# Numerical Model of a Variable-Combined-Cycle Engine for Dual Subsonic and Supersonic Cruise

^{*}

## Abstract

**:**

## 1. Introduction

Mach Range | Regime | Mode | Bypass Burner | Bypass Nozzle |
---|---|---|---|---|

0.0–0.9 | Subsonic Acceleration | Turbofan | On | Open |

0.9 | Subsonic Cruise | Turbofan | Off | Fully Open |

0.9–2.5 | Supersonic Acceleration | Turbofan | On | Open |

2.5–5.0 | Supersonic Acceleration | Ramjet + ATR | On | Open |

5.0 | Supersonic Cruise | ATR | Off | Closed |

## 2. Numerical Model

#### 2.1. Intake

#### 2.2. Combustion Chambers and Nozzle

#### 2.3. Heat Exchangers

**Figure 4.**Scheme of the air turbo-rocket numerical model: the station and component labeling are shown; the engine control devices are enclosed in circles.

#### 2.3.1. Precooler

**Figure 5.**Frontal view of the precooler module [17].

**Figure 7.**Computational domain for the periodic flow field: thermal field in counterflow disposition (

**a**), longitudinal cut (

**b**) and transversal cut (

**c**) of the tube matrix.

^{th}root of the sum of the ${n}^{th}$ powers of the limiting solutions of the independent variable [18]. The following ad hoc power-mean combination was used to define the laminar-turbulent transitional zone:

**Figure 9.**Comparison of different correlations for the Nusselt number for the precise geometry of the precooler. The operational range of the current application is shown between red lines.

#### 2.3.2. Reheater

**Figure 10.**Frontal and side view of the reheater module [17].

#### 2.3.3. Regenerator

**Figure 13.**Geometry of the regenerator units (

**left**) and cross section of the periodic domain representative of the overall unit (

**right**).

#### 2.4. Turbomachinery

**Figure 14.**Compressor-turbine matching: compressor and turbine input data are shown respectively in blue and red, dependent variables are shown in black.

${\overrightarrow{B}}^{d}$ | |||||||||||||

${p}_{{t}_{in}}^{d}$ | ${T}_{{t}_{in}}^{d}$ | ${\dot{m}}^{d}$ | ${\eta}^{d}$ | ${N}^{d}$ | $\left|{T}_{q}^{d}\right|$ | ${I}_{sq}$ | Scaling factors | ||||||

Label | [bar] | [K] | [kg s ${}^{-}$] | ${\pi}^{d}$ | [%] | ${\beta}^{d}$ | ${\tilde{N}}^{d}$ | [rpm] | [hN m] | [kg m ${}^{2}$] | ${K}_{\dot{m}}$ | ${K}_{\pi}$ | ${K}_{\eta}$ |

C | 2.9 | 648 | 172.6 | 4.2 | 86 | 0.74 | 0.75 | 10,471 | 594 | 0.51 | 1.54 | 2.27 | 0.99 |

C1 | 129.6 | 296 | 11.2 | 1.5 | 94 | 0.63 | 0.72 | 48,340 | 7 | 0.01 | 0.40 | 0.39 | 1.11 |

C2 | 129.5 | 371 | 11.2 | 1.5 | 94 | 0.63 | 0.72 | 48,340 | 9 | 0.01 | 0.45 | 0.39 | 1.11 |

C3 | 129.4 | 458 | 11.2 | 1.5 | 94 | 0.63 | 0.72 | 48,340 | 11 | 0.01 | 0.50 | 0.39 | 1.11 |

C4 | 129.3 | 559 | 11.2 | 1.5 | 94 | 0.63 | 0.72 | 48,340 | 13 | 0.01 | 0.55 | 0.39 | 1.11 |

C5 | 129.2 | 676 | 11.2 | 1.5 | 94 | 0.63 | 0.72 | 48,340 | 16 | 0.01 | 0.61 | 0.39 | 1.11 |

C6 | 50.9 | 33 | 11.2 | 3.9 | 90 | 0.63 | 0.65 | 48,357 | 4 | 0.01 | 0.45 | 2.71 | 1.08 |

C7 | 51.4 | 75 | 11.2 | 3.9 | 90 | 0.63 | 0.65 | 48,357 | 8 | 0.01 | 0.62 | 2.67 | 1.08 |

C8 | 51.3 | 151 | 11.2 | 3.9 | 90 | 0.63 | 0.65 | 48,357 | 15 | 0.01 | 0.86 | 2.68 | 1.08 |

T1 | 195.7 | 1000 | 89.6 | 1.5 | 91 | 0.20 | 1.00 | 10,471 | 594 | 0.51 | 1.03 | 1.05 | 0.96 |

T2 | 130.1 | 863 | 22.5 | 2.5 | 89 | 0.35 | 1.00 | 48,340 | 56 | 1.00 | 2.40 | 1.00 | 1.02 |

T3 | 130.1 | 863 | 11.1 | 2.5 | 88 | 0.35 | 1.00 | 48,357 | 27 | 1.00 | 1.18 | 1.00 | 1.01 |

## 3. Engine Control

## 4. Results

**Figure 15.**Station total pressures (${\text{blue numerals, bar}}$), total temperatures (${\text{red numerals, K}}$) and mass flows (black numerals, kg/s) during acceleration (

**A**and

**B**). Fluid lines and components working with air, helium, hydrogen and combustion gases are drawn respectively in blue, green, brown and red colors. The ambient (static) conditions are labeled with (*).

**Figure 16.**Station total pressures (${\text{blue numerals, bar}}$), total temperatures (${\text{red numerals, K}}$) and mass flows (black numerals, kg/s) during acceleration (

**C**and

**D**). Fluid lines and components working with air, helium, hydrogen and combustion gases are drawn respectively in blue, green, brown and red colors. The ambient (static) conditions are labeled with (*).

**Figure 17.**Station total pressures (${\text{blue numerals, bar}}$), total temperatures (${\text{red numerals, K}}$) and mass flows (black numerals, kg/s) during acceleration (

**E**) and cruise (

**F**). Fluid lines and components working with air, helium, hydrogen and combustion gases are drawn respectively in blue, green, brown and red colors. The ambient (static) conditions are labeled with (*).

**Figure 19.**Turbomachinery performance: adiabatic efficiency (η), pressure ratio (π) and corrected speed ($\tilde{N}$) vs. flight Mach number (Ma${}_{\infty}$) and throttling level (${\dot{m}}_{11}$).

**Figure 20.**Scimitar core operational range: uninstalled thrust and specific impulse vs. fuel consumption and flight condition.

## 5. Conclusions

## Acknowledgements

## Nomenclature:

A | = transversal area [m${}^{2}$] |

${A}_{t}$ | = cross section of the reheater helium channels [m${}^{2}$] |

a | = speed of sound [m s${}^{-1}$] |

${a}_{ji}$ | = total mass of chemical element j in the chemical species i |

b | = blockage ratio of the tubes in crossflow |

${b}_{j}$ | = total mass of chemical element j in the gas mixture |

C | = heat capacity [m${}^{2}$ K${}^{-1}$ s${}^{-2}$] |

${C}_{p}$ | = specific heat at constant pressure [m${}^{2}$ K${}^{-1}$ s${}^{-2}$] |

D | = diameter [m] |

${D}_{h}$ | = hydraulic diameter [m] |

e | = tube minimum distance to diameter ratio or total specific energy
($e=u+0.5\phantom{\rule{4pt}{0ex}}{v}^{2}$) [m${}^{2}$ s${}^{-2}$] |

${F}_{u}$ | = uninstalled thrust [kg m s${}^{-2}$] |

$\mathcal{F}$ | = mathematical function whose root defines the constraint which the turbomachinery design parameters ${\pi}^{d}$, ${T}_{q}^{d}{\Omega}^{d}$ and ${\dot{m}}^{d}$ must satisfy |

G | = Gibbs potential [kg m${}^{2}$ s${}^{-2}$] |

${\mathcal{G}}_{\dot{m}}$ | = mathematical function whose root defines the locus of turbomachinery operational points which satisfy the fluid dynamical constraint imposed by the turbomachine discharge duct |

${\mathcal{G}}_{\Omega}$ | = mathematical function whose root defines the locus of turbomachinery operational points which satisfy the mechanical constraint imposed by the turbomachine shaft |

$\mathcal{H}$ | = adiabatic efficiency characteristic of the unscaled turbomachine |

h | = specific enthalpy [m${}^{2}$ s${}^{-2}$] |

${h}_{c}$ | = convective heat transfer coefficient [kg K${}^{-1}$ s${}^{-3}$] |

${I}_{sh}$ | = shaft inertia [kg m${}^{2}$] |

${I}_{sp}$ | = specific impulse [m s${}^{-1}$] |

k | = thermal conductivity [kg K${}^{-1}$ m s${}^{-3}$] or controller sensitivity |

${K}_{\dot{m}}$ | = mass flow scaling factor |

${K}_{\eta}$ | = efficiency scaling factor |

${K}_{\pi}$ | = pressure ratio scaling factor |

L | = length [m] |

${L}_{c}$ | = characteristic length [m] |

${L}_{s}$ | = axial length of the reheater module strips [m] |

${l}_{th}$ | = characteristic thermal entry length [m] |

Ma | = Mach number |

${M}_{c}$ | = number of coolant channels per row in the regenerator module |

${M}_{h}$ | = number of heatant channels per row in the regenerator module |

${M}_{m}$ | = number of modules in each regenerator unit |

$\mathcal{M}$ | = mass flow characteristic of the unscaled turbomachine [kg s${}^{-1}$] |

m | = mass [kg] |

$\dot{m}$ | = mass flow rate [kg s${}^{-1}$] |

$\tilde{\dot{m}}$ | = corrected mass flow rate [Equation (38)] [kg s${}^{-1}$] |

N | = number of mols or rotational speed [rpm] |

${N}_{p}$ | = number of plates of the reheater module |

${N}_{r}$ | = number of rows of heatant / coolant channels per regenerator module |

${N}_{t}$ | = number of helium channels per strip of the reheater module |

$\tilde{N}$ | = corrected speed [Equation (37)] |

n | = number of nodes |

${n}_{z}$ | = number of strips per plate of the reheater module |

Nu | = Nusselt number |

Nu${}_{tur}$ | = Nusselt number in turbulent flow |

Nu${}_{lam}^{q}$ | = Nusselt number in laminar fully developed flow with uniform heat flux boundary condition |

Nu${}_{lam}^{T}$ | = Nusselt number in laminar fully developed flow with isothermal boundary condition |

Pr | = Prandtl number |

p | = pressure [kg m${}^{-1}$ s${}^{-2}$] |

$\dot{q}$ | = heat flux [kg s${}^{-3}$] |

R | = ideal gas constant [K${}^{-1}$ m${}^{2}$ s${}^{-2}$] |

${R}_{c}$ | = curvature radius [m] |

Re | = Reynolds number |

Sh | = Strouhal number |

${\mathcal{S}}_{\dot{m}}$ | = mathematical function whose root defines the locus of the turbomachinery steady operational points which satisfy the fluid dynamical constraint imposed by the turbomachine discharge duct |

${\mathcal{S}}_{\Omega}$ | = mathematical function whose root defines the locus of turbomachinery operational points which satisfy the mechanical constraint imposed by the turbomachine shaft |

s | = tangential pitch between the reheater plates [m] |

T | = temperature [K] |

${T}_{q}$ | = torque [kg m${}^{2}$ s${}^{-2}$] |

t | = reheater plate thickness [m] or time [s] |

${t}_{c}$ | = characteristic time [s] |

v | = velocity [m s${}^{-1}$] |

${x}_{l}$ | = ratio of precooler tube longitudinal pitch to external diameter |

${x}_{t}$ | = ratio of precooler tube transversal pitch to external diameter |

Greek | |
---|---|

β | = coordinate of the turbomachine map parametrization |

Γ | = perimeter [m] |

γ | = ratio of specific heats |

$\Delta x$ | = increment of x |

δ | = dimensionless turbomachine equivalent inlet pressure ($p/{p}_{std}$) |

${\u03f5}_{r}$ | = rugosity [m] |

η | = adiabatic efficiency |

${\eta}_{k}$ | = intake kinetic efficiency |

${\eta}_{n}$ | = nozzle efficiency |

Θ | = dimensionless turbomachine equivalent inlet temperature [Equation (39)] |

λ | = tube stagger angle [rad] |

μ | = viscosity [kg m${}^{-1}$ s${}^{-1}$] |

ξ | = friction factor [m${}^{-1}$] |

Π | = pressure ratio characteristic of the unscaled turbomachine |

π | = turbomachine compression (compressor) or expansion (turbine) ratio |

ρ | = density [kg m${}^{-3}$] |

σ | = constant of Stefan-Boltzmann [kg K${}^{-4}$ s${}^{-3}$] |

τ | = valve response time [s] |

Ω | = rotational speed [rad s${}^{-1}$] |

Subscripts | |
---|---|

c | = relative to the compressor |

i | = relative to the i ^{th} grid node or element in the set |

$in$ | = relative to the inlet |

$out$ | = relative to the outlet |

s | = corresponding to an isentropic evolution |

$st$ | = relative to the stream tube |

$std$ | = standard |

t | = stagnation quantity or relative to the turbine |

$th$ | = relative to the nozzle throat |

w | = relative to the wall |

∞ | = free stream conditions |

Superscript | |
---|---|

$\dot{x}$ | = time derivative |

${x}^{d}$ | = design value |

${x}^{0}$ | = reference value |

Acronyms | |
---|---|

BB | = bypass burner |

BN | = bypass nozzle |

CC | = main combustion chamber |

DASSL | = differential algebraic system solver algorithm |

ESPSS | = European Space Propulsion System Simulation |

F | = by-pass fan |

HT | = hub turbine |

MR | = mixture ratio, i.e., ratio of air to fuel mass flows |

NIST | = National Institute of Standards and Technology |

PB | = preburner |

PC | = precooler |

RG | = regenerator |

TPR | = intake total pressure recovery (${p}_{{t}_{out}}/{p}_{{t}_{in}}$) |

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## Share and Cite

**MDPI and ACS Style**

Fernandez-Villace, V.; Paniagua, G. Numerical Model of a Variable-Combined-Cycle Engine for Dual Subsonic and Supersonic Cruise. *Energies* **2013**, *6*, 839-870.
https://doi.org/10.3390/en6020839

**AMA Style**

Fernandez-Villace V, Paniagua G. Numerical Model of a Variable-Combined-Cycle Engine for Dual Subsonic and Supersonic Cruise. *Energies*. 2013; 6(2):839-870.
https://doi.org/10.3390/en6020839

**Chicago/Turabian Style**

Fernandez-Villace, Victor, and Guillermo Paniagua. 2013. "Numerical Model of a Variable-Combined-Cycle Engine for Dual Subsonic and Supersonic Cruise" *Energies* 6, no. 2: 839-870.
https://doi.org/10.3390/en6020839