Jet Propulsion

Concept

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.

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

1.2 Energy Balance

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

Propulsive power: $P = F V_0.$

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

$η_p = \frac{2V_0}{V_e + V_0}.$

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

2.4 Overall Efficiency

$η_o = η_p η_{th}.$ Where $η_{th}$ is thermal efficiency of the cycle.

3. The Ideal Brayton Cycle for Jet Propulsion

3.1 Process Steps

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

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

3.2 Jet Velocity and Thrust

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

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

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 $β$ : $β = \frac{\dot{m}_{bypass}}{\dot{m}_{core}}.$

5.2 Effective Jet Velocity

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

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

5.3 Efficiency and Trade-offs

High $β$ : higher propulsive efficiency, lower specific thrust.
Low $β$ : higher specific thrust, lower efficiency.

Typical $β ≈ 5–10$ for modern turbofans.

6. Turbojet vs. Turbofan

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

7. Ramjet and Scramjet Engines

7.1 Ramjet (Subsonic Combustion)

Compression achieved by inlet shock system.
No rotating machinery.

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

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.

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.

9. Component Efficiencies and Real Effects

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

Total-to-total efficiency: $η_{overall} = η_c η_t η_n η_b.$

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: $\frac{A_e}{A^*} = \frac{1}{M_e}\left[\frac{2}{γ+1}(1 + \frac{γ-1}{2}M_e^2)\right]^{(γ+1)/2(γ-1)}.$

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

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

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.

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	$η_p = 2V_0/(V_e + V_0)$	Optimal when $V_e ≈ V_0$
Brayton thermal efficiency	$η_{th} = 1 - 1/r_p^{(γ-1)/γ}$	Ideal gas model
Bypass ratio	$β = \dot{m}_{bypass}/\dot{m}_{core}$	Defines turbofan class
Exergy destruction	$E_D = T_0 S_{gen}$	Irreversibility measure

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.