User:TSRL/sandbox/Hangered/Range equation

The range equation provides an approximate analytic expression for the range of powered aircraft, useful in understanding the relevant parameters and in initial design. There are two versions, one for jet powered aircraft and the other for propeller aircraft, but the essential caculations are the same. Both describe constant fuel consumption flights and ignore climb, descent and other manoeuvres. The analysis involves weights (forces), typically measured in units such as Newtons (N), pounds force/weight (lbf/lbwt) etc. Mass is measured in Kg, lb etc. The equations do not depend on which self-consistent set of units is used.

Suppose the total weight of the aircraft at a particular time $$t$$ is

$$W$$ = $$W_e + W_f$$, where $$W_e$$ is the zero-fuel weight and $$W_f$$ the weight of the fuel. If the weight of fuel burned per unit time is $$F$$, then $$F$$ = $$-\frac{dW_f}{dt} = -\frac{dW}{dt}$$.

The rate of change of aircraft weight with distance $$R$$ is

$$\frac{dW}{dR}=\frac{\frac{dW}{dt}}{\frac{dR}{dt}}= - \frac{F}{V}$$,

where $$V$$ is the speed, so that

$$\frac{dR}{dt}=-\frac{V}{F}{\frac{dW}{dt}}$$.

It follows that the range is obtained from the definite integral below, with $$W_1$$ and $$W_2$$ the initial and final aircraft weights

$$R = \int_{W_2}^{W_1}\frac{V}{F}dW$$.

Jet aircraft
In a jet engine, the fuel weight loss rate $$F$$ is proportional to the thrust $$T$$, that is

$$F= {c_T} T$$,

where $${c_T}$$ is a constant with the dimesion of time.

For flights at constant height and speed, where thrust $$T$$ = drag $$D$$ and lift $$L$$ = weight $$W$$, the range integral becomes

$$R=\int_{W_2}^{W_1}\frac {L}{D}\frac {V}{c_T} \frac 1 W dW$$.

If $$V$$, $$ \frac {L} {D}$$ and $${c_T}$$ are constant throughout the flight, then

$$R=\frac {L}{D}\frac {V}{c_T}\int_{W_2}^{W_1}\frac 1 W dW=\frac {L}{D}\frac {V}{c_T}ln \frac{W_1} {W_2} $$.

This last is the range equation for jet propelled aircraft.