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Optimization of the propulsion for multistage solid rocket motor launchers
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PII:
Acta Astronautica Vol. 50, No. 4, pp. 201–208, 2002 ? 2002 International Astronautical Federation. Published by Elsevier Science Ltd. All rights reserved. Printed in Great Britain S0094-5765(01)00164-3 0094-5765/02/$ - see front matter
OPTIMIZATION OF THE PROPULSION FOR MULTISTAGE SOLID ROCKET MOTOR LAUNCHERS† M. CALABRO‡, A. DUFOUR and A. MACAIRE EADS-Launch Vehicles, 66, Rte de Verneuil, 78133 - Les Mureaux, France (Received 25 August 1999)
Abstract—Some tools focused on a rapid multidisciplinary optimization capability for multistage launch vehicle design were developed at EADS-LV. These tools may be broken down into two categories, those related to propulsion design optimization and a computer code devoted to trajectories and under constraints optimization. Both are linked in order to obtain optimal vehicle design after an iterative process. After a description of the two categories tools, an example of application is given on a small space launcher. ? 2002 International Astronautical Federation. Published by Elsevier Science Ltd. All rights reserved
1. INTRODUCTION
EADS-LV has been designing launchers for many years, so computer codes focused on a rapid multidisciplinary optimization had to be developed. From computer codes dedicated to weight and sizing of solid rocket boosters, a special one was conceived and written in order to optimize the main parameters of a solid rocket motor (SRM) under constraints [1]. The propulsion issues are a catalog of SRMs with various propellant masses, used as data by the trajectory optimization code. Let us look at the scheme in Fig. 1. From derivatives based on know-how, and technical data, geometrical constraints as well as mission requirements, the tool named PROPULSE Erst delivers a catalog of motors [2]. The trajectory optimization computer code MAXTOM will then optimize the staging of the launcher, the thrust law shape of the Erst stage and of the upper stage, burning time of all stages and Enally will give a new set of derivatives. The optimal thrust law shape will be used to search with dedicated internal ballistic computer codes, the better grain design versus thrust law requirements and a Erst complete optimization is
†Paper IAF97-S.l.07 presented at the 48th International Astronautical Congress, October 6 –10, 1997, Turin, Italy. ‡Corresponding author. 201
done; issues of propulsion tools are re-injected as data for a next optimization loop. So the global process may be described as indicated in Fig. 2.
2. TRAJECTORY ANALYSIS AND VEHICLE OPTIMIZATION : MAXTOM
2.1. General description of the tool (inputs=outputs)
This tool is based on the conventional architecture of an optimization program. The optimizer (MAX) is separated from the simulation tool (TOM) that takes into account the characteristics of the launcher. This architecture (Fig. 3) provides capabilities to change the optimizer in order to compare other programming algorithms. 2.1.1. Optimizer characteristics. The optimizer is based on a parametric optimization with gradient method using partial derivatives. These derivatives are assessed from some evaluations of the function with appropriate increment on input optimization parameters. This method implies that the parameters, described below, may be changed continuously on their intervals. In fact, in order to work properly, the constraints and the criterion function must have continuous derivatives versus the optimization parameters. This requirement has some consequences on the interfaces with other tools, specially with the propulsion analysis program.
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• the inert mass ejected law, • the speciEc impulse evolution.
Fig. 1. Propulse code inputs and outputs.
and some additional data from propulsion design (weight, nozzle area, etc.) and from launcher conEguration: aerodynamic coeJcients, atmospheric model, mission information). MAXTOM has to be adapted in order to use the propulsive optimization tool PROPULSE. The aerodynamic coeJcients and the reference area (Sref ) have to be changed as well, if the shape of the launcher or simply the diameter is modiEed. These aerodynamic coeJcients are coupled with the volume and the density of the optimized staging at each iteration of the algorithm and designed thanks to the semi-empirical formulae described in [3]. 2.2. Constraints description
We will distinguish below the constraints usually required on the determination of the optimal trajectory for a given launcher and the ones that had to be added in order to introduce propulsive characteristics in the optimization process. 2.2.1. Usual constraints. Four topics are con-
cerned by the constraints: Fig. 2. Maxtom and propulse coupling.
• the trajectory deEnition concerning the orbital parameters reached at the end of each stage (apogee, perigee, etc.), • the visibility of the launcher from ground stations, • the locations of the impact of re-entry stages (bound to fall in the sea in general), • the limits on dynamic pressure, thermal Iux, dynamic pressure at the separation of the Erst stage, etc.
Particular constraints are used for optimizing the propulsion characteristics of the launcher. If the propulsive characteristics are modiEed, some special treatments have to be done: 2.1.2. Simulator characteristics. The three degreesof-freedom equations of motion are numerically (1) for the disengagement of the launcher at integrated using Runge–Kutta algorithm. The lift-oL, a speciEcation of non-collision imlauncher is considered as a point mass; Earth plies in general a constraint on the accelrotation and oblateness are modeled and a U.S. eration and so a constraint on the thrust standard atmosphere is used. The atmospheric level=total mass ratio at lift-oL, drag and lift and the propulsive forces are applied (2) constraints should be met on control: modon the launcher (the rotational motion around the ifying the propulsive characteristics has center of mass is supposed to have little impact on an impact on controllability. This modiEthe trajectory). cation concerns especially the maximum Basically, the simulator takes into account for gimballing angle of each nozzle but also one stage: the thrust level. Fig. 3. MAXTOM architecture.
• geometrical data, • the SRMs mass Iow rate law,
(3) constraints on general loads: For a given sizing wind, the maximum normal force
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should be constrained in order to be coherent with the structural deEnition (basically, the constraint is a maximum dynamic pressure). (4) constraints on maximum acceleration with regards to payload. 2.3. Parameters description
As for the constraints, some modiEcations have to be carried out to take into account the modiEcation on propulsive characteristics. 2.3.1. Usual parameters. Three types of parame-
Fig. 4. Optimal mass Iow rate law research.
ters are usually necessary:
• attitude law parameter, that is the characteristic of the evolution of azimuth and Iight path angle of the launcher, • durations of ballistic phases if it turns out to be interesting, • particular parameters used for optimizing the propulsive characteristics. A propulsive stage is characterized by the following main parameters: • Fi =Fm ratio (initial thrust=mean thrust) which has an implication on the shape of the law correlated to SRM mass, • combustion time, • propellant mass, • tail-oL duration, • nozzle area ratio, • external diameter, • maximum pressure (MEOP). These parameters are obviously correlated. The global optimization process is designed to take into account the internal constraints of the propulsive stage and the external implication on the characteristics of the launcher (the aerodynamic reference area, etc.). From a reduced number of parameters, a multivariable interpolation is performed to rebuild what is necessary to compute the trajectory (Iow rates and speciEc impulse laws — see above). For this process, the propulsive analysis tool PROPULSE generates a “catalog” of stages corresponding to a class of stages optimized for the general conEguration of the launcher. 2.4. Outputs of the tool
The propulsive catalog is made from the partial derivatives of the total mass at lift-oL versus the mean Iow rate and mean speciEc impulse which are the output of the MAXTOM tool.
This optimization was for example made for the ARIANE 5 boosters (EAP); about 15 catalogs were calculated. For the preliminary design study, the problem was more to deEne a thrust template rather than the nominal law itself. The mass Iow rate law obtained after calculation is optimized by segments (Fig. 4). 3. SRM SIZING AND OPTIMIZATION PROCESS
The inputs for the sizing and the optimization of solid rocket motors are mainly: • the propellant mass, • the outer diameter, • the maximum gimballing angle, and sometimes the overall length or other parameters. At Erst, inputs are diLerent depending on the category of the stage: • Erst stage is provided with a nozzle taking into account a Iow separation criterion for the nozzle operating at ground level and with a special thrust law shape (use of previous study results), • upper stage with a decrease in the last part of the thrust law to be compliant with the maximum acceleration requirement, • intermediate stage with pressure versus time as constant as possible (optimization of the performance of the stage). For each of these categories, the design takes into account an a priori adapted design of the grain characterized by a Paverage =MEOP ratio. The choice of technologies has to be made among the following possibilities: • wounded or metallic case, • upstream or downstream pivot point Iexseal,
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Fig. 5. Upstream pivot point Iexseal.
• extendible exit cone (one or two rings, panels, etc.) or not, • conventional nozzle (carbon phenolic with carbon=carbon ITE) or completely thermostructural nozzle.
Fig. 6. Downstream pivot point Iexseal.
DiLerent materials may be used. After a selection of design rules (safety coeJcients, margin, etc.) the SRM will be designed and sized to determine the mass budget and the speciEc impulse (and also CG and inertiae versus time). • The case is always computed directly using a simple method: — metallic case: stress computation with experimental corrective factors depending on the chosen technology, — wounded case: “MOethode en Elet” giving the Elament=epoxy mass, the design of polar bosses and their masses come from the consultation of a database made with the results of Enite elements computations. The mass itself is obtained by a polynomial formula. • The Iexseal (Figs. 5 and 6) is also computed using a semi-empirical method: use of an analytical computation of stresses under MEOP without deIection and experimental corrections to take into account the stresses due to the gimballing angle (the shims can be metallic or composite). • The Exed housing is directly computed (buckling and internal pressure) [4]. • The sizing of the other components needs to create databases. • For the more frequently used composite propellants (HTPB, CTPB, PBAN, etc.) the database gives: CD (Iow rate coeJcient), density, combustion temperature, range of burning rate, burning rate exponent, mechanical property, ODE speciEc impulse, etc.
Fig. 7. Accuracy of the Isv calculation.
• The thicknesses of the thermal protection of the nozzle are tabulated functions of the propellant, the materials, the area ratio. The tables contain several phenolic materials, several carbon=carbon, thickness results from a thermomechanical behavior computation. • The thicknesses of the Internal Protection are tabulated versus exposure time and pressure, the material (ablation rate, density, diLusivity, etc.) for a generic kind of propellant grain (Finocyl, Axial slots, etc.). • The inner bore diameter results from an interpolation in a database taking into account the grain mechanical behavior (thermal stresses, Ering) and the internal Iow. • The speciEc impulse (Fig. 7) is computed using a semi-empirical method like the method developed by Landsbaum, Salinas and Leary [5]. The main diLerence is that, in our method, the shape of the nozzle is accurately taken into account (database on the losses function of e ; ; ). The shape parameters are e the mean angle of the nozzle, the deIection of the nozzle (diLerence between the maximum angle and the minimum angle of the divergent wall), the area ratio.
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Fig. 9. Example of grain design.
So the next step will be using speciEc computer codes (Fig. 9), the design of the grain and the choice of the initial shape (Finocyl, conocyl, star, radial slots, etc.) making a trade-oL between best performance and the industrial knowledge of the company in charge of the grain manufacture: • available propellants (mechanical properties, range of burning rate, etc.), • technical constraints (background, manufacturing tools, etc.). Fig. 8. PROPULSE methodology.
So the practical speciEc impulse results from the removal from the ODE speciEc impulse of aerodynamical losses and of other empirical determined losses function of aluminum amount, e ; , throat diameter. These terms come from a statistical analysis on the results of more than 200 SRM Erings. • The throat erosion (C=C, graphite, or pyrographite) is obtained using the following formula: 1:9 ; d where k1 is a characteristic coeJcient of the throat material and k2 is an erosivity coeJcient function of the propellant. 1:35 0:3 P = k1 k2 pm col tCu ×
With all these data (Fig. 8) with a set of derivatives (dCu =dMp ; dCu =dMi ; dCu =dlsv ; dCu ; dAs ; etc.), with the gimballing required angle ( max ), with the burning time obtained from the trajectory analysis (MAXTOM), the computer code will optimize for each stage the main characteristics: for a given propellant mass: MEOP, outer diameter (if not imposed), geometry of the nozzle, etc. After the Erst MAXTOM=PROPULSE loop the requirements of optimal thrust laws (translated in terms of a pressure versus time) are obtained for the Erst and the upper stage. For an intermediate stage the pressure has to be as constant as possible.
Presently, we have some simpliEed models for the cost of SRMs. Due to the fact that the cost optimization is not yet taken into account in MAXTOM, we did not implement them in PROPULSE. DiLerent possible thrust laws are designed (corresponding to a feasible grain) and become an input for MAXTOM that has to choose and adjust the results of the Erst loops. 4. EXAMPLE ON A SMALL LAUNCHER WITH SOLID PROPELLANT STAGES
The overall architecture optimization on a small launcher was made with the following basic hypothesis: • a three-stage launcher (as the result of a good performance=cost eLectiveness), • use of an HTPB propellant 86-18-14 (ARIANE 5 type) with a possible exception for the upper stage, • Elament wounded case (carbon=epoxy), • grains: Enocyl or star grain for the lower stages, tubular or slotted for the upper stage, • nozzle with conventional downstream pivot point Iexseal for the Erst stage, upstream pivot point Iexseal if the grain is under the 10 tons of mass, • increased margins and sizing coeJcients (1.7 for the lower stages, 1.55 for the upper stage), • maximum MEOP: 10:0 MPa, • three conEgurations PX, PY, PZ (optimal) compared to PX, PX, PY and PX, PY, PY for payload in the range 300 –1200 kg on LEO. In the case of an architecture with two stages of
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Fig. 10. Comparison of architectures. Fig. 12. Comparison of optimal and real Erst stage motor Iow rate laws. Table 1.
PCu Max.dyn.P. Dyn.P.sep. max
Fig. 11. Evolution of the recurring cost vs payload mass.
the same propellant mass, the two stages PX or PY have to be strictly identical. 4.1. Choice of the architecture
The Erst choice was the choice of architecture PX, PY, PZ or PX, PY, PY. The study shows that in terms of performance a conEguration with two identical stages has a lower performance than an optimal one (PX, PY, PZ) but the diLerence is not so great (Fig. 10). In terms of costs (development and recurring costs), two identical stages lead to save money: • only two stages to develop instead of three, • series eLects very sensitive even with a small series. For smaller payloads, the best architecture is PX, PY, PY and for bigger payloads, the best architecture is PX, PX, PY. Figure 11 shows the evolution of the cost (price of the kilogram of payload in orbit); the diLerence (less than 10%) is easily compensated by the series eLect only.
Pure Finocyl
Star=Finocyl
0 38,000 4100 2.7
+100 50,000 4000 4.8
For example, putting 800 kg on LEO, P40 . P40 . P7 is equivalent to a P51 . P19 . P6. For a 600 kg, there are three possibilities: P42 . P13 . P4, P33 . P33 . P5 and P51 . P7 . P7. If stages are available from the shelf a conEguration with three diLerent stages is always better in terms of cost=performance eLectiveness. 4.2. Optimization of the grains
On a selected conEguration with a total propellant mass of approximately 110 tons (P50 . P50 . P8) a study of the grain was done: 4.2.1. First stage: PX. After comparative studies leading to rejection of pure star and radial slots grains, a comparison was done between aft Finocyl grain and star=Finocyl grain. Figure 12 shows that the optimal law issued from MAXTOM is diJcult to obtain taking into account the industrial possibilities (propellant, grain shape, etc.). In this case, the use of an existing propellant leads us not to respect the optimal law, specially in terms of combustion time, due to combustion rate and maximum propellant thickness. Comparison in terms of performance on a 700=700 km LEO is given in Table 1. 4.2.2. Upper stage: PZ. The choice was between a tubular grain and diLerent kinds and numbers of radial slots.
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Fig. 15. InIuence of second stage motor nozzle angle.
Fig. 13. InIuence of grain design on mass Iow rate law.
Fig. 16. P50 design (Erst and second stages).
Fig. 14. InIuence of second stage motor nozzle length.
Tubular grain (the cheapest one) provides too much acceleration (Fig. 13); in fact the selection was done only according to industrial criteria; in such a stage, the thrust law can be easily tailored to meet the acceleration requirement. 4.3. Optimization of the nozzle
In order to increase the performance of the launcher, some studies were performed on the motors. For instance, if the nominal architecture has two SRMs PX–PX, we look at the length of the nozzle of the second stage. The skirt mass variation is taken into account in this study. One can see in Fig. 14 that longer the nozzle, bigger is the payload. Remember that in this case, we do not have the constraint of Iow separation, because it is a second stage. The same study is also performed on the mean angle of the nozzle. We can see below that the nominal value close to 20 deg is Erst very near the optimal value and then that the result in terms of payload does not vary signiEcantly (see Fig. 15), when the mean divergent angle remains close to the optimal value, but strongly decreases for values less than 19 or greater than 22.
Fig. 17. P7 design (third stage).
Finally, we obtain the SRMs shown in Fig. 16 and 17 (nominal design), it means: • P50=P50: same conEgurations for the two Erst stages with a Finocyl grain and • P7: conEguration with downstream pivot point with radial slots grain. 5. CONCLUSION
EADS-LV began to develop optimization tools 20 years ago. Today, the tools are fully operational and validated for industrial projects. The tools have been conceived for preliminary design of launchers, so they are fast and eJcient, but they do not fully take into account manufacturing problems. For the design of the grain, a special study loop is always needed to be compliant with the industrial capabilities of manufacturers. For the Erst stage, the optimal thrust law required by trajectory optimization is, from this point of view, sometimes a diJcult problem to solve.
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1. Astorg, J. M., Calabro, M. and Donguy, P., The European small launcher: SRM technical choices. IAF 94-S2-4-09. 2. Trarieux, D., Astorg, J. M., Grosicchio, G. and Gros, J. P., Propulsion in space transportation, May 1996 Solid Propulsion for Small Launchers. AAAF 11-19.
3. Girard, H., Magne, J., Simultaneous Optimization of Vehicle Trajectory and Design under Controllability Constraints. AIAA 94-4405. 4. Weingarten, V. I., Buckling of thin wall circular cylinders. NASA SP 8007. 5. Landsbaum, Salinas and Leary, Journal of Spacecraft, 17(5) Paper 80-4096.
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Acta Astronautica Vol. 50, No. 4, pp. 201–208, 2002 ? 2002 International Astronautical Federation. Published by Elsevier Science Ltd. All rights reserved. Printed in Great Britain S0094-5765(01)00164-3 0094-5765/02/$ - see front matter
OPTIMIZATION OF THE PROPULSION FOR MULTISTAGE SOLID ROCKET MOTOR LAUNCHERS† M. CALABRO‡, A. DUFOUR and A. MACAIRE EADS-Launch Vehicles, 66, Rte de Verneuil, 78133 - Les Mureaux, France (Received 25 August 1999)
Abstract—Some tools focused on a rapid multidisciplinary optimization capability for multistage launch vehicle design were developed at EADS-LV. These tools may be broken down into two categories, those related to propulsion design optimization and a computer code devoted to trajectories and under constraints optimization. Both are linked in order to obtain optimal vehicle design after an iterative process. After a description of the two categories tools, an example of application is given on a small space launcher. ? 2002 International Astronautical Federation. Published by Elsevier Science Ltd. All rights reserved
1. INTRODUCTION
EADS-LV has been designing launchers for many years, so computer codes focused on a rapid multidisciplinary optimization had to be developed. From computer codes dedicated to weight and sizing of solid rocket boosters, a special one was conceived and written in order to optimize the main parameters of a solid rocket motor (SRM) under constraints [1]. The propulsion issues are a catalog of SRMs with various propellant masses, used as data by the trajectory optimization code. Let us look at the scheme in Fig. 1. From derivatives based on know-how, and technical data, geometrical constraints as well as mission requirements, the tool named PROPULSE Erst delivers a catalog of motors [2]. The trajectory optimization computer code MAXTOM will then optimize the staging of the launcher, the thrust law shape of the Erst stage and of the upper stage, burning time of all stages and Enally will give a new set of derivatives. The optimal thrust law shape will be used to search with dedicated internal ballistic computer codes, the better grain design versus thrust law requirements and a Erst complete optimization is
†Paper IAF97-S.l.07 presented at the 48th International Astronautical Congress, October 6 –10, 1997, Turin, Italy. ‡Corresponding author. 201
done; issues of propulsion tools are re-injected as data for a next optimization loop. So the global process may be described as indicated in Fig. 2.
2. TRAJECTORY ANALYSIS AND VEHICLE OPTIMIZATION : MAXTOM
2.1. General description of the tool (inputs=outputs)
This tool is based on the conventional architecture of an optimization program. The optimizer (MAX) is separated from the simulation tool (TOM) that takes into account the characteristics of the launcher. This architecture (Fig. 3) provides capabilities to change the optimizer in order to compare other programming algorithms. 2.1.1. Optimizer characteristics. The optimizer is based on a parametric optimization with gradient method using partial derivatives. These derivatives are assessed from some evaluations of the function with appropriate increment on input optimization parameters. This method implies that the parameters, described below, may be changed continuously on their intervals. In fact, in order to work properly, the constraints and the criterion function must have continuous derivatives versus the optimization parameters. This requirement has some consequences on the interfaces with other tools, specially with the propulsion analysis program.
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• the inert mass ejected law, • the speciEc impulse evolution.
Fig. 1. Propulse code inputs and outputs.
and some additional data from propulsion design (weight, nozzle area, etc.) and from launcher conEguration: aerodynamic coeJcients, atmospheric model, mission information). MAXTOM has to be adapted in order to use the propulsive optimization tool PROPULSE. The aerodynamic coeJcients and the reference area (Sref ) have to be changed as well, if the shape of the launcher or simply the diameter is modiEed. These aerodynamic coeJcients are coupled with the volume and the density of the optimized staging at each iteration of the algorithm and designed thanks to the semi-empirical formulae described in [3]. 2.2. Constraints description
We will distinguish below the constraints usually required on the determination of the optimal trajectory for a given launcher and the ones that had to be added in order to introduce propulsive characteristics in the optimization process. 2.2.1. Usual constraints. Four topics are con-
cerned by the constraints: Fig. 2. Maxtom and propulse coupling.
• the trajectory deEnition concerning the orbital parameters reached at the end of each stage (apogee, perigee, etc.), • the visibility of the launcher from ground stations, • the locations of the impact of re-entry stages (bound to fall in the sea in general), • the limits on dynamic pressure, thermal Iux, dynamic pressure at the separation of the Erst stage, etc.
Particular constraints are used for optimizing the propulsion characteristics of the launcher. If the propulsive characteristics are modiEed, some special treatments have to be done: 2.1.2. Simulator characteristics. The three degreesof-freedom equations of motion are numerically (1) for the disengagement of the launcher at integrated using Runge–Kutta algorithm. The lift-oL, a speciEcation of non-collision imlauncher is considered as a point mass; Earth plies in general a constraint on the accelrotation and oblateness are modeled and a U.S. eration and so a constraint on the thrust standard atmosphere is used. The atmospheric level=total mass ratio at lift-oL, drag and lift and the propulsive forces are applied (2) constraints should be met on control: modon the launcher (the rotational motion around the ifying the propulsive characteristics has center of mass is supposed to have little impact on an impact on controllability. This modiEthe trajectory). cation concerns especially the maximum Basically, the simulator takes into account for gimballing angle of each nozzle but also one stage: the thrust level. Fig. 3. MAXTOM architecture.
• geometrical data, • the SRMs mass Iow rate law,
(3) constraints on general loads: For a given sizing wind, the maximum normal force
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203
should be constrained in order to be coherent with the structural deEnition (basically, the constraint is a maximum dynamic pressure). (4) constraints on maximum acceleration with regards to payload. 2.3. Parameters description
As for the constraints, some modiEcations have to be carried out to take into account the modiEcation on propulsive characteristics. 2.3.1. Usual parameters. Three types of parame-
Fig. 4. Optimal mass Iow rate law research.
ters are usually necessary:
• attitude law parameter, that is the characteristic of the evolution of azimuth and Iight path angle of the launcher, • durations of ballistic phases if it turns out to be interesting, • particular parameters used for optimizing the propulsive characteristics. A propulsive stage is characterized by the following main parameters: • Fi =Fm ratio (initial thrust=mean thrust) which has an implication on the shape of the law correlated to SRM mass, • combustion time, • propellant mass, • tail-oL duration, • nozzle area ratio, • external diameter, • maximum pressure (MEOP). These parameters are obviously correlated. The global optimization process is designed to take into account the internal constraints of the propulsive stage and the external implication on the characteristics of the launcher (the aerodynamic reference area, etc.). From a reduced number of parameters, a multivariable interpolation is performed to rebuild what is necessary to compute the trajectory (Iow rates and speciEc impulse laws — see above). For this process, the propulsive analysis tool PROPULSE generates a “catalog” of stages corresponding to a class of stages optimized for the general conEguration of the launcher. 2.4. Outputs of the tool
The propulsive catalog is made from the partial derivatives of the total mass at lift-oL versus the mean Iow rate and mean speciEc impulse which are the output of the MAXTOM tool.
This optimization was for example made for the ARIANE 5 boosters (EAP); about 15 catalogs were calculated. For the preliminary design study, the problem was more to deEne a thrust template rather than the nominal law itself. The mass Iow rate law obtained after calculation is optimized by segments (Fig. 4). 3. SRM SIZING AND OPTIMIZATION PROCESS
The inputs for the sizing and the optimization of solid rocket motors are mainly: • the propellant mass, • the outer diameter, • the maximum gimballing angle, and sometimes the overall length or other parameters. At Erst, inputs are diLerent depending on the category of the stage: • Erst stage is provided with a nozzle taking into account a Iow separation criterion for the nozzle operating at ground level and with a special thrust law shape (use of previous study results), • upper stage with a decrease in the last part of the thrust law to be compliant with the maximum acceleration requirement, • intermediate stage with pressure versus time as constant as possible (optimization of the performance of the stage). For each of these categories, the design takes into account an a priori adapted design of the grain characterized by a Paverage =MEOP ratio. The choice of technologies has to be made among the following possibilities: • wounded or metallic case, • upstream or downstream pivot point Iexseal,
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Fig. 5. Upstream pivot point Iexseal.
• extendible exit cone (one or two rings, panels, etc.) or not, • conventional nozzle (carbon phenolic with carbon=carbon ITE) or completely thermostructural nozzle.
Fig. 6. Downstream pivot point Iexseal.
DiLerent materials may be used. After a selection of design rules (safety coeJcients, margin, etc.) the SRM will be designed and sized to determine the mass budget and the speciEc impulse (and also CG and inertiae versus time). • The case is always computed directly using a simple method: — metallic case: stress computation with experimental corrective factors depending on the chosen technology, — wounded case: “MOethode en Elet” giving the Elament=epoxy mass, the design of polar bosses and their masses come from the consultation of a database made with the results of Enite elements computations. The mass itself is obtained by a polynomial formula. • The Iexseal (Figs. 5 and 6) is also computed using a semi-empirical method: use of an analytical computation of stresses under MEOP without deIection and experimental corrections to take into account the stresses due to the gimballing angle (the shims can be metallic or composite). • The Exed housing is directly computed (buckling and internal pressure) [4]. • The sizing of the other components needs to create databases. • For the more frequently used composite propellants (HTPB, CTPB, PBAN, etc.) the database gives: CD (Iow rate coeJcient), density, combustion temperature, range of burning rate, burning rate exponent, mechanical property, ODE speciEc impulse, etc.
Fig. 7. Accuracy of the Isv calculation.
• The thicknesses of the thermal protection of the nozzle are tabulated functions of the propellant, the materials, the area ratio. The tables contain several phenolic materials, several carbon=carbon, thickness results from a thermomechanical behavior computation. • The thicknesses of the Internal Protection are tabulated versus exposure time and pressure, the material (ablation rate, density, diLusivity, etc.) for a generic kind of propellant grain (Finocyl, Axial slots, etc.). • The inner bore diameter results from an interpolation in a database taking into account the grain mechanical behavior (thermal stresses, Ering) and the internal Iow. • The speciEc impulse (Fig. 7) is computed using a semi-empirical method like the method developed by Landsbaum, Salinas and Leary [5]. The main diLerence is that, in our method, the shape of the nozzle is accurately taken into account (database on the losses function of e ; ; ). The shape parameters are e the mean angle of the nozzle, the deIection of the nozzle (diLerence between the maximum angle and the minimum angle of the divergent wall), the area ratio.
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Fig. 9. Example of grain design.
So the next step will be using speciEc computer codes (Fig. 9), the design of the grain and the choice of the initial shape (Finocyl, conocyl, star, radial slots, etc.) making a trade-oL between best performance and the industrial knowledge of the company in charge of the grain manufacture: • available propellants (mechanical properties, range of burning rate, etc.), • technical constraints (background, manufacturing tools, etc.). Fig. 8. PROPULSE methodology.
So the practical speciEc impulse results from the removal from the ODE speciEc impulse of aerodynamical losses and of other empirical determined losses function of aluminum amount, e ; , throat diameter. These terms come from a statistical analysis on the results of more than 200 SRM Erings. • The throat erosion (C=C, graphite, or pyrographite) is obtained using the following formula: 1:9 ; d where k1 is a characteristic coeJcient of the throat material and k2 is an erosivity coeJcient function of the propellant. 1:35 0:3 P = k1 k2 pm col tCu ×
With all these data (Fig. 8) with a set of derivatives (dCu =dMp ; dCu =dMi ; dCu =dlsv ; dCu ; dAs ; etc.), with the gimballing required angle ( max ), with the burning time obtained from the trajectory analysis (MAXTOM), the computer code will optimize for each stage the main characteristics: for a given propellant mass: MEOP, outer diameter (if not imposed), geometry of the nozzle, etc. After the Erst MAXTOM=PROPULSE loop the requirements of optimal thrust laws (translated in terms of a pressure versus time) are obtained for the Erst and the upper stage. For an intermediate stage the pressure has to be as constant as possible.
Presently, we have some simpliEed models for the cost of SRMs. Due to the fact that the cost optimization is not yet taken into account in MAXTOM, we did not implement them in PROPULSE. DiLerent possible thrust laws are designed (corresponding to a feasible grain) and become an input for MAXTOM that has to choose and adjust the results of the Erst loops. 4. EXAMPLE ON A SMALL LAUNCHER WITH SOLID PROPELLANT STAGES
The overall architecture optimization on a small launcher was made with the following basic hypothesis: • a three-stage launcher (as the result of a good performance=cost eLectiveness), • use of an HTPB propellant 86-18-14 (ARIANE 5 type) with a possible exception for the upper stage, • Elament wounded case (carbon=epoxy), • grains: Enocyl or star grain for the lower stages, tubular or slotted for the upper stage, • nozzle with conventional downstream pivot point Iexseal for the Erst stage, upstream pivot point Iexseal if the grain is under the 10 tons of mass, • increased margins and sizing coeJcients (1.7 for the lower stages, 1.55 for the upper stage), • maximum MEOP: 10:0 MPa, • three conEgurations PX, PY, PZ (optimal) compared to PX, PX, PY and PX, PY, PY for payload in the range 300 –1200 kg on LEO. In the case of an architecture with two stages of
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Fig. 10. Comparison of architectures. Fig. 12. Comparison of optimal and real Erst stage motor Iow rate laws. Table 1.
PCu Max.dyn.P. Dyn.P.sep. max
Fig. 11. Evolution of the recurring cost vs payload mass.
the same propellant mass, the two stages PX or PY have to be strictly identical. 4.1. Choice of the architecture
The Erst choice was the choice of architecture PX, PY, PZ or PX, PY, PY. The study shows that in terms of performance a conEguration with two identical stages has a lower performance than an optimal one (PX, PY, PZ) but the diLerence is not so great (Fig. 10). In terms of costs (development and recurring costs), two identical stages lead to save money: • only two stages to develop instead of three, • series eLects very sensitive even with a small series. For smaller payloads, the best architecture is PX, PY, PY and for bigger payloads, the best architecture is PX, PX, PY. Figure 11 shows the evolution of the cost (price of the kilogram of payload in orbit); the diLerence (less than 10%) is easily compensated by the series eLect only.
Pure Finocyl
Star=Finocyl
0 38,000 4100 2.7
+100 50,000 4000 4.8
For example, putting 800 kg on LEO, P40 . P40 . P7 is equivalent to a P51 . P19 . P6. For a 600 kg, there are three possibilities: P42 . P13 . P4, P33 . P33 . P5 and P51 . P7 . P7. If stages are available from the shelf a conEguration with three diLerent stages is always better in terms of cost=performance eLectiveness. 4.2. Optimization of the grains
On a selected conEguration with a total propellant mass of approximately 110 tons (P50 . P50 . P8) a study of the grain was done: 4.2.1. First stage: PX. After comparative studies leading to rejection of pure star and radial slots grains, a comparison was done between aft Finocyl grain and star=Finocyl grain. Figure 12 shows that the optimal law issued from MAXTOM is diJcult to obtain taking into account the industrial possibilities (propellant, grain shape, etc.). In this case, the use of an existing propellant leads us not to respect the optimal law, specially in terms of combustion time, due to combustion rate and maximum propellant thickness. Comparison in terms of performance on a 700=700 km LEO is given in Table 1. 4.2.2. Upper stage: PZ. The choice was between a tubular grain and diLerent kinds and numbers of radial slots.
Optimization of the propulsion
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Fig. 15. InIuence of second stage motor nozzle angle.
Fig. 13. InIuence of grain design on mass Iow rate law.
Fig. 16. P50 design (Erst and second stages).
Fig. 14. InIuence of second stage motor nozzle length.
Tubular grain (the cheapest one) provides too much acceleration (Fig. 13); in fact the selection was done only according to industrial criteria; in such a stage, the thrust law can be easily tailored to meet the acceleration requirement. 4.3. Optimization of the nozzle
In order to increase the performance of the launcher, some studies were performed on the motors. For instance, if the nominal architecture has two SRMs PX–PX, we look at the length of the nozzle of the second stage. The skirt mass variation is taken into account in this study. One can see in Fig. 14 that longer the nozzle, bigger is the payload. Remember that in this case, we do not have the constraint of Iow separation, because it is a second stage. The same study is also performed on the mean angle of the nozzle. We can see below that the nominal value close to 20 deg is Erst very near the optimal value and then that the result in terms of payload does not vary signiEcantly (see Fig. 15), when the mean divergent angle remains close to the optimal value, but strongly decreases for values less than 19 or greater than 22.
Fig. 17. P7 design (third stage).
Finally, we obtain the SRMs shown in Fig. 16 and 17 (nominal design), it means: • P50=P50: same conEgurations for the two Erst stages with a Finocyl grain and • P7: conEguration with downstream pivot point with radial slots grain. 5. CONCLUSION
EADS-LV began to develop optimization tools 20 years ago. Today, the tools are fully operational and validated for industrial projects. The tools have been conceived for preliminary design of launchers, so they are fast and eJcient, but they do not fully take into account manufacturing problems. For the design of the grain, a special study loop is always needed to be compliant with the industrial capabilities of manufacturers. For the Erst stage, the optimal thrust law required by trajectory optimization is, from this point of view, sometimes a diJcult problem to solve.
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