Propulsive efficiency is the ratio between propulsive power and the rate of production of propulsive kinetic energy. 

Propulsive power is defined as the thrust produced by the engine, T, multiplied by the speed of the aircraft, V0. 

The rate of change of propulsive kinetic energy is the influence of the combustion process as air is mixed with fuel in the combustion chamber. Before entering the combustion chamber the air has a mass flow rate of m0 and is travelling at V0, so the kinetic energy is m0.V0^2/2.

When the air enters the combustion chamber it is combined with fuel, the enthalpy of the air mixture increases and it is passed through the turbine and accelerated through the nozzle. The mass flow rate at the exit of the nozzle is m1. In a closed system mass is conserved, based on Bernoulli’s equation, m0=m1=m.

Propulsive efficiency = (T.V0)/(1/2)*((m1.V1^2)-m0.V0^2))
T=m(V1-V0)
Propulsive efficiency =mV0(V1-V0)/(m/2)*(V1^2-V0^2)
=2*V0*(V1-V0)/(V1^2-V0^2)
=2*V0*(V1-V0)/(V1+V0)*(V1-V0)
=2*V0/(V1+V0)



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