Propulsion and Jet Engines — Thermodynamics, Fluid Dynamics, and Cycle Analysis

Scope: a first-principles treatment of airbreathing propulsion systems. Includes thrust generation, turbojet/turbofan cycles, performance metrics, and thermodynamic irreversibility.

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1. Fundamentals of Propulsive Flow

1.1 Control Volume Formulation

For a steady one-dimensional flow through a propulsion device: $F = \dot{m}(V_e - V_0) + (p_e - p_0)A_e.$

Where:

• $F$: net thrust
• $V_0$: freestream velocity
• $V_e$: exhaust velocity
• $p_e, A_e$: exit static pressure and area

If $p_e \approx p_0$, pressure thrust is negligible: $F \approx \dot{m}(V_e - V_0).$

1.2 Energy Balance

Total energy change per unit mass: $h_0 = h + \frac{V^2}{2} = \text{constant (adiabatic flow)}.$

Propulsive power: $P = F V_0.$

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2. Specific Impulse and Efficiency Metrics

2.1 Specific Impulse

$I_{sp} = \frac{F}{\dot{m}_f g_0}.$

2.2 Thrust Specific Fuel Consumption (TSFC)

$\text{TSFC} = \frac{\dot{m}_f}{F}.$

2.3 Propulsive Efficiency

$\eta_p = \frac{2V_0}{V_e + V_0}.$

Maximum $\eta_p$when $V_e \approx V_0$— thus high mass flow, low jet velocity engines (turbofans) are more efficient.

2.4 Overall Efficiency

$\eta_o = \eta_p \eta_{th}.$ Where $\eta_{th}$is thermal efficiency of the cycle.

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3. The Ideal Brayton Cycle for Jet Propulsion

3.1 Process Steps

1. Isentropic compression in inlet and compressor.
2. Constant-pressure heat addition in combustor.
3. Isentropic expansion in turbine and nozzle.

Thermal efficiency: $\eta_{th} = 1 - \frac{1}{r_p^{(\gamma-1)/\gamma}},$ where $r_p = p_2/p_1$is compressor pressure ratio.

3.2 Jet Velocity and Thrust

From energy conservation: $\frac{V_e^2 - V_0^2}{2} = c_p T_0 (\eta_{overall})(1 - T_4/T_0).$

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4. Turbojet Engine

4.1 Configuration

• Diffuser → Compressor → Combustor → Turbine → Nozzle.

4.2 Thrust Equation

$F = \dot{m}_a (V_e - V_0) + (p_e - p_0)A_e.$

For ideal expansion $p_e = p_0$: $F = \dot{m}_a (V_e - V_0).$

4.3 Energy Flow

Compressor work: $w_c = c_p (T_{t2} - T_{t1}).$ Turbine work: $w_t = c_p (T_{t3} - T_{t4}).$ Combustor heat addition: $q_{add} = c_p (T_{t3} - T_{t2}).$

Turbine drives compressor: $w_t = w_c.$

4.4 Efficiency Relations

Thermal efficiency: $\eta_{th} = 1 - \frac{T_{t4} - T_{t1}}{T_{t3} - T_{t2}}.$ Overall efficiency: $\eta_o = \frac{F V_0}{\dot{m}_f q_{add}}.$

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5. Turbofan Engine

5.1 Bypass Concept

A turbofan divides airflow into:

• Core stream (through turbine and nozzle)
• Bypass stream (fan-accelerated air)

Bypass ratio $\beta$: $\beta = \frac{\dot{m}_{bypass}}{\dot{m}_{core}}.$

5.2 Effective Jet Velocity

$V_{e,eff} = \frac{\beta V_{bypass} + V_{core}}{\beta + 1}.$

Thrust: $F = \dot{m}_{core}(V_{core} - V_0) + \dot{m}_{bypass}(V_{bypass} - V_0).$

5.3 Efficiency and Trade-offs

• High $\beta$: higher propulsive efficiency, lower specific thrust.
• Low $\beta$: higher specific thrust, lower efficiency.

Typical $\beta \approx 5-10$for modern turbofans.

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6. Turbojet vs. Turbofan

Parameter	Turbojet	Turbofan
Bypass ratio	0	5–10
Efficiency	Moderate	High
Noise	High	Low
Best for	Supersonic	Subsonic

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7. Ramjet and Scramjet Engines

7.1 Ramjet (Subsonic Combustion)

• Compression achieved by inlet shock system.
• No rotating machinery.

Ideal efficiency: $\eta_{th} = 1 - \frac{T_0}{T_{comb}}.$

Thrust: $F = \dot{m}(V_e - V_0).$

Operates efficiently at $M = 3-6.$

7.2 Scramjet (Supersonic Combustion)

• Flow remains supersonic throughout combustion.
• Requires precompression via oblique shocks.

Energy balance: $\dot{Q}_{comb} = \dot{m} c_p (T_{t4} - T_{t3}).$

Efficiency depends on combustion residence time and mixing.

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8. Rocket Engines (Contrast)

Thrust generation independent of ambient air: $F = \dot{m}_p V_e + (p_e - p_0)A_e.$

Specific impulse: $I_{sp} = \frac{V_e}{g_0}.$

Key distinction: mass flow from onboard propellant, not atmosphere.

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9. Component Efficiencies and Real Effects

Component	Efficiency	Description
Compressor	$\eta_c = (T_{t2s} - T_{t1}) / (T_{t2} - T_{t1})$	Polytropic loss
Turbine	$\eta_t = (T_{t3} - T_{t4}) / (T_{t3} - T_{t4s})$	Expansion inefficiency
Combustor	$\eta_b = (h_{t3} - h_{t2})/(f h_{fuel})$	Incomplete combustion
Nozzle	$\eta_n = (V_e/V_{e,isentropic})^2$	Kinetic energy recovery

Total-to-total efficiency: $\eta_{overall} = \eta_c \eta_t \eta_n \eta_b.$

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10. Afterburning and Variable Geometry

10.1 Afterburner

Reheats exhaust gases downstream of turbine: $q_{ab} = c_p (T_{t5} - T_{t4}).$ Increases thrust but reduces efficiency dramatically.

10.2 Variable-Area Nozzle

Maintains optimal expansion across flight regimes: $\frac{A_e}{A^*} = \frac{1}{M_e}\left[\frac{2}{\gamma+1}(1 + \frac{\gamma-1}{2}M_e^2)\right]^{(\gamma+1)/2(\gamma-1)}.$

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11. Exergy and Irreversibility in Propulsion Systems

Entropy generation in control volume: $\dot{S}_{gen} = \dot{m}(s_e - s_0) - \frac{\dot{Q}}{T_0}.$

Exergy destruction rate: $\dot{E}_D = T_0 \dot{S}_{gen}.$

Irreversibilities dominate in:

• Combustion (chemical disequilibrium)
• Turbine/compressor inefficiencies
• Shock-induced entropy increase in high-Mach inlets

Exergy efficiency: $\eta_{ex} = 1 - \frac{\dot{E}_D}{\dot{E}_{input}}.$

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12. Example Performance Parameters

Engine Type	Bypass Ratio	Typical Isp (s)	Flight Mach	η_o
Turbojet	0	2000–2500	1–3	0.25–0.35
Turbofan	5–10	3000–4000	0.8–1.2	0.35–0.45
Ramjet	—	1000–1200	2–6	0.20–0.30
Scramjet	—	1200–1500	6–15	0.15–0.25

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13. Design Trade-offs and Trends

• Turbofan evolution: ultra-high bypass ratios → reduced fuel burn.
• Variable cycle engines: adaptive geometry for Mach 0–5 operation.
• Combined-cycle systems: turbine-based + airbreathing + rocket modes.
• Electric and hybrid propulsion: integration with battery or fuel-cell sources.

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14. Summary of Key Equations

Concept	Equation	Notes
Thrust	$F = \dot{m}(V_e - V_0) + (p_e - p_0)A_e$	Momentum–energy balance
Specific impulse	$I_{sp} = F/(\dot{m}_f g_0)$	Measure of fuel efficiency
Propulsive efficiency	$\eta_p = 2V_0/(V_e + V_0)$	Optimal when $V_e \approx V_0$
Brayton thermal efficiency	$\eta_{th} = 1 - 1/r_p^{(\gamma-1)/\gamma}$	Ideal gas model
Bypass ratio	$\beta = \dot{m}_{bypass}/\dot{m}_{core}$	Defines turbofan class
Exergy destruction	$E_D = T_0 S_{gen}$	Irreversibility measure

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15. Cross-Links

• Fluid_Dynamics/11_Turbomachinery_and_Compressible_Devices.md — compressor and turbine fundamentals.
• Thermodynamics/10_NonEquilibrium_Thermodynamics.md — entropy production and chemical exergy.
• Fluid_Dynamics/09_Compressible_and_Supersonic_Flow.md — shock and expansion dynamics.
• Aero_Thermodynamics/HighSpeed_Propulsion.md — advanced hypersonic and combined-cycle systems.