware, and (4) boundary layer loss-

0 m o t o r 1, 1,000 p s io m o t o r 2, 1,000 p s i0 m o t o r 2, 685 ps iA m o t o r 2, 550 p s i

m o t o r 3, 75 p s im o t o r 3, 150 p s i

A m o t o r 3, 1,000 p s i

es in the nozzle. The losses from ideal performancefor specific impulse c an be broken u p int o two types;(1) losses upstream of the nozzle throat character-ized by c* (characteristic velocity) com bustion loss-es represented by the c* efficiency factor (qe),and(2) losses in t he nozzle from inefficient expansion ofthe gas in the nozzle characterized by thrust coeffi-cient losses rep resented by t he divergence correction(h) (divergence losses, which are easy to separateout and quantifjr) and the CF efficiency factor (q,)(all of the other t hru st coefficient losses).

In other words, if a m otor is delivering a low spe-cific impulse, the motor development engineerwould loo k for two possible sources; either the noz-zle isn't functioning as efficiently as desired (thenozzle thrust coefficient is low, likely source a lowCF efficiency factor) , or there are com bustion lo sses(c* losses) ups tream of the nozzle th roa t (a low c*efficiency factor). Two-phase flow losses, heat

10 20 30 40 50M e a n r e s i d e n ce t i m e i n m o t o r , m s e c

Figure 9 - ffect of Residence Time onCompleteness of Metal Combustion.transfer into the nozzle structure, and bounlayer losses can lower the thrust coefficient (lth e CF efficiency factor). Combustion inefficiinternal two-phase flow losses, and heat traninto the motor structure can lower the c* efficifactor. A low residence time in the m otor for mlized (typically aluminum) propellants can resuunburned metal exiting the motor, as showFigure 9 from NASA SP-8 064 (page 52 of Ref. 2 , inal reference Ref. 13) show ing unb urne d alum ias a function of residence time for an aluminpropellant. Unburned metal is lost energy from bustio n, res ultin g in combu stion inefficiency, loing the c* efficiency factor.Figure 10 , from NASA SP-8064 (page 5 2 of Rorigin al reference Ref. 14 ), presen ts Isp efficiencpercent) versus motor residence time for an minized propellant (ALPBAN) (Footnote 1). tha t when comparing with Figure 9, when the dence time is increased, less of the aluminuunburned, and the motor delivers a higher speimpulse efficiency. Note also from the indiv

Footnote 1- 1 = aluminum; PBAN = polybutadiene-acrylic acid-acrylonitrile terpolymer40 High Power Roc


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o 1-1b m o to r sQ 4-lb m o to r sA 10-lb m o to r s0 50-ib m o to r s'47 350-lb m o to r s

5 10 15 20 25 30 35 40 45 Average res idence t ime , msec

Figure 10- ffect of Residence Time on lW Efficiency in an AI-PBAN Propelladata points in Figure 10 that larger motors general-ly have longer residence times du e to generally hav-ing longer cores. Larger motors also have lower hea tlosses and boundary layer losses. One can concludetherefore that in general, larger motors are moreefficient than smaller motors, and will have highervalues of the c* efficiency factor. One can see thisfrom the d ata presented in Figure 10. If the d ata wasplotted versus motor propellant weight rather thanmotor residence time; the 1 Ib propellant weightmotors have an average Isp efficiency of approxi-mately 87.5% (qe - 0.875), the 10 Ib propellantweight motors have an average Isp efficiency ofapproximately 92% (qe- .92), the 50 lb propellantweight motors have an average Isp efficiencyapproaching 92.5% (qe- .925), and the 350 lb pro-pellant weight motors have an average Isp efficiencyof approximately 95% (qe= 0.95).While non-metallized propellants will not havethe problem of unburned metal for short residencetimes shown in Figure 9, or the kinetic portion oftwo-phase flow losses, small motors will still havethermal equilibrium losses , combustion inefficiencylosses, and heat losses that on a percentage basiswill be higher than for large motors, and will showthe sam e basic trend of increasing Isp efficiency withmotor size seen in Figure 1 0.

October 2004

Note the low l, fficiency values for 1 lb plant weight motors in Figure 10, and the trenindicating even lower I, fficiency for motoless than 1 lb of propellant. It's clear from Figthat for composite aluminized (or metallizedpellant high power rocket motors and modelmotors with propellant weights under 1 lb thaes from the theoretical specific impulse predicPROPEP-type programs can be 15% or Experimentallamateur rocketeers making with alum inized (or metallized) propellant alsto be aware of losses from the theoretical simpulse predicted by PROPEP-type prograapproximately 12.5% for 1 lb propellant motors, 8% for 1 0 lb propellant weight m otorfor 50 lb propellant weight motors, an d 5% flb or greater propellant weight motors (basedefficiency and qe values from Figure 10 aprevious p aragraph s), when their propellinstalled in an actual motor. Rather than delia theoretical specific impulse for the propeldelivers the a ctual delivered specific impulse motor.Normally in propellant characterization teexperimentallamateur rocketeer characterizes(propellant burning surface area divided byarea) versus chamber pressure, burn rate


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I I I I solution (no condensed phaseI Y = 1 . 2 ------.)I I I presenrIR*/R. = 1.0 ,1 2 4 6 10 20 40 60 100 200

Area ratio, 6Figure I 1- ivergence Efficiency for Conical Nozzles.

chamber pressure, and erosive burning characteris-tics of their propellant. The present author recom-mends that as an additional "characterization" oftheir propellant, that experimentallamateur rocke-teers track delivered specific impulse as a percent-age (0-100%)of the theoretical specific impulse, andc* efficiency factor (qe ), both as a function of m otorsize. This will allow the experimental/amateur rock-eteer to make more accurate predictions for motorperformance as motor size is increased from 1 lbpropellant weight to 5 0 lb propellant weight, to evenup to 350-f- lb propellant weight, with a continuingincrease in delivered specific impulse and c* effi-ciency factor as the motor size is increased.Comparison of the Conical NozzleDivergence Correction Factor Equationwith the Method of Characteristics

Figure 11 from NASA SP-8039 (Ref. 7, original ref-erence Ref. 15) shows a comparison of the nozzledivergence correction factor (A) (labeled as diver-gence efficiency in Figure 11) predicted using theconical nozzle divergence correction factor equation(Eq. (13)) for a series of conical nozzles with d iffer-ent divergence half angles, compared with predic-tions using the Method of Characteristics, the pri-mary nozzle design method used in the profession-al liquid rocket engine and solid rocket motor indus-tries. As can be seen from Figure 11, despite thesimplified form of Eq. (13) it compares quite wellwith the much more sophisticated Method ofCharacteristics method.

Applicability to Hybrid Rocket Motors Liquid Rocket EnginesThe previous sections on determining the dered specific impulse from the theoretical speimpulse obtained from codes such as the JANTherm ochem ical Equilibrium Code, the NASA LCode, the Air Force Chemical Equilibrium SpeImp ulse Code (the USAF ISP Code), and the PROprogram, presented techniques specifically tailto solid rocket motors. The present autho r propthat with similar thermal equilibrium losses, bustion inefficiency losses, heat los ses to the m

hardware, and for metallized fuel two-phase losses, that hybrid rocket motors can be modusing similar techniques, i.e., the Standard Medelivered specific impulse equation, Eq. (31).difference is the combustion efficiency loincluded in the c* efficiency factor (ye) will alsofunction of the m ixing efficiency of the hyb rid lioxidizer and the solid fuel. A hybrid rocket mwith an oxidizer injector with a high mixing efficy will have a higher c* efficiency factor, indepent of motor size.The delivered specific impulse material presein previous sections is not applicable to liquid ret engines. Different JANNAF techniques are for the specific impulse of liquid rocket engines

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CF Efficiency Factor for Straight-CutThroats and Rounded Throats - HistoricalThiokol Solid Rocket Motor Data

CF efficiency factor (q,) da ta for historical Thiokolsolid rocket motors from Ref. 16 is presented inFigure 12. The historical Thiokol solid rocket motordata from Ref. 16 was of interest, because the dataprovided for each motor included the actual thrustcoefficient (CF ,,,, ) based on actual static test data,and the predicted ideal thrust coefficient (eFcal-culated using Eq . (9).The nozzle geometry data pro-vided for each motor also allowed the nozzle diver-gence correction factor (h) to be determined. Thusfrom the T hiokol da ta from Ref. 16 , the CF efficien-cy factor can be determined from Eq . (28) ,Eq . (28): CF ,act = h q~L d ~resulting in Eq . (36).

The actual thrust coefficient, ideal thrust coeffi-cient, diverg ence correction facto r an d the CF effi-ciency factor for selected Thiokol solid rocket moto rsfrom Ref. 16 is presented in Figure 12 . Note that allof the Thiokol solid rocket motors included in Ref.16 were reviewed; the da ta in Figure 1 2 is for all of

the motors with conical nozzles and straightrounded thro ats where sufficient details of thzle geometry was presented. A detailed cudrawing for each motor showing d etails of thzle geometry is also included in Figure 1motors with straight-cut throats the th roat stcut section Leng th-to-Diameter (LID) ratio isAdditionally, since there a re thr ust coefficiees from two-phase flow effects which wodependent on w hether the propellant was metor not, and larger motors have lower thrust cient losses from heat losses into the nozzleture and boundary layer losses in the nozzmotor length, propellant weight, and the percmetallization of the propellant are also preseFigure 12.

Experimental Measurement of ThrusCoefficient, Chamber Pressure, andEfficiency Factor

In addition to the historical Thiokol solid motor data from Ref. 16, experimental datstatic tests of high power solid rocket motoanalyzed to provide additional data on me(actu al) thru st co efficients and CF efficiency fThe high power solid rocket motor experidata w as also intended to provide a compartypical th ru st coefficien ts and CF efficiency

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October 2004


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Histo rical Thiokol Solid Rocket Motor Data (Ref. 16)Actual Thrust Coefficient (CF,,,.)

ldeal Thrust Coefficient (OF)Nozzle Divergence Correction Fac tor (A)

CF Efficiency Factor (qF)Application of the CF efficiency factor (qF)and the divergence correction factor (A)based on the Standard Method:

All dimensions in in

TA-M-I 156-inch Booster

Length = 1206.1 in Conical D ivergent Section Nozzle Performance (Sea Leve l):Prop Wt = 798,074 Ib Half Angle = 17.5 deg Theoretical CF : CF = 1.49 (Note 1)p, (avg) = 668 psia (Note 5) Expansion Ratio = 7.12 Actual (Measured) CF : CF = 1.46 (NCompo site APIAL Rounded Throat Divergence C orrection: h = 0.97686 (N16% AL Nozzle CF Correction: q~ = 1.003 (Notey (chamber) = NAy (nozzle exit) = I . 8

C.G. - LOADED 98.7 -4C.G.-EMPTY 138.0

TX-354-3 Castor IILength = 247.01 inProp Wt = 8,220 Ibp, (avg) = 640 psiaPBAAIAPIAL20% ALy (chamber) = NAy (nozzle exit) = 1.16Figure 12 - Historical

Conical Divergent SectionHalf Angle = 22.62 degExpansion Ratio = 21 -22Rounded Throat

Thiokol Solid Rocket

Nozzle Performance (Vacuum):Theoretical CF : COF = 1.86Actual (Measured) CF : CF =Divergence C orrection: h = 0.96Nozzle CF Correction: q~ = 0.99

Motor Data (Ref. 16); Actual ThCoefficient (CF,,,,), ldeal Thrus t Coefficient (OF), Nozzle Divergence Correction Fa(A), and CF Efficiency Factor (qF).44 High Power Roc


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TX-I 2 SergeantLength = 195.3 in Conical D ivergent Section Nozzle Performance (Sea Leve l):Prop Wt = 5,925 Ib Half Angle = 15.0 deg Theoretical CF : CF = 1.46p, (avg) = 527 psia Expansion Ratio = 5.37 Actua l (Measure d) CF: CF = 1.3763% AP, binder and Rounded Throat Divergence C orrection: h = 0.98296fuel not specified Nozzle CF Correction: qF= 0.955y (chamber) = 1.26y (nozzle exit) = NA

Length = 186.95 in Conical Divergent Section Nozzle Performance (Sea Level):Prop'Wt = 1,200 Ib Half Angle = 15.0 deg Theoretical CF: CF = 1.58p, (avg) = 1,150 psia Expansion Ratio = 6.04 Actual (Measured) CF : CF = 1.47PolysulfideIAPlAL Rounded Throat Divergence Correction: h = 0.982962% AL Nozzle CF Correction: q~ = 0.947y (chamber) = NAy (nozzle exit) = 1.22

C ~ G . 08.1 (LOADED)rnC.G. 147.8 (EMPTY)\

TX-33-39 ScoutLength = 233.1 0 in Conical Divergent Section Nozzle Performance (Sea Lev el):Prop Wt = 7,313 Ib Half Angle = 15.0 deg Theoretical CF : COF = 1.49p, (avg) = 512 psia Expansion Ratio = 5.87 Actual (Measured) CF: CF , = 1.38PBAAIAPIAL Rounded Throat Divergence Correction: h = 0.9829614% AL Nozzle CF Correction: qF= 0.942y (chamber) = NAy (nozzle exit) = 1.16

October 2004Figure 12 (continued)


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TX-M-330-0 ArbalistLength = 37.41 in Conical Divergent Section Nozzle Performance (Sea Le vel):Prop Wt = 7.518 Ib Half Angle = 15.0 deg Theoretical CF : COF= 1.61p, (avg) = 1,500 psia Expansion Ratio = 5.24 Actual (Measured) CF CF ac t= 1 4 2MAPO IAPIAL Rounded Throat Divergence Correction: h = 0.9829616% AL Nozzle CF Correction: qF= 0.897y (chamber) = NAy (nozzle exit) = 1.14

Length = 77.537 inProp Wt = 84.70 Ibpc (avg) = 2,014 psiaCompositeIAPlAL10% ALy (chamber) = 1 I64y (nozzle exit) = 1.2

k

TE-M-483 Zap MotorConical Divergent Section Nozzle Performance (Sea Level):Half Angle = 12.0 deg Theoretical CF COF = 1.56Expansion Ratio = 3.51 Actual (Measured) CF : CF ,act= 1.46Straight-Cut Throat Divergence C orrection: h = 0.98907Throat LID = 0.037 Nozzle CF Correction: qF= 0.946Rounded Entrance

TE-M-I46Length = 62.80 in Conical Divergent SectionProp Wt = 52.12 Ib Half Angle = 15.0 degp, (avg) = 1,775 psia Expansion Ratio = 5.89PolysulfideIAPlAL Straight-Cut Throat2% AL Throat LID = 0.118y (chamber) = 1.178 Rounded Entrancey (nozzle exit) = 1.185

CherokeeNozzle Performance (Sea Leve l):

Theoretical CF : CF= 1.63Actual (Measured) C F : CF ,act= 1.58Divergence Co rrection: h = 0.98296Nozzle CF Correction: qF= 0.986

Figure 12 (continued)46 High Power Ro


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TE-M-344 Cheyenne

Length = 13.021 in Conical Divergent Section Nozzle Performance (Vacuum):Prop Wt = 4.56 Ib Half Angle = 15.0 deg Theoretical CF: CF = 1.84pc (avg) = 1,230 psia Expansion Ratio = 18.0 Actual (Measured) CF : CF ,act= 1.69Composite APIAL Straight-Cut Throat Divergence Correction: h = 0.9829616% AL Throat LID = 0.267 Nozzle CF Correction: qF= 0.934y (chamber) = 1.14 Rounded Entrancey (nozzle exit) = 1.18

TU-223 Mace BoosterLength = 128.80 in Conical Divergent Section Nozzle Performance (Sea Leve l):Prop Wt = 1,365 Ib Half Angle = 15.0 deg Theoretical CF : CF = 1.54pc (avg) = 725 psia (Note 6) Expansion Ratio = 5.2 Actual (Measured) CF : CF ,act= 1.48PolysulfideIAP Straight-Cut Throat Divergence Correction: h = 0.982960% AL Throat LID = 0.303 Nozzle CF Correction: qF= 0.978y (chamber) = NA Rounded Entrancey (nozzle exit) = 1.23

TE-M-82-4 CajunLength = 107.98 inProp Wt = 119 Ibpc (avg) = 1,077 psiaPolysulfidelAP0% ALy (chamber) = 1.25y (nozzle exit) = NA

October 2004

Conical Divergent Section Nozzle Performance (100 K ft):Half Angle = 15.0 deg Theoretical CF : CF = I67Expansion Ratio = 6.3 Actual (Measured) CF : CF ,act = 1.60Straight-Cut Throat Divergence C orrection: h = 0.98296Throat LID = 0.423 Nozzle CF Correction: q~ = 0.975Rounded Entrance

Figure 12 (continued)


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- 107.9104 53.11.4 LOADEDRED 1 I

Length = 107.910 inProp Wt = 132.9 Ibp, (avg) = 685 psiaUrethaneIAPlAL20% ALy (chamber) = 1.16y (nozzle exit) = NA

TE-M-307-3ApacheConical D ivergent Section Nozzle Performance (70K ft):

Half Angle = 15.0 deg Theoretical CF : COF = 1.69Expansion Ratio = 6.32 Actual (Meas ured) CF : CF ,act= 1.60Straight-Cut Throat Divergence C orrection: h = 0.98296Throat LID = 0.431 Nozzle CF Correction: qF= 0.963Rounded Entrance

TE-M-29-1RecruitLength = 105.28 in Conical D ivergent Section Nozzle Performance (Sea Level):Prop Wt = 267 Ib Half Angle = 15.0 deg Theoretical CF : COF = 1.60p, (avg) = 1,710 psia Expansion Ratio = 7.06 Actual (M easured) CF: CF ,act= 1.42PolysulfidelAP Straight-Cut Throat Divergence Correction: h = 0.982960% AL Throat LID = 0.44 Nozzle CF Correction: qF= 0.903y (chamber) = 1.19 Rounded Entrancey (nozzle exit) = NA

TE-M-473 SandhawkLength = 201.0 in Conical Divergent Section Nozzle Performance (Sea Level):Prop Wt = 1,106 Ib Half Angle = 16.5 deg Theoretical CF : COF = 1.49p, (avg) = 1 168 psia Expansion Ratio = 10.9 Actual (Meas ured) CF : CF ,act= 1.46Composite APIAL (Note 7) Straight-Cut Throat Divergence C orrection: h = 0.9794118% AL Throat LID = 0.50 Nozzle CF Correction: qF= 1.001y (chamber) = 1 I46 Rounded Entrancey (nozzle exit) = NA

Figure 12 (continued)High Power Ro


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CF efficiency factor (qF)calculated based on Eq. (36):C~ actrl F =-CFBurn time average chamber p ressure. Based on Thiokol burn time definition; starts when chamberpressure has risen to 10%of maxim um value, ends when chamber pressure h as dropped to 75% ofmaximum value.

Average chamber pressure for Mace Booster based on action time. Thiokol action time definition; swhen chamber pressure has risen to 10%of maximum value, en ds when chamber pressure has droto 10%of maximum value .Present author believes binder probably HTPB.NA = Not AvailableAL = aluminumAP = ammonium perchloratePBAA = polybutadiene-acrylic acid polymerHTPB = hydroxyl-terminated polybutadieneMAP0 = tris-[I- (2-methyl) aziridinyl] phosphine oxide

for high power and experimentallamateur solidrocket motor nozzles relative to the Thiokol data forprofessional solid rocket motors. A review of thisexperimental data is also useful for reviewing themeasurement techniques and analysis techniquesfor the accurate measurement of thrust coefficient,chamber pressure, and C, efficiency factor.From Eq. (1) it's clear that to make a n experimen-tal measurement of thru st coefficient, the thru stneeds to be measured (using a load cell, the basicrequirement for an y instrumented static test), and ameasurement needs to be made of the chamber pres-sure.

FEq. (1): CF = ----Pc Ath

Figure 1 3 shows the typical location of the sure tap used to measure chamber pressure solid rocket motor, the specific exam ple shown ba high power rocket motor. Figures 14 and 15 schamber pressure instrumentation mounted iaerospike solid rocket motor (from Ref. 17inflight measurement of chamber pressure. NoFigures 14 and 1 5 tha t the chamber pressure inmentation is offset from the center of the motoreads pressure from a plenum area at the headChamber PressureMeasurementPressure Tap

poa= Core Aft-End Stagnation (Total) Pressure

(pd,, = Nozzle Stagnation (Total) Pressure pOh= Head-End Chamber Pressure"Measured" Chamber Pressure(pd,, Assumed to be Equal to poa

ion for Pressure Tap for Measurement of Chamber PresHead-End Chamber Pressure (pOh),Core Aft-End Stagn

Nozzle Stagnation (Total) Pressure ((50 High Power Roc


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ulkhead with C

Figure 15 - DART Aerospike Solid Rocke t Motor wi th Chamber nstrumentation and Head-End Ignitor Ready for Installation into DARTRocket.October 2004


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of the motor) in order to leave room for the head-en dignitor shown installed in Figures 14 and 15.Experimental measurem ent of nozzle thrust coef-ficient at first appears to be straightforward. UsingEq. (1) the load cell meas urem ent for thm st is divid-ed by the p roduct of the m easured cham ber pressureand the throat area; the result is an experimentallymeasured thrus t coefficient. The problem is that th epressure tap shown in Figure 1 3 meas ures the head-end stagnation (total) pressure in the motor. Thehead-end stagnation pressure is not the chamberpressure, it needs to be corrected to become chamberpressure.Just what precisely is chamber pressure? FromNASA SP- 125 (Ref. 18 , portions repr inted in Ref. 1 9) ,the actual chamber pressure is the nozzle stagna-tion (total) pressure, the total (stagnation) pressurejust prior to entering the convergent section of thenozzle (shown in Figure 13).

Where:pc ,act = actual chamber pressure, Pa (Ib/in2)PC),, = nozzle stagnation (total) pressure, Pa(lbJin2)The assumption can be made that the s tagnation(total) pressure at the end of the core (pea , shownin Figure 13) is equal to the nozzle stagnation pres-sure.

Where:

po a = stagnation (total) pressure at aft end ofcore, Pa (lbJin2)The measured chamber pressure is the stagnation(total) pressure at the head end of the core @oh,shown in Figure 13).

Where:Poh = stagnation (total) pressure athea d en d of core, Pa (lbIin2)pc ,measured = measured chamber pressure,Pa (lbJin2)Thus the measured chamber pressure (poh) eedsto be corrected to determine the core aft-end s tagn a-tion pressure (pea), which is the actual chamberpressure of the motor.

There is a loss in stagnation (total) pressure dthe core of the motor, which is the correction tocore head-end stagnation pressure required to dmine the core aft-end stagnation pressure. (Of iest beyond chamber pressure measurement cotions, this stag nation pressure loss down the coone of the performance losses from erosive bing.) Spa ce Propulsion Analysis a nd Design 10) presents a method for determining the losstagnation (total) pressure down the core. Firscore Mach number as a function of the core csectional area divided by the throat area is dmined from Eq. (40).

Where:Ap = port area (core cross-sectional aream2 (in2)Ma = core Mach number a t aft end of codimensionlessEq. (40) gives the core Mach number at th e afof the core. The core Mach number is of courseat the hea d end of the core (zero velocity). TheMach number at the aft end of the core is the important core Mach number, as it is the higMach number in the core and hence the locatiothe h ighest velocity-based erosive burning. TheMach number at the aft end of the core also dmines the stagnation (total) pressure loss downcore. The term "core Mach number a t the a ft enthe core" is often truncated to the commonly term "core Mach number" when describing a m

design.Using Eq. (40) if the desired core Mach numbknown, and the port to throat area ratio (APneeds to be determined, the core Mach numbesimply be entered into Eq. (40) to determine theto throat area ratio. The more common situatithat the port to throat are a ratio is known, andthe core Mach number t ha t needs to be determSimple iterative methods can be used to solve(40) for the core Mach number given the pothroat area ratio, with the simplest method bsimply iterating the core Mach number fromusing an increment of 0.01 until the port to tarea ratio is matched.Once the core Mach number for a given pothro at are a ratio is determined, Eq. (41) can beto determine the ratio of the stagnation (total) sure at the aft end of the core Po,) divided bstagnation (total) pressure at the head end o

@oh).High Power Roc


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Thus the measured chamber pressure from the pres-sure tap in Figure 13,poh , is converted to the coreaft-end stagnation pressure pea) using Eq. (4 I) ,since the core aft-end stagnation pressure isassumed to be equal to the nozzle stagnation pres-sure (the actual chamber pressure), the core aft-endstagnation pressure is used as the chamber pressurein Eq. (1) to get the measured (actual) thrust coeffi-cient.

Note that since Eqs. (40) and (41) are for the flowdown the core inside the motor, if several value s forthe ratio of specific heats are available, the chambervalue should be used, versus using the throat ornozzle exit values.Note tha t the d erivation of Eqs. (40) and (41) fromRef. 10 assumes a c onstan t port area, i.e. a con stantcross-sectional area of the core. A more complex

method for tapered cores is presented in S6.5.2 of Ref. 10. All cores that have erosive buat ignition will have a tapered core later in thebut the present author recommends that Eqsan d (4 1) be used for all solid rocket motochamber pressure corrections, with the caveathe equations will be exact for constant crostional area cores and non-erosive motorsapproximate for tapered cores and erosive moEqs. (40) and (41) are somewhat comalthough the equations can be built into the mated data reduction programs which are used for recording, plotting, and manipulatingfrom static tests. Additionally, often times ofew hand calculations are required at select palong the thrust and chamber pressure time ries to determine the actual thrust coefficientfew points along the thrust curve. Eqs. (40) ancan be used for hand calculations, but to simthe process the present author has created F16. Figure 16 is a plot of poa / pOh as a functA, /Ath based on Eqs. (40) and (41) assum ing aof specific heats (y) equal to 1.2 (a good repretive value for solid rocket motors). Experimamateur rocketeers can use Figure 16 for

0 1 2 3 4 5 6 7 8AP-fh

Figure 16 - o, /pOh s a function of A, /Ath, = 1.2.October 2004


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ure I 7- pc)n,I(pc)injs a function of Ao/Ath, = I 2 and I 4.determination of poa/poh for a given Ap /Ath to cor-rect measured chamber pressure to the actua l cham-ber pressure for chamber pressure a nd thr ust coeffi-cient measurements.Note from Figure 16 th at it's pro bably a good ideafor motors designed specifically for thrust coeffi-cient measurement tests to have an initial port-to-throa t are a ratio (Ap /Ath) of at le ast 3 , and perhapseven as high a s 5 , to minimize the poa/poh correc-tion to the measured chamber pressure to obtainhigher quality chamber pressure and thrust coeffi-cient data , and to allow for rapid reduction of cham-ber pressure/thrust coefficient data using Figure 16with hand calculations.Finally, to ob tain CF efficiency factor (qF) da tafrom static test data, for either the entire thrustcurve or selected points along the thrust curve,using the Standard Method the measured actualthr ust coefficient (CF ,act) based on the measu redthrust and the corrected chamber pressure, is divid-ed by the n ozzle diverge nce correction factor (A) andthe calculated ideal thrust coefficient (PF to deter-mine the experim entally measu red CF efficiency fac-tor (1?,).Using the Standard M ethod (Eq. (28)):

Results in Eq. (43):

Note that since the actual thrust coefficient(CF,act) s based on the corrected chamber pressure(p the ideal thrust coefficient (PF)sed in Eq.(43) must also be based on the corrected chamberpressure. The ideal thrust coefficient is calculatedusing atmospheric pressure, so it is important tomeasure the atmospheric pressure (or obtain atmos-pheric pressure data from a nearby airfield orweather station) when performing static tests for

thrust coefficient measurements.It's important to note that the chamber prescorrection method described above, for correthe measured chamber pressure to the actual chber pressure, is for chamber pressure measure mfor solid rocket motors. Based on the derivatioRef. 10 for the poa/Poh correction, using a covolume analysis based on mass addition downmotor core, the present author proposes that (37)-(43) can a lso be used for hybrid rocket moFor liquid rocket engines a different methodcorrecting the measured chamber pressure toactual chamber pressure is used. The actual chber pressure is the nozzle stagnation (total) psure, (p) , the total (stagnation) pressure prior to entering the convergent section of the zle. For a pressure tap drilled through the injface of a liquid rocket engine the measured chapressure is the injector stagnation (total) presFigure 17, from NASA SP-125 (Ref. 18,tions reprinted in Ref. 19),presents the correctithe injector stagnation (total) pressure to obtainnozzle stagnation (total) pressure, (pc)ns/(pc)infunction of the ratio of the combustion chacross-sectiona l are a (A,) to the thr oat ar ea (Attwo ratios of specific heats.Where:

A, = combustion cham ber cross-sectionarea, m2 (in2)@Ainj = injector stagnation (total) pressurePa (lb/in2)Note th at the dat a plotted in Figure 17 is for c

drical combustion chambers, the combustion cber sha pe used on almost all liquid rocket engi

(This article will be continued in the next iVolume 35 Number 8, November 2004, and include a glossary an d references.)High Power Roc


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(Editor's note: Continued from the Volume 35,Number 7, October 2004 issue of High PowerRocketry .)

To verify the trends from the Thiokol solid rocketmotor d ata from Ref. 16 presented in Figure 12, andto provide additional data from tests of high powersolid rocket motors (versus professional motorsfrom the Thiokol data ), experimental da ta from s tat-ic tests of two high power solid rocket motors wasanalyzed to obtain CF efficiency factor (qF)data forhigh power solid rocket motors with straight-cutthroats. The analysis of the experimental data fromthe two static tests also serve as examples of theapplication of the chamber pressure correction andthe thrust coefficient and CF efficiency factor meas-urement techniques/analysis methods presented inthe previous section (Eqs. (37)-(4 3),Figures 13 and16).

EAC CSXT 75m m KN-484 PropellantCharacterization Test MotorAs part of the motor developm ent program unde r-taken by the Environmental Aerosciences Corpora-tion (EAC) for the Civilian Space exploration Team(CSXT) experimental/amateur rocket, the first non-professionallnon-governmental rocket launch ed intospace which reached an altitude of 379 ,900 feet (72

miles), a series of propellant characterization testswere performed by Derek Deville of EAC, with con-sultation by the present author, to characterize thepropellant for the CSXT solid rocket motor. Erosiveburning characterization tests were also performed,Copyright 02004 by Charles E. Rogers. All rights reserved. Publishedwith permission.

but the tests of interest for determining nothru st coefficient were the propellant K, (propeburning surface area divided by throat area) vechamber pressure tests, due to the higher portthroat area ratio (APIArh)sed on those tests (reing in smaller corrections to the measured champressure). One particular test was chosen fothru st coefficient and CF efficiency factor analythe 75m m diameter motor KN-484 propellant cacterization test. Figure 18 shows the nozzle uon the 75mm KN=484 Propellant Characteriza

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EAC CSXT 75mm KN=484 Propellant Characterization Test Motor NozzleMotor Data: Conical Divergent Section Nozzle Performance (30.09 in-Hg,Length = 22.0 in Half Angle = 15.0 deg approximately Sea Level):Prop Wt = 3.74 Ib Expansion Ratio = 8.35 Divergence C orrection: h = 0.98HTPBIAPIMGIAL (Note I Straight-Cut Throat Nozzle CF Correction: qf= 0.908% MGIAL Throat LID = 0.833 (measured)y = I 2363 Sharp Entrance(non-rounded)Note:1) MG = magnesiumFigure 18- AC CSXT 75mm KN=484 Propellant Characterization Test Motor No

November 2004


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easured" Chamber Pressure

For ~a plete productdescri t rate charts,to:TO P FLIGHT RECOVERY, LLC

S l 2 6 2 l Donald RoadSpring Green, Wl 53588

Test Motor. The nozzle drawing is in two parts, th e main pnolic part of the nozzle, and the graphite throat insert. Aincluded in Figure 18 is data on the nozzle performanincluding the results from the CF efficiency factor analwhich follows.Figure 19 presents the thrust and chamber pressure thistories for the EAC CSXT 75mm KN=484 PropelCharacterization Test Motor. The chamber pressure dshown in Figure 19 was m easured using a pressure tap a thead end of the motor. Six points along the thrus t and chber pressure time histories during the first 2 seconds of burn were analyzed using Eq. (28) (the Standard M ethod) Eqs. (37)-(4 3) to (1) correct the m easured head-end champressure to the actu al chamber pressure, (2) determine actual thrust coefficient, (3) calculate the ideal thrust coecient based on the actual chamber pressure and atmosphpressure, and (4) calculate the measured CF efficiency fa(qF).Table 2 presents the thrust coefficient and CF efficiefactor analysis results for the EAC CSXT 75mm KN=Propellant Characterization Test Motor for the six points althe first 2 seconds of the th rust curve. Note for the results sented in Table 2 that a single ratio of specific heat s w as ufor both the chamber pressure correction calculations andnozzle ideal thr ust coefficient calculations.The analysis for the EAC CSXT 75mm KN=484 PropelCharacterization Test Motor presented in Table 2 is an exple of doing hand calculations at selected points along thrust and chamber pressure time histories to determine rresentative values for the thrust coefficient and CF efficiefactor, without having to do the calculations for the en

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CSXT 75mm KN=484 ropellant CharacterizationTest Motor CF Efficiency Factor Analysis ResultsDth = 0.45 inAth = 0.15904 in2D, = 1.3 inA , = 1.32732 in2Nozzle Expan sion Area R atio; E = 8.35Nozzle Divergence Half An gle; a = 15.0 de gRatio of Specific He ats; y = 1.2363Atmospheric Pressure; p , = 30.09 in-HgStandard MethodCF ,act = ?F @F

Time Thrust Poh ApIAth Ma Poaf..~, Pop CF ,act PF h q~(sec) (lb) (psia) (P W0.17 215.33 978.0 3.84 0.1556 0.9856 963.92 1.4046 1.5456 .98296 .92450.29 212.67 975.33 4.60 0.1293 0.9899 965.48 1.3850 1.5458 .98296 .9110.50 214.0 979.33 5.91 0.1003 0.9939 973.36 1.3824 1.5468 .98296 .go921.0 216.67 999.33 9.05 0.0653 0.9974 996.73 1.3824 1.5498 .98296 .go741.5 217.33 1013.33 12.18 0.0485 0.9986 1011.91 1.3504 1.5516 .98296 .8852.0 216.67 1020.67 15.32 0.0385 0.9991 1019.75 1.3360 1.5525 .98296 .875

Table 2 - EAC CSXT 75mm KN=484CF Efficiency Factor Analysis Results.thrust curve.The C, efficiency factor results from Table 2 forthe six points during the first 2 seconds of thethru st curve are plotted versus time in Figure 20. Ascan be see n from Table 2 and Figure 20, th e C effi-ciency fac tor (q,) for the EAC CSXT 75mmKN=484Propellant Characterization Test Motor Nozzle isapproximately 0.90. A C, efficiency factor of 0.90represents a 10% oss in thru st coefficient, which ispoor performance for a no zzle.

DART Pro iect Conical Nozzle Control MotorThe NASA Dryden Aerospike Rocket Test (DART)aerospike rocket project, where an aerospike nozzlewa s retrofitted on to a high power solid rocket motor(documen ted in Ref. IT), had two solid rocket motor

configurations that were static tested with thrustmeasurements and chamber pressure measure-ments; an aerospike motor, and a conical nozzlemotor know n as th e "control" motor. The purpose ofthe conical nozzle control motor was to be identicalin every way to the aerospike nozzle motor, (samepropellant formulat ion, same propellant grainNovember 2004

Propellant Characterization Test Mo

geometry, same throat area), with the only dience being that a conventional conical nozzle installed in place of the aerospike nozzle to althe performance differences between the aerospike nozzle and the conventional conical nzle to be quantified. The aerospike nozzle motor the con ical nozzle control motor where both builCesaroni Technology Incorporated (CTI), and wbased on the CTI 05100 reloadable high power ret motor. Due to concerns about two-phase feffects in the aerospike nozzle, the propellant uin the aerospike nozzle motor an d the con ical nocontrol motor had a reduced aluminum contentaluminum).Figure 21 shows the nozzle used on the DAProject Conical Nozzle Control Motor, and incluadditional data on the motor, and the results fth e CF efficiency factor analys is wh ich follows.

In the thrust coefficient and CF efficiency faanalysis for the DART Project Conical Nozzle CoMotor an automated approach was undertawhere Dr. Trong Bui from NASA-Dryden integrEq. (28) (the Standard Method) and Eqs. (37)into a n auto mated MATLAB langu age program ufor data reduction from m otor static tests to (1)


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haracterization Test Motor

1 2

TIME (sec)Figure 20 - CF Efficiency Factor (qF)versus Time for First 2 Seconds of Thrust Cfor EAC CSXT 75mm KN=484 Propellant Characterization Test Motor Nozzle.

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rect the measured head-end chamber pressure toactual chamber pressure, (2) determine the actual thcoefficient, (3) calculate th e ideal thru st coefficient bon the actual chamber pressure and atmospheric psure, and (4) calculate the measured C' efficiency faNote for the analysis results which will be presenteth e DART Project Conical

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