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In this paper, the effects of different grain shapes of a hybrid rocket motor (HRM) and different payload mass/orbit heights on the design of small launch vehicles (SLVs) are systematically discussed. An integrated overall design model for the hybrid rocket motor-powered small launch vehicle (HPSLV) is established, and two groups of three-stage SLVs capable of sending small payloads to the low earth orbit (LEO) are designed and optimized. In the first group, the SLVs with different grain shapes and different numbers of chambers in HRMs at the 1st and the 2nd stages are optimized and analyzed. In the second group, the SLVs capable of sending different payload mass to different orbit heights are optimized and analyzed. Pareto graphs of the design results show that the design of HRM at the 1st stage has the greatest impact on the take-off mass, total velocity increment, and maximum axial overload of the SLV. Self-organizing maps show that the take-off mass, maximum diameter, overall length, and velocity increment of the SLVs have the same variation tendency. For the 1-chamber HRM at the 1st stage, the wheel-shaped grain is better than circle-shaped and star-shaped grains in terms of reducing the total mass and length of the SLV, and the 4-chamber parallel HRM has more advantages over all 1-chamber designs for the same reason. The theoretical velocity increments are calculated by the Tsiolkovsky formula, and the actual velocity increments are obtained based on the trajectory simulation data. The results indicate that the HPSLV has a regular distribution in terms of the ratio of theoretical (actual) velocity increments at three different stages, and the estimated distribution ratio is around 1 : 1.55 : 1.69 (1 : 1.9 : 2.39), which can provide some reference for future development of HPSLV.

With the increasing demands for low earth orbit (LEO) payloads, microsatellites are paid much attention in recent years. As a result, these satellites (especially formation-flying satellites [

A good propulsion system for SLV can further reduce cost, improve reliability, and increase payload capacity at the same time. The hybrid rocket motor (HRM) is such an excellent propulsion system. Figure _{2}O/paraffin HRMs and proved the efficiency by comparing the proposed model with hot-fire test data [

Framework of the HRM working principle.

Various applications of HRM in transportation systems are also explored around the world. The application of HRMs in manned lunar landing has been studied, and the uncertain factors during the design phase are also considered [_{2}O_{2}/paraffin is used for the air/ground-launched SLV to send a 20 kg payload into the 300 km circle orbit [_{2}O_{2}/HTPB propellant [

However, most of the applications above focus on the overall design of HRM-powered LVs. In fact, the working process of HRM follows the law of diffusion combustion, and the internal ballistic performance of the motor depends on the mass flux of a liquid oxidizer, which leads to a strong coupling relationship between the fuel regression rate, oxidizer mass flux, and combustion chamber pressure during the operation time. This mechanism makes the design of HRM more complicated. Moreover, there are different shapes of fuel grains and chamber configurations, which lead to different flight characteristics of the SLV. To obtain a comprehensive understanding of the design criteria of the HRM used in SLV, the design parameters for different types of HRMs need to be compared. Thus, the HRM performances are simulated in this paper, and the overall scheme of the HRM-powered SLV (HPSLV) is designed and optimized. The impacts of design parameters on SLV overall performances, the influence of different grain shapes on the 1st- and 2nd-stage propulsion performances, and the relationship between the desired orbit height/payload mass and the velocity increment of HPSLV are studied. This paper is devoted to revealing the characteristics of the HRM used in SLV.

This paper is organized as follows. In Section

In this study, three-stage SLVs with tandem configuration are designed to send small payloads (from 100 kg to 200 kg) to the LEO (from 300 to 700 km circle orbit). The baseline configuration of SLV is shown in Figure

SLV configuration.

The main parts of the SLV are the three stages of HRM. Therefore, a preliminary design code of typical HRMs, which consist of an oxidizer feed system and several (one or four) thrusters with solid fuel grains, is developed. The main propulsion performances of each HRM, such as the propellant mass, specific impulse, and thrust, are computed by this code, as shown in Figure

Framework of the propulsion system design module.

In this study, three types of fuel grain are considered: the circle-shaped grain, star-shaped grain, and wheel-shaped grain, as shown in Figure

Parameters of different grain shapes: (a) circle-shaped grain; (b) star-shaped grain; (c) wheel-shaped grain.

Definition of initial grain shape parameters.

Grain shape | Burning perimeter length | Grain port area |
---|---|---|

Circle-shaped grain | ||

Star-shaped grain | Related to the outer diameter | |

Wheel-shaped grain | Related to the outer diameter |

A 98% mass concentration of H_{2}O_{2} and a solid fuel, which consists of 60% of HTPB, 28% of Al, 10% of Mg, and 2% of carbon, are selected as the propellant combination in this study. A thermal calculation code is used to obtain the performance parameters of the propellant combinations. These parameters, such as the specific impulse

At the initial time of motor working, the flux of propellant, grain size, and area of nozzle throat can be all determined by the initial force

As a critical parameter affecting the internal ballistic performance, the regression rate of the fuel grain depends on the oxidizer mass flux

Along with the process of grain shape design, thermodynamic calculation provides necessary values needed in the phase of internal ballistic simulation, which is conducted to calculate the internal ballistic parameters and main dimensions of the thruster [

The purpose of this phase is to obtain the HRM performance parameters, such as the

A series of long-time working HRM firing tests is conducted [

Experimental and predicted curves: (a) chamber pressure; (b) thrust.

Chamber pressure

Thrust

By the calculation of propulsion, HRM performance parameters are exported as outputs, which provides the parameters needed in calculation of the mass and dimension of HRM.

The rocket take-off mass is a sum of the mass of all components. As shown in Figure

The main components of an HRM are divided into two parts: solid part (solid fuel grain, chamber, nozzle, etc.) and liquid part (liquid oxidizer, oxidizer tank, gas bottle, valves and tubes, etc.), as shown in Table _{2}O_{2} is stored in the oxidizer tank and transported into the chamber by a turbopump feed system or a gas pressure feed system. When it comes to the gas pressure feed system, the gas movement in the gas bottle is considered an adiabatic expansion process, and that in the oxidizer tank is considered a constant pressure process. The mass of pressurizing gas and the volumes of the oxidizer tank and gas bottle are obtained according to the first law of thermodynamics, the mass conservation law, and ideal gas state equation. The chamber is used to pack fuel grain, and the outer diameter of a single chamber consists of a diameter of fuel grain, thickness of heat insulating layer, and thickness of a chamber shell. A catalyst bed is also adopted in this study, which is used to decompose H_{2}O_{2} into oxygen and water gas with high temperature to enhance combustion efficiency.

Main structure of the SLV propulsion system.

Main structure | Material | Function |
---|---|---|

Liquid oxidizer | 98% H_{2}O_{2} | Provide chemical energy |

Solid fuel | HTPB-based grain | Provide chemical energy |

Gas bottle | CF wound aluminum liner | Contain pressurization gas |

Oxidizer tank | CF wound aluminum liner and oxidizer sac | Contain liquid oxidizer |

Chamber and nozzle | CF wound shell and high-silica insulation | Contain fuel grain and high-pressure burn gas |

Valves and tubes | Mainly aluminum alloy | Feed liquid propellant |

Catalyst bed | Stainless-steel shell and silver mesh | Decompose H_{2}O_{2} |

Turbopump | Details refer to Reference [ | Increase the pressure of oxidizer |

Thrust vector control actuator | Given by experience | Control the thrust vector direction |

The mass and dimension of these components are calculated based on the results of the propulsion design module. All the thickness of the chamber, nozzle, oxidizer tank, and gas bottle is calculated based on the maximum stress intensity theory, while the mass and dimension of some components are given by experience, such as the catalyst bed, turbopump, actuator of thrust vector control, and valves and tubes.

In addition, the layout of these components in the HRM at different stages (1st/2nd/3rd stage) needs to be arranged properly.

Different types of HRM in the 1st/2nd stage of SLV: (a) 1-chamber HRM; (b) 4-chamber HRM.

1-chamber HRM

4-chamber HRM

Configuration of 3rd-stage HRM.

The mission of the SLV in this paper is to send a small payload to a LEO circle orbit, and the schematic diagram of the flight process is shown in Figure

Schematic diagram of the flight process.

According to the flight process of SLV, different flight phases correspond to different control strategies. The pitch angle

Besides, the above phases are powered by three HRMs, a free glide phase without power is considered to turn the velocity direction from radial direction to tangential direction of the required orbit, and the control equation is

Moreover, separation between stages occurs after shutdown of the 1st- (or 2nd-) stage HRM, as the 2nd- (or 3rd-) stage HRM fires several seconds later. The pitch angle during separation can be considered a constant, and the control equation is

The three-degree-of-freedom (3DOF) mass point trajectory considering the rotation of the earth is used in this paper. The dynamics equations and kinematics equations of the SLV flight are shown as Equations (

The equations for aerodynamic force are only a rough approximation based on small attack/sideslip angle hypothesis and are acquired as shown in Equation (

Approximation aerodynamic coefficients of the “Titan II” rocket.

Aerodynamic coefficients | Velocity range | Formula |
---|---|---|

0.29 | ||

2.8 | ||

3.55 |

Ignoring rarefaction gas and other negligible factors, the status at the moment of 3rd-stage shutdown directly determines the parameters of payload orbit. One method for acquiring these orbital characteristics is to use turnoff velocity, height, and local trajectory tilt angle, as shown in

Based on difference of the grain shape, payload mass, and orbit height, two groups of different SLVs are designed and optimized. The SLVs in the first group are designed to deliver a 100 kg payload to a 300 km attitude orbit. Six cases with different 1st/2nd stages are considered in the first group to study how different grain shapes at different stages influence rocket performances. The SLVs in the second group are designed to deliver payloads with different mass (from 100 kg to 200 kg) to the orbits with different attitudes (from 300 km to 700 km). Five cases with the same SLV design scheme are considered. The purpose of the second group is to study how payload mass and orbit height influence HPSLV. All the cases mentioned above use the same solving method given in Section

Table

Description and ranges of variables.

Variable | Description | Unit | Lower bound | Upper bound |
---|---|---|---|---|

DV1 | Diameter of the 1st stage | m | 1 | 2.5 |

DV2 | Diameter ratio of the 2nd stage to1st stage | — | 0.7 | 1 |

DV3 | Ratio of the envelope diameter of the 2nd HRM to the diameter of the 2nd stage | — | 0.5 | 1 |

DV4 | Diameter ratio of the 3rd tank to the 3rd stage | — | 0.3 | 0.7 |

DV5 | Ratio of the envelope diameter of the 3rd HRM to the diameter of the 3rd stage | — | 0.3 | 1 |

DV6 | Initial thickness of the 1st grain | m | 0.05 | 0.4 |

DV7 | Initial thickness of the 2nd grain | m | 0.05 | 0.4 |

DV8 | Initial thickness of the 3rd grain | m | 0.05 | 0.4 |

DV9 | Initial thrust at the 1st stage | kN | 40 | 1250 |

DV10 | Initial thrust at the 2nd stage | kN | 2 | 650 |

DV11 | Initial thrust at the 3rd stage | kN | 0.25 | 85 |

DV12 | Initial pressure at the 1st chamber | MPa | 2 | 7 |

DV13 | Initial pressure at the 2nd chamber | MPa | 2 | 7 |

DV14 | Initial pressure at the 3rd chamber | MPa | 2 | 7 |

DV15 | Initial oxidizer-fuel ratio at the 1st stage | — | 2 | 7 |

DV16 | Initial oxidizer-fuel ratio at the 2nd stage | — | 2 | 7 |

DV17 | Initial oxidizer-fuel ratio at the 3rd stage | — | 2 | 7 |

DV18 | Expansion ratio at the 1st stage | — | 10 | 50 |

DV19 | Expansion ratio at the 2nd stage | — | 10 | 100 |

DV20 | Expansion ratio at the 3rd stage | — | 10 | 100 |

DV21 | Maximum attack angle during the program-turn phase | ° | 0.1 | 3 |

DV22 | Control parameter of attack angle during the program-turn phase | — | 0.15 | 1 |

DV23 | Glide time between 2nd and 3rd stages | s | 0 | 300 |

As shown in Figure

Envelope diameter of the 2nd HRM and 3rd HRM: (a) cross section of the 2nd stage (four-chamber); (b) cross section of the 3rd stage.

^{2}).

According to Section

This section analyzes the influence of different grain shapes on the main performance of SLV. All cases are aimed at sending a 100 kg payload into 300 km circle orbit. Table

SLV cases with different grain shapes and numbers of chambers.

Case | 1st stage | 2nd stage | ||
---|---|---|---|---|

Grain shape | Number of chambers | Grain shape | Number of chambers | |

Circle | 1 | Circle | 4 | |

Circle | 1 | Circle | 1 | |

Circle | 1 | Wheel | 1 | |

Circle | 4 | Circle | 4 | |

Wheel | 1 | Circle | 4 | |

Star | 1 | Circle | 4 |

As the mathematical model (as shown in Section

Optimal results of SLV cases with different grain shapes.

Parameter | Description | ||||||
---|---|---|---|---|---|---|---|

DV1 | 1.466 | 1.477 | 1.458 | 1.824 | 1.687 | 1.499 | |

DV2 | 0.994 | 0.851 | 0.999 | 0.718 | 0.773 | 0.983 | |

DV3 | 0.936 | 0.891 | 0.580 | 0.968 | 0.978 | 0.943 | |

DV4 | 0.498 | 0.402 | 0.465 | 0.415 | 0.428 | 0.418 | |

DV5 | 0.989 | 0.843 | 0.928 | 0.943 | 0.960 | 0.903 | |

DV6 | 0.058 | 0.062 | 0.059 | 0.242 | 0.055 | 0.087 | |

DV7 | 0.254 | 0.058 | 0.310 | 0.276 | 0.235 | 0.253 | |

DV8 | 0.283 | 0.329 | 0.308 | 0.291 | 0.333 | 0.284 | |

DV9 | 359.836 | 312.910 | 346.020 | 331.422 | 321.285 | 378.552 | |

DV10 | 92.776 | 69.271 | 92.970 | 83.086 | 85.685 | 102.660 | |

DV11 | 16.448 | 15.384 | 15.397 | 7.613 | 10.192 | 11.159 | |

DV12 | 5.288 | 6.997 | 5.552 | 6.440 | 4.657 | 6.764 | |

DV13 | 5.928 | 5.326 | 3.760 | 6.325 | 7.000 | 5.874 | |

DV14 | 3.772 | 4.952 | 2.808 | 2.086 | 2.368 | 2.123 | |

DV15 | 7.000 | 5.789 | 5.376 | 3.021 | 2.284 | 4.558 | |

DV16 | 3.151 | 4.277 | 2.551 | 2.619 | 3.116 | 2.700 | |

DV17 | 2.829 | 2.236 | 2.488 | 2.442 | 2.533 | 2.307 | |

DV18 | 20.916 | 17.647 | 20.861 | 19.930 | 14.215 | 26.059 | |

DV19 | 85.460 | 92.558 | 77.360 | 68.275 | 99.873 | 76.668 | |

DV20 | 81.554 | 100.000 | 99.252 | 87.288 | 97.569 | 100.000 | |

DV21 | 2.189 | 2.166 | 2.130 | 1.866 | 2.142 | 0.981 | |

DV22 | 0.516 | 0.690 | 0.470 | 0.733 | 0.693 | 0.256 | |

DV23 | 83.3 | 76.0 | 74.3 | 0 | 26.7 | 11.1 | |

Target | 16489 | 16844 | 16257 | 15438 | 15705 | 15820 | |

Constraint | ^{2})) | 98 | 83 | 89 | 265 | 77 | 86 |

Constraint | ^{2})) | 105 | 21 | 23 | 128 | 93 | 108 |

Constraint | ^{2})) | 59 | 109 | 68 | 28 | 60 | 33 |

Constraint | 13.1 | 15.4 | 14.9 | 8.9 | 10.5 | 13.2 | |

Constraint | 15.8 | 15.1 | 16.0 | 15.9 | 15.9 | 15.9 | |

Constraint | 0.96 | 0.62 | 0.95 | 0.99 | 0.98 | 0.98 | |

Constraint | 0.066 | 0.052 | 0.065 | 0.054 | 0.057 | 0.069 | |

Constraint | 306 | 300 | 302 | 307 | 306 | 313 |

Figure

Comparison of structures with different grain shapes.

Take-off mass and overall dimension of different cases.

Case | Take-off mass (t) | Diameter (m) | Length (m) | Length-diameter ratio |
---|---|---|---|---|

16.489 | 1.466 | 19.163 | 13.071 | |

16.844 | 1.477 | 22.800 | 15.436 | |

16.257 | 1.458 | 21.662 | 14.859 | |

15.438 | 1.824 | 16.241 | 8.906 | |

15.705 | 1.687 | 17.692 | 10.486 | |

15.820 | 1.499 | 19.731 | 13.161 |

The comparison between cases A, B, and D shows that the take-off mass and total length of the SLV are reduced when the 4-chamber configuration is adopted, but the diameter becomes larger. For instance, compared with those in case B, the take-off mass and total length in case D are reduced by 8.35% and 28.77%, respectively, and the diameter in case D increases by 23.49% correspondingly. This implies that the 4-chamber configuration makes the HRM more compact, which can effectively reduce the structural mass and shorten the overall length of the motor.

To evaluate the impact of different stages on the overall design of SLV, cases A, C, and E are selected for comparison. Compared with that in case A, the wheel-shaped grain at the 2nd stage is adopted in case C, and the take-off mass reduces by 1.41%. Meanwhile, the take-off mass of case E, in which the wheel-shaped grain is adopted at the 1st stage, reduces by 4.75% compared with case A. This means that compared with the situation when the wheel-shaped grain is adopted at the 1st stage, the impact of the 2nd grain is relatively small. Therefore, the mass of the 1st HRM is the main part affecting the overall mass of SLV. In general, the adoption of more complex grain and multicombustion chamber structure will help to achieve the key goal of reducing take-off mass but also increases the complexity of the propulsion system.

Figure

Thrust and oxidizer-fuel ratio with time of the 1st stage in cases A, E, and F: (a) thrust; (b) oxidizer-fuel ratio.

Table

Flight sequence of different cases of SLV.

Phase | ||||||
---|---|---|---|---|---|---|

1st motor work (s) | 65.2 | 81.2 | 71.0 | 78.7 | 78.5 | 79.1 |

1st-stage separation (s) | 8.0 | 8.0 | 8.0 | 8.0 | 8.0 | 8.0 |

2nd motor work (s) | 145.9 | 182.1 | 127.5 | 137.2 | 127.0 | 135.2 |

2nd-stage separation (s) | 8.0 | 8.0 | 8.0 | 8.0 | 8.0 | 8.0 |

Glide time (s) | 83.3 | 76.0 | 74.3 | 0 | 26.7 | 11.1 |

3rd motor work (s) | 168.5 | 134.6 | 188.1 | 314.9 | 261.7 | 290.4 |

Total time (s) | 478.9 | 489.9 | 476.9 | 546.8 | 509.9 | 531.8 |

Height-time curve of different cases: (a) whole flight; (b) glide phase and 3rd-stage working phase.

Pareto graphs of the main performance parameters in case A: (a)

The result shows that the most affecting factors of

Self-organizing map of case A: (a)

To analyze the influence of different payloads and different orbit heights on SLV performance, 5 SLV cases are set up for optimization design based on case A. The parameters of each case are shown in Table

Payload mass and orbit height of each case.

Case | Payload mass (kg) | Object height (km) |
---|---|---|

100 | 300 | |

100 | 500 | |

100 | 700 | |

150 | 300 | |

200 | 300 |

Using the parameterized design model established in Section

Optimal results of SLV cases with different payload mass/orbit heights.

Parameter | Description | |||||
---|---|---|---|---|---|---|

DV1 | 1.466 | 1.426 | 1.510 | 1.394 | 1.518 | |

DV2 | 0.994 | 0.994 | 0.981 | 0.980 | 1.000 | |

DV3 | 0.936 | 0.958 | 0.979 | 0.977 | 0.979 | |

DV4 | 0.498 | 0.473 | 0.392 | 0.405 | 0.477 | |

DV5 | 0.989 | 0.979 | 0.927 | 0.949 | 0.989 | |

DV6 | 0.058 | 0.082 | 0.071 | 0.069 | 0.061 | |

DV7 | 0.254 | 0.342 | 0.229 | 0.257 | 0.222 | |

DV8 | 0.283 | 0.228 | 0.319 | 0.299 | 0.243 | |

DV9 | 359.836 | 364.718 | 465.276 | 350.354 | 464.752 | |

DV10 | 92.776 | 91.446 | 116.690 | 100.868 | 116.972 | |

DV11 | 16.448 | 10.369 | 11.016 | 10.958 | 19.369 | |

DV12 | 5.288 | 5.529 | 6.994 | 6.902 | 6.081 | |

DV13 | 5.928 | 7.000 | 6.288 | 6.952 | 6.501 | |

DV14 | 3.772 | 5.656 | 2.037 | 5.579 | 2.095 | |

DV15 | 7.000 | 4.343 | 4.890 | 3.701 | 4.444 | |

DV16 | 3.151 | 3.014 | 2.923 | 3.289 | 3.046 | |

DV17 | 2.829 | 2.072 | 2.351 | 2.755 | 2.250 | |

DV18 | 20.916 | 22.300 | 30.020 | 17.212 | 32.685 | |

DV19 | 85.460 | 62.732 | 88.940 | 84.065 | 93.787 | |

DV20 | 81.554 | 100.000 | 97.231 | 91.644 | 99.179 | |

DV21 | 2.189 | 1.682 | 0.958 | 2.734 | 2.168 | |

DV22 | 0.516 | 0.563 | 0.269 | 0.524 | 0.437 | |

DV23 | 83.285 | 131.649 | 131.352 | 27.845 | 84.972 | |

Target | 16489 | 19675 | 22149 | 17256 | 20890 | |

Constraint | ^{2})) | 98 | 105 | 115 | 100 | 105 |

Constraint | ^{2})) | 105 | 260 | 95 | 125 | 90 |

Constraint | ^{2})) | 59 | 25 | 40 | 40 | 45 |

Constraint | 13.1 | 14.9 | 16.0 | 15.8 | 15.2 | |

Constraint | 15.8 | 15.7 | 16.0 | 15.8 | 15.2 | |

Constraint | 0.96 | 0.38 | 0.49 | 0.96 | 0.86 | |

Constraint | 0.066 | 0.048 | 0.057 | 0.066 | 0.066 | |

Constraint | 306 | 510 | 728 | 309 | 321 |

Obviously, Table

Velocity increments and velocity distribution ratios of several typical three-stage SLVs.

Name | Stage | Theoretical velocity increment (m/s) | Actual velocity increment (m/s) | Theoretical velocity distribution ratio | Actual velocity distribution ratio |
---|---|---|---|---|---|

Ро́кот (liquid) [ | 1st | 3987 | 3139 | 44.12% | 41.43% |

2nd | 2607 | 2327 | 28.85% | 30.72% | |

3rd | 2443 | 2110 | 27.03% | 27.85% | |

Total | 9037 | 7576 | — | — | |

Pegasus (solid) [ | Carrier | 700 | 236 | 7.42% | 2.90% |

1st | 2984 | 2307 | 31.64% | 28.35% | |

2nd | 2927 | 2787 | 31.04% | 34.23% | |

3rd | 2818 | 2810 | 29.89% | 34.52% | |

Total | 9429 | 8140 | — | — | |

HPSLV of Purdue University (hybrid) [ | 1st | 2773 | — | 26.76% | — |

2nd | 3734 | — | 36.03% | — | |

3rd | 3851 | — | 37.20% | — | |

Total | 10358 | — | — | — |

Figure

Velocity increment and velocity distribution ratio: (a) theoretical and actual velocity increment; (b) theoretical and actual velocity distribution ratio.

Theoretical and actual velocity increment

Theoretical and actual velocity distribution ratio

A comparison of case

From the perspective of the velocity distribution ratio at each stage, the theoretical/actual distribution ratio has slight change with the increase in payload mass and orbit height, and the estimated ratio is around 1 : 1.55 : 1.69/1 : 1.9 : 2.39. The actual velocity distribution ratio of the three-stage HPSLV is roughly 16% to 21% for the 1st stage, 32% to 38% for the 2nd stage, and 42% to 49% for the 3rd stage. This result can be used to estimate velocity increments of HPSLV in the future.

In all the cases, the velocity loss at each stage shows the same distribution law. The velocity at the 1st stage accounts for the largest proportion (23% to 35%), the 2nd stage follows (8% to 25%), and the 3rd stage is the smallest (1% to 10%). The loss of the 1st stage mainly comes from the atmospheric drag and gravity when the rocket flights through the dense atmosphere. At the 2nd stage, the SLV passes through the atmosphere during flight and is still greatly affected by atmospheric drag. At the 3rd stage, the SLV mainly works in the vacuum environment, and the loss mainly comes from the gravity when the altitude changes.

This paper carries out optimization design of HPSLVs with different grain shapes, chamber configurations, payload mass, and orbit heights. The results show that the star-shaped or wheel-shaped grains are more effective than circle-shaped grains in reducing the take-off mass and dimension of the SLV. The adoption of four combustion chambers makes the SLV design more compact, which has a significant effect on reducing the take-off mass and the total length. The analysis of SA and SOM shows that the optimal target of the take-off mass mainly depends on the design of the 1st stage. Furthermore, the influence of different payload mass and different orbit heights on the performance parameters of the HPSLV is investigated, and the velocity increments, velocity losses, and velocity distribution ratios at each stage are obtained. Compared with that in Purdue’s SLV, the velocity distribution ratio of the HPSLV in this paper shows the same regularity, which can provide some reference for future HPSLV design and development.

Grain port area

Nozzle throat area

Regression rate coefficient, acceleration

Acceleration of Coriolis inertial force

Acceleration of centrifugal inertial force

Orbital semimajor axis

Drag coefficient

Thrust coefficient

Derivative of lift coefficient to attack angle

Characteristic velocity

Drag, diameter of the rocket

Outer diameter of grain

Centre hole diameter of wheel-shape grain

Wheel channel inner diameter of wheel-shaped grain

Wheel channel outer diameter of wheel-shaped grain

Hydraulic diameter

Grain thickness

Orbital ellipticity

Thrust

Universal gravitational constant, mass flux

Acceleration of gravity

Final orbit height at perigee

Specific impulse

Specific heat ratio

lift, length of the rocket

Length of the fuel grain

Take-off mass

The mass of the earth

The mass of the rocket

Mass flow rate

Number of combustion chambers, aerodynamic force

Axial overload

Normal overload

Number of star angle/wheel hole, flux rate exponent

Object height

Chamber pressure

Equilibrium pressure

Dynamic pressure

Specific gas constant

Radius of star tip arc, radius of chamfer of wheel-shaped grain, distance from the turnoff point to the earth’s core

Regression rate

Apogee altitude

Perigee altitude

Radius of star root arc

Burning perimeter length

Reference area

Glide time between 2nd and 3rd stages

Adiabatic combustion temperature

Time

Velocity increment

Effective volume of chamber

Velocity

Turnoff velocity of the third stage

Position in the

Position in the

Lateral force

Position in the

Oxidizer to fuel ratio, attack angle

Control parameter of attack angle during the program-turn phase, sideslip angle

Star angle coefficient, nozzle expansion ratio

Angle of star root, trajectory tilt angle

Local trajectory tilt angle at the turnoff point

Effective propellant mass fraction

Diameter ratio coefficient in design variables

Pitch angle

Pitch angular velocity

Density

Velocity loss

Combustion gas

Envelope

Initial

Oxidizer

Fuel

Maximum value

Current time

Launch point

First stage

Second stage

Third stage

The data used to support the findings of this study are included within the article.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this article.

_{2}O

_{2}hybrid rockets for launching nanosats