User:TSRL/sandbox/Hangered/Range

Range
The range of a powered aircraft is ultimately limited by the amount of fuel it carries and the rate at which it burns that fuel. In constant speed. level flight the is determined determined by the requirement of thrust ($$T$$) to balance the drag $$D$$ at the cruise speed $$V$$.

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

$$W$$ = $$W_e + W_f$$, where $$W_e$$ is the empty weight and $$W_e$$, the fuel consumption rate $$\frac{dW_f}{dt} = -\frac{dW_f}{dt}$$.

The fuel consumption rate may be written in terms $$T$$ by defining the specific fuel consumption

$${c_T} =-\frac {1}{T}\frac{dW}{dt}$$,

whether the engine is a jet turbine or drives a propeller, and this can be rewritten as

$${dt} = -\frac{dW}{c_T T}$$,

the short time in which fuel weight $$-dW$$ is burned. During that time the aircraft flies a distance

$${dr}={V}{dt} = {-V}\frac{dW}{c_T T}$$

and the range $$R$$ is the definite integral of this equation between the start at finish of the steady, level flight, which ends at $$t_r$$ say, so, with initial and final fuels weights $$W_0$$  and $$W_f$$

$$R= \int_{0}^{R} dr = -\int_{W_0}^{W_f}\frac {V}{c_T T}dW=\int_{W_f}^{W_0}\frac {L}{D}\frac {V}{c_T} dW$$.

The second form of the integral follows because $$T=D$$ and $$L=W$$, the two conditions for steady state flight. If $$V$$, $$ \frac {L} {D}$$ and $${c_T}$$ are constant throughout the flight, then

$$R=\frac {L}{D}\frac {V}{c_T}\int_{W_f}^{W_0}\frac 1 W dW=\frac {L}{D}\frac {V}{c_T}ln \frac{W_0} {W_f} $$.

This last is sometimes known as the Breguet equation, albeit not quite in its original form. Breguet, naturally in the pre-jet days, was interested in propeller driven aircraft for which constant power $$P$$, not $$T$$, required constant fuel consumption. The specific fuel consumption is

$${c} =-\frac 1\frac{dW}{dt}$$

and the rate of fuel consumption can be written as either $${c_T}{T}$$ or $${c}{P}$$. These quantities must be equal and since the thrust from a propeller is $${\eta_p}\frac{P}{V}$$, defining the propeller efficiency $${\eta_p}$$, $${c}={c_T}\frac {\eta_p} {V}$$.

Substituting this into the range equation above gives

$${R_{prop}}=\frac{\eta_p} {c}\frac {L}{D}\int_{W_f}^{W_0}\frac 1 W dW=\frac{\eta_p} {c}\frac {C_L}{C_D}ln \frac{W_0} {W_f} $$, assuming $${\eta_p}$$, $${c}$$ and $${ \frac {L}{D}}$$ are constant over the flight. This approximation is a useful in preliminary calculations.

Conditions for maximum range - propeller driven aircraft
The expression for $${R_{prop}}$$ above shows that to fly the greatest distance the aircraft needs the most efficient propeller setting, a fuel economical engine and the use of as much of the maximum weight of fuel, consistent with safety, that it will carry. It also requires flight at the highest lift to drag ratio possible. The total drag coefficient $${C_D}$$ of an aircraft is often written as the sum of two components

$${{C_D}= {C_{D0}} + {K} {{C_L}^2}}$$

The first term is the total lift independent drag and the second the total lift induced drag. With this form the condition for minimum $$\frac{C_L}{C_D}$$ is that $${{K} {{C_L}^2= C_{D0}}}$$ so maximum range requires the aircraft to be flown at this $${C_L}$$ value which also determines the speed required to maintain level flight. As the fuel is burned and the weight falls, the $${C_L}$$ maintaining velocity will decrease unless the aircraft climbs into less dense air.

Substitution of the maximising value $${C_L}=\sqrt{\frac{C_{D0}} {K}}$$ into the equation for level flight and the definition of $${C_L}$$ gives

$${W}={L}={\frac{\rho_0} 2{S}{V^2}({\frac{C_{D0}} {K}})^{1/2}}$$,

where $${\rho_0}$$ is the air density and $${S}$$ the surface area. On rearrangement this yields the speed for the greatest range

$$({\frac{2WS}{\rho_0}})^{1/2}({\frac {C_{D0}} {K})^{1/4}}$$.

Using $$(\frac {L} {D})_{max}=(\frac {C_L} {C_D})_{max}={\frac {1} {2}{D_0}}({\frac{D_0} {K}})^{1/2}=  $$,

so the greatest range is

$${R_{prop-max}}=\frac{\rho_p} ln \frac{W_0} {W_f}$$,

so long as $${\rho_p}$$ and $${c}$$ are constant over the flight. This is the Breguet in its original form.

Conditions for maximum range - jet driven aircraft
The expression for the range given earlier in terms of $${c_T}$$,

$$R=\int_{W_f}^{W_0}\frac {L}{D}\frac {V}{c_T} dW$$

is appropriate for the jet engine, which has thrust approximately independent of speed. As before the fraction $$\frac {L} {D}=\frac {c_L} {C_D}$$; $${V}$$ can be written in terms of $${C_L}$$$$ {V}$$ because in steady state flight $${W}={L}=\frac{\rho_0} 2{S}{V^2}{C_L}$$ as before. Substituting this and the equivalent expression for $${D}$$ into the expression for $${R}$$ gives

$$R=\frac {1} {c_T} ({\frac{2}{\rho_0S}})^{1/2}\frac {{C_L}^{1/2}} {C_D} \int_{W_f}^{W_0}\frac{dW} {{W}^{1/2}}= \frac {1} {c_T} ({\frac{2}{\rho_0S}})^{1/2}\frac {{C_L}^{1/2}} {C_D} ({{W_0}^{1/2} -{W_f}^{1/2}})$$.

It shows that to achieve the greatest distance a jet aircraft needs to fly at the maximum of $$\frac{{C_L}^{1/2}} {C_D}$$ rather than $$\frac{C_L} {C_D}$$ Using the same expression for the drag as before, this occurs when the profile drag is three times the induced drag, rather than equal to it as for the propeller driven aircraft. Since the profile drag increases with speed, this implies that the jet powered version achieves its maximum range at a higher speed. Arguments similar to those above this speed is greater by a factor of 31/4. The maximum jet range is

$$R=\frac {3} {c_T} ({\frac{2}{rho_0S}})^{1/2}({\frac {1} {3K{C_{D0}}^3}})^{1/4}({{W_0}^{1/2} -{W_f}^{1/2}})$$.

Endurance
The endurance of an aircraft is the maximum time it can stay aloft without refuelling, perhaps subject to named constraints such as the speed, altitude or safety fuel reserves. Since $${dt} = -\frac{dW} $$

gives the time taken to burn a weight of fuel $${c_T T}$$ while producing a thrust of $${T}$$, it integration over the whole fuel load provides the endurance

$${E} =\int_{W_f}^{W_0}\frac{W} \frac{dW} {W}= \int_{W_f}^{W_0}\frac{1} \frac{L}{D}\frac {dW} {W}$$,

since $$\frac{W} {T}=\frac{L} {D}$$ in steady state flight. If $${c_T}$$ and $$\frac{C_L} {C_D}$$ are constant during the flight, a useful assumption for preliminary calculations, then

$${E} =\frac{L} {ln \frac{W_0} {W_f}} $$.

Graphical Approach
The analytical treatment given above is based on a simplified drag polar and provides a useful design tool. Once an aircraft is flying, direct measurements of the power or thrust need to maintain a given velocity a particular load and altitude can be made and presented graphically. The speeds for maximum range and endurance can be easily read off them. The velocities at the power minima of the thrust and power curves provide the optimum for endurance for jet and propeller driven aircraft respectively. If the two axes are multiplied by time, the plot becomes a scaled fuel consumption versus distance flown plot without a change in curve shape and a tangent from the origin locates the maximum range, or on he original plot the speed at which it should be flown. The coincidence of the maximum range on the power plot with the maximum endurance speed on the thrust plot follows from from the $$P=T.V$$ relationship and remains for any form of $$T$$ for the same aircraft/engine combination.