Propulsion question

Question

A medium sized aircraft with two turbojet engines installed is cruising at an altitude of 10000m at a Mach number of 0.8. At this flight condition, each engine has a net specific work output of 250 kJ/kg and the fuel to air ratio by mass is 0.02.

calculate the engine’s thermal efficiency, propulsion efficiency and overall efficiency

Dry air: Cp=1010J/kgK, γ = 1.4 Combustion products: Cp=1100J/kgK, γ = 1.33 Fuel: QR = 44, 000 × 103J/kg.

Answer 1

Hi Terence,

Step #1 - Find the Temperature & Velocity at Given Altitude

1. At 10,000m the temperature is 223.1 K And to find the velocity use the equation v = M*c, where c is √γRT.
2. v_i = 0.8(1.4*286*223.1)^(1/2) = 239 m/s

Step #2 - Find the Outlet Temperature & Velocity

1. We know that the outlet specific work is 250 kJ/kg and is defined as ^Δh = cp*^ΔT.
2. To find the exit temperature: T_i = ^Δh/cp + T_f = 250/1.1 + 223.1 = 450 K
3. To find the exit velocity: ^Δh = v_e²/2 = 250,000 = v_e²/2, ∴ ve = 707 m/s

Step #3 - Find Propulsive Efficiency

1. η_(prop) = 2/(1 + v_e/v_i) = 2/(1 + 707/239) = 0.5053 = 50.53%

Step #4 - Find the Thermal Efficiency via Carnot's Efficiency

1. η_(therm) = 1 - Tc/Th = 1 - 223.1/450 = 0.5042 = 50.42%

Step #5 - Find Overall Efficiency

1. η_(overall) = η_(prop)η_(therm) = 0.5053*0.5042 = 0.2548 = 25.48%

Notes & Comments

The exit velocity and temperature found in Step #2 is at the exit of the turbine. Since we don't know what the combustion temperature is or the temperature drop across the turbine, we cannot use the commonly used efficiency equation of heat in work out equation.