# Hybrid Rocket Engine Design Optimization at Politecnico di Torino: A Review

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Hybrid Rocket Engine Modeling

#### 2.1. Propellant Combination

#### 2.2. Feed System

#### 2.3. Grain Geometry

#### 2.4. Engine Modeling

## 3. Optimization

**r**(radius, latitude and longitude), velocity

**v**(radial, eastward and northward components) and rocket mass M, as reported in a vectorial form by Equation (6):

**F**and

**D**are the non-dimensional thrust and drag vector, respectively. Equation (7) presents the inverse-square gravity field assumed for

**g**, where G is the gravitational constant and ${M}_{*}$ is the planet mass.

#### 3.1. Robustness

**p**, ${g}_{j}$ is the j-th inequality constraint, ${\mathit{b}}_{L}$ and ${\mathit{b}}_{U}$ are, in the general case, the lower and upper boundary of the design parameters, respectively.

## 4. Applications

#### 4.1. Sounding Rocket for Micro-Gravity Platform

#### 4.2. Hypersonic Accelerator

#### 4.3. Mars Ascent Vehicle

#### 4.4. Small Launcher

#### 4.5. Upper Stage

- 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.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AIAA | American Institute of Aeronautics and Astronautics |

BD | Blow Down |

DET | DETerministic |

EE(s) | Elementary Effect(s) |

ESA | European Space Agency |

FO | Feed system Option |

GTOC | Global Trajectory Optimization Competition |

HP | Hydrogen Peroxide |

HTPB | Hydroxyl-Terminated PolyButadiene |

ILS | Iterated Local Search |

JPL | Jet Propulsion Laboratory |

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

LOX | Liquid OXygen |

MAV | Mars Ascent Vehicle |

NASA | National Aeronautics and Space Administration |

NOX | NiTrous Oxide |

N20 | Nitrous oxide |

PE | Polyethylene |

POLITO | POLItecnico di TOrino |

PS | Ports Shapes |

PSO | Particle Swarm Optimization |

QP | Quadrangular Ports |

R | Regulated |

ROB | ROBust |

SP | Self Pressurization |

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

TP | Triangular Ports |

VLV | Vega Launch Vehicle |

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${}^{n-1}$ |

b | design variables vector |

${\mathit{b}}_{L}$ | lower bound vector |

${\mathit{b}}_{U}$ | upper bound vector |

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

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

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

D | rocket outer diameter, m |

E | nozzle area ratio |

${E}_{e}$ | electric energy, Wh |

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

F | thrust, N |

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

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

${g}_{j}\left(\mathit{b}\right)$ | j-th inequality constraint |

h | specific enthalpy, J/kg |

${I}_{SP}$ | 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 |

${M}_{*}$ | planet mass, kg |

m | mass, kg |

n | mass-flux exponent |

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

p | uncertain variables vector |

p | pressure, bar |

${R}_{g}$ | grain outer radius, m |

${R}_{i}$ | grain initial inner radius, m |

${R}_{th}$ | throat radius, m |

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

s | eroded distance, mm |

t | time, s |

T | temperature, K |

u | specific internal energy, J/kg |

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

v | velocity, m/s |

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

w | web thickness, m |

y | burning distance, m |

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

${\mathit{z}}^{p}$ | noise vector of p |

$\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 |

$\Delta $ | altitude violation, m |

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

$\mu $ | payload, kg |

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

$\Phi $ | objective function, kg |

Superscripts | |

$\dot{}$ | time derivative |

* | characteristic |

Subscripts | |

0 | ambient |

1 | combustion chamber at head-end |

A | set A electric properties |

$Al$ | Aluminum |

a | auxiliary gas |

$abl$ | ablative |

$avg$ | average |

B | set B electric properties |

$BD$ | beginning of blow down phase |

b | batteries |

$burn$ | engine burn |

c | combustion chamber at nozzle entrance |

$case$ | engine casing |

$cc$ | combustion chamber |

d | discharge |

$dry$ | dry |

e | nozzle exit |

$ep$ | electric motor and pump |

$ev$ | evaporation |

F | fuel |

f | final |

$fs$ | feed system |

g | pressurizing gas |

i | initial value |

$lin$ | liner |

$max$ | maximum |

$min$ | minimum |

$nz$ | nozzle |

O | oxidizer |

p | overall propellant (oxidizer + fuel) |

$ref$ | reference |

$rep$ | repressurization |

$res$ | residual |

$sl$ | sliver |

t | oxidizer propellant tank |

$th$ | throat |

$tot$ | total |

$ull$ | ullage |

v | vapor |

⊕ | standard astronomical symbol for planet Earth |

## Appendix A. Gas Pressure Feed System

#### Appendix A.1. Blow down Feed System and Repressurization

#### Appendix A.2. Self-Pressurizing Oxidizer

#### Appendix A.3. Regulated Feed System

## Appendix B. Electrical Turbo Pump Feed System

Design Set | ${\mathit{\delta}}_{\mathbf{bp}}$ | ${\mathit{\delta}}_{\mathbf{be}}$ | ${\mathit{\delta}}_{\mathbf{ep}}$ | ${\mathit{\eta}}_{\mathbf{ep}}$ | ${\mathit{t}}_{\mathbf{burn}}^{*}$ |
---|---|---|---|---|---|

kW/kg | Wh/kg | kW/kg | - | s | |

A | 3.00 | 90.00 | 1.25 | 0.68 | 108 |

B | 6.95 | 198.50 | 3.92 | 0.53 | 103 |

## Appendix C. Multi-Port Fuel Grain Design

## Appendix D. Engine Modeling

## Appendix E. Robust-Based Method

#### Appendix E.1. Sensitivity Analysis and Screening

Input Parameter | Low | Nom | High | ||
---|---|---|---|---|---|

1 | ${\dot{s}}_{ref}$ | $\mathrm{m}/\mathrm{s}$ | 0.8510-4 | $1\times {10}^{-4}$ | $1.15\times {10}^{-4}$ |

2 | ${p}_{c,ref}$ | $\mathrm{bar}$ | 9.7 | 10 | $10.3$ |

3 | ${R}_{th,ref}$ | $\mathrm{m}$ | 0.198 | $0.200$ | $0.202$ |

4 | a | - | $9\times {10}^{-5}$ | $9.1\times {10}^{-5}$ | $9.2\times {10}^{-5}$ |

5 | n | - | $0.68$ | $0.69$ | $0.70$ |

6 | ${\rho}_{Al}$ | $\mathrm{kg}/{\mathrm{m}}^{3}$ | 2758 | 2800 | 2842 |

7 | ${\rho}_{abl}$ | $\mathrm{kg}/{\mathrm{m}}^{3}$ | 1724 | 1750 | 1776 |

8 | ${\rho}_{F}$ | $\mathrm{kg}/{\mathrm{m}}^{3}$ | 926 | 940 | 954 |

9 | ${\rho}_{O}$ | $\mathrm{kg}/{\mathrm{m}}^{3}$ | 1123 | 1140 | 1157 |

10 | ${T}_{g}$ | $\mathrm{K}$ | 278 | 298 | 318 |

11 | ${s}_{lin}$ | $\mathrm{m}$ | $5.6\times {10}^{-3}$ | $6.0\times {10}^{-3}$ | $6.4\times {10}^{-3}$ |

12 | ${s}_{cc}$ | $\mathrm{m}$ | $4.67\times {10}^{-4}$ | $5.00\times {10}^{-4}$ | $5.33\times {10}^{-4}$ |

13 | J | - | $0.495$ | $0.500$ | $0.505$ |

14 | ${p}_{t,i}$ | $\mathrm{bar}$ | 24.25 | 25 | $25.75$ |

15 | $\frac{\left|\right|{v}_{i}\left|\right|}{\left|\right|{v}_{i,ref}\left|\right|}$ | % | 97 | 100 | 103 |

16 | $\frac{\left|\right|{r}_{i}\left|\right|}{\left|\right|{r}_{\oplus}\left|\right|}$ | - | $1.0133$ | $1.0136$ | $1.0139$ |

17 | ${V}_{ull}$ | % | $2.85$ | $3.00$ | $3.15$ |

18 | ${\eta}_{{C}_{F}}$ | - | $0.975$ | $0.980$ | $0.985$ |

19 | ${\eta}_{{c}^{*}}$ | - | $0.955$ | $0.960$ | $0.965$ |

Parameter | Value |
---|---|

Number of steps for design parameters | 100 |

Number of input parameters | 19 |

Number of starting points | 100 |

Number of repetitions | 10 |

Number of function evaluations | $100\times (19+1)\times 10=$ 20,000 |

- (1)
- parameters with negligible or no effect on model output (i.e., both ${\mu}_{j}^{*}$ and ${\sigma}_{j}$ are negligible or zero): ${s}_{cc}$, $\frac{\left|\right|{\mathit{v}}_{i}\left|\right|}{\left|\right|{\mathit{v}}_{i,ref}\left|\right|}$, $\frac{\left|\right|{\mathit{r}}_{i}\left|\right|}{\left|\right|{\mathit{r}}_{\oplus}\left|\right|}$, ${\eta}_{{C}_{F}}$ and ${\eta}_{{c}^{*}}$ (marked in red in Figure A3). Hence, these input parameters may be regarded as model constants in the robust-based design and optimization and their values will be set equal to nominal ones;
- (2)
- parameters with linear effect on model output (i.e., ${\mu}_{j}^{*}$ and ${\sigma}_{j}$ are both significant and ${\mu}_{j}^{*}\simeq {\sigma}_{j}$): ${\rho}_{Al}$, ${\rho}_{abl}$, ${\rho}_{O}$, ${T}_{g}$, ${s}_{lin}$, ${V}_{ull}$ (marked in black in Figure A3). The effect of these input parameters is expected to be the same everywhere in the design space. Thus, they may be considered as model constant during the robust-based design and optimization, in order to minimize the number of uncertain parameters. However, their actual effect on the optimal solutions found out should be checked “a posteriori”;
- (3)
- parameters with non-linear effect on model output (i.e., both ${\mu}_{j}^{*}$ and ${\sigma}_{j}$ are significant): ${\dot{s}}_{ref}$, ${p}_{c,ref}$, ${R}_{th,ref}$, a, n, ${\rho}_{F}$, ${p}_{t,i}$ and J (given in light blue and green in Figure A3). These input parameters exhibit strong, not uniform and unpredictable effects on system output among the design space. Hence, such sources of uncertainty cannot be neglected and have to be taken into account in the robust-based design and optimization in each objective function evaluation. One can notice that the parameters marked in light blue (${\dot{s}}_{ref}$, ${p}_{c,ref}$ and ${R}_{th,ref}$) exhibit smaller (but still significant and not linear) effects with respect to the parameters marked in green (a, n, ${\rho}_{F}$, ${p}_{t,i}$ and J). Since these three parameters are involved only in the nozzle throat erosion model (see Equation (A40)), the authors opt for the definition of a synthetic erosion uncertain input parameters ${K}_{ero}$, as reported in Equation (A44), in order to maintain the number of uncertain parameters as low as possible.$${K}_{ero}={\dot{s}}_{ref}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{R}_{th,ref}^{0.2}/{p}_{c,ref}^{0.8}$$In this way, the rate of throat erosion $\dot{s}$, can be rewritten as:$$\dot{s}={K}_{ero}\frac{{p}_{c}^{0.8}}{{R}_{th}^{0.2}}$$

**p**collects six parameters and can be written as:

Input Parameter | ${\mathit{\mu}}_{\mathit{j}}^{*}$ | ${\mathit{\sigma}}_{\mathit{j}}$ | |
---|---|---|---|

1 | ${\dot{s}}_{ref}$ | 52.71 | 99.07 |

2 | ${p}_{c,ref}$ | 42.30 | 79.43 |

3 | ${R}_{th,ref}$ | 11.14 | 20.41 |

4 | a | 11,267.57 | 20,119.48 |

5 | n | 28,938.43 | 43,635.39 |

6 | ${\rho}_{Al}$ | 699.06 | 699.01 |

7 | ${\rho}_{abl}$ | 375.11 | 376.62 |

8 | ${\rho}_{F}$ | 9054.35 | 12,355.32 |

9 | ${\rho}_{O}$ | 463.84 | 463.88 |

10 | ${T}_{g}$ | 25.31 | 25.34 |

11 | ${s}_{lin}$ | 95.59 | 95.54 |

12 | ${s}_{cc}$ | 0.00 | 0.00 |

13 | J | 3868.66 | 6609.87 |

14 | ${p}_{t,i}$ | 4006.43 | 6401.29 |

15 | $\frac{\left|\right|{v}_{i}\left|\right|}{\left|\right|{v}_{i,ref}\left|\right|}$ | 0.00 | 0.01 |

16 | $\frac{\left|\right|{r}_{i}\left|\right|}{\left|\right|{r}_{\oplus}\left|\right|}$ | 0.29 | 1.10 |

17 | ${V}_{ull}$ | 8.11 | 8.11 |

18 | ${\eta}_{{C}_{F}}$ | 0.01 | 0.02 |

19 | ${\eta}_{{c}^{*}}$ | 0.01 | 0.02 |

#### Appendix E.2. Robust Design Model

1 | 2 | 2 | 2 | 2 | 2 |

3 | 2 | 2 | 2 | 2 | 2 |

2 | 1 | 2 | 2 | 2 | 2 |

2 | 3 | 2 | 2 | 2 | 2 |

2 | 2 | 1 | 2 | 2 | 2 |

2 | 2 | 3 | 2 | 2 | 2 |

2 | 2 | 2 | 1 | 2 | 2 |

2 | 2 | 2 | 3 | 2 | 2 |

2 | 2 | 2 | 2 | 1 | 2 |

2 | 2 | 2 | 2 | 3 | 2 |

2 | 2 | 2 | 2 | 2 | 1 |

2 | 2 | 2 | 2 | 2 | 3 |

1 | 1 | 1 | 1 | 1 | 1 |

1 | 2 | 2 | 2 | 2 | 2 |

1 | 3 | 3 | 3 | 3 | 3 |

2 | 1 | 1 | 2 | 3 | 3 |

2 | 2 | 2 | 3 | 1 | 1 |

2 | 3 | 3 | 1 | 2 | 2 |

3 | 1 | 2 | 3 | 2 | 3 |

3 | 2 | 3 | 1 | 3 | 1 |

3 | 3 | 1 | 2 | 1 | 2 |

1 | 1 | 3 | 2 | 2 | 1 |

1 | 2 | 1 | 3 | 3 | 2 |

1 | 3 | 2 | 1 | 1 | 3 |

2 | 1 | 2 | 1 | 3 | 2 |

2 | 2 | 3 | 2 | 1 | 3 |

2 | 3 | 1 | 3 | 2 | 1 |

3 | 1 | 3 | 3 | 1 | 2 |

3 | 2 | 1 | 1 | 2 | 3 |

3 | 3 | 2 | 2 | 3 | 1 |

1 | 1 | 2 | 1 | 2 | 2 |

1 | 1 | 2 | 3 | 2 | 2 |

1 | 3 | 2 | 1 | 2 | 2 |

1 | 3 | 2 | 3 | 2 | 2 |

3 | 1 | 2 | 1 | 2 | 2 |

3 | 3 | 2 | 1 | 2 | 2 |

3 | 1 | 2 | 3 | 2 | 2 |

3 | 3 | 2 | 3 | 2 | 2 |

2 | 1 | 1 | 2 | 1 | 2 |

2 | 1 | 3 | 2 | 1 | 2 |

2 | 1 | 3 | 2 | 3 | 2 |

2 | 1 | 1 | 2 | 3 | 2 |

2 | 3 | 3 | 2 | 3 | 2 |

2 | 3 | 3 | 2 | 1 | 2 |

2 | 3 | 1 | 2 | 3 | 2 |

2 | 3 | 1 | 2 | 1 | 2 |

2 | 2 | 1 | 1 | 2 | 1 |

2 | 2 | 1 | 1 | 2 | 3 |

2 | 2 | 1 | 3 | 2 | 1 |

2 | 2 | 1 | 3 | 2 | 3 |

2 | 2 | 3 | 1 | 2 | 1 |

2 | 2 | 3 | 1 | 2 | 3 |

2 | 2 | 3 | 3 | 2 | 1 |

2 | 2 | 3 | 3 | 2 | 3 |

1 | 2 | 2 | 1 | 1 | 2 |

1 | 2 | 2 | 1 | 3 | 2 |

1 | 2 | 2 | 3 | 1 | 2 |

1 | 2 | 2 | 3 | 3 | 2 |

3 | 2 | 2 | 1 | 1 | 2 |

3 | 2 | 2 | 1 | 3 | 2 |

3 | 2 | 2 | 3 | 1 | 2 |

3 | 2 | 2 | 3 | 3 | 2 |

2 | 1 | 2 | 2 | 1 | 3 |

2 | 1 | 2 | 2 | 3 | 1 |

2 | 1 | 2 | 2 | 3 | 3 |

2 | 3 | 2 | 2 | 1 | 1 |

2 | 3 | 2 | 2 | 1 | 3 |

2 | 3 | 2 | 2 | 3 | 1 |

2 | 3 | 2 | 2 | 3 | 3 |

1 | 2 | 1 | 2 | 2 | 1 |

1 | 2 | 1 | 2 | 2 | 3 |

1 | 2 | 3 | 2 | 2 | 1 |

1 | 2 | 3 | 2 | 2 | 3 |

3 | 2 | 1 | 2 | 2 | 1 |

3 | 2 | 1 | 2 | 2 | 3 |

3 | 2 | 3 | 2 | 2 | 1 |

3 | 2 | 3 | 2 | 2 | 3 |

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**Figure 1.**Comparison of NOX tank pressures: calculated vs. experimental data [27].

**Figure 3.**Thrust and mixture ratio histories for blow down, regulated and pump-fed optimized solutions.

**Figure 5.**Deterministic optimization (800 km altitude): thrust (F), longitudinal acceleration ($F/m$), specific impulse (${I}_{SP}$) and mixture ratio ($\alpha $) histories of deterministic optimal design A (left sub-figures) and B (right sub-figures).

**Figure 6.**Tank pressure (${p}_{t}$), thrust level (F) and mixture ratio ($\alpha $) during upper stage operation.

**Figure 7.**Trust and mixture ratio histories for the deterministic and robust optimal design (2 uncertain parameters).

**Table 1.**HREs propellant combinations. PE and HTPB stand for PolyEthylene and Hydroxyl-Terminated PolyButadiene, respectively.

Propellants | a | n | Ref. |
---|---|---|---|

${\mathbf{m}}^{1+2\mathbf{n}}$${\mathbf{kg}}^{-\mathbf{n}}{\mathbf{s}}^{\mathbf{n}-1}$ | - | - | |

HP/PE | $7.00\times {10}^{-6}$ | $0.800$ | [19,20] |

HP 90%/HTPB | $2.47\times {10}^{-5}$ | $0.666$ | [21] |

LOX/HTPB | $9.29\times {10}^{-6}$ | $0.852$ | [22] |

LOX/Wax | $9.10\times {10}^{-5}$ | $0.690$ | [13] |

NOX/PE | $1.04\times {10}^{-4}$ | $0.352$ | [23] |

NOX/HTPB | $1.87\times {10}^{-4}$ | $0.347$ | [24] |

**Table 2.**Mass budget comparison. The first column reports the feed system employed: regulated (R) or turbo-pump (TP). (A) and (B) refer to the electrical properties used in the mass evaluation (see Table A1).

Case | $\mathit{\mu}$ | ${\mathit{m}}_{\mathit{p}}$ | ${\mathit{m}}_{\mathit{c}\mathit{c}}$ | ${\mathit{m}}_{\mathit{t}}$ | ${\mathit{m}}_{\mathit{n}\mathit{z}}$ | ${\mathit{m}}_{\mathit{c}\mathit{a}\mathit{s}\mathit{e}}$ | ${\mathit{m}}_{\mathit{g}}$ | ${\mathit{m}}_{\mathit{a}}$ | ${\mathit{m}}_{\mathit{b}}$ | ${\mathit{m}}_{\mathit{e}\mathit{p}}$ |
---|---|---|---|---|---|---|---|---|---|---|

kg | kg | kg | kg | kg | kg | kg | kg | kg | kg | |

R | 2070 | 10,768 | 160 | 299 | 351 | 166 | 24 | 150 | - | - |

TP${}_{A}$ | 2322 | 10,800 | 147 | 12 | 310 | 164 | 0.032 | - | 98 | 134 |

TP${}_{B}$ | 2468 | 10,795 | 162 | 12 | 240 | 158 | 0.031 | - | 77 | 74 |

**Table 3.**Optimal design and performance for different port shapes (PS): triangular ports (TP) and quadrangular ports (QP). The second column reports the feed system option (FO): blow down (BD) and regulated (R). The design variables are ${F}_{i}$, ${\alpha}_{i}$, ${\left({p}_{t}\right)}_{i}$ and E. ${R}_{g}$, w, ${R}_{t}$ and L stand for grain outer radius, web thickness, throat radius and engine length, respectively. $F/m$ is the maximum longitudinal acceleration imparted to the payload $\mu $.

PS | FO | N | ${\mathit{F}}_{\mathit{i}}$ | ${\mathit{\alpha}}_{\mathit{i}}$ | ${\left({\mathit{p}}_{\mathit{t}}\right)}_{\mathit{i}}$ | E | ${\mathit{R}}_{\mathit{g}}$ | w | ${\mathit{R}}_{\mathit{t}}$ | L | $\mathit{F}/\mathit{m}$ | $\mathit{\mu}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathrm{kN}$ | − | $\mathrm{bar}$ | − | $\mathrm{m}$ | $\mathrm{m}$ | $\mathrm{m}$ | $\mathrm{m}$ | $\mathrm{g}$ | $\mathrm{kg}$ | |||

TP | BD | 6 | 365 | 7.08 | 20.2 | 11.8 | 0.520 | 0.039 | 0.288 | 12.3 | 5.77 | 1796 |

QP | BD | 6 | 359 | 7.09 | 20.7 | 12.1 | 0.515 | 0.040 | 0.281 | 12.2 | 5.67 | 1814 |

TP | BD | 8 | 414 | 7.02 | 19.8 | 11.3 | 0.558 | 0.034 | 0.310 | 12.0 | 6.64 | 1775 |

QP | BD | 8 | 388 | 7.01 | 20.8 | 11.9 | 0.537 | 0.037 | 0.293 | 11.9 | 6.18 | 1811 |

TP | R | 6 | 209 | 6.22 | 13.0 | 11.9 | 0.508 | 0.042 | 0.271 | 9.8 | 5.49 | 1989 |

QP | R | 6 | 203 | 6.21 | 13.5 | 12.4 | 0.501 | 0.044 | 0.262 | 9.6 | 5.35 | 2012 |

TP | R | 8 | 231 | 6.16 | 12.3 | 11.2 | 0.546 | 0.037 | 0.293 | 9.4 | 6.33 | 1971 |

QP | R | 8 | 213 | 6.11 | 13.3 | 12.1 | 0.522 | 0.041 | 0.271 | 9.3 | 5.82 | 2013 |

**Table 4.**Mass budget for different port shapes (PS): triangular ports (TP) and quadrangular ports (QP). The second column reports the feed system option (FO): blow down (BD) and regulated (R). The average mixture ratio and specific impulse are reported as ${\alpha}_{avg}$ and ${\left({I}_{SP}\right)}_{avg}$, respectively. ${m}_{res}$ is the residual propellant mass, ${m}_{g}$ is the pressurizing gas mass, ${m}_{t}$ is the tank mass, ${m}_{gt}$ is the pressurizing gas tank mass, ${m}_{nz}$ is the nozzle mass, ${m}_{cc}$ is the combustion chamber mass and ${m}_{hc}$ is the engine casing mass.

PS | FO | N | $\mathit{\mu}$ | ${\mathit{m}}_{\mathit{p}}$ | ${\mathit{m}}_{\mathit{r}\mathit{e}\mathit{s}}$ | ${\mathit{m}}_{\mathit{g}}$ | ${\mathit{m}}_{\mathit{t}}$ | ${\mathit{m}}_{\mathit{g}\mathit{t}}$ | ${\mathit{m}}_{\mathit{n}\mathit{z}}$ | ${\mathit{m}}_{\mathit{c}\mathit{c}}$ | ${\mathit{m}}_{\mathit{h}\mathit{c}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathrm{kg}$ | $\mathrm{kg}$ | $\mathrm{kg}$ | $\mathrm{kg}$ | $\mathrm{kg}$ | $\mathrm{kg}$ | $\mathrm{kg}$ | $\mathrm{kg}$ | $\mathrm{kg}$ | |||

TP | BD | 6 | 1796 | 10,966 | 43.6 | 22.4 | 445.0 | - | 330.4 | 125.1 | 207.0 |

QP | BD | 6 | 1814 | 10,925 | 31.0 | 23.0 | 456.3 | - | 328.5 | 121.6 | 204.9 |

TP | BD | 8 | 1775 | 10,984 | 57.2 | 22.3 | 440.3 | - | 333.8 | 121.5 | 201.5 |

QP | BD | 8 | 1811 | 10,931 | 31.8 | 23.3 | 459.3 | - | 329.8 | 117.6 | 200.2 |

TP | R | 6 | 1989 | 10,971 | 51.7 | 21.7 | 164.8 | 139.4 | 329.7 | 104.1 | 163.7 |

QP | R | 6 | 2012 | 10,921 | 37.3 | 22.7 | 170.4 | 146.1 | 327.2 | 100.2 | 161.1 |

TP | R | 8 | 1971 | 10,990 | 67.6 | 21.3 | 156.4 | 136.8 | 335.5 | 100.1 | 157.3 |

QP | R | 8 | 2013 | 10,926 | 38.6 | 23.0 | 167.5 | 148.0 | 329.8 | 95.6 | 155.5 |

**Table 5.**Example of design of experiment techniques: Taguchi’s ${L}_{18}$ orthogonal array. Each row reports a combination of uncertain parameters to be evaluated. The 1, 2 and 3 are the discrete levels that represent respectively lower than nominal, nominal and higher than nominal values. Each column corresponds to one of the uncertain parameters and shows its value expressed by the aforementioned levels.

1 | 1 | 1 | 1 | 1 | 1 |

1 | 2 | 2 | 2 | 2 | 2 |

1 | 3 | 3 | 3 | 3 | 3 |

2 | 1 | 1 | 2 | 3 | 3 |

2 | 2 | 2 | 3 | 1 | 1 |

2 | 3 | 3 | 1 | 2 | 2 |

3 | 1 | 2 | 3 | 2 | 3 |

3 | 2 | 3 | 1 | 3 | 1 |

3 | 3 | 1 | 2 | 1 | 2 |

1 | 1 | 3 | 2 | 2 | 1 |

1 | 2 | 1 | 3 | 3 | 2 |

1 | 3 | 2 | 1 | 1 | 3 |

2 | 1 | 2 | 1 | 3 | 2 |

2 | 2 | 3 | 2 | 1 | 3 |

2 | 3 | 1 | 3 | 2 | 1 |

3 | 1 | 3 | 3 | 1 | 2 |

3 | 2 | 1 | 1 | 2 | 3 |

3 | 3 | 2 | 2 | 3 | 1 |

**Table 6.**Optimal design and performance for the considered propellant combinations. ${m}_{p}$, ${R}_{t}$, D and L are the propellants mass boarded, the nozzle throat radius, the engine overall diameter and length, respectively. ${\left({V}_{g}\right)}_{i}$ and ${\left({p}_{t}\right)}_{i}$ are optimization variables only for HP/PE and LOX/HTPB combinations.

Propellants | ${\mathit{F}}_{\mathit{i}}$ | ${\mathit{\alpha}}_{\mathit{i}}$ | E | ${\left({\mathit{p}}_{\mathit{t}}\right)}_{\mathit{i}}$ | ${\left({\mathit{V}}_{\mathit{g}}\right)}_{\mathit{i}}$ | ${\mathit{m}}_{\mathit{p}}$ | ${\mathit{R}}_{\mathit{t}}$ | D | L | ${\mathit{t}}_{\mathit{\mu}\mathit{g}}$ |
---|---|---|---|---|---|---|---|---|---|---|

$\mathrm{kN}$ | − | − | $\mathrm{bar}$ | ${\mathrm{m}}^{3}$ | $\mathrm{kg}$ | $\mathrm{m}$ | $\mathrm{m}$ | $\mathrm{m}$ | $\mathrm{s}$ | |

HP/PE | 24.8 | 8.57 | 4.98 | 60.1 | 0.094 | 339 | 0.048 | 0.42 | 5.01 | 299 |

LOX/HTPB | 23.5 | 3.20 | 4.53 | 59.9 | 0.097 | 328 | 0.047 | 0.46 | 5.52 | 219 |

NOX/HTPB | 16.2 | 10.92 | 4.14 | 45.0 | – | 340 | 0.046 | 0.40 | 4.75 | 177 |

**Table 7.**Optimal design and performance of the hybrid-powered accelerator. D and L are the engine diameter and length. In the first column (Cfg.) the configurations are reported as: (1) single stage, (1+1) two stage/single engine per stage, (2+1) two stage/two engines in the first stage.

Cfg. | $\mathit{\mu}$ | ${\mathit{\alpha}}_{\mathit{i}}$ | ${\mathit{F}}_{\mathit{i}}$ | ${\left({\mathit{p}}_{\mathit{t}}\right)}_{\mathit{i}}$ | ${\left({\mathit{V}}_{\mathit{g}}\right)}_{\mathit{i}}$ | E | D | L | ${\mathit{M}}_{\mathit{f}}$ |
---|---|---|---|---|---|---|---|---|---|

$\mathrm{kg}$ | − | $\mathrm{kN}$ | $\mathrm{bar}$ | ${\mathrm{m}}^{3}$ | − | $\mathrm{m}$ | $\mathrm{m}$ | − | |

1 | 100 | 7.43 | 26.64 | 58.20 | 0.265 | 5.89 | 0.431 | 8.62 | 7.51 |

1 | 200 | 7.39 | 29.57 | 70.86 | 0.191 | 5.88 | 0.407 | 8.15 | 5.34 |

1+1 | 100 | 8.62 | 34.73 | 86.27 | 0.082 | 4.67 | 0.468 | 4.68 | 7.50 |

1+1 | 200 | 8.96 | 37.10 | 93.77 | 0.079 | 5.11 | 0.454 | 4.54 | 4.80 |

2+1 | 100 | 9.10 | 18.10 | 64.41 | 0.076 | 6.71 | 0.408 | 4.08 | 7.68 |

2+1 | 200 | 8.49 | 18.02 | 75.64 | 0.058 | 6.35 | 0.390 | 3.90 | 5.32 |

**Table 8.**Hybrid engine design and performance: sample return mission. The first column (Cfg.) reports the launcher configuration, e.g., 2+1 stands for two engines in the first stage and one engine in the second stage. The second column reports the feed system, blow down (BD) or regulated (R). ${V}_{He}$ is the volume of pressurizing gas D and L are engine diameter and length. ${m}_{d}$ is the engine dry mass and ${m}_{p}$ is the propellant mass.

Cfg. | FO | ${\mathit{\alpha}}_{\mathit{i}}$ | ${\left({\mathit{p}}_{\mathit{t}}\right)}_{\mathit{i}}$ | ${\left({\mathit{V}}_{\mathit{g}}\right)}_{\mathit{i}}$ | ${\mathit{V}}_{\mathit{H}\mathit{e}}$ | E | D | L | ${\mathit{m}}_{\mathit{d}}$ | ${\mathit{m}}_{\mathit{p}}$ | $\mathit{\mu}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

− | $\mathrm{bar}$ | ${\mathrm{m}}^{3}$ | ${\mathrm{m}}^{3}$ | − | $\mathrm{m}$ | $\mathrm{m}$ | $\mathrm{kg}$ | $\mathrm{kg}$ | $\mathrm{kg}$ | ||

2+1 | BD | 1.53 | 21.38 | 0.071 | - | 17.6 | 0.31 | 2.98 | 79.2 | 345.8 | 75.0 |

3+1 | BD | 1.44 | 21.51 | 0.055 | - | 18.8 | 0.36 | 2.63 | 85.4 | 339.8 | 74.8 |

4+1 | BD | 1.39 | 20.86 | 0.044 | - | 18.1 | 0.33 | 2.39 | 89.2 | 337.8 | 72.3 |

2+1 | R | 1.47 | 16.90 | - | 0.0097 | 17.2 | 0.24 | 2.77 | 67.8 | 346.0 | 86.3 |

3+1 | R | 1.38 | 17.92 | - | 0.0074 | 18.6 | 0.22 | 2.37 | 71.9 | 340.6 | 86.7 |

4+1 | R | 1.32 | 19.00 | - | 0.0058 | 19.5 | 0.21 | 2.13 | 76.2 | 338.2 | 84.8 |

**Table 9.**Hybrid engine design and performance: manned mission. The first column (Cfg.) reports the launcher configuration, e.g., 2+1 stands for two engines in the first stage and one engine in the second stage. The second column reports the feed system, blow down (BD) or regulated (R). D and L are engine diameter and length. ${m}_{d}$ is the engine dry mass and ${m}_{p}$ is the propellant mass.

Cfg. | FO | ${\mathit{\alpha}}_{\mathit{i}}$ | ${\left({\mathit{p}}_{\mathit{t}}\right)}_{\mathit{i}}$ | ${\left({\mathit{V}}_{\mathit{g}}\right)}_{\mathit{i}}$ | ${\mathit{V}}_{\mathit{H}\mathit{e}}$ | E | D | L | ${\mathit{m}}_{\mathit{d}}$ | ${\mathit{m}}_{\mathit{p}}$ | $\mathit{\mu}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

− | $\mathrm{bar}$ | ${\mathrm{m}}^{3}$ | ${\mathrm{m}}^{3}$ | − | $\mathrm{m}$ | $\mathrm{m}$ | $\mathrm{tons}$ | $\mathrm{tons}$ | $\mathrm{tons}$ | ||

2+1 | BD | 2.11 | 18.29 | 10.05 | - | 16.2 | 1.37 | 21.27 | 4.80 | 43.15 | 12.05 |

3+1 | BD | 2.05 | 17.06 | 7.76 | - | 15.2 | 1.25 | 18.94 | 4.83 | 42.89 | 12.38 |

4+1 | BD | 2.03 | 15.76 | 6.01 | - | 13.1 | 1.17 | 16.84 | 4.55 | 43.03 | 12.42 |

2+1 | R | 1.98 | 16.03 | - | 0.79 | 16.6 | 1.15 | 17.62 | 4.44 | 42.61 | 12.95 |

3+1 | R | 1.93 | 15.74 | - | 0.56 | 16.6 | 1.03 | 15.61 | 4.40 | 42.25 | 13.35 |

4+1 | R | 1.89 | 15.54 | - | 0.43 | 16.4 | 0.94 | 14.32 | 4.40 | 42.14 | 13.46 |

**Table 10.**Performance and design summary: deterministic optimization (800 km altitude). ${R}_{f}$ stands for the final grain burning distance, ${m}_{F}$ is the total fuel mass burned and ${m}_{d}$ is the engine dry mass.

Des. | ${\mathit{F}}_{\mathit{i}}$ | ${\left({\mathit{V}}_{\mathit{g}}\right)}_{\mathit{i}}$ | ${\mathit{\alpha}}_{\mathit{i}}$ | w | ${\mathit{R}}_{\mathit{f}}$ | D | E | ${\mathit{m}}_{\mathit{O}}$ | ${\mathit{m}}_{\mathit{F}}$ | ${\mathit{m}}_{\mathit{d}}$ | $\mathit{\mu}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathrm{kN}$ | ${\mathrm{m}}^{3}$ | − | $\mathrm{m}$ | $\mathrm{m}$ | $\mathrm{m}$ | − | $\mathrm{kg}$ | $\mathrm{kg}$ | $\mathrm{kg}$ | $\mathrm{kg}$ | |

A | 11.5 | 0.253 | 1.86 | 0.140 | 0.211 | 0.481 | 14.0 | 291.5 | 134.2 | 61.0 | 48.0 |

B | 27.2 | 0.108 | 2.58 | 0.105 | 0.213 | 0.549 | 12.5 | 297.2 | 129.1 | 57.8 | 73.6 |

**Table 11.**Performance and design comparison: deterministic design A (DET${}_{A}$) vs. robust design A (ROB${}_{A}$) (800 km altitude). ${\Delta}_{avg}$ is the average altitude violation with respect to the target altitude, whereas ${\Phi}_{avg}$ is the optimization merit function.

Des. | ${\mathit{\alpha}}_{\mathit{i}}$ | ${\mathit{R}}_{\mathit{g}}$ | ${\left({\mathit{V}}_{\mathit{g}}\right)}_{\mathit{i}}$ | ${\mathit{m}}_{\mathit{O},\mathit{t}\mathit{o}\mathit{t}}$ | E | ${\mathit{m}}_{\mathit{F}}$ | ${\mathit{m}}_{\mathit{d}}$ | $\mathit{\mu}$ | ${\mathbf{\Delta}}_{\mathit{a}\mathit{v}\mathit{g}}$ | ${\mathbf{\Phi}}_{\mathit{a}\mathit{v}\mathit{g}}$ |
---|---|---|---|---|---|---|---|---|---|---|

− | $\mathrm{m}$ | ${\mathrm{m}}^{3}$ | $\mathrm{kg}$ | − | $\mathrm{kg}$ | $\mathrm{kg}$ | $\mathrm{kg}$ | $\mathrm{km}$ | $\mathrm{kg}$ | |

DET${}_{A}$ | 1.86 | 0.211 | 0.253 | 291.5 | 14.00 | 134.2 | 61.0 | 48.0 | 500.75 | −9967.1 |

ROB${}_{A}$ | 1.87 | 0.212 | 0.261 | 291.8 | 14.00 | 133.3 | 63.0 | 34.3 | 0.00 | 34.3 |

**Table 12.**Hybrid upper stage design and performance. The first column reports the propellant combination. E column reports the initial nozzle area ratio when erosion effects are considered.

PC | ${\mathit{F}}_{\mathit{i}}$ | ${\mathit{\alpha}}_{\mathit{i}}$ | ${\left({\mathit{p}}_{\mathit{t}}\right)}_{\mathit{i}}$ | E | ${\mathit{R}}_{\mathit{g}}$ | w | L | ${\mathit{m}}_{\mathit{p}}$ | $\frac{{\mathit{m}}_{\mathit{p}}}{{\mathit{m}}_{\mathit{p}}+{\mathit{m}}_{\mathit{d}}}$ | $\mathit{\mu}$ |
---|---|---|---|---|---|---|---|---|---|---|

$\mathrm{kN}$ | − | $\mathrm{bar}$ | ${\mathrm{m}}^{3}$ | $\mathrm{m}$ | $\mathrm{m}$ | $\mathrm{m}$ | $\mathrm{kg}$ | − | $\mathrm{kg}$ | |

HP/PE | 231 | 6.16 | 12.3 | 11.2 | 0.546 | 0.037 | 9.4 | 10,989 | 0.924 | 1971 |

LOX/Wax | 185 | 1.98 | 13.3 | 10.8 | 0.619 | 0.258 | 9.7 | 10,699 | 0.933 | 2311 |

**Table 13.**Normalized ratios for the reference (VLV) and robust optimized solutions. GP2 and EP2 are the solutions for gas pressurized and electric turbo pump feed system employing simpler 2-parameter robust-based approach. GP6 reports the mass ratios for the 6-parameter robust-based approach.

Case | $\mathit{\mu}$ | $\frac{\mathit{\mu}}{{\mathit{m}}_{\mathit{i}}}$ | $\frac{{\mathit{m}}_{\mathit{p}}}{{\mathit{m}}_{\mathit{i}}}$ | $\frac{{\mathit{m}}_{\mathit{p}}}{{\mathit{m}}_{\mathit{p}}+{\mathit{m}}_{\mathit{d}\mathit{r}\mathit{y}}}$ |
---|---|---|---|---|

- | - | - | ||

VLV | 1430 | 0.094 | 0.731 | 0.840 |

GP2 | 2070 | 0.143 | 0.741 | 0.904 |

EP2 | 2322 | 0.160 | 0.744 | 0.926 |

GP6 | 2001 | 0.140 | 0.746 | 0.905 |

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**MDPI and ACS Style**

Casalino, L.; Masseni, F.; Pastrone, D.
Hybrid Rocket Engine Design Optimization at Politecnico di Torino: A Review. *Aerospace* **2021**, *8*, 226.
https://doi.org/10.3390/aerospace8080226

**AMA Style**

Casalino L, Masseni F, Pastrone D.
Hybrid Rocket Engine Design Optimization at Politecnico di Torino: A Review. *Aerospace*. 2021; 8(8):226.
https://doi.org/10.3390/aerospace8080226

**Chicago/Turabian Style**

Casalino, Lorenzo, Filippo Masseni, and Dario Pastrone.
2021. "Hybrid Rocket Engine Design Optimization at Politecnico di Torino: A Review" *Aerospace* 8, no. 8: 226.
https://doi.org/10.3390/aerospace8080226