# Optimization of the Conceptual Design of a Multistage Rocket Launcher

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## Abstract

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## 1. Introduction and State of the Art

- The use of a multistage rocket, as proposed by, among others, Tsiolkovsky. A SSTO (Single Stage to Orbit) would consume more fuel, but if complete reusability could be achieved, as in the case of an airplane, the cost savings would be significant. Unfortunately, the efficiency of the engines prevents their use in these scenarios. Multistage rockets increase performance, eliminating part of the mass that is no longer necessary during the ascent to orbit: By dividing the rocket into multiple zones or stages, we can reduce the size of these losses. Each stage will, thus, have a fraction of the total propellant, which can be optimized to maximize performance. Another advantage comes from the type of propellant: the first stages, being subjected to higher atmospheric pressures, have to withstand a greater aerodynamic drag, together with a substantial modification in the propulsive performance of the nozzle. High pressures present in the early stages require a shorter nozzle geometry, while in vacuum or at high altitudes, a more elongated shape is preferable for optimal performance. Therefore, this is one of the most remarkable advantages of the use of several stages since it allows to use different engines, depending on the external conditions, thus improving the overall efficiency of the rocket. Finally, a multistage rocket allows for the use of different propellants, depending on the stage. The first stages tend to use solid or liquid fuel based on hydrocarbons, which can achieve the great thrusts necessary to overcome the gravitational attraction, while being relatively low cost. For the following stages, the required thrust is substantially lower, as sub-orbital speeds are reached. In this way, fuels are chosen that admit a large specific impulse with a low volume. Therefore, in these stages, high efficiencies can be achieved without an excessive increase in cost [3]. On the contrary, for multistage rockets, one or more engines will be required at each stage, thus increasing the cost and structural fraction of the whole. The complexity of design and manufacturing also increases, leading to additional reliability issues.
- Optimization of launch parameters, among others: the mass distribution between the stages, the most suitable engine for each of them, the size and mass of the resulting rocket, the amount of propellant required, and the ascent path. All these factors interact with each other, so the optimal solution will be given by an adequate balance between all the elements.

## 2. Materials and Methods

#### 2.1. System Modeling

#### 2.1.1. Cost Model

#### 2.1.2. Propulsive Model

#### 2.1.3. Atmospheric and Gravitational Model

#### 2.1.4. Mass Model

#### 2.1.5. Aerodynamical Model

- The angle of attack is small, typically less than 10° in atmospheric flight.
- The flow is uniform around the rocket.
- The rocket is axisymmetric, which is the case in serial staged vehicles without fins. Thus, only serial rockets are considered on the trajectory analysis. However, if the vehicle has fins, an additional term can be added on (15) to take into account the additional drag, as described in the study of Caporaso [27].
- The lift is negligible.
- Induced drag is considered negligible.
- The variation in drag with the angle of attack is also negligible.

_{Dfb}is the body drag, and C

_{Db}is the base drag. C

_{Dbd}is the component resulting from the sum of pressure drag and skin friction drag. The first can be expressed according to Caporaso [27]:

#### 2.2. Algorithm Design

#### 2.2.1. Optimization Algorithm

#### 2.2.2. Optimal Stage Distribution

#### 2.2.3. Design Optimization

#### 2.2.4. Trajectory Optimization

- 6 state variables: r, $\varphi $, θ, V, γ, χ
- 2 control variables: α, β
- 1 independent monotonically increasing variable: t

_{f}

## 3. Results and Discussion

#### 3.1. Mass Optimization Validation

#### 3.2. Design Algorithm Application

#### 3.3. Trajectory Optimization Application

- The angle of attack and slip angle is limited, for structural reasons, to 6 degrees at altitudes below 100 km and 15 degrees for higher altitudes [36].
- The aerodynamic load is the product of the dynamic pressure and the angle of attack, and it affects the integrity of the rocket. The considered limit is 230,000 Pa degrees [36].

## 4. Conclusions and Future Works

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Diagram of a launch vehicle with parallel stages [31].

**Figure 5.**Altitude in the optimized trajectory. The red marks indicate the staging or coasting phase.

**Figure 6.**Velocity in the optimized trajectory. The red marks indicate the staging or coasting phase.

**Figure 7.**Flight path angle in the optimized trajectory. The red marks indicate the staging or coasting phase.

**Figure 8.**Angle of attack in the optimized trajectory. The red marks indicate the staging or coasting phase.

**Figure 9.**Sideslip angle in the optimized trajectory. The red marks indicate the staging or coasting phase.

**Figure 10.**Mass evolution in the optimized trajectory. The red marks indicate the staging or coasting phase.

Category | Mass (t) |
---|---|

Small | 0–2 |

Midsize | 2–20 |

Heavy | 20–50 |

Super heavy | >50 |

Engines | Propellant | Thrust [kN] | Specific Impulse (vac) [s] | Mixture Ratio | Expansion Ratio | Height [m] | Diameter [m] | Mass [kg] |
---|---|---|---|---|---|---|---|---|

11D58M | LO2/Kerosene | 79.5 | 353 | 2.48 | 189 | 2.27 | 1.17 | 230 |

RD-0210 | N2O4/UDMH | 582 | 327 | 1.95 | 81.3 | 2.33 | 1.47 | 566 |

AESTUS | N2O4/MMH | 30 | 325 | 2.05 | 84 | 2.2 | 1.32 | 111 |

J-2 | LO2/LH2 | 890 | 426 | 5.5 | 28 | 3.38 | 2.01 | 1438 |

YF-75 | LO2/LH2 | 79 | 440 | 5 | 80 | 2.8 | 1.5 | 550 |

LE-5B | LO2/LH2 | 137 | 447 | 5 | 110 | 2.78 | 2.15 | 269 |

HM7-B | LO2/LH2 | 70 | 447 | 5.14 | 83.1 | 1.8 | 1 | 155 |

VINCI | LO2/LH2 | 180 | 465 | 4.83 | 240 | 4.2 | 2.15 | 550 |

RL-10B | LO2/LH2 | 110 | 462 | 4.83 | 250 | 4.15 | 2.13 | 301 |

Engine | Propellant | Thrust [MN] | Specific Impulse (sl.) [s] | Mixture Ratio | Expansion Ratio | Height [m] | Diameter [m] | Mass [kg] |
---|---|---|---|---|---|---|---|---|

RD-170 | LO2/Kerosene | 7.65 | 310 | 2.6 | 36.87 | 3.78 | 4.02 | 9750 |

RD-180 | LO2/Kerosene | 3.82 | 311 | 2.72 | 36.4 | 3.56 | 3.15 | 5480 |

RD-107 | LO2/Kerosene | 0.81 | 257 | 2.06 * | 18.86 | 2.578 | 1.85 | 1250 |

F-1 | LO2/RP1 | 6.91 | 264 | 2.27 | 16 | 5.64 | 3.72 | 8391 |

MA-5A | LO2/RP1 | 1.84 | 263 | 2.25 | 8 | 3.43 | 1.19 | 1610 |

RS-27 | LO2/RP1 | 0.91 | 263 | 2.245 | 8 | 3.63 | 1.07 | 1027 |

RD-253 | N2O4/UDMH | 1.47 | 285 | 2.67 | 26.4 | 3 | 1.5 | 1300 |

YF-20 | N2O4/UDMH | 0.76 | 259 | 1.95 * | 10 | 2 * | 0.84 | 712.5 * |

Viking 6 | N2O4/UH25 | 0.68 | 249 | 1.71 | 10.5 | 2.87 | 0.99 | 826 |

RS-68 | LO2/LH2 | 2.89 | 360 | 4.83 * | 21.5 | 5.2 | 2.43 | 6600 |

RD-108 | LO2/Kerosene | 0.78 | 252 | 2.77 * | 18.9 | 2.86 | 0.67 | 1250 |

Viking 5C | N2O4/UH25 | 0.68 | 249 | 1.7 | 11 | 2.87 | 2.22 | 826 |

YF-20B | N2O4/UDMH | 0.73 | 259 | 1.95 * | 10 | 2 * | 0.84 | 712.5 |

RS-68 | LO2/LH2 | 2.89 | 360 | 4.83 * | 21.5 | 5.2 | 2.43 | 6597 |

SSME | LO2/LH2 | 1.82 | 364 | 6 | 77.5 | 4.24 | 1.63 | 3177 |

RD-0120 | LO2/LH2 | 1.51 | 359 | 6 | 85.7 | 4.55 | 2.42 | 3450 |

LE-7A | LO2/LH2 | 0.84 | 338 | 5.9 | 51.9 | 3.67 | 2 * | 1800 |

Vulcain 2 | LO2/LH2 | 0.94 | 320 | 6.7 | 61.5 | 3.6 | 2.1 | 811 |

$\Delta {V}_{orbit}$ [m/s] | 7788.5 |

$\mathsf{\Delta}{V}_{r}$ [m/s] | −441.3 |

$\mathsf{\Delta}{V}_{g}$ [m/s] | 1200 |

$\mathsf{\Delta}{V}_{d}$ [m/s] | 100 |

$\mathsf{\Delta}{V}_{p}$ [m/s] | 35 |

Hohmann transfer [m/s] | 3934.5 |

$\mathsf{\Delta}{V}_{mission}$ [m/s] | 12,616.7 |

Simulation | Real Data | Error [%] | |
---|---|---|---|

$GLOW\left[\mathrm{kg}\right]$ | 401.640 | 402.800 | 0.288 |

${n}_{0}$ | 3.885 | 3.844 | 1.076 |

${n}_{1}$ | 1.083 | 1 | 8.298 |

n_{2} | 3.024 | 2.773 | 9.051 |

${n}_{3}$ | 3.497 | 3.841 | 8.978 |

Falcon 9 Full Thrust Reviewed | Atlas V 501 Reviewed | |
---|---|---|

Engine 1st stage | RD-180 (2 units) | RD-180 |

Engine 2nd stage | VINCI (4 units) | 11D58M (5 units) |

$TW{R}_{1}$ | 1.112 | 1.25 |

$TW{R}_{2}$ | 0.512 | 0.538 |

${n}_{1}$ | 2.901 | 2.708 |

${n}_{2}$ | 3.353 | 5.241 |

**Table 7.**Comparison of relevant quantities of the revised Falcon 9 v1.2 and Atlas V 501, with respect to the real rockets.

Falcon 9 v1.2 | Atlas V 501 | |||
---|---|---|---|---|

Simulation | Real | Simulation | Real | |

$GLOW\left[\mathrm{kg}\right]$ | 465.070 | 572.000 | 228.680 | 337.887 |

Radius [m] | 3.78 | 1.83 | 1.896 | 1.905 |

Length [m] | 68.858 | 71 | 41.591 | 32.46 |

${\u03f5}_{1}$ | 0.0736 | 0.0579 | 0.0674 | 0.068 |

${\u03f5}_{2}$ | 0.157 | 0.0854 | 0.06 | 0.0842 |

${m}_{0,1}$ [kg] | 465.070 | 549.000 | 228.680 | 337.887 |

${m}_{0,2}$ [kg] | 136.100 | 134.300 | 74.023 | 33.044 |

$TW{R}_{1}\left(real\right)$ | 1.675 | 1.363 | 1.704 | 1.155 |

$TW{R}_{2}\left(real\right)$ | 0.540 | 0.709 | 0.548 | 0.306 |

${S}_{s}nosecone\left[{\mathrm{m}}^{2}\right]$ | 182.258 | Unknown | 45.870 | Unknown |

Stage 1 | Stage 2 | Stage 3 | Stage 4 | |
---|---|---|---|---|

Engine | P80 | Zefiro 23 | Zefiro 9 | AVUM |

${I}_{sp}$ [s] | 280 | 287.5 | 295.5 | 314.6 |

$T$ [kN] | 2261 | 871 | 260 | 245 |

${m}_{p}$[kg] | 87,710 | 23,814 | 10,567 | 577 |

${m}_{0}$ [kg] | 137,798 | 41,535 | 15,235 | 2695 |

Constraint | Limits |
---|---|

$r$ | $r>{R}_{E}$ |

$\alpha $ | $-6\xb0<\alpha <6\xb0$$\mathrm{for}\mathrm{h}100\mathrm{km}$ $-15\xb0<\alpha <15\xb0$ for h ≥ 100 km |

$\beta $ | $-6\xb0<\beta <6\xb0$$\mathrm{for}\mathrm{h}100\mathrm{km}$ $-15\xb0<\beta <15\xb0$ for h ≥ 100 km |

$q$ | $\mathrm{55,000}\mathrm{Pa}$ |

$q\alpha $ | $230,000\mathrm{Pa}\mathrm{deg}$ |

$\dot{q}$ | 1135 W/m^{2} (fairing)40 MW/m ^{2} (structure) |

Orbital Element | Simulated Value | Goal Value | Absolute Error | Percentage Error |
---|---|---|---|---|

Altitude (instead of a) [km] | 699.99 | 700 | 0.01 | $8.71\times {10}^{-5}$% |

Eccentricity | 0.053 | 0 | 0.053 | - |

Inclination [°] | 90.01 | 90 | 0.01 | 0.005% |

Longitude of ascending node [°] | 6.20 | - | - | - |

Perigee argument [°] | 35.57 | - | - | - |

True anomaly [°] | 180.20 | - | - | - |

Constraint | Trajectory Value | Constraint LIMIT |
---|---|---|

$\alpha $ [°] | 6 | 15 (>100 km) |

$\beta $ [°] | 0.34 | 15 (>100 km) |

$q$ [Pa] | 43,651 | 55,000 |

$q\alpha $ [Pa deg] | 88,450 | 230,000 |

$\dot{q}$ [W/m^{2}] | 28,910,000 | 40,000,000 |

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**MDPI and ACS Style**

Orgeira-Crespo, P.; Rey, G.; Ulloa, C.; Garcia-Luis, U.; Rouco, P.; Aguado-Agelet, F.
Optimization of the Conceptual Design of a Multistage Rocket Launcher. *Aerospace* **2022**, *9*, 286.
https://doi.org/10.3390/aerospace9060286

**AMA Style**

Orgeira-Crespo P, Rey G, Ulloa C, Garcia-Luis U, Rouco P, Aguado-Agelet F.
Optimization of the Conceptual Design of a Multistage Rocket Launcher. *Aerospace*. 2022; 9(6):286.
https://doi.org/10.3390/aerospace9060286

**Chicago/Turabian Style**

Orgeira-Crespo, Pedro, Guillermo Rey, Carlos Ulloa, Uxia Garcia-Luis, Pablo Rouco, and Fernando Aguado-Agelet.
2022. "Optimization of the Conceptual Design of a Multistage Rocket Launcher" *Aerospace* 9, no. 6: 286.
https://doi.org/10.3390/aerospace9060286