# Optimal Design of Electrically Fed Hybrid Mars Ascent Vehicle

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## Abstract

**:**

## 1. Introduction

## 2. Engine Design and Optimization

## 3. Rocket Configurations and Trajectory Optimization

## 4. Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

BD | Blow Down |

LRE(s) | Liquid Rocket Engine(s) |

LOX | Liquid OXygen |

MAV | Mars Ascent Vehicle |

R | Regulated |

RHS | Right-Hand Side |

SRM(s) | Solid Rocket Motor(s) |

TP | Turbo Pump |

Nomenclature | |

${A}_{b}$ | burning surface area, m${}^{2}$ |

${A}_{p}$ | port area, m${}^{2}$ |

${A}_{th}$ | nozzle throat area, m${}^{2}$ |

a | regression constant, m${}^{1+2n}$ kg${}^{-n}$ s${}^{n-1}$ |

${C}_{F}$ | thrust coefficient |

${c}^{\ast}$ | characteristic velocity, m/s |

$\mathit{D}$ | drag vector, N |

d | rocket outer diameter, m |

$\mathit{F}$ | thrust vector, N |

F | thrust, N |

G | gravitational constant, Nm${}^{2}$/kg${}^{2}$ |

$\mathit{g}$ | gravity acceleration, m/s${}^{2}$ |

H | Hamiltonian |

h | altitude, km |

${I}_{SP}$ | mean specific impulse, s |

J | throat area to initial port area ratio |

L | overall engine length, m |

${L}_{b}$ | fuel grain length, m |

M | rocket mass, kg |

m | mass, kg |

${N}_{1}$ | engines number in the first stage |

n | mass-flux exponent |

${P}_{e}$ | electric power, kW |

p | pressure, bar |

$\mathit{r}$ | position vector, m |

t | time, s |

T | temperature, K |

V | volume, m${}^{3}$ |

$\mathit{v}$ | velocity vector, m/s |

w | web thickness, m |

y | burning distance, m |

Z | hydraulic resistance, 1/(kg m) |

$\alpha $ | mixture ratio |

$\gamma $ | specific heat ratio |

${\delta}_{ep}$ | electric motor and pump power density, kW/kg |

${\delta}_{be}$ | batteries energy density, Wh/kg |

${\delta}_{bp}$ | batteries power density, kW/kg |

$\u03f5$ | nozzle area ratio |

${\eta}_{ep}$ | electric motor and pump efficiency |

$\lambda $ | adjoint variable |

$\mu $ | payload, kg |

$\rho $ | density, kg/m${}^{3}$ |

Superscripts | |

$\dot{}$ | time derivative |

* | characteristic |

Subscripts | |

1 | combustion chamber at head-end |

$atm$ | atmospheric |

$avg$ | average |

$cc$ | combustion chamber |

d | discharge |

$dry$ | dry |

e | nozzle exit |

$ep$ | electric motor and pump |

F | fuel |

f | final |

$fs$ | feed system |

i | initial value |

$max$ | maximum |

$min$ | minimum |

n | nozzle |

O | oxidizer |

p | overall propellant (oxidizer + fuel) |

$rel\pounds $ | relative |

$res$ | residual |

t | oxidizer propellant tank |

$th$ | throat |

$tot$ | total |

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**Figure 2.**Specific impulse ${I}_{SP}$ and mixture ratio $\alpha $ histories for the best solution of each feed system. The BD solution was obtained with ${N}_{1}=2$ and R and TP solutions with ${N}_{1}=3$.

**Figure 3.**Thrust F and longitudinal acceleration $F/m$ histories for the best solution of each feed system. The BD solution was obtained with ${N}_{1}=2$ and R and TP solutions with ${N}_{1}=3$.

**Figure 4.**Trajectory for the best solution of each feed system. The BD solution was been obtained with ${N}_{1}=2$ and R and TP solutions with ${N}_{1}=3$. The dots and crosses represent the stages ignition and burnout, respectively.

**Table 1.**Overall MAV mass budget and performance. BD, R and TP stand for blow down, regulated and turbo pump feed system, respectively.

Case | $\mathit{\mu}$ | ${\mathit{m}}_{\mathit{p}}$ | ${\mathit{m}}_{\mathit{d}\mathit{r}\mathit{y}}$ | ${\mathit{\alpha}}_{\mathit{a}\mathit{v}\mathit{g}}$ | ${\mathit{I}}_{{\mathit{s}\mathit{p}}_{\mathit{a}\mathit{v}\mathit{g}}}$ | ${\mathit{V}}_{\mathit{l}\mathit{o}\mathit{s}\mathit{s}\mathit{e}\mathit{s}}$ |
---|---|---|---|---|---|---|

- | kg | kg | kg | - | s | km/s |

2 + 1 BD | 75.03 | 345.81 | 79.16 | 2.062 | 298.1 | 0.794 |

3 + 1 BD | 74.77 | 339.81 | 85.42 | 2.038 | 298.7 | 0.710 |

4 + 1 BD | 72.29 | 337.79 | 89.17 | 2.024 | 298.0 | 0.671 |

2 + 1 R | 86.25 | 345.95 | 67.80 | 2.071 | 296.9 | 0.672 |

3 + 1 R | 86.73 | 340.60 | 71.85 | 2.053 | 297.7 | 0.591 |

4 + 1 R | 84.82 | 338.17 | 76.15 | 2.038 | 298.3 | 0.554 |

2 + 1 TP | 93.04 | 348.78 | 58.18 | 2.128 | 290.9 | 0.665 |

3 + 1 TP | 100.91 | 335.24 | 63.85 | 2.152 | 293.2 | 0.487 |

4 + 1 TP | 99.81 | 331.70 | 68.49 | 2.191 | 294.4 | 0.425 |

**Table 2.**Hybrid engine design. BD, R and TP stand for blow down, regulated and turbo pump feed system, respectively. The sixth column (${m}_{e}$) reports the sum of electrical components masses (motor, pump and batteries) when the turbo pump feed system is employed.

Case | ${\mathit{\alpha}}_{\mathit{i}}$ | ${\left({\mathit{p}}_{\mathit{f}\mathit{s}}\right)}_{\mathit{i}}$ | ${\left({\mathit{V}}_{\mathit{g}}\right)}_{\mathit{i}}$ | ${\mathit{V}}_{\mathit{H}\mathit{e}}$ | ${\mathit{m}}_{\mathit{e}}$ | $\mathit{\u03f5}$ | d | ${\mathit{L}}_{\mathit{c}\mathit{c}}$ | ${\mathit{L}}_{\mathit{t}}$ | ${\mathit{L}}_{\mathit{n}}$ |
---|---|---|---|---|---|---|---|---|---|---|

- | - | bar | m${}^{3}$ | m${}^{3}$ | kg | - | m | m | m | m |

2 + 1 BD | 1.53 | 21.4 | $7.1\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | - | - | 17.8 | 0.31 | 0.61 | 1.97 | 0.40 |

3 + 1 BD | 1.44 | 21.5 | $5.5\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | - | - | 18.8 | 0.36 | 0.53 | 1.74 | 0.36 |

4 + 1 BD | 1.39 | 20.9 | $4.4\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | - | - | 18.1 | 0.33 | 0.48 | 1.58 | 0.33 |

2 + 1 R | 1.47 | 16.9 | - | $9.7\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | - | 17.2 | 0.24 | 0.67 | 1.68 | 0.42 |

3 + 1 R | 1.38 | 17.9 | - | $7.4\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | - | 18.6 | 0.22 | 0.58 | 1.42 | 0.37 |

4 + 1 R | 1.32 | 19.0 | - | $5.8\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | - | 19.5 | 0.21 | 0.52 | 1.28 | 0.33 |

2 + 1 TP | 1.40 | 22.7 | - | $6.0\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 2.93 | 18.8 | 0.30 | 0.66 | 1.14 | 0.40 |

3 + 1 TP | 1.30 | 48.2 | - | $6.0\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 7.36 | 34.7 | 0.26 | 0.61 | 1.07 | 0.38 |

4 + 1 TP | 1.37 | 46.4 | - | $6.0\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 8.78 | 35.5 | 0.24 | 0.58 | 1.01 | 0.37 |

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**MDPI and ACS Style**

Casalino, L.; Masseni, F.; Pastrone, D.
Optimal Design of Electrically Fed Hybrid Mars Ascent Vehicle. *Aerospace* **2021**, *8*, 181.
https://doi.org/10.3390/aerospace8070181

**AMA Style**

Casalino L, Masseni F, Pastrone D.
Optimal Design of Electrically Fed Hybrid Mars Ascent Vehicle. *Aerospace*. 2021; 8(7):181.
https://doi.org/10.3390/aerospace8070181

**Chicago/Turabian Style**

Casalino, Lorenzo, Filippo Masseni, and Dario Pastrone.
2021. "Optimal Design of Electrically Fed Hybrid Mars Ascent Vehicle" *Aerospace* 8, no. 7: 181.
https://doi.org/10.3390/aerospace8070181