Optimization of a Hybrid Rocket Upper Stage

with Electric Pump Feed System

Lorenzo Casalino∗

and Dario Pastrone∗

Politecnico di Torino, 10129 Torino, Italy.

An electrical pump is considered to feed the oxidizer into the combustion chamber of ahybrid rocket motor, which is used as the third stage of a three-stage launcher. The motoruses hydrogen peroxide as the oxidizer and polyethylene as the fuel; advanced Lithiumbatteries are adopted to power the pump. The design of the hybrid rocket motor and thetrajectory are simultaneously optimized by means of a nested direct/indirect procedure.Direct optimization of the parameters that affect the motor design is coupled with indirecttrajectory optimization to maximize the launcher payload for assigned characteristics ofthe first and second solid propellant stages and final orbit. The optimization provides theoptimal values of the main engine design parameters, the corresponding grain and enginegeometry, and the control law. A mission profile based on the Vega launcher is considered.The performance obtained using an electrical pump feed system is compared with pressure-gas feed systems. Results show the relevant improvements that can be obtained when apresent-technology electrical pump feed system is adopted.

Nomenclature

Ab = burning surface area, m2

Ap = port area, m2

Ath = nozzle throat area, m2

a = regression constant, m1+2n kg−n sn−1

CDj = j-th stage drag coefficientCF = thrust coefficientc∗ = characteristic velocity, m/sD = drag vector, NEe,tot = total electric energy, JF = thrust vector, NF = thrust magnitude, Ng = gravity acceleration, m/s2

h = initial port dimension, mIsp = specific impulse, sJ = throat area to initial port area ratioL = overall length, mLb = grain length, mM = rocket mass, kgm = mass, kgN = number of portsn = mass-flux exponentp = pressure, barP = burning perimeter, mPe = electric power, WR = radius, mRg = grain outer radius, m

Rth = throat radius, mr = position vector, mSj = j-th stage reference area, m2

t = time, stb = HRM burning time, sV = volume, m3

v = velocity vector, m/sw = web thickness, mx = port angular fractiony = burned distance, mZ = hydraulic resistance, (kg m)−1

α = mixture ratioβ = angle (see Fig. 1), radγ = specific heat ratioδbe = battery energy density, J/kgδbp = battery power density, kW/kgδep = electric drive system and pump

power density, kW/kgε = nozzle area-ratioρ = density, kg/m3

Superscripts

˙ = time derivative

∗Associate Professor, Dipartimento di Energetica, Corso Duca degli Abruzzi, 24 Torino. AIAA Senior Member.

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46th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit25 - 28 July 2010, Nashville, TN

AIAA 2010-6954

Copyright © 2010 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Subscripts

1 = combustion chamber at head-enda = auxiliary gasatm = atmosphericavg = averageb = batteriesc = combustion chamber at nozzle entranced = pump dischargecc = combustion chambere = nozzle exitep = electric drive system and pumpF = fuelg = pressurizing gas

gt = pressurizing gas tanki = initial valuemax = maximum valuen = nozzleO = oxidizerp = overall propellant (oxidizer + fuel)rc = rocket casingrel = relativeres = residuals = hybrid rocket structural masst = oxidizer tankth = throatu = payload

I. Introduction

The performance of a hybrid rocket motor (HRM) is similar to that of storable liquid propellant enginesand solid propellant motors. HRMs exhibit favorable properties similar to those of solid propellant rockets(safety, reliability, and low cost) and have a reduced environmental impact. Since the oxidizer is stored apartfrom the fuel, HRMs can be shut-off and restarted like liquid rocket engines, and can be throttled withina wide thrust range. In comparison to storable liquid propellants, special safety steps needed for chemicalssuch as NTO, MMH and UDMH are eliminated and operation costs fall as a result. Hybrid propellantrockets have already been proposed to replace solid propellant rockets, with particular reference to boosterstages;1–5 moreover, a hybrid motor was used for a manned suborbital flight to a 100-km altitude,6 and willbe probably employed for commercial space flights.7 The features of hybrid propellants make HRMs alsosuitable to replace solid rocket motors and storable liquid rocket engines used in launchers upper stages.8,9

Their use as the third and final stage of a low-cost launcher is particularly attractive and is here investigated.In a previous study10 the authors optimized the design of a hybrid rocket upper stage to maximize the

payload inserted into a given final orbit. The goal was to evaluate the performance improvement that canbe obtained by using hybrid rocket motors to upgrade existing/under-development launchers, which employsolid rocket motors or storable liquid propellants for powering upper stages. A three-stage launcher, withsolid-propellant first and second stage and a hybrid-propellant third stage was considered. The design of thehybrid-propellant upper stage and the whole ascent trajectory were simultaneously optimized by means of anested direct/indirect procedure, for given characteristics of the first and second stages. A pressurized feedsystem (blow-down or partially regulated system) was adopted with the aim of keeping costs low. The testcase showed that a good improvement of the launcher performance can be obtained. Results also showedthat the mass of the oxidizer tank and auxiliary tank, for the regulated feed system, are relevant, due to therequirement of a large tank pressure to keep the thrust magnitude and regression rate sufficiently large.

In the present paper a different feed system for the liquid propellant is considered to further improve thelauncher performance: pumps are used for the liquid propellant feed system, so the tank pressure can belower, thus greatly reducing the tank mass. Due to the fact that a HRM has only one liquid propellant, it isdifficult to have high-energy fluid to feed a turbine to move the pump. Solutions such as decomposition of amonopropellant (e.g. HP), heating of the liquid oxidizer, thrust chamber tap off or use of an auxiliary liquidpropellant, introduce complexity, reduce reliability and increase costs. For these reasons, an electrical drivesystem powered by batteries is here proposed. The analysis is pursued via the multidisciplinary optimizationapproach, which couples the optimization of propulsion system and trajectory, used in Ref 10. Reference ismade to the Vega launcher11 and the results which have been obtained in the case of a hybrid third stagewith a gas pressurized feed system.10

II. Grain Geometry and Ballistic Model

A multi-port grain geometry is adopted, while considering a uniform regression rate along the port axis.A simple circular-section grain12 with either 6 or 8 ports is adopted; different grain geometries could provideimproved performance and packaging efficiency13 but are not considered here. The grain geometry, described

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in Fig. 1, is defined by the number of ports N , the web thickness w and the grain outer radius Rg. One has

x = (N/π) sin−1[w/(Rg − w)] (1)

h =√

(Rg − w)2 − w2 − w tan(π/2 − π/N) (2)

β = π/2 + xπ/N (3)

The initial port area is then easily determined

(Ap)i = 2N[

(Rg − w)2(1 − x)π/(2N) − hw/2)]

(4)

For a given burning distance y (0 ≤ y ≤ w), one easily computes the burning perimeter

P = 2N [(Rg − w + y)(1 − x)π/N + βy + h + (π/2 − π/N)y] (5)

and the port area

Ap = (Ap)i + 2N{[

(Rg − w + y)2 − (Rg − w)2]

(1 − x)π/(2N) + βy2/2 + hy + (π/2 − π/N)y2/2}

(6)

Rg

ww

w h

x π / N

(1-x) π / Nβ

Figure 1. Grain geometry

The regression rate is determined by the oxidizer massflux and grain geometry

y = a (mO/Ap)n (7)

Hydrogen-peroxide (HP) / polyethylene (PE) is the pro-pellant combination, with14,15 a = 7 · 10−6 and n = 0.8,when SI units are used. The mass fluxes obtained in thisarticle always fall in the validity range (20−1000 kg/s/m2)of this curve fit. No pyrolysis of the lateral ends is consid-ered. Pressure losses inside the combustion chamber aretaken into account by relating the chamber head-end pres-sure p1 to the chamber nozzle-stagnation pressure pc. Anapproximate relation, similar to that proposed by Barrereet al.16 for solid-propellant side-burning grains, is used

p1 =

[

1 + 0.2

(

Ath

Ap

)2]

pc (8)

The pressure loss is typically between 2 and 5 %, consistent with HRM operation (see, for instance, Ref.2). The hydraulic resistance Z in the oxidizer flow path from the pump exit to the combustion chamberdetermines the oxidizer flow rate. Under the assumption of incompressible turbulent flow

mO =√

(pd − p1)/Z (9)

The value of Z is assumed to be constant during motor operation. The fuel mass flow is obtained as

mF = ρF yAb = ρF yLbP (10)

and the mixture ratio is

α =mO

mF

∝ m1−nO An

p/Ab (11)

An isentropic expansion in the nozzle is assumed, and the chamber nozzle-stagnation pressure pc is deter-mined by

pc =(mO + mF )c∗

Ath

(12)

The performance of the propellant combination is evaluated17 as a function of the mixture ratio α,assuming pc = 10bar. Even though the actual pressure in the combustion chamber can span over a widerange during motor operations, the error is small for chamber pressures and mixture ratios considered inthis paper. Frozen equilibrium expansion is assumed; the exhaust gas maintains throughout the nozzle thecomposition that it has in the combustion chamber. This conservative assumption of frozen equilibriumexpansion is adopted to account for the low combustion efficiency of HRMs; in addition a 0.96 c∗-efficiency18

is introduced. Third-degree polynomial curves fitting the characteristic velocity and specific heat ratio areembedded in the code to compute the proper values as the mixture ratio changes during motor operations.

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III. Electric Pump Feed System

An electric system is used to drive the pump which feeds the oxidizer to the combustion chamber. Theneeded energy comes from batteries. The masses of these main components (pump, electrical drive systemand batteries) have to be evaluated. Existing literature provides typical values of power density (power tomass ratio) for liquid propellant turbopump systems19 and expected values for the electric drive system.20

The mass of the electric drive system and of the pump is then evaluated as

mep =Pe,max

δep

(13)

where Pe,max is the maximum electrical power required. In the following, a power density δep = 1.25 kW/kghas been assumed for the electric drive system plus pump. The electrical power use to drive the pump is

Pe =mO(pd − pt)

ρOηep

(14)

where pd is the pump discharge pressure and pt is the propellant tank pressure. The tank pressure is assumedto be constant during operation. A value of pt = 1 bar is assumed, which guarantees, in the most demandingcases, a suction parameter of about 0.05 consistent with present-technology centrifugal pumps. The massused to keep pt constant is negligible and is not considered in the following. The overall efficiency ηep takesinto account the conversion of electrical energy stored in the batteries into flow head rise and ηep = 0.64 isassumed.

The mass of the batteries depends on the most stringent requirement between the maximum electricalpower required Pe,max, and the electrical energy needed to drive the pump during the burning time Ee,tot.In the case considered here, the discharge time is the overall HRM burning time tb, and the needed electricalenergy is

Ee,tot =

∫ tb

0

Pedt (15)

For any given kind of battery and discharge time, typical values of power density δbp and energy densityδbe (energy to mass ratio) can be found in literature, and, assuming a safety factor of 1.2, the mass of thebattery is evaluated as

mb = 1.2max

(

Pe,max

δbp

,Ee,tot

δbe

)

(16)

Due to their small size and light weight compared to all other technologies, Lithium batteries, proposed since1976,21 are considered for the present application. A number of different cathode, anode and electrolytematerials offer both advantages and disadvantages. The HRM requires high power levels for a relativelyshort time, so that high rate capability is required. Recently developed batteries, such as those based onLithium titanate, presents such a feature. They have very high power density, while their energy density islower when compared to other lithium-ion batteries, but it is still higher than lead acid and NiCd batteries.Based on Ragone plots,22 δbp = 3.0 kW/kg and δbe = 90 Wh/kg have been assumed. The pump is operatedat constant power Pe = Pe,max and

Ee,tot = Pe,maxtb (17)

The value of (pd)i at the beginning of HRM operation is determined by the optimization procedure, thusdetermining the value of Pe,max.

IV. Motor Design and Operation

According to the chosen ballistic model, the design of the HRM is defined by

• initial thrust level Fi,

• initial mixture ratio αi,

• nozzle expansion ratio ε,

• initial pump discharge pressure (pd)i,

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• initial value of chamber pressure (pc)i,

• ratio J of the throat area to the initial port area.

The initial chamber pressure is assigned by imposing (pc)i = 0.4 (pd)i; actually, the ratio pd/pc variesduring operation, but the assumed initial ratio is sufficient to guarantee pd/pc > 1.5 and to avoid couplingbetween the hybrid motor and the oxidizer feed system. The initial port area to throat area ratio J influencesboth motor geometry (and weight) and operation (namely, the mixture ratio shift and the correspondingchange of specific impulse); preliminary results showed that the optimization would require quite large valuesof J ; in the present article, its value is fixed at J = 0.5 to avoid excessive pressure losses and nonuniformgrain regression, typically related to larger values of J . The remaining design parameters are optimized asshown in section V.

Given the set of design parameters, the motor geometry is first determined. The relevant properties ofthe combustion gases can be computed, owing to the fact that the initial values of c∗ and γ can be calculatedfrom αi via the aforementioned curve fittings. The thrust coefficient CF can then be evaluated by assumingan isentropic one-dimensional expansion with constant γ, provided the ambient pressure is known. With a0.98 thrust correction factor introduced to modify the vacuum thrust coefficient, one has

CF = 0.98

√

√

√

√

√

2γ2

γ − 1

(

2

γ + 1

)

γ + 1γ − 1

1 −

(

pe

pc

)

γγ − 1

+ εpe

pc

− εpatm

pc

(18)

where the term related to the atmospheric pressure is always small, as the third stage always flies at highaltitude. The mass flow rates at rocket ignition (i.e., at t = 0) are found from the initial thrust Fi

(mp)i = (1 + αi)(mF )i =1 + αi

αi

(mO)i =Fi

c∗i (CF )i

(19)

The throat and initial port areas Ath and (Ap)i are then determined

Ath =(mp)i

(pc)ic∗i; (Ap)i =

Ath

J(20)

The nozzle throat area Ath is considered to be constant during operation. One also finds

(Ab)i =(Ap)

ni

aρF

(mF )i

(mO)ni

(21)

The grain geometry can then be derived once a tentative value is assumed for Rg. Equations (1)-(4) arenumerically solved for x, β, h, and w given the required initial port area. Equation (5) at ignition (y = 0)gives the initial perimeter Pi to compute the grain length Lb = (Ab)i/Pi. Equation (7) is integrated upto burnout during the optimization of the ascent trajectory. The ascent optimization procedure correctsthe tentative value for Rg to match the necessary condition yf = w at burnout. The head-end pressure iscomputed with Eq. (8) and, knowing the initial pump discharge pressure, also the hydraulic resistance Zcan be determined by applying Eq. (9) at t = 0. The motor geometry is completely defined and the motorperformance can be evaluated during operation.

The pump discharge pressure rules motor operation. Numerical integration of Eqs. (7), (9), and (10),allows the evaluation of the fuel grain geometry, the exhausted masses of oxidizer and fuel. At each instantt, an iterative procedure is required to determine the pump discharge pressure; once a tentative value isassumed for pd and the motor geometry is known, the regression rate, the propellant flow rates (and theirratio α), c∗, pc and p1 are computed by numerically solving Eqs. (7)-(12) while the curve fit for c∗ as afunction of α is used. The oxidizer flow rate and pd allow to evaluate the required electrical power, whichmust be constant; a procedure based on Newton’s method is used to correct the tentative value for pd untilPe = Pe,max Then, the thrust level F = pcAthCF is determined by evaluating CF at the actual altitude viaEq. (18), in order to integrate the trajectory equations. At burnout the overall propellant mass is finallyevaluated, and an estimation of the structural masses can be obtained.

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V. Optimization

The optimization procedure aims at finding the motor design parameters and the corresponding trajectorythat maximize the mission performance index, which is, in this paper, the payload inserted into a prescribedorbit. As the number of motor design parameters is low (four, in this case), their optimization is easily carriedout by a direct method. Also, the relations, which determine the motor behavior, cannot be written explicitlyand indirect methods cannot be used. On the other hand, the trajectory optimization is characterized bycontinuous controls (namely, the thrust direction), which would either require a discretization by means ofa large number of parameters or the use of indirect methods. A mixed optimization procedure24 is hereadopted. An indirect method25 optimizes the trajectory for each choice of the motor parameters. These areinstead optimized by means of a direct procedure.26 Both methods have been developed at the Politecnicodi Torino.

Tentative values are initially assumed for the design parameters, i.e., Fi, αi, ε, (pd)i. For each set ofparameters the fast and accurate indirect procedure provides the optimal trajectory and the correspondingperformance index; few seconds are required when a 2 GHz PC is used. The design parameters are thenvaried by small quantities to numerically evaluate the derivatives of the performance index with respect tothe design parameters. To find the maximum performance index, a procedure based on Newton’s methodis used to determine the set of design parameters which simultaneously nullify the index partial derivatives.Only a few minutes are sufficient to obtain the optimal design and the corresponding trajectory.

A point mass rocket is considered for the trajectory optimization. The state equations provide thederivative of: Position r (radius, latitude and longitude), velocity v (radial, eastward, and northward com-ponents) and rocket mass M . The equations of motion are written in non-dimensional form to improve theintegration’s numerical accuracy. In a vectorial form one has

dr

dt= v

dv

dt= g +

F − D

M

dM

dt= −

|F |

c∗CF

(22)

An inverse-square gravity field is assumed and D = (1/2)ρatmCDjSjv2rel is the aerodynamic drag magnitude;

each subrocket has an assigned reference area Sj and the drag coefficient CDj is a known function of theMach number; the relative velocity is evaluated taking Earth’s rotation into account. The thrust is writtenas a function of the vacuum thrust F = Fvac − ǫAthpatm. Numerical fits of the US Standard Atmosphereare used to determine pressure, density and temperature as functions of the altitude, for the determinationof drag and thrust.

The performance of a rocket with characteristics similar to those of the European small launcher Vega11

are evaluated and used as reference case. Gross and propellant masses and thrust profiles of the first threesolid-propellant stages and the fourth liquid-propellant stage are assigned; the payload is maximized fora 700-km circular polar orbit. The same initial mass and trajectory constraints of the reference case areassumed for the three-stage launchers with a hybrid upper stage; also, the characteristics of the first twosolid-propellant stages are unchanged. Given the initial mass, the gross masses of the first two stages, andthe fairing mass, the remaining mass budget is split between hybrid-stage mass and payload; for assignedcharacteristics of the hybrid rocket, the trajectory is optimized in order to maximize the final mass, and thepayload is obtained by subtracting the dry mass of the hybrid rocket (combustion chamber, nozzle, tanks,case, feed system, propellant residual).

The trajectory is split into phases; the first stage burn is divided into 1) vertical ascent, 2) kick phase,and 3) zero-lift gravity-turn ascent; then the first stage is jettisoned and a 4) coast phase follows. Thesecond stage burn is a single 5) zero-lift gravity-turn phase followed by the stage jettisoning and a 6) coastarc. The third stage performs two burns with optimal thrust direction; during the first burn the fairing isjettisoned and the following phases are introduced: 7) HRM first burn (with fairing), 8) HRM first burn(without fairing), 9) coast arc, and 10) HRM second burn. The time-length of phases 1, 2, 3, 4, 5, 6 and7 are assigned. The time-length of the remaining phases is determined during the trajectory optimization.The vacuum thrust magnitude and the propellant flow rate are assigned functions of time for the solid-propellant rockets, whereas they are computed from the HRM design optimization variables for the thirdstage. The thrust direction is vertical during phase 1, parallel to the relative velocity in phases 3 and 5, freeand optimized during phases 2, 7, 8, and 10.

Boundary conditions define the mission to be performed: initial mass (137276 kg), position and velocityare assigned (launch from Kourou is here assumed). The masses of the exhausted stages and fairing are also

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given. The free molecular heat flux is fixed at 0.1BTU/ft2/s = 1135W/m2 at the end of phases 7 (i.e., whenthe fairing is jettisoned) and 8 (first shutdown of the third stage); also, horizontal flight is imposed at thelatter point, to prevent the rocket to reenter the lower layers of the atmosphere, where the heat flux wouldbecome larger. The final orbit is specified by assigning altitude (700 km), eccentricity (0, i.e., circular orbit),and inclination (90 deg, i.e., polar orbit); the longitude of the ascending node is left free.

The theory of optimal control is applied to optimize the trajectory, given the characteristics of thehybrid-propellant third stage. The details of the indirect optimization procedure can be found in Refs. 10,24and are here only summarized. An adjoint variable is associated to each equation; the theory of optimalcontrol provides the Euler-Lagrange equations for the adjoint variables, algebraic equations that determinethe control variables (i..e., the thrust direction), and the boundary conditions for optimality, which alsoimplicitly define the hybrid motor switching times. The multipoint boundary value problem, which arisesfrom the application of the theory of optimal control, is solved by a procedure27 based on Newton’s method.Tentative values are initially chosen for the problem unknowns and progressively modified to fulfill theboundary conditions. The grain outer radius is and additional unknown.

VI. Numerical Results

The optimization procedure, which has been described in the previous section, is now applied to the designof a hybrid propellant third stage with the aim of maximizing the payload delivered into a 700-km polarorbit. For the sake of comparison, the trajectory and performance of a launcher with three solid propellantstages and a liquid propellant fourth stage, with characteristics similar to those of the Vega launcher,11 havebeen computed and are used as a baseline. The payload of the all-solid launcher is about 1400 kg, consistentwith the performance of Vega. The same constraints on the trajectory are assumed for the launcher with ahybrid-propellant third stage. Reference is made to the results presented in Ref. 10, concerning the use ofpressurized feed systems. Grains with either six or eight ports are considered.

The indirect trajectory optimization maximizes the final mass (initial mass minus exhausted propellant)given the engine geometry. The payload is then evaluated by subtracting the mass of the propulsion system,i.e., the masses of combustion chamber, nozzle, tanks, rocket casing, propellant sliver, pump, electric drivesystem and batteries from the final mass (additional feed systems masses, e.g., valves and plumbings, areneglected); these are estimated by means of suitable assumptions and approximations. The combustionchamber has a 6-mm insulating liner (with density equal to that of the solid fuel) and an aluminum alloycylindrical wall. The diameter of the aluminum cylindrical oxidizer tank is fixed at 1.9 m, matching thediameter of Vega’s third stage; its wall thickness is fixed at 0.5 mm. The HRM is encapsulated by a 1-mmthick cylindrical aluminum casing. A 45-deg convergent and a 20-deg divergent nozzle with a phenolic silicaablative layer is considered. A uniform thickness is assumed and is evaluated according to Ref. 28, usingaverage values of the transport properties and an estimation of the heat flux at ignition; these assumptionsare conservative and the nozzle mass is therefore overestimated; a more detailed analysis is left for futurework. The nozzle structural mass is small compared to the ablative layer mass and is thus neglected. Themass of pump, electric drive system and batteries are evaluated according to Eqs. (13) and (16). For theconsidered cases energy storage is the most stringent requirement to determine the mass of the batteries and

mb = 1.2Pe,maxtb

δbe

(23)

The optimization procedure determines the nozzle expansion area ratio and the initial values of thrust,mixture ratio, and pump discharge pressure. Their values are dictated by the opposite requirements ofkeeping the system dry mass low, while assuring a large specific impulse and low velocity losses during theascent to orbit in order to increase the final mass. The mixture ratio influences the required propellant massand the motor dry mass; the propellant mass is minimum when the value corresponding to the maximumspecific impulse (about 6.7, in the present case) is adopted, but larger values of mixture ratio reduce thesystem dry mass; the choice of αi must also consider that α varies during operation. A large nozzle expansionarea ratio improves the specific impulse (the third stage always flies in vacuum-like conditions) but increasesthe nozzle mass.

The choice of initial thrust magnitude and pump discharge pressure greatly influences performance. Alarge thrust value is usually preferable to reduce gravitational losses and increase the regression rate, butit implies larger dry mass. Large values of pump discharge pressure allow for larger regression rates but

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Table 1. HRM optimal design.

Feed N Fi αi (pt)i ǫ αavg (Isp)avg Rg w Rth L

system kN bar s m m m m

blowdown 6 365.1 7.08 20.15 11.83 7.57 270 0.520 0.039 0.288 12.3

blowdown 8 414.1 7.02 19.77 11.30 7.65 271 0.558 0.034 0.310 12.0

press.-reg. 6 209.2 6.22 13.02 11.91 7.39 272 0.508 0.042 0.271 9.8

press.-reg. 8 230.5 6.16 12.33 11.17 7.47 271 0.546 0.037 0.293 9.4

Feed N Fi αi (pd)i ǫ αavg (Isp)avg Rg w Rth L

system kN bar s m m m m

pump-fed 6 173.1 5.43 20.60 17.86 7.54 282 0.453 0.061 0.194 9.2

pump-fed 8 187.8 5.46 18.12 15.83 7.72 281 0.491 0.052 0.216 8.7

Table 2. Mass budget.

Feed N mu mp mp/(mp + ms) mres mg mt mgt mn mcc mrc

system kg kg kg kg kg kg kg kg kg

blowdown 6 1796.1 10966.0 0.919 43.6 22.4 445.0 - 330.4 125.1 207.0

blowdown 8 1774.9 10984.0 0.918 57.2 22.3 440.3 - 333.8 121.5 201.5

press.-reg. 6 1989.4 10970.9 0.931 51.7 21.7 164.8 139.4 329.7 104.1 163.7

press.-reg. 8 1970.6 10989.9 0.930 67.6 21.3 156.4 136.8 335.5 100.1 157.3

Feed N mu mp mp/(mp + ms) mres mb mt mep mn mcc mrc

system kg kg kg kg kg kg kg kg kg

pump-fed 6 2221.8 10872.1 0.928 97.9 76.9 26.28 95.0 301.0 91.0 153.5

pump-fed 8 2182.3 10900.2 0.927 121.5 67.3 26.39 90.7 312.1 86.8 148.2

increase the battery and pump mass. When the gas pressure feed systems are used,10 the regression ratetends to become too small and a constraint on its final value must be introduced to avoid grain cooking; incontrast when the pump-fed system is adopted, the regression rate remains, as discussed later, larger thanthe critical value of 0.2 mm/s, above which cooking typically does not occur. No additional constraint istherefore necessary, and both Fi and (pd)i are free optimization variables. Results are summarized in Tables1 and 2. The “motor only” propellant fraction mp/(mp +ms) is shown to allow for comparison with existingrockets.

Since the pump is operated at constant power, the oxidizer mass flow and pd are almost constant, asshown by Figures 2 and 3 for the 6-port and 8-port grains, respectively (note that the final burn has a veryshort duration, 5-6 s, and has a limited influence of the rocket performance; even though it is accounted for

20.0

20.2

20.4

20.6

20.8

0 50 100 150 2005.70

5.75

5.80

5.85

.

.m

o

pd

Time from ignition t, s

Pum

p di

scha

rge

pres

sure

pd ,

bar

Oxi

dixe

r m

ass

flow

mO ,

kg/s

Figure 2. Pump discharge pressure and oxidizer mass

flow: 6-port grain.

17.8

17.9

18.0

18.1

18.2

0 50 100 150 2006.20

6.25

6.30

6.35

6.40

.

.m

o

pd

Time from ignition t, s

Pum

p di

scha

rge

pres

sure

pd ,

bar

Oxi

dixe

r m

ass

flow

mO ,

kg/s

Figure 3. Pump discharge pressure and oxidizer mass

flow: 8-port grain.

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200

300

400

0 50 100 150 200

6 ports8 ports

pump-fed

press.-reg.

blowdown

Time from ignition t, s

Thr

ust

F, k

N

Figure 4. Thrust.

5

6

7

8

0 50 100 150 200

6 ports8 ports

pump-fed

press.-reg.

blowdown

Time from ignition t, s

Mix

ture

rat

io α

Figure 5. Mixture ratio.

during the performance evaluation, it is not shown in the figures). The oxidizer flow rate actually exhibitsa moderate growth, whereas the fuel flow rate diminishes; this behavior is determined by the port areagrowth, which offsets the opposite effect caused by an increasing burning area. In fact, the overall flowrate remains almost constant due to the limited variations in pc and c∗. As a result, the thrust is almostconstant whereas a remarkable increase in the the mixture ratio occurs. Figures 4 and 5 present the thrustand mixture-ratio histories during the first burn of the HRM. The mixture ratio shifting is larger comparedto the pressure-fed system cases due to the larger variation of Ap (lower outer diameter, larger w), and theinitial value of mixture ratio is lower in order to keep the average mixture ratio at a proper value. The initialthrust required by the pump-fed system is therefore lower than in the pressure-fed system cases, as thrustand regression rate remain at sufficiently high values (see Figures 4 and 6).

The conditions (height and velocity) at the end of the HRM first burn are almost the same of pressure-fedsystems, since they are dictated by the imposed heat flux value and the requirement of leaving the rocket onan ellipse tangent to the final orbit, in order to perform the orbit insertion with minimum propellant cost.The lower thrust of the pump-fed case determines a less steep ascent, longer duration and lower longitudinalacceleration (Figure 7).

The lower thrust, associated with high chamber pressures (comparable with the maximum values forthe blowdown cases) determines a low throat area. Even if the burning time is longer, higher values of thenozzle expansion ratio are now possible and the average specific impulse is improved. This fact determinesa saving of about 90 kg of propellant. This is partially offset by a larger sliver (almost doubled compared tothe gas pressurized feed systems), due to the reduced grain outer diameter. Different grain geometries couldalleviate this problem and improve performance. This issue is left for future work.

The largest influence on performance improvement is due to the reduction of the feed system mass. Theoxidizer tank weight becomes very small compared to gas pressurized cases; the weight reduction is morethan 400 kg with respect to the blowdown cases, and 100 kg with respect to the partially regulated cases.The weight of batteries and pump is instead comparable to the auxiliary tank, and the net mass saving isabout 250 kg and 100 kg compared to the blowdown and partially regulated cases, respectively.

As in the case of pressurized feed systems, a larger number of ports allows for an increase of the burningperimeter and determines a reduction of the grain length, with a small benefit on the system dry mass (thelength reduction also improves structural robustness). However, results show that the use of a 8-port grainreduces the performance in terms of payload. In fact, it causes a larger mixture-ratio shifting during engineoperation and, in turn, a lower average specific impulse and a larger propellant consumption. A largerpropellant sliver is also experienced.

Even though the results obtained here suffer from simplifications, they clearly show that the introductionof a single hybrid rocket as a replacement for the solid third stage and liquid fourth stage can remarkablyimprove the launcher performance. The margin ranges from about 400 to almost 800 kg, depending on graingeometry and feed system, and is in particular relevant when an electrical pump is used to feed the oxidizerinto the combustion chamber.

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0.2

0.3

0.4

0.5

0.6

0 50 100 150 200

6 ports8 ports

.

Time from ignition t, s

Reg

ress

ion

rate

y ,

mm

/s

Figure 6. Regression rate.

1

3

5

0 50 100 150 200

6 ports8 ports

press.-reg.

pump-fedblowdown

Time from ignition t, s

Long

itudi

nal a

ccel

erat

ion

T/ M

, g

Figure 7. Thrust acceleration.

VII. Conclusions

An optimization procedure, which couples direct and indirect optimization methods has been appliedto the design optimization of a hybrid rocket to be used as the third stage of a three-stage launcher. Theoptimization of the parameters that affect the design of the motor is coupled with the trajectory optimization.The coupling is fundamental, as there is a mutual dependence of the mission requirements and engine optimalcharacteristics. The optimization method shows to be fast and reliable and provides the motor design andthe corresponding trajectory to maximize the payload for given final orbit. Few minutes are sufficient on astandard PC.

The analysis is applied to analyze the use of an electrical pump feed system for the oxidizer. Even thoughan exact comparison is not possible, the results show that a hybrid third stage can significantly improvethe performance compared, for instance, to the use of the combination of a solid propellant stage with aliquid propellant upper stage for orbit insertion, while also not increasing (or possibly reducing) the systemsimplicity and reliability. The use of the electric pump feed system remarkably improves the payload incomparison to gas-pressurized feed systems. Other propellant combinations, in particular liquid oxygen andhydroxyl-terminated polybutadiene, grain geometries (quad port design grains or grains with multiple rowsof ports) and pump power throttling could be considered to further improve performance and packagingefficiency.

References

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Spacecraft and Rockets, Vol. 32, No. 6, 1995, pp. 964-969.3Schoonover, P.L., Crossley, W.A., and Heister, S.D., “ Application of a Genetic Algorithm to the Optimization of Hybrid

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