Aircraft Performance and Propulsion Efficiency

General Equations of Motion

• To calculate aircraft performance, solve the general equations of motion.
• Two-dimensional symmetric flight involves numerous variables.
• Equations with more than two unknowns are typically unsolvable.

Aerodynamic Drag and Propulsive Force

• Drag can be expressed as a function of airspeed assuming lift equals weight.
• Aim to express propulsive force as a function of airspeed to simplify motion equations.

Refresher on Propulsion

• Fundamental Thrust Equation: Applies to all air-breathing engines (propellers, turbofans, jets, helicopter rotors).
• Principle: Accelerate a mass of air to create force (momentum equation from Newton's second law).

Propulsion System Overview

• Steady Conditions:
  • Inflow velocity = flight speed
  • Jet velocity = velocity behind the aircraft
• Mass Flow (mdot): Air mass flow through the engine; includes fuel mass flow at exit.
• Thrust Equation:
  • Thrust = change in momentum rate
  • Simplified to ignore fuel mass flow and pressure terms

Options for Thrust Creation

• Jet Engines: Small mdot, high jet velocity
• Propellers: Large mdot, low acceleration

Efficiency Considerations

• Objective: Fly from A to B at specific velocity V.
• Power Available: Thrust x velocity
• Thermal Power (Q): Energy from fuel mass flow x energy per kg of fuel (H)
• Total Efficiency: Ratio of power available to thermal power
• Efficiency < 100% due to kinetic and heat energy left in the atmosphere

Propulsive Efficiency

• Formula: Propulsive efficiency (η_j) = Power available / Jet power
• Simplified Equation: 2 / (1 + Vj/V)
• Propulsive efficiency depends on the jet velocity to flight velocity ratio.
• Efficiency decreases as the airspeed to jet velocity ratio increases.

Example Calculations

• Scenario A: Aircraft at 100 m/s, thrust = 1000 N, efficiency = 66%
• Scenario B: Aircraft at 200 m/s, thrust = 1000 N, efficiency = 80%

Observations

• High jet velocity engines are inefficient at low speeds but efficient at high speeds.
• Explanation for jet engines in high-speed flights and propellers in low-speed applications.