Propulsion: air-breathing cycles, thrust and efficiency, and rocket nozzle analysis
Propulsion spans air-breathing engines (turbojet, turbofan, ramjet as Brayton-cycle machines) and rockets, with heavy emphasis on thrust, specific fuel consumption, efficiencies and nozzle behaviour.
Core rules
- Thrust: F = ṁ[(1+f)u_e − u] + (p_e − p_a)A_e; the pressure term vanishes when the nozzle is ideally expanded.
- Ideal turbojet: specific thrust = (1+f)u_e − u; TSFC = ṁf/F. Propulsive efficiency η_p = 2u/(u + u_e) for a simple jet, thermal efficiency from the Brayton cycle, overall = η_th × η_p.
- A high bypass ratio raises propulsive efficiency (lower exhaust velocity) but limits high-speed thrust; an afterburner adds thrust at a large TSFC penalty.
- Rocket: Δv = Isp·g0·ln(m0/mf). Characteristic velocity c* = p0A*/ṁ measures combustion performance; thrust coefficient CF measures nozzle performance.
- Nozzle: choked at the throat, area ratio from the isentropic area–Mach relation; underexpansion when p_e > p_a, overexpansion when p_e < p_a, with flow separation possible in the latter case.
Worked example
A rocket with Isp = 300 s and mass ratio m0/mf = 4: exhaust velocity = 300 × 9.81 = 2943 m/s, so Δv = 2943 × ln 4 = 2943 × 1.386 = 4079 m/s ≈ 4.08 km/s. Doubling the mass ratio to 8 adds only ln 2 × 2943 ≈ 2.04 km/s - a reminder that delta-V is logarithmic in propellant mass.
Traps
- Forgetting to convert Isp in seconds to exhaust velocity by multiplying by g0.
- Ignoring the pressure-thrust term for a nozzle operating off-design.
- Using static thrust expressions in flight, or applying the propulsive-efficiency formula outside a simple jet.
- Assuming higher expansion ratio always helps: in the atmosphere it can cause separation.