Quest for Performance: The Evolution of Modern Aircraft



Appendix E

Mass-Flow Rate, Thrust, and Propulsive Efficiency



[515] Among the advantages of the turbofan are that, for a given energy addition per unit time (fuel-flow rate), the turbofan will produce more thrust and have a higher propulsive efficiency than will a turbojet with a gas generator of the same size and level of technical sophistication as the turbofan. These advantages can be explained by the following highly simplified analysis of an idealized turbofan engine. Assume the air that enters the inlet of the engine to have a free-stream velocity of Vi and a uniform exhaust velocity from the fan and gas generator of Ve. The mass flow entering the inlet per unit time is m. The thrust produced by the engine can then be expressed by the following simple relationship if the static pressure in the exhaust is assumed to have the free-stream static value where the exhaust velocity is measured and the momentum of the fuel itself is neglected:


formula for determining engine thrust(E 1)


where T is the thrust.

The amount of energy added to the flow by the fuel may be expressed as the difference between the kinetic energy per unit time entering and exhausting from the engine and is given as follows:


 formula to deduce energy added by fuel flow(E2)


or with the use of equation (E1),


another formula to deduce energy added by fuel flow(E3)


[516] The propulsive efficiency is defined as that fraction of the kinetic energy added to the mass flow m that is usefully employed in propelling the aircraft. The propulsive efficiency , propulsive efficiency, percentcan be expressed in the following form:


equation to determine propulsive efficiency(E4)


Substitution of equations (El) and (E3) for the thrust and kinetic energy gives


propulsive efficiency formula(E5)


Equation (E3) clearly shows that for a given thrust level, the required rate of energy input is reduced as the value of [Delta] V/Vi is decreased. For a given thrust level, equation (E1) shows that as the value of [Delta] V is decreased, the mass-flow rate m must increase correspondingly. The most thrust is therefore obtained for a given energy input rate from the addition of a small velocity increment to a large mass-flow rate; and the propulsive efficiency, given by equation (E5), is increased as the value of [Delta] V is reduced. The turbofan engine therefore provides higher efficiency and more thrust than a turbojet engine with the same rate of energy input and having the same component (compressor, burner, and turbine) efficiencies.