The overall effective efficiency of a jet-propelled flight may be defined as the net useful work that is done in accelerating the vehicle, raising it to height and combating its drag, divided by the calorific value of the propellants used. The analysis presented here shows that for a single-stage vehicle this overall efficiency is equal to the integral of the fraction
with respect to each element of propellants used. In this expression u'is the effective jet velocity (allowing for jet deflection, this is √(1 + k2) times the actual exhaust velocity), υ the vehicle velocity, q the fuel/air ratio and H the calorific value of the fuel in dynamic units. The mean lift/drag ratio of the vehicle is taken as 1/k, so that the last three terms of the numerator represent the work lost in dragging, raising and accelerating each element of propellant to the range R, height h and flight velocity υ at which it is burnt. The fraction may therefore be considered as the efficiency with which that element of propellant is used with respect to the whole flight plan.
For advanced air-breathing projects—by-pass engines up to aircraft speeds around Mach 2 and ramjets between Mach 4 and 9—the analysis shows that the sum of the υ-dependent terms in the efficiency factor is practically constant. The potential energy term is insignificant in the normally accepted flight corridor. Thus the overall efficiency with which propellants are consumed in covering a range R1 at flight speed υ js closely approximated to by the fraction
and the useful work done by the mass mm of propellants is
The vehicle mass (including engine, payload, and so on) that can be dragged over the range R1 by unit mass of propellants can then be calculated.
Although such simplified expressions cannot substitute for the final detailed analysis of a projected flight plan, it is hoped that they will prove more useful in preliminary analysis than the figures based entirely on cruise performance which have often been quoted in the past.
