4.1. Sounding Rocket for Micro-Gravity Platform
One of the key features of HREs is their relatively low cost when compared to the achievable performance, making a hybrid powered sounding rocket a viable option to transport small payloads to high altitudes, where the absence of the atmosphere allows for micro-gravity conditions that can be experienced in free flight. In such kind of application, a suitable optimization merit function to be maximized is the time spent above 100 km (later called
), when the initial and payload mass,
respectively, are given. In this context, the authors considered a 500-kg sounding rocket as a case study [16
]. The initial mass includes a 100-kg payload, fixed masses, propulsion system and propellants. The optimal mass split between propellants and propulsion, the optimal grain geometry and the corresponding optimal trajectory are sought in order to maximize
. A single-port grain design and blow down feed system are adopted to reduce the complexity and the cost of the propulsion system. Several options are taken into consideration as propellant combination, namely HP/PE, LOX/HTPB and NOX/HTPB. The design variables are three when self pressurizing NOX is employed as oxidizer: the initial thrust level
, the initial mixture ratio
and the nozzle expansion area ratio E
. Erosion effects are here neglected, thus
. On the other hand, a total of five design parameters are required when HP and LOX are used: the initial tank pressure
and the ullage volume
alongside with the aforementioned
. The use of NOX fixes the initial tank pressure due to the liquid-vapor equilibrium (see Appendix A.1
for the details), whereas the ullage volume is fixed at 3% of the tank volume.
The results are reported in Table 6
, showing the higher performance of the HP/PE combination. The LOX solution exhibits a low initial mixture ratio, requiring a larger and heavier grain which increases propulsion system mass at the expense of the propellant mass on board. On the contrary, the use of the NOX combination results in worst performance due to its lower specific impulse, albeit interesting features in terms of overall rocket length and diameter are obtained thanks to the larger mixture ratio. Moreover, looking at Figure 4
, one can notice that the optimal initial thrust and thrust history are quite different depending on the propellants employed.
The complex influence of the design parameters on the thrust history (i.e., on the optimal trajectory) is well exemplified switching the initial thrust of HP/PE and NOX/HTPB. In these cases, presents a relevant reduction to 150 s for NOX/HTPB () and to 267 s for HP/PE (). Concerning the tank pressure histories, the NOX solution is able to maintain an high pressure level longer than HP and LOX, due to its self-pressurization capability, resulting in a more regular thrust level. Such behavior holds until the liquid phase exist inside the oxidizer tank ( s); later the thrust drops and one can observe a change in tank pressure derivative due to the residual gas phase which is fed into the combustion chamber acting as a propellant.
4.2. Hypersonic Accelerator
Hypersonic propulsion and the development of hypersonic-capable vehicles is a growing trend in recent years. The test conditions of such systems, subsystems and devices require Mach number greater than five at an altitude in the range 25–50 km. Thus, one more time, a HRE is a viable option for this kind of missions for the same reasons explained in Section 4.1
. Ref. [42
] considered the following test case: initial mass
fixed and equal to 1000 kg, payload mass
equal to either 100 kg or 200 kg and 30-km target altitude. HP/PE, single circular port and blow down are the considered propellants, grain design and feed system, respectively. The optimization aims at the maximization of the Mach number
at the target altitude. Despite the high altitude operation considered as target, the rocket spends a relevant amount of time at lower altitude, thus requiring an accurate modeling of its aerodynamic interactions (i.e., drag) in order to obtain significant results. A single stage design has been considered, which employs a single HRE, alongside with two different two stage designs: the first uses a single HRE in the first and second stage (1+1 configuration), whereas the second design employs two HRE in the first stage and a single engine in the second stage (2+1 configuration). To reduce development cost and system complexity, the same HRE is used for each multi-engine configuration. In this context, the design parameters are the initial thrust
, the initial mixture ratio
, the initial tank pressure
, the initial volume of pressurizing gas
and the nozzle expansion area ratio E
. The value of
is evaluated in order to grant a sufficient regression rate at burnout (i.e., avoid grain cooking), despite the blow down operation. Table 7
reports the optimal design and performance of the HRE.
To reduce propulsion system mass, small thrust-low tank pressure designs should be preferred; however such optimization strategy fails in the 100-kg single stage case and in the 2+1 case (both payload) because the resulting thrust level tends to become too small and unfeasible solutions with negative coast arcs would be obtained. Thus, the initial thrust is dropped from the set of design parameters for such cases and determined by the trajectory optimization procedure itself, allowing for a feasible solution. The results show that a hybrid powered rocket can effectively accelerate a payload in the range 100–200 kg to the hypersonic corridor. Looking at the last column of Table 7
, one can notice that, counter intuitively, the two stage design does not result in Mach number
higher than the single stage configuration (first two rows), with the exception of the 2+1 configuration when the lowest payload is taken into account (second-to-last row).
4.3. Mars Ascent Vehicle
HREs are particularly well suited for missions which require shut down and restart capabilities and are less sensitive to low temperature, with respect to liquid rocket engines. These characteristics make a hybrid powered Mars Ascent Vehicle (MAV) a viable alternative to the heritage options for such application. In particular, the use of LOX/Wax propellant combination is really promising, since the oxidizer could be produced in situ from CO2 in Mars atmosphere. In Ref. [43
] the authors analyzed a two-stage vehicle for the return of samples from Mars’ surface and for a manned mission. A cluster of engines is used in the first stage, whereas a single HRE powers the second stage. The same engine is used in both stages, in order to limit the development costs of the MAV. Both a simple blow down feed system and a more complex regulated system are taken into account as possible architectures. The number of engine design parameters is four, three of them being in common between the two feed system alternatives, namely initial mixture ratio
, initial tank pressure
and nozzle area ratio E
. The fourth parameter is the initial volume of pressurizing gas
, when the blow down feed system is employed, or the exhausted oxidizer mass during regulated operation
, when the regulated system is considered. The number of engines used in the first stage must also be selected, since different configuration are possible, in particular
are analyzed. The optimization aims at the maximization of the MAV payload
at given initial mass, namely 500 kg in the sample return mission and 60 tons in the manned mission. A reference target orbit of 170-km altitude is taken into consideration for both the test cases. The most significant results are reported in Table 8
and Table 9
for the sample return and manned mission, respectively.
Concerning the sample return mission, the maximum payloads are obtained with the use of two and three engines, when the simple blow down feed or the regulated system are respectively used. The use of a regulated system always improves the resulting payload mass. On the other hand, in the manned mission, the greatest payload is achieved when four engines are used in the first stage since the larger and sustained thrust allows for a more efficient trajectory and, alongside with the dry mass reduction, increases the payload mass. For the same reasons, the payloads of the regulated optimal solutions are always greater than the corresponding blow down ones, analogously to the sample return mission optimal solutions.
These results are comparable, in terms of payload, to those provided by traditional liquid rocket eninges envisaged for a Mars return mission. However, the low cost and simplicity of HREs makes the described launcher options worth of consideration.
4.4. Small Launcher
Since the early 2000s, the demand for small launch vehicles is constantly growing due to the spread of the small satellite market. Until recently, operative launcher especially designed for such application are an exception and HREs, due to their features, are a promising propulsion system for a small satellite launcher. For these reasons, in Refs. [44
] the authors investigated the optimal design of a hybrid-powered 5000-kg small launcher. LOX/paraffin-based wax is considered as propellant combination for its high regression rate, which allows for a simpler single port grain design. Concerning the feed system, a blow down type is taken into account. The launcher employs the same HRE in different number in each stage: six, three and one in the first, second and third stage, respectively. An airborne launch is considered at given altitude and speed (e.g., Pegasus Launcher).
The optimization is carried out from both a deterministic and a robust point of view, and the number of design parameters are respectively two (
) and five (
, total oxidizer mass boarded
). Different trajectory setups are taken into consideration, namely given initial thrust (called design A) and free initial thrust/fixed maximum acceleration (called design B). In the latter cases
is an additional design parameters which is optimized. Only the regression rate is regarded as uncertain in the robust-based optimization. The optimization aims at the maximization of the payload inserted in a given-altitude, circular and polar orbit. The optimal designs for the deterministic optimization are reported in Table 10
, whereas the time histories of their key performance indexes are shown in Figure 5
In this application, the optimal ascent trajectory prescribes a large acceleration for a suitably long time, in order to reduce gravitation and aerodynamic losses. Thus, the deterministic design A exhibits a large value of because its initial thrust is fixed, whereas the deterministic design B is free to follow a steep trajectory by means of a larger initial thrust despite a smaller volume of pressurizing gas. This behavior allows for an engine dry mass reduction, albeit a greater propellant load is required due to the less efficient trajectory and lower nozzle expansion area ratio E (i.e., lower specific impulse ), with respect to the deterministic design A. Thus, the deterministic design B is capable of inserting a 73.6-kg payload into the target orbit, i.e., a 25-kg increase with respect to the deterministic design A.
In the context of the robust-based optimization, the authors decided to maintain the same design philosophy, first seeking for a fixed initial thrust solution (here called robust design A) and then trying to find out an optimized thrust solution (here called robust design B). The hope was to obtain a performance improvement, analogously at what happened in the deterministic optimization. In the context of the robust-based optimization, the performance index to be maximized was
, a linear combination of the payload
inserted in the target orbit and the average altitude violation
due to uncertainty with respect to such orbit. A large value is here used for the linear combination coefficient in order to force the altitude violation to zero. A detailed definition of the robust-based merit function can be found in Ref. [46
]. The number of design parameters grows up to four when robust-based procedure is employed, namely the initial mixture ratio
, the grain outer radius
, the initial pressurizing gas volume
and the total oxidizer mass boarded. The optimal robust design A is reported in Table 11
and compared to the optimal deterministic design A. One can observe that
is equal to 500 km for the deterministic solution in the face of a 800-km target orbit, i.e., more than half of the considered uncertain parameters combinations lead to a rocket crash (insertion altitude null) when a proper robust-based design is not carried out. On the contrary, the ROB
solution exhibits a null altitude violation, granting robustness in the design, at the cost of a relevant payload reduction (−28% with respect to deterministic design A). To improve this result, a robust design B (ROB
) is sough, by freeing the initial thrust, which become an additional trajectory unknown, and adding a constraint on the final acceleration, as previously done in the deterministic procedure. This approach results in payload increases, with respect to ROB
optimal solution, of 10 kg and 20 kg, when maximum accelerations of 6 g and 5 g are imposed in the indirect trajectory optimization. However, the average altitude violation of these solution is not passably close to zero (131 km and 50 km when 6 g and 5 g are respectively used) and unacceptable in a robust-based optimization.
4.5. Upper Stage
In the context of small and low-cost launchers, SRMs have been used extensively used: one can consider the Pegasus launcher and the European Vega launcher as an example for two different payload class. In Pegasus all the stages are powered by SRMs. However, the lack of shut down/restart capability of SRMs imposes the use of at least one stage for each burn required by the ascent trajectory, whereas the absence of throttling determines relevant scattering in the target orbit insertion parameters (e.g., the altitude). Moreover, SRMs are characterized by a lower specific impulse than their liquid counterparts, which results in penalties in the final part of the ascent. These issues are solved in Vega by employing an additional small liquid-powered stage, which is devoted to the final orbital maneuvers but results in higher costs and worse structural efficiency. HREs represents an alternative and viable solution for an upper stage in this context, because they are able to deliver the high performance of a SRM and the fine control features of the aforementioned additional small LRE, but at a lower cost. Furthermore, the use of typical hybrid propellant combinations avoids safety issues related to the chemicals used in the LRE final stage, such as Nitrogen TetrOxide (NTO) and MonoMethylHydrazine (MMH).
For these reasons, the authors focused on the design and optimization of a hybrid powered upper stage suitable for the replacement of the third and fourth stages of a launcher based on the characteristic of the European Vega Launch Vehicle (VLV). The optimization aims at the maximization of the payload inserted into a 700-km polar orbit with a launch from Kourou. A baseline value for the payload mass using three solid-propellant stage and a liquid-powered upper stage is 1430 kg, consistent with the performance of the Vega [47
]. The initial mass of the launcher is given and the lower stages are unmodified.
Back in 2010 the authors approached to this topic for the first time [48
]. The trajectory is optimized from lift-off to orbit insertion and the same constraints of the reference launch are taken into account, such as heat flux limitation after fairing jettisoning. The considered propellant combination is HP/PE, which requires a multiple-port grain design in order to deliver the required high thrust levels. Several options for grain geometry and feed system are compared. The engine design parameters are four, namely the initial thrust
, the initial mixture ratio
, the initial tank pressure
and the nozzle area ratio E
(erosion effect are neglected). The initial ullage volume
(blow down case only), the oxidizer mass at the start of the blow down and its final value (regulated case only), and the grain outer radius
are additional unknowns of the ascent trajectory optimization. Figure 6
depicts tank pressure, thrust and mixture ratio histories for a blow down and a partially regulated feed system. These results are obtained assuming eight quadrilateral ports for the fuel grain.
The use of the blow down feed system results in a large thrust level variation during engine burn. Thus, a really high initial tank pressure is mandatory, in order to maintain a sufficient thrust until the end of the engine burn, also avoiding an increase in pressurizing gas mass. A performance improvement is obtained employing a partially regulated feed system: the introduction of an auxiliary pressurizing gas tank is balanced, in terms of engine dry mass, by the reduction of the oxidizer tank mass due to the lower pressurization. However, a stronger mixture ratio shifting penalizes the performance of the HRE during the constant pressure operation, with respect to the blow down optimal solution. Further performance improvement are achievable employing an electrically driven turbo-pump feed system [29
]. The effect of grain geometry (number of ports and shapes) on launcher performance has been already described at the end of Section 2.3
In 2014, the authors considered alternative propellant combinations for the same upper stage application, namely HP90%/HTPB, LOX/HTPB and LOX/paraffin-based wax [33
]. Moreover, the authors took into account nozzle throat erosion effects by means of the Bartz method (see Appendix D
). The use of paraffin-based wax allows for a single port grain design, thanks to its relatively high regression rate value. The optimal solution presents a payload increase of about 350 kg with respect to the best performing conventional propellant combination, as shown in Table 12
, being at the same time only slightly affected by performance deterioration due to throat erosion.
In 2015 the authors started to shift their attention from design trade-off between different HRE configuration, regarded strictly from a deterministic point of view, to uncertainty based approaches in the context of the optimization of the aforementioned upper stage. The first attempt to perform a robust-based optimization is presented in Ref. [11
]. Regression rate correlation coefficient a
and exponent n
are varied around their nominal values for the optimal solutions and off-design launcher performance are evaluated. The results show that the mission is unfeasible when the actual regression rate is greater than the nominal one, because a relevant amount of unburned oxidizer remains as an inert mass after fuel grain burn out. On the other hand, off-nominal launcher performance of the optimal design are satisfying when a regression rate larger than nominal is assumed during the optimization procedure. In this case the average altitude is increased and the standard deviation reduced (from 250 km to 12 km).
This management of the intrinsic uncertainties represents a preliminary approach to robust design, which the authors improved in Ref. [49
] embedding uncertain parameters variations inside the optimization procedure, in place of an “a posteriori” check of optimal solutions off-design performance. Thus, a new optimization merit function is introduced, combining linearly the payload mass, which is unaffected by the uncertainties, with the average altitude violation with respect to target altitude, which, on the contrary, is affected by uncertainty because the actual insertion orbit depends on the actual value of the regression rate. The number of design parameters rises due to the robust-based formulation of the optimization problem. In the deterministic approach oxidizer and fuel masses are such that they end at the same moment at burnout and are optimized (i.e., minimized) by the indirect procedure to reach exactly the target orbit. On the other hand, in the robust-based optimization, the actual value of regression rate varies resulting in three possible situations:
the regression rate is higher than nominal; in this case all the fuel burns whereas a certain amount of unburned oxidizer remains in the tank;
the regression rate is lower than nominal; in this case all the oxidizer is exhausted whereas a fuel sliver is present in the combustion chamber;
the regression rate is the nominal one; in this case fuel and oxidizer end at the same time.
The surplus in fuel/oxidizer mass required to reach (at least) target orbit, even in the higher/lower than nominal operation, have to be fixed before trajectory optimization, in order to compute launcher payload and define its design. Thus, the fuel mass
(given by grain outer radius
only for circular port grain,
and web thickness w
for multiport grain design), and the total oxidizer mass boarded are no longer trajectory unknowns but become engine design parameters. These surplus in propellant masses result in payload penalties for the robust optimal solutions with respect to deterministic one. Obviously the robust designs are able to reach higher altitude than the target when nominal regression rate occurs. The results prove that robustness in the design can be achieved at the expense of a small payload reduction of 82 kg, i.e., less than 5% of the launcher payload, with respect to the deterministic optimal solution for the same grain geometry, propellants and feed system. Figure 7
reports the thrust and mixture ratio histories for the deterministic and optimal design. The robust design shows a slightly larger thrust magnitude variation and mixture ratio shifting during engine burn, due to the different grain geometry which causes enhanced changes of port and burning area. The obtained robust optimal solution is able to grant the fulfillment of launcher mission goals (i.e., 100% target orbit reaching) despite the presence of regression rate uncertainties. In Ref. [30
] this robust-based approach is applied to a slightly different upper stage architecture, which employs electrically driven pump to feed the oxidizer into the combustion chamber. Mission goals and all the other aspects of the problem are unmodified. The results show that this feed system option is viable, even when uncertainties in the design are taken into account. The use of electric systems is able to grant a relevant performance boost with respect to conventional gas pressurized feed system, which in terms of payload provides a 12% increase (+250 kg).
In the robust-based optimization new core aspects become more relevant, namely the number of uncertain parameters and the algorithm to be used to lead the engine design optimization. The first determines the number of uncertain parameters combination that have to be considered in the robust-based procedure, i.e., the number of trajectory integration required to compute one value of the optimization merit function. The latter is devoted to the choice of the engine designs to be evaluated during the optimization in order to reach an optimal solution. Thus, both this aspects have a huge impact on the computational effort of the whole robust-based optimization method. For these reasons, the authors in Ref. [35
] proposed a comparison of the performance of different optimization algorithm, namely a Genetic Algorithm (GA), a Particle Swarm Optimization algorithm (PSO), a hybrid genetic-Taguchi algorithm and an Iterated Local Search (ILS). The optimization of the hybrid powered upper stage is the test case, one more time. PSO and ILS outperformed the other methods, showing similar performance, despite their completely different nature: the first is a population-based algorithm, whereas the second employs a meta-heuristic approach.
In Ref. [40
] the authors described an approach, which takes advantage of a sensitivity analysis performed by means of Morris Method, to select the model parameters to be regarded as uncertain [41
]. Details about this technique are reported in Section 3.1
. The results show that even small variations of six model parameter (regression rate correlation coefficient a
and exponent n
, throat erosion synthetic parameter
, fuel density
, initial tank pressure
and initial throat to port area ratio J
) have a relevant impact on system performance and thus have to be considered as uncertain in the upper stage test case. In terms of payload, this uncertainty model determines a 12% payload reduction, with respect to the deterministic optimal solution obtained for the same grain geometry and propellants. This reduction is more than twice the one imposed by the simpler uncertain model, which considered only the regression rate uncertainties, mentioned in the previous paragraph, but it is worth taking the robustness gain into account. For the sake of comparison, Table 13
reports the normalized mass ratios for the reference and the robust optimized solutions.