Propulsion Systems Summary

Cheatsheet Content

### Introduction to Propulsion Systems A **propulsion system** is a device that generates **thrust** to push an object **forward**. This is achieved by accelerating a **working fluid** (like gas) through the engine, and the reaction to this acceleration produces a force on the engine. **Newton's Third Law:** When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body. This principle is fundamental to how propulsion systems work. ### Categorization of Propulsion Systems Propulsion systems can be broadly categorized: * **Air-breathing engines:** * Propeller (Reciprocating engine) * Jet Propulsion: Thrust equation for jet propulsion * Turbo engine * Turbofan engine * Ramjet engine * **Rocket engines:** * Calculating rocket engine performance * Rocket equation * Multistage rockets * Propellants * **Advanced space propulsion devices:** Electric propulsion ### Fundamental Thrust Equation The **Fundamental Thrust Equation** applies to both air-breathing and rocket engines. It describes the net force produced by the engine. $$T = (\dot{m}_{air} + \dot{m}_{fuel})V_e - \dot{m}_{air}V_\infty + (p_e - p_\infty)A_e$$ Where: * $T$: **Thrust force** (N) - The propulsive force generated by the engine. * $\dot{m}_{air}$: **Mass flow rate of air** (kg/s) - The mass of air entering the engine per unit time. For rocket engines, $\dot{m}_{air} = 0$. * $\dot{m}_{fuel}$: **Mass flow rate of fuel** (kg/s) - The mass of fuel consumed by the engine per unit time. * $V_e$: **Exhaust gas speed** (m/s) - Velocity of the hot gas mixture exiting the back of the engine. * $V_\infty$: **Free-stream air velocity** (m/s) - The velocity of the incoming air relative to the engine (or aircraft speed). * $p_e$: **Gas static pressure at exit** (Pa) - Static pressure of the exhaust gas at the nozzle exit. * $p_\infty$: **Ambient pressure** (Pa) - Pressure of the surrounding atmosphere (free-stream static pressure). * $A_e$: **Cross-sectional area of nozzle exit** (m²) - The area of the engine's exhaust nozzle. **Simplified Interpretations:** * **Momentum Thrust:** $(\dot{m}_{air} + \dot{m}_{fuel})V_e - \dot{m}_{air}V_\infty$ represents the change in momentum of the working fluid. * **Pressure Thrust:** $(p_e - p_\infty)A_e$ accounts for any pressure difference between the exhaust and the ambient environment at the nozzle exit. **Limitations/Considerations:** * The equation assumes steady-state flow. * For jet engines, $V_e >> V_\infty$ is desired for high thrust. * For rocket engines, there is no incoming air, so $\dot{m}_{air} = 0$. ### Jet Propulsion Jet engines generate thrust by exhausting gas out the back end faster than it comes in through the front. $$T = \dot{m}_{air} (V_e - V_\infty) + (p_e - p_\infty)A_e$$ Where: * $\dot{m}_{air}$: mass flow of air (kg/s) * $V_\infty$: free-stream air velocity (relative local wind velocity in front of the engine) (m/s) * $V_e$: velocity of hot gas mixture exiting at back of the jet engine (m/s) * $p_\infty$: the ambient pressure (Pa) * $p_e$: gas static pressure at the exit of engine (Pa) * $A_e$: area of the exit (m²) **Principle:** The net force is produced by pressure and shear stress distributions over the surface of the engine duct. **Integral Form of Thrust (for duct of jet engine):** The x-component of the net force on the gas inside the controlled volume is: $$F = p_\infty A_i + \int(p_s dS)_x - p_e A_e$$ Where $A_i$ is the inlet area and $\int(p_s dS)_x$ is the integral of static pressure over the internal surface. The force on the control volume due to momentum change is: $$F = (\dot{m}_{air} + \dot{m}_{fuel})V_e - \dot{m}_{air}V_\infty$$ By equating these, and considering the integral term for force at internal surface: $$\int(p_s dS)_x = (\dot{m}_{air} + \dot{m}_{fuel})V_e - \dot{m}_{air}V_\infty + p_e A_e - p_\infty A_i$$ Substituting this into the earlier equation for T (from the duct surface, not the gas): $$T = \int(p_s dS)_x + p_\infty \int(dS)_x = \int(p_s dS)_x + p_\infty (A_i - A_e)$$ This gives the fundamental thrust equation. ### Turbojet Engine A turbojet engine is a type of jet engine that uses a turbine to drive a compressor, which compresses incoming air. The air is then mixed with fuel, ignited, and the hot exhaust gases are expelled to generate thrust. **Thrust Equation for Turbojet (Simplified):** $$T = \dot{m}_{air} (V_e - V_\infty) + (p_e - p_\infty)A_e$$ (Note: $\dot{m}_{fuel}$ is often considered negligible compared to $\dot{m}_{air}$ for simplification in some contexts). **Key Components:** * **Diffuser:** Slows down incoming air. * **Compressor:** Increases air pressure. * **Burner/Combustion Chamber:** Fuel is injected and ignited. * **Turbine:** Extracts energy from hot gases to drive the compressor. * **Nozzle:** Accelerates exhaust gases to create thrust. **Example Problem Strategy:** 1. **Identify Given Values:** $\dot{m}_{air}$, $V_\infty$, $V_e$, $p_e$, $A_e$. 2. **Determine Missing Values:** Often $p_\infty$ needs to be found from standard atmosphere tables based on altitude. 3. **Check Units:** Ensure all units are consistent (e.g., convert km/h to m/s, kPa to Pa). 4. **Apply Thrust Equation:** Substitute values into the equation to calculate $T$. ### Turbofan Engine A turbofan engine is a type of jet engine that incorporates a large ducted fan. The fan accelerates a large mass of air (bypass air) that flows around the engine core, contributing significantly to thrust. The turbine drives both the fan and the compressor. **Thrust Generation:** Thrust is a combination of the thrust produced by the fan blades and the jet from the exhaust nozzle. **Bypass Ratio:** * **Definition:** The ratio of the mass of air passing outside of the core (bypass air) to the mass flow through the core engine. * **Significance:** Higher bypass ratios generally mean better fuel efficiency. * GE90 turbofan (Boeing 777): Bypass ratio of 9 (high bypass). * Pratt & Whitney F135 (F35 fighter): Bypass ratio of 0.2 (low bypass). **Thrust-Specific Fuel Consumption (TSFC):** * **Definition:** The rate of fuel mass flow divided by the thrust produced by the engine. It represents the fuel consumed per unit thrust per unit time. * **Efficiency Metric:** Lower TSFC indicates better fuel efficiency. * Turbojet: TSFC $\approx 1.0$ lb of fuel per pound of thrust per hour. * Turbofan: TSFC $\approx 0.6$ lb of fuel per pound of thrust per hour. * **Conclusion:** Turbofan engines are more fuel-efficient than turbojets, which is why modern commercial aircraft use them. ### Rocket Engine Rocket engines carry both their fuel and oxidizer, making them independent of the atmosphere for combustion. This allows them to operate in the vacuum of space. **Differences from Jet Engines:** * **Oxidizer Source:** Rockets carry their own oxidizer (liquid tank or mixed in solid fuel); jet engines use atmospheric air. * **Operating Environment:** Rockets can operate in space (vacuum); jet engines require air. * **Thrust Direction:** Rockets often provide upward thrust for launch; jet engines typically provide forward thrust. **Fundamental Thrust Equation for Rocket Engines:** Since $\dot{m}_{air} = 0$: $$T = \dot{m}_{fuel}V_e + (p_e - p_\infty)A_e$$ Where: * $\dot{m}_{fuel}$: **Total mass flow rate of propellants** (kg/s) - Sum of fuel and oxidizer mass flow rates ($\dot{m}_{fuel} = \dot{m}_{fuel\_actual} + \dot{m}_{oxidizer}$). * $V_e$: **Exhaust gas speed** (m/s). * $p_e$: **Gas static pressure at exit** (Pa). * $p_\infty$: **Ambient pressure** (Pa). In space, $p_\infty \approx 0$. * $A_e$: **Cross-sectional area of nozzle exit** (m²). **Nozzle Design (De Laval Nozzle):** * **Throat:** The narrowest section of the nozzle where the flow reaches Mach 1 (sonic flow). * **Supersonic Expansion:** After the throat, the exhaust gases expand supersonically, further increasing velocity and generating thrust. * **Equation for Varying Cross-Sectional Area:** $$\frac{dA}{A} = \frac{dV}{V}(M^2 - 1)$$ Where: * $A$: Cross-sectional area of the nozzle. * $V$: Flow velocity. * $M$: Mach number ($M = V/c$, where $c$ is the speed of sound). * **Subsonic Flow ($M 1$):** To increase $V$, $A$ must increase (diverging nozzle). * **Sonic Flow ($M = 1$):** Occurs at the throat where $dA/A = 0$. ### Engine Efficiency **Propulsive Efficiency ($\eta_p$):** * **Definition:** The ratio of the useful power provided by the propulsive device to the total power generated by the device. It measures how effectively the kinetic energy of the exhaust is converted into useful work (thrust). * **Formula:** $$\eta_p = \frac{\text{Useful Power}}{\text{Total Power}} = \frac{T_A V_\infty}{\frac{1}{2}\dot{m}(V_e^2 - V_\infty^2)}$$ Substituting $T_A = \dot{m}(V_e - V_\infty)$ (simplified thrust for efficiency analysis, assuming $p_e=p_\infty$ and $\dot{m}_{fuel}$ is included in $\dot{m}$): $$\eta_p = \frac{\dot{m}(V_e - V_\infty)V_\infty}{\frac{1}{2}\dot{m}(V_e^2 - V_\infty^2)} = \frac{2V_\infty(V_e - V_\infty)}{(V_e - V_\infty)(V_e + V_\infty)} = \frac{2V_\infty}{V_e + V_\infty} = \frac{2}{V_e/V_\infty + 1}$$ **Analysis of Propulsive Efficiency:** * **Maximum Efficiency:** $\eta_p$ reaches its maximum (100%) when $V_e = V_\infty$. However, at this point, the thrust $T_A = \dot{m}(V_e - V_\infty) = 0$. * **Relationship with Thrust:** As $V_e$ increases relative to $V_\infty$, thrust increases, but propulsive efficiency decreases. * **Trade-off:** There is a fundamental trade-off between maximizing thrust and maximizing propulsive efficiency. This trade-off justifies the existence of various types of propulsive devices, each optimized for different operating conditions (e.g., turbofans for high efficiency at subsonic speeds, rockets for high thrust in vacuum).