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AOO-36527

AIAA 2000-3291Optimal Performance of aTripropellant Single-Stage RocketL. Casalino, and D. PastronePolitecnico di TorinoTorino, Italy

36th AIAA/ASME/SAE/ASEEJoint Propulsion Conference and Exhibit

17-19 July 2000Huntsville, Alabama

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AIAA-2000-3291

OPTIMAL PERFORMANE OF A TRIPROPELLANTSINGLE-STAGE ROCKET

Lorenzo Casalino* and Dario Pastrone*Politecnico di Torino, 10129 Torino, Italy

Abstract

The theory of optimal control is used to maxi-mize the performance of a single-stage rocket whichis moved by a tripropellant engine that burns LOX,RP-1 and LH2 in the combustion chamber. For thesake of simplicity and because of the theoretical na-ture of the paper, the rocket motion is analyzed inthe absence of gravitational and aerodynamic forces.The tank mass is assumed to be proportional to thepropellant volume and either the vehicle gross massor the dry mass is minimized for the assigned pay-load and velocity increment. The analysis providesthe optimal values of mixture ratio and hydrogenfraction, which are varied during the operation. Par-ticular characteristics of the optimal solutions arepointed out. An estimation of the benefit that canbe obtained using a tripropellant engine compared toother advanced propulsion systems is also provided.

Nomenclature

ABcDHmrnrtTVa

constant, see Eqs. (14) and (15)constant, see Eq. (16)effective exhaust velocityconstant, see Eq. (21)Hamiltonianmassmass flow-ratehydrogen-engine propellant-fractiontimethrustvelocityoxygen/fuel mixture ratio

* Researcher, Dipartimento di Energetica. Member AIAA.t Associate Professor, Dipartimento di Energetica, Senior

Member AIAA.Copyright ©2000 by the American Institute of Aeronauticsand Astronautics, Inc. All rigths reserved.

€ = structural densityA = adjoint variablep — densityif = performance indexX = hydrogen/overall-fuel ratio

Subscripts

+ = just after the switching point— = just before the switching point1 = switch to tripropellant mode2 = switch to hydrogen modee = engine/ = finali = initialh = hydrogenk = keroseneo = oxygenp = propellantt = tanku = payload

Introduction

The successful operation of a single-stage-to-orbit (SSTO) requires a substantial improvement incurrent technology concerning rocket structures andengines: Any device that could possibly improvesystem performance deserves attention. It is wellknow that a high density specific impulse is prefer-able during the first phase of the ascent trajectory,whereas the highest specific impulse must be ex-ploited in the final phase. This goal can be obtainedby using two different fuels, namely, kerosene (RP-1) and liquid hydrogen (LH2), with liquid oxygen(LOX) as the oxidizer, or varying the mixture-ratioof the same pair of propellants (LOX/LH2 being themost suitable). As far as the use of two fuels is

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concerned, at least two different approaches can beconsidered. In the dual-fuel engine, kerosene andhydrogen are burnt in separate combustion cham-bers, while the tripropellant engine uses a singlemain combustion chamber wherein both fuels areburnt with oxygen. Over the past years a number ofresearchers 1~7 investigated the possible advantagesof these approaches and obtained sometimes oppo-site conclusions6'7: The practical usefulness of de-veloping these innovative propulsion systems has notyet been conclusively asserted.

In previous papers the authors applied the the-ory of optimal control (OCT) to maximize the per-formance of a single-stage rocket either consideringa variable-mixture-ratio hydrogen engine8'9 or thesimplest dual-fuel system, which uses a LOX/RP-1engine in parallel with a LOX/LH2 engine.10 Therocket motion was analyzed in the absence of gravi-tational and aerodynamic forces for the sake of sim-plicity and with the aim of obtaining a theoreticalinsight into the problem. The tank mass was as-sumed to be proportional to the propellant volume,and the analysis provided the optimal strategy tominimize either the gross mass or the dry mass forthe assigned payload and velocity increment.

In the present paper, this simple model is re-tained to investigate the performance of a tripropel-lant rocket that can burn a mix of oxygen, hydro-gen and kerosene in the same combustion chamber.Two controls define the engine operation, i.e., theoxygen to overall fuel mass-flow ratio and the hy-drogen to overall fuel mass-flow ratio. Both controlvariables continously change during operation. Theapplication of OCT and the numerical solution of theresulting boundary value problem provide the opti-mal values of the controls during the rocket accel-eration. Particular features of the optimal solutionsare pointed out.

A comparison of actual propulsion systems is be-yond the scope of the present paper as a wide num-ber of parameters should be taken into account. Arough estimation of the benefit provided by a tripro-pellant engine in comparison to a hydrogen engineand a dual-fuel system is instead given.

The coupling of the present analysis to the opti-mization of the ascent trajectory would be requiredto obtain more realistic results. The authors' ap-proach to trajectory optimization11 is analogous andcan be easily integrated with the optimization of thepropulsion system.

Statement of the problem

The tripropellant engine which is considered inthe present paper has a single main combustionchamber where two different fuels, namely, liquid hy-drogen and kerosene can burn with liquid oxygen asthe oxidizer. Two controls determine the relativequantities of the three propellant mass-flows: themixture ratio a = m0/(mh -i-m^), and the hydrogenfraction x — mft/(mft + m^). By introducing thetotal propellant mass-flow mp = m0 + rhft + m^ oneeasily obtains

X drap

+ a dt

1 -dt 1 + a dt

(1)

(2)

In a vacuum, the equation of motion for a rocket,which is only subject to its thrust, is

dV_dt

T_m

c dm,,m dt (3)

where m — 1 — mp is the instantaneous rocket mass(the initial mass has been arbitrarily assumed as uni-tary, i.e., masses are expressed as fractions of therocket gross mass). The effective exhaust velocity isa function of the relative mass-flows of the propel-lants, i.e., c = c(a,x), as the chamber pressure andnozzle geometry are considered to be assigned.

It is convenient to choose the mass of the ejectedpropellant mp as the new independent variable andreduce Eqs. (l)-(3) to

Xdm

dmk

dmc

dVdm

1 + a

c1 — mv

(4)

(5)

(6)

The control variables of the problem are the mixtureratio and hydrogen fraction, and OCT is used tomaximize the rocket performance.

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The Hamiltonian, which is defined as

H - \hX ~ X)•a 1 + a (7)

is an explicit function of the independent variableand is not constant (the process is not autonomous).The state variables do not appear explicitly in H;therefore, the adjoint variables result to be constant,on the basis of the Euler-Lagrange equations.12

Boundary conditions

The problem is completed by the boundary con-ditions. The initial values (rapj = 0) for Eqs. (4)-(6)are nihi — mki = Vi = 0. The final value of one statevariable is prescribed (V/ = AV) while the otherboundary conditions are provided by the theory ofoptimal control and depend on the performance in-dex ip.

The dry (final) mass of the vehicle and the pro-pellant mass constitute the unitary gross mass. Thedry mass is

• mt + me (8)

The engine mass also includes all other massesthat do not change as the propulsion requirementschange. The tank mass is linearly related to thepropellant volume

e emt = — rnhf + —

Ph Pk Po0t (9)

for a given payload. The former is important to as-sure the feasibility of a mission; the latter is moredirectly related to the costs of the launch system.Both analyses are equivalently carried out as maxi-mization problems of a proper performance index.

Minimum gross mass

The minimization of the gross mass per unitpayload is obtained via the maximization of the pay-load ratio (i.e., the non-dimensional payload). Theperformance index is expressed, by means of Eq. (9),as a function of the independent and state variables

if = mu = I — me — mt —

= 1 - me — Ahmhf - Akmkf -

where the parameters

(13)

ePh

€

Po

5 = 1 +

Pk Po

€Po

(14)

(15)

(16)

only depend on the tank and propellant characteris-tics. By using Eqs. (10)-(12) one obtains

\h = -Ah

\k = -Ak

(17)

(18)

The masses of the exhausted propellants are freeand the necessary conditions for optimality are12

dp

dpdmkf

dp

(10)

dm•pf(12)

H t — B

Minimum dry mass

(19)

In this case, the payioad/dry-mass ratio is max-imized; the performance index is written as

mu Ahmh} +Akmkf + Dif = —— = ti — - ——————————

1 — mpf

where

Two analyses are carried out in this paper, whichminimize either the system gross mass or dry mass D = — + me

Po(21)

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Equations (10)-(12) provide

Ah

Ak

\ — m.•pf

„ _ Ahmhf +f~ -

+ D

(22)

(23)

(24)

Equation (7) is homogeneous in the adjoint variablesthat can therefore be scaled up or down by means ofan arbitrary constant. The numerical solution of theproblem is equivalentiy and more easily achieved ifall the right hand sides of Eqs. (22)-(24) are multi-plied by 1 — mpf. Thus, one obtains the same con-ditions A/i = —Ah and Aj = — Ak, as in the caseof the gross-mass minimization, whereas Eq. (24) isreplaced by

cro"coI03

0 0.5 1.0

Hydrogen/overall-fuei mass-flow ratio

1.5

Fig. 1 - Typical profiles of the Hamiltonian as afunction of the hydrogen fraction.

Ht = DI -mpf

(25)

Optimal controls

Since both A^ and Aj; are assigned constants,which only depend on propellants and tank charac-teristics, Eq. (7) is a function of the control vari-ables and of the ratio \v/(I — mp), where \v is anunknown constant. Pontryagin's Maximum Princi-ple states that optimal controls must maximize theHamiltonian over the sets of admissible controls.13

Therefore, the optimal controls are determined oncethe ratio Xy /(I — mp) has been specified.

In the absence of constraints, the optimal con-trols are found by nulling the partial derivatives ofH relative to the controls

(1 + a)2 (1 + a)2

A

1- -£ = oda

l-mpdx£ = 0

(26)

(27)

In the present case, the control x is constrained bythe condition 0 < x < 11 and the absolute maximumof H may occur on the boundary of the admissible

control region. To discuss the problem, typical pro-files of the Hamiltonian are plotted in Fig. 1 as afunction of only the control variable x, by assumingdifferent values of the ratio Ay/(l — mp); for eachvalue of x, the optimal control a is computed fromEq. (26), which always provides an admissible valuea > 0. Three different situations occur, dependingon the numerical value of the ratio Av/( l — rnp). Astationary point corresponding to the absolute max-imum may be found for 0 < x < 1 (curve A); in thiscase, the optimal controls are provided by the systemof Eqs. (26) and (27). If the parameter Ay/( l — mp)is increased, the stationary point moves out of theadmissible control region (curve B) and the absolutemaximum is on the boundary x = 1; with a providedby Eq. (26). On the contrary, if \v/(l — rnp) is de-creased, the absolute maximum of the Hamiltonianis on the boundary x = 0 of the admissible controlregion (curve C); the corresponding mixture ratio ais again provided by Eq. (26).

For velocity increments in the range of interestfor SSTO vehicles (AV = 8 -H lOkm/s), in the ini-tial phase of the acceleration (i.e., for the lowest mp)the absolute maximum of H is obtained for x = 0(kerosene mode). Then, a transition to tripropellantmode (0 < x < 1) occurs as mp decreases, and thehydrogen fraction jumps from 0 to a value whichis usually greater than 0.5 (i.e., the controls arediscontinuous at the switch point which is called acorner13). As mp continues to decrease, x gradually

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increases until the hydrogen mode (x = 1) becomesoptimal; in this case, the control variables are con-tinuous at the switching point. Some of these phasesmay be absent; in particular, the kerosene mode isabsent for the lowest AV when the gross mass isminimized; the hydrogen mode is not exploited forthe lowest AV when the dry mass is minimized.

Optimal solution

Particular features of the optimal solution arefound for both optimization indexes. To illustratethese characteristics providing numerical results andfigures, € = 16kg/m3 and me = 0.05 have been as-sumed. The engine effective exhaust velocity, whichhas been computed assuming 200 bar for the com-bustion chamber pressure and 90 for the nozzle ex-pansion area ratio (one-dimensional frozen equilib-rium flow), is approximated with the quadratic re-lation

c(a, x) = 0.446 + 0.004a + 0.198*-0.00342a2-0.129x2 + (28)

(throughout the paper, velocities are normalized byusing the circular velocity at the Earth surface asthe reference value). Equation (28) is quite ac-curate for the hydrogen and tripropellant modes,when x > 0.5; on the contrary, the approxi-mation is poor for x = 0; in this case, c(a,0) =0.313 + O.lOOa - 0.0212a2 is used. A numerical solu-tion can however be obtained for any different func-tion c(a,x)-

When the gross mass is minimized, the finalvalue of the controls are independent of the finalmass (i.e., of the velocity increment which is sought).The final values of the controls, a/ and Xf and theratio Ay /(I — mp/) are in fact determined by solvingthe algebraic system which is obtained by combin-ing Eq. (19), Eq. (26), and the proper equation todetermine x (usually * = 1, as the rocket accel-eration ends with the hydrogen mode) at the finalpoint. The solution of the algebraic system only de-pends on constant quantities, which are determinedby the propellant and tank characteristics. The op-timal control law during the acceleration is only afunction of the ratio (1 — mp)/(l - mp/). Figure 2compares the control history for different values ofthe final velocity.

A similar situation occurs for the minimizationof the dry mass. It is easily proven that the equality

stated by Eq. (25) holds not only at the final pointbut also during the whole trajectory (the derivativesrelative to the independent variable of both sides co-incide); in particular, at the initial point, one obtains

Ht = D (29)

The system of Eq. (29), Eq. (26), and the properequation for x (usually x = 0) at the initial pointprovides Ay and the initial controls a; and Xi whichare fixed if the propellant, tank and engine charac-teristics have been assigned. As Ay is fixed, the con-trol variables only depend on mp, that is, the controlhistory during the acceleration is always the sameand is independent of the final velocity; the engineis turned off when the required velocity is reached.Figure 3 shows the control history highlighting thefinal point for different values of the velocity incre-ment.

The switching from kerosene mode, x = 0, totripropellant mode, with x provided by Eq. (27),presents a particular characteristic. The previoussection has shown that the transition occurs whenthe local maximum of the Hamiltonian for x = 0assumes the same value as at the stationary point,that is, when

Ay

(30)

The mixture ratio just before the switching pointai_ is provided by Eq. (26) with x = 0, whereasthe control values just after the switch ai+ and xi+are obtained solving the system of Eqs. (26) and(27). The addition of Eq. (30) determines the spe-cific value of the parameter Ay/( l — mpi) which de-pends only on the propellant and tank characteris-tics. Therefore, the values «i_ = 2.412, xi+ = 0.646and «i+ = 5.225 are fixed and do not depend on thefinal velocity and performance index; moreover, theswitch occurs when the same propellant mass hasbeen used (i.e., mpi = 0.301) in the case of mini-mum dry mass, whereas for the same value of theratio (1 — mpi)/(l — mp/) = 8.478 in the case of theminimum gross mass. In a similar way, the hydrogenmode starts when the system of Eqs. (26) and (27)provides x — 1> that is, when 0:2 = 5.225 and eithermp2 = 0.652, in the case of minimum dry mass, or(1 — mp2)/(l — mp/) = 4.222, for minimum grossmass.

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.0

to—— .

— —

AV=-\.3Av=^.2ZlV=1.1AV=1.0

1.0 -

0.8

co'•G 0.6

•o>^

0.4

0.2

0

Xxx

0 0.2 0.4 0.6 0.8 1.0

Propellant mass m

Fig. 2 - Optimal control laws for gross-massminimization.

A numerical solution of the optimization prob-lem can be easily obtained for different values of thepropellant consumption mpf ; once Ay and the pro-pellant masses corresponding to the transition pointshave been computed, the system of Eqs. (4)-(6) canbe integrated from mp = 0 to mp = mpj; the fi-nal velocity and the corresponding optimal value ofthe performance index are obtained as a function of

The benefit provided by the tripropellant en-gine in comparison to the LOX/LH2 engine with thesame chamber pressure, nozzle expansion area ratioand engine mass me, is presented in Figs. 4 and 5.

.o'(0

o

5

4

oo

o

1 1

X' /

, E . 1 ,

/*\\

AV=<(

i t

\\ v+AV=1.3 -

—— 7̂/y_ _ _ _ _ ££

i . i

•"

,

1.0

0.8

0.6

0.4

0.2

0

N,

0"tsCO

c0)o>0

iT

0 0.2 0.4 0.6 0.8 1.0

Propeliant mass m

Fig. 3 - Optimal control laws for dry-massminimization.

The percentage increment of the performance index(either mu or mu/nv) is shown for velocity incre-ments in the range of interest for an SSTO vehicle.The results for the LOX/LH2 engine have been ob-tained assuming x = 1 and considering a as theonly control variable. The benefit is higher for largevelocity increments and in the case of dry-mass min-imization.

A rough estimation of the performance of a dual-fuel propulsion system is also shown in Figs. 4 and 5.In the present analysis, only the kerosene engine isused in the initial phase of the acceleration, whereasonly the hydrogen engine is used after the switch. Toreduce the engine mass for given liftoff thrust, actualdual-fuel systems should use both engines in the ini-tial phase of the rocket acceleration and then turnthe RP-1 engine off.14 The complete optimization ofa dual-fuel system is beyond the scope of the presentpaper and has been carried out elsewhere.10 A sim-plified analysis is provided in the Appendix, wherethe performance of the dual-fuel system is estimated,assuming the same values of me, chamber pressureand nozzle expansion area ratio as the tripropellantengine; it must be noted that RP-1 engines usuallypresent quite different values. The performance isquite close to the performance of the tripropellantengine; other issues, such as the engine complexity,will probably drive the choice of the propulsion sys-tem.

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o>o>

COcoCOEv>COo

DL

1.1 1.2

Velocity increment AV

1.3

Fig. 4 - Benefit of the tripropellant engine forgross-mass minimization.

o>oc,g03

COCOcoE

Q

co

COD.

Velocity increment AV

Fig. 5 - Benefit of the tripropellant enginedry-mass minimization.

for

Conclusions here considered. The Hamiltonian is defined as

The theory of optimal control has been appliedto the analysis of a tripropellant propulsion system.The performance of such a system depends on agreat number of parameters and a complete anal-ysis is beyond the scope of this paper. However,the simple model which has been adopted has pro-vided a preliminary but documented insight into theproblem. For the dry-mass minimization, the controlhistory does not depend on the velocity increment;for the gross-mass minimization, the final controlsare fixed; in both cases the switchs from keroseneto tripropellant mode and from tripropellant to hy-drogen mode occur under the same circumstances.The benefit provided by a tripropellant engine rel-ative to a LOX/LH2 engine is high for velocity in-crements in the range of interest for a SSTO rocket,in particular as far as the dry mass is concerned.The performance of a dual-fuel system is compara-ble; other considerations should drive the choice ofthe propulsion system.

Appendix

A dual-fuel propulsion system which is com-posed of two separate engines with a different pairof propellants, namely LOX/LH2 and LOX/RP-1, is

H =\vc- m

(A-l)

where a^ and a^ are the ratios of the oxygen mass-flow to the corresponding fuel mass-flow, in the hy-drogen and kerosene engine, respectively; r is thefraction of the mass flow in the hydrogen engineto the overall mass flow. The effective exhaust ve-locity of the system is a linear combination of theeffective exhaust velocities of each engine, whichonly depend on the corresponding mixture ratioc = rch(ah) + (1 — r)ck(ak). As H is linear withr, a bang-bang control is optimal and r assumesits minimum value (r = 0 in the present simplifiedanalysis, which assumes the use of the engines in se-quence) at the beginning of the rocket accelerationand is switched to the maximum value (r = 1) whenits multiplier in Eq. (A-l) becomes positive. Oneshould note that r = 0 and r = 1 correspond ex-actly to x = 0 and x = 1> respectively; the optimalsolution can therefore be found using the same pro-cedure as for the tripropellant rocket, but excludingthe tripropellant mode (0 < x < !)• The engine isswitched from x = 0 to x — 1 when the local max-ima of the Hamiltonian for x = 0 and \ = \ assumethe same value.

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References

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2 Salkeld, R., "Mixed-Mode Propulsion for theSpace Shuttle," Astronautics and Aeronautics,Vol. 9, No. 8, 1971, pp. 52-58.

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