doug talley and ed coy- the constant volume limit of pulsed propulsion for a constant y ideal gas

8/3/2019 Doug Talley and Ed Coy- The Constant Volume Limit of Pulsed Propulsion for a Constant y Ideal Gas

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The Constant Volume Limit of Pulsed Propulsion for a Constant Ideal GasPost rint

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Doug Talley and Ed Coy5d. PROJECT NUMBER

3058RF9A

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MAA- AOO-36467

AIAA 2000-3216The Constant Volume Limit of PulsedPropulsion for a Constant y Ideal GasDoug Talley and Ed CoyAir Force Research LaboratoryAFRL/PRSA10 E. Saturn Blvd.Edwards AFB,CA 93524-7660


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AIAA-2000-3216The Constant Volume Limit of Pulsed P ropulsion for a Constant 7 Ideal Gas

Doug TalleyAir Force Research Laboratory, AFRL/PRSA10 E. Saturn Blvd .Edwards AFB, CA 93524-7660AbstractThe constant volume (CV) l imit of pulsed propulsion was explored theoretically, where the combustionchamber was approximated as being t ime-varying bu t spatially uniform, while the nozzle flow was approximated as

being one dim ensional but quasi-steady. The CV limit was explored for the isentropic blow down of a constant yideal gas for fixed expansion ratios and for variable expansion ratios which were adjusted to match the pressure ratioat al l times. The CV limit calculations were notable in that all the fixed expansion ratio results could be expressed asanalytical solutions. The CV l imit was then compared with tw o relevant constant pressure (CP) cycles. A mo ng theseveral conclusions were that using a fixed expansion ratio nozzle during a CV b low down did not exert more than a3% performance penalty over using a variable expansion ratio nozzle as long as the fixed expansion ratio nozzle wasoptimized for the blow down. Another major conclusion was that, while the impulse produced by a CV device couldsignificantly exceed that of a CP device op erating at the fill pressure of the CV device at elevated ambient pressures(e .g . , I atm), the impulse produced by a CP device could actually exceed, albeit only slightly, the impulse producedby a CV device when the ambient pressure was near zero, such as would occur in the near vacuum conditions inspace.

NomenclatureA - areac - speed of soundC F - thrust coefficientcp - specific heat at constant pressureF - thrust

T - t/cQA*V ; dimensionless timeSubscriptscp - pertaining to a constant pressure cyclee - nozzle exit/ - fill conditionIc - pertaining to l imit cycle operation


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l imit are explored here for the isentropic blow down ofa constant y ideal gas at fixed expansion ratios. Theresults ar e compared with the case where the expansionratio is continuously varied to match the pressure ratioat al l times during the blow down.

2 . Formulat ion2 .1 GeneralThe impulse produced by the unsteady blow downof the combustion gases in the CV l imi t is the integralof the thrust F = mv e + (P e - P^)A e over time, wherem is the mass flow rate, P ^ is the ambient pressure,and ve , P e , an d A e are the velocity, pressure, and area,respectively, at the exit plane. U nder the transformationdt - (dt I dp)dp = - (V I m)dp and the furthertransformation r = p I p 0 , where V is the combustionchamber volume an d p0 is the initial chamber densityafter combustion bu t before blow down at time t = 0 ,the total impulse / and blow down t ime t may beexpressed as

( 2 )2.2 F ixed NozzlesF or isentropic flow, the instantaneous exit velocityis ve =J2cp (T - Te ) , where T and Te are the

Equation (4) has two solutions fo r re corresponding tosubsonic an d supersonic flow at the exit plane. Thesesolutions will depend only on e and y . F or supersonicflow, the density ratio re will therefore remain constantduring blow down as long as compression waves do notenter the nozzle, and eq. (3) will be analytically inte-grable when inserted into eq. (1). Compres-sion/expansion waves will remain external to the nozzlefor al l r such that 4f

>Jwhere 00 = P 01 P x an d < j) e=P e / P m are the ratios of theinitial and exit pressures, respectively, to the ambientpressure, and where $ ' e is the critical value of < t > e belowwhich a compression wave enters the nozzle. The criti-cal value of 0e is given by ( see eqs. 2.48a and 5.2 ofref. 2 )

(5)

+ (Pe-P^Ae/m)dr, (1) wherey + 1

M=-7-1 - -1(6)

(7)While eq. (5) holds, the flow will also be choked,and the throat conditions will be functions only of y .

For a throat density and velocity given by ( see eq . 2.35a1/(r 1)


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wherea(r)=l-r ( r+1) /2 b(r) =(1/r) '2-1 . (12)

The quantity t/ T will be related to the average thrustW = I / t . Noting that C Q = y#r0 an d / , = P 0RT0 , thencombining constants after dividing leads to

y i /T = F/ P0A*scF, (13)where C F is the average thrust coefficient for the blowdown.2 .3 V ariable NozzlesFixed expansion ratio nozzles will be optimized forat most only a single value of r during the blow down;losses will occur at all other values. To assess what theperformance would be if these losses were not present,a variable nozzle can be envisioned where the expan-sion ratio is continuously adjusted to match the pressureratio at all times. The variable nozzle limit produces themaximu m possible impulse, and is therefore also usefulas a reference case.

W ith the exit pressure always matched to the ambi-

slant throat area are still given by eqs. (8) and (11), re -spectively. If the throat area is varied, then the massflow an d blow down time for a fixed exit areaar em = peAeve =p0c0Ae 7-1

(15)1/2 s(y-l)/r

where Te = tc0Ae/V is defined this time in terms of theexit area. Eq. (16) also cannot be integrated analyticallybut may easily be integrated numerically. Inasmuch asshocks an d expansion waves will no t occur when theexpansion ratio is always matched to the instantaneouspressure ratio, eqs. (14)-(16) above for the variablenozzle case will apply until the expansion ratio reducesto unity and the flow becomes unchoked, i . e . , for all rsuch that s i / ( r - i ) / \ i / rr>\-^\ hr (17)I 2 J U o J2 .4 Limit Cycle OperationIn repetitive operation, intake valves would beopened at some point during the blow down in order tointroduce fresh propellants for the next cycle. Any re-maining combustion products not yet expelled would bepushed out, or purged, by the incoming fresh propel-lants. If the purge of the remaining combustion prod-


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y + l f j _r

C r - 0 / 2-1

where the subscript "/c" stands for "limit cycle." F orl imit cycle operation with variable nozzles, the totalcycle impulse is given instead by the sum of the con-stant pressure impulse with th e impulse calculated byeq . (14) , and the total cycle blow down t ime is the sumof the constant pressure t ime plus the blow down timecalculated by eq. (11) or eq. (16) . Th e expansion ratioalways matches the pressure ratio in the variable nozzlecase, so the exit density ratio must be set to

The density ratio r cannot be specified arbitrarily iflimit cycle operation is to be achieved. Limit cycle op-eration requires the blow down to proceed at least untilthe chamber pressure reaches the f i l l pressure at whichfresh propellants are introduced. This givesr < ( P f / P 0 ) l / Y , where P f is the fill pressure. The ine-quality sign indicates that th e blow down ca n proceedto a pressure lower than the fill pressure if the freshreactants undergo compression during the fill. Giventhat p0 = P f for CV heat addition, the ratio P f IP 0could be on the order of 1000 times smaller for propel-lants in an initial liquid state compared to an initialgaseous state.For a given density ratio r, the blow down of theremaining combustion products in a CP mode producesa greater impulse than if the blow down had been per-

wherecp

( 2 2 )

( 23 )and where icp = I cp I p 0c0V , Tcp = tcpc0A IV ,c p . c p = FI P o A * > and the values of p0, c0 , P 0, and Vare those of the constant volume process. Th e quantityTcp in this context can be interpreted to be the dimen-sionless time required by the constant pressure processto expel the same mass as the initial mass of the con-stant volume process.

3. Results and Discussion3.1 Effect of ThermochemistryThe dimensionless impulse and blow down time,eqs. (10)-(12), depend explicitly on r and implicitlyonly on y, r e (or e) , and 00. Of these, only y de-pends on the thermochemistry, bu t this dependency isweak. The major influence of thermochemistry comesthrough the initial speed of sound c0 used to normalizethe impulse. Thus the specific impulse Is = lc0 ismaximized when c0 is maximized. This in turn impliesthat the optimum thermochemistry is that which maxi-mizes the initial combustion temperature T0 and mini-mizes the molecular weight. The same general guidanceis known to also apply to constant pressure devices.Thus constant volume devices should opt imize at ap-


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expansion rat io producing the maximum thrust for agiven isobar.Th e blow down of a fixed nozzle CV device pro-ceeds along a vertical path of constant between tw oisobars. The thrust produced will be some average be-tween the two isobars. Picking any two isobars in F ig.1, say between 00 = 50 and 00 = 20, and following thevertical l ine between them for various e , it can be en-visioned that the average thrust will reach a maximumneither at large e nor at e = 1, but at some optimum ein between. However, the average thrust cannot be cal-culated directly from Fig. 1, because the thrust coeffi-cient is proportional to the thrust divided by the cham-be r pressure, not the thrust alone. A blow down from0o = 50 to 00 = 20 is replotted in F ig. 2 where thecurve for 00 = 20 is normalized by the same pressureas for 0o - 50, making all curves proportional to thethrust. This is accomplished by mult iplying the thrustcoefficient of Fig 1. by 0.4 for 00 = 20. The curve la-beled "b.d." gives the average thrust coefficient for theblow down and is seen to be an average of the curvesfor 00 = 50 and 00 = 2 0, as expected. The variablenozzle case shown in Fig. 2 will be discussed later.Average thrust coefficients ar e plotted as functionsof 00 and for blow downs to r = 0.75 , 0.5, an d 0.25in Figs. 3, 4, and 5, respectively. Dimensionless blowdown times are plotted as a function of r for two valuesof y in Fig . 6. As can be seen from eq . (11) and Fig. 6,

tional as they were in earlier figures, where r was fixed.The optimum thrust coefficient is plotted in Fig. 9.3.3 V ariable Nozzles

Because the expansion ratio is always optimizedfor variable nozzles, the blow down of variable nozzlesproceeds along the curve of maximum thrust in Fig. 1.The dimensionless impulse produced can be computedas a function of 00 and r by integrating eq . (14) , an dcompared with the performance of fixed nozzles oper-ating at the optimum expansion ratio. The results arenot easily viewed in the form of Fig . 8, however, sothey are plotted instead here in Fig. 10 as the impulsepenalty in using optimized fixed nozzles compared tothe variable nozzle case, based on the percentage of thevariable nozzle impulse. The penalty is shown not toexceed about 3% for all cases calculated. The reasonthe penalty is not more severe can be traced to the flat-ness of the curves in Fig . 1 near the points of maximumthrust. This is illustrated more clearly in Fig. 2 , wherethe difference between the variable and optimized fixednozzle blow down paths is shown. As can be seen, themaximum instantaneous thrust coefficient is not thatmuch higher for the variable nozzle case than for opti-mized fixed nozzles, leading to calculated averagethrust coefficients which ar e very close.3.4 Comparison with Constant P ressure DevicesBy normalizing with the same constants as used forthe CV device, the correction factors Q an d < as de-fined in eq. ( 23 ) an d used in eqs. ( 2 0 ) - ( 2 2 ) allow the


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the fill pressure of the CV device, namely < P = P 01 P / .This would allow a comparison of two devices havingequivalent feed systems. Assuming the purge pressureis the same as the feed pressure, limit cycle operationgives r = (P f I P 0 ) l / r . Given further that P0=p0RT0and that p 0 = p / for a CV device, where p/ is thedensity after propellant fill but before combustion, thenPO/P/ = 10 would be representative of propellants inan initial gaseous state, while PQ/P/ =1000 wouldapproach propellants in an initial liquid state. A CPdevice operating at the same pressure as the fill pres-sure of the CV device will be referred to here as a "CP/"device fo r brevity.The second possibility will be to compare the per-formance of combustion chambers experiencing thesame material stresses. A very rough approximation ofthis would be a CP device operating at the same pres-sure as the peak pressure of a CV device, namely0 =1. A CP device operating at the same pressure asthe peak pressure of the CV device will be referred tohere as a "C P P" device, again for brevity.The ratio of the dimensionless impulse producedby a CP/ device to the dimensionless impulse of a CVdevice is presented in Fig. 11, and the ratio of the di-mensionless impulse produced by the CP^, device to thedimensionless impulse of a CV device is presented inF ig . 12 . Because the same quantities are used to nondi-mensionalize the impulse in all cases, these plots also

and the present formulation no longer applies. The lefthand extent of the curves as drawn in the figures is lim-ited solely for reasons of grap hical clarity; the curves inboth figures actually all extend to 1/00 = 0. The limit-in g values of the curves at l / ( j > 0 =0 are given in thefigures. These limiting values are exactly the same forboth the CP f and CP P cases. The reason for this is thateq . (20) becomes independent of the chamber pressurewhen 1/00 =0.

Away from 1/00 = 0, the impulse of a CP/ deviceis shown in Fig. 11 to be inferior to that of the CV de-vice, and the impulse of a CP P device is shown in Fig .12 to be superior to that of the CV device. The relativedifferences diminish as 1/00 approaches zero. At largeenough 1/00, the relative differences ar e significantlylarger than the 3% maximum performance penaltyshown in Fig. 10 attributed to using an optimized fixednozzle instead of a variable nozzle. This implies that, atlarge enough 1/00 , the performance penalty associatedwith the practical simplicity of using an optimized fixednozzle might be tolerable compared with the complica-tions of developing a variable nozzle. To assesswhether a practically attainable device could fall withina regime where such losses might be tolerable, a refer-ence case consisting of the approximate operating pointof a stoichiometric kerosine/air (jet fuel/air) systemoperating at 1 atm ambient pressure and a 3 atm fillpressure is plotted as a black dot in Fig. 11. The oper-ating point is shown to be well within a regime where


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(24)

The maximum value of the expression occurs asP0/Pf-*, which fo r 2=y1'2 an d y =1.2 isi / i ^p =1.004 . As P^IPf becomes smaller, the magni-tude of the impulse ratio also becomes smaller, untileventually it is reduced to a value less than unity. F orP 0IP f = 1 0 , this value becomes ilicp = 0 . 9 9 2 . Limitvalues of i/icp at 1/00 =0 shown in Figs 11 and 12were calculated using eq . ( 2 4 ) .

4 .0 ConclusionsAnalytical solutions of the constant volume limit ofpulsed propulsion for fixed expansion ratios have beenexplored and compared with the case where a variablenozzle is used to match the pressure ratio at all timesduring the blow down. Th e solutions apply for all su-personic flows where compression waves remain exte-rior to the nozzle. It was found that:1. CV devices should optimize at approximately thesame mixture ratio as CP devices, namely the mix-ture ratio which maximizes the initial temperaturean d minimizes the molecular weight.2. The thrust of CV devices is maximized in the samewa y as for CP devices, namely by maximizing theinitial pressure and the throat area.3. The blow down time depends on r = p/p0, only

c. At large enough 1/00, the relative differencebetween a CV device and the two CP devicesconsidered is significantly larger than the im-pulse penalty associated with using an opti-mized fixed expansion ratio instead of a vari-able nozzle on the CV device.d. The regime where the relative difference be -tween a CV device and a CP device is signifi-cantly larger than the impulse penalty associ-ated with using an optimized fixed expansionratio on the CV device appears to be practicallyachievable for at least the one reference caseconsidered, which involved kerosine/air opera-tion at 1 atm am bient pressure.e. Near 1/00 = P ^/ P 0 =0 , which is an importantlimit in that it represents operation at the nearzero ambient pressure of space, the differencebetween the two CP devices considered van-ishes, and the impulse of the CP device can be-come either slightly higher or slightly lowerthan the impulse of the CV device, dependingon the magnitude of P 0 IP f .This work was supported by the Propulsion Directorateo f the Air Force Research Laboratory. Many valuablediscussions with Dr. Ed Coy and Dr. Phil Kesse l con-cerning this work are gratefully acknowledged .

References1. Kailasanath, K., Patnaik, G., and Li, C.,


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Figure 1. Steady state thust coefficientse = AJ A*

Figure 2. Blow d own from 00 = 50 to < /> 0 = 20 , where"b.d." curve is for the blow down. Arrows indicate pathstaken for fixed and variable nozzles.


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2.5 -,

2 -

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1 -

0.5 -

y=1.2 10000.

shock=5

0.2 0.4 0.6 0.8r

Figure 9. Thrust coefficient at the optimum expansion ratio

7=1.2

o .o0.4 0.6 0.8

Figure 10 . Impulse penalty of an optimized fixed nozzleCompared with a variable nozzle

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doug talley and ed coy- the constant volume limit of pulsed propulsion for a constant y ideal gas

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