Design and optimization of a novel supersonic rocket with small caliber

Hao Yan; Xiaobing Zhang

doi:10.3934/jimo.2022110

Article Contents

2023, Volume 19, Issue 5: 3794-3818. Doi: 10.3934/jimo.2022110

This issue Previous Article Traffic flow modeling and optimization control in the approach airspace Next Article Performance evaluation and bi-objective optimization for $ F $-policy queue with alternating service rates

Design and optimization of a novel supersonic rocket with small caliber

Hao Yan and
Xiaobing Zhang^,

School of Energy and Power Engineering, Nanjing University of Science and Technology, Nanjing, 210094, China

^*Corresponding author: Xiaobing Zhang
^*Corresponding author: Xiaobing Zhang

Received: November 2021

Revised: May 2022

Early access: July 2022

Published: May 2023

Abstract / Introduction Full Text(HTML) Figure(13) / Table(10) Related Papers Cited by

Abstract

It's difficult to meet the design requirements of a rocket with sensitive parameters using the traditional methods. This paper develops a design and optimization concept for a lightweight solid-propellant rocket with a small caliber and a high Mach number. The structural design and numerical modeling depend on the interior ballistic methodology. The model is validated by comparing pressure-time data from simulation and experiment. A modified evolutionary algorithm with constraints is applied because of the interactive design parameters and conflicting objects. The launch performance is improved by single- and multi-objective optimization. Under unchanged interior ballistic performance conditions, the peak pressure is reduced by 46.4%, and the erosive peak ratio is reduced by 46.4%. Despite the constraints, the Mach number is improved by 4.9%, the total impulse is improved by 6.3%, the peak pressure is reduced by 92.5%, and the erosive effects are reduced by 38.1% using different optimal solutions. A Pareto front is obtained by a constrained NSGA-Ⅱ, which reveals non-linear and non-uniform relations among design objects. A tidying method is proposed for a clear Pareto front. It indicates that, despite the sensitive parameters, launch safety and higher velocity are possible. The results help designers choose the best design schemes.

Keywords:

Mathematics Subject Classification: Primary: 90B50, 90C29; Secondary: 68M20.

Citation:

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Figure 1. A design framework of the MDO problem

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Figure 2. An integrated solution flow for rocket design

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Figure 3. Structural sketch for small-caliber rocket

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Figure 4. The three combustion stages in solid-propellant rocket chamber

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Figure 5. Procedure of NSGA-Ⅱ with two modules

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Figure 6. Comparisons of P-t traces between numerical calculation and experiment

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Figure 7. Initial P-t and F-t curves of numerical simulation

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Figure 8. Convergence curves of single-object optimization when $ A $ = 5%

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Figure 9. Initial Pareto front of multi-objective optimization

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Figure 10. Pareto front after tidying up

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Figure 11. Comparison of special Pareto-optimal points

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Figure 12. Motor interior ballistic performance

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Figure 13. Three degree-of-freedom trajectory validation

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Table 1. Propellant ingredients in validation case

Ingredients	Value
Ammonium perchlorate (AP)	83%
Hydroxy-terminated polybutadiene (HTPB)	11.9%
Oxamide	5%
Carbon black	0.1%

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Table 2. Charge & motor parameters in validation case

Parameters	Value
Burning rate coefficient (mm/s/Pa$ ^n $)	0.0038
Burning rate pressure index	0.461
Nozzle throat diameter(mm)	26.416
Specific heat	1.217
Propellant web thickness(mm)	22.86
Propellant length(mm)	692

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Table 3. Initial charge parameters

Parameters	Value	Parameters	Value
Burning rate coefficient (mm/s/Pa$ ^n $)	0.034	Pressure index	0.358
Characteristic velocity(m/s)	1341	Temperature factor	1.0
Propellant density(kg/m$ ^3 $)	1610	Flow rate factor	0.95
Propellant web thickness(mm)	27.3535	Specific heat	1.25

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Table 4. Initial motor parameters

Parameters	Value	Parameters	Value
Length-to-diameter proportion	18.75	Nozzle throat diameter(mm)	18.0
Nozzle expansion ratio	1.6	Nozzle exit diameter(mm)	28.8

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Table 5. Parameters of initial performance with erosive effects

Parameters	Value	Parameters	Value
Equilibrium pressure (MPa)	31.3275	Average thrust(kN)	12.8273
Peak pressure(Mpa)	105.3515	Total impulse(kN $ \cdot $ s)	10.2872
Max thrust(kN)	40.5700	Burning time(s)	0.801
Erosive peak ratio	3.3629	Max Mach number	2.5510

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Table 6. Parameters of initial performance theoretically

Parameters	Value	Parameters	Value
Equilibrium pressure (MPa)	31.3275	Average thrust(kN)	11.2740
Peak pressure(Mpa)	31.3275	Total impulse(kN $ \cdot $ s)	10.2815
Max thrust(kN)	12.0287	Burning time(s)	0.911
Erosive peak ratio	1.0000	Max Mach number	2.5517

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Table 7. Optimal design parameters. The size of nozzle throat becomes smaller to enhance the interior ballistic performance, with reduced erosive effects

$ A $	$ r_p $	$ {P_{\rm{eq}}} $/MPa	$ B $	$ e $/mm	$ d_{\rm{t}} $/mm	$ A_{\rm{t}} $/mm$ ^2 $
0.1%	1.8018	31.3586	+ 0.1%	19.8536	19.7633	306.7656
1%	1.7993	31.6407	+ 1%	19.8536	19.7043	304.9377
5%	1.7884	32.8938	+ 5%	19.8536	19.4502	297.1239
10%	1.7748	34.4605	+ 10%	19.8536	19.1497	288.0125
15%	1.7619	36.0264	+ 15%	19.8536	18.8663	279.5513
20%	1.7492	37.5927	+ 20%	19.8536	18.5984	271.6681

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Table 8. Optimal rocket parameters. The peak pressure and maximum propulsion force are increased but the change of average thrust ($ {F_{\rm{ave}}} $) and maximum Mach number ($ M_{\rm{max}} $) fluctuate because of different burn time. $ {F_{\rm{m}}} $ is the maximum thrust

$ A $	$ {P_{\rm{m}}} $/MPa	$ {F_{\rm{m}}} $/kN	$ {F_{\rm{ave}}} $/kN	$ I $/kN$ \cdot $s	$ t $/s	$ M_{\rm{max}} $
0.1%	56.5028	26.2026	15.2652	9.0061	0.589	2.3231
1%	56.9306	26.2440	15.1632	9.0065	0.593	2.3221
5%	58.8283	26.4259	15.4245	9.0075	0.583	2.3236
10%	61.1599	26.6331	15.3745	9.0091	0.585	2.3225
15%	63.4747	26.8312	15.5617	9.0098	0.578	2.3232
20%	65.7580	27.0144	15.4561	9.0105	0.582	2.3219

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Table 9. Relevant optimal motor parameters. The ratio of nozzle throat area to sectional area of charging channel is decreased and the ratio of burning surface area of solid propellant to nozzle throat area is increased

$ A $	$ m $/kg	$ \Delta $/(g/cm$ ^3 $)	$ {\text{æ}}_0 $	$ J $	$ K_{\rm{N}} $
0.1%	4.2276	1.2890	388.4005	0.4504	862.2808
1%	4.2276	1.2890	388.4005	0.4478	867.4496
5%	4.2276	1.2890	388.4005	0.4363	890.2617
10%	4.2276	1.2890	388.4005	0.4229	918.4257
15%	4.2276	1.2890	388.4009	0.4105	946.2236
20%	4.2276	1.2890	388.4005	0.3989	973.6809

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Table 10. Primary optimum solutions. The four special points, A, B, C, and D, represent the boundary points of Pareto front. The goal for minimum peak pressure, minimum erosive effects, and maximum Mach number can be attained by different points of the Pareto front, and their threshold is determined by these special points

	$ {P_{\rm{m}}} $/MPa	$ r_p $	$ I $/kN$ \cdot $s	$ e $/mm	$ d_{\rm{t}} $/mm	$ d_{\rm{e}} $/mm	$ t $/s	$ t_{\rm{max}} $/s	$ M_{\rm{max}} $
A	7.9447	2.2001	8.1926	19.854	40	50	1.161	1.161	2.0913
B	59.977	1.7818	9.1996	19.854	19.301	33.745	0.594	0.578	2.3668
C	59.993	4.3926	10.93	28.251	23.372	50	1.024	1.021	2.6762
D	18.792	2.082	7.7942	19.854	29.745	32.239	0.849	0.849	2.0185

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