RANGE

The range of an aircraft is the disfance that the aircraft can fly. Range is generally defined subject to other requirements. In the case of military aircraft, one usually works to a mission profile that may specify a climb segment, a cruise segment, a loiture, an enemy engagement, a descent to unload cargo, a climb, a return cruise, a hold, and a descent. In the case of civil aircraft, the range is usually taken to mean the maximum distance that the airplane can fly on a given amount of fuel with allowance to fly to an alternate airport in case of bad weather.

‘ Let us put aside the range profile for the present and consider only the actual distance that an airplane can fly at cruising altitude and airspeed on a given amount of fuel. For a propeller-driven airplane, the rate at which fuel is

Wf = (BSFQ(bhp) lb/hr W, = (SFC)(kW) N/s

where SFC is in units of newtons per kilowatt per second.

Using the SI notation, the total fuel weight consumed over a given time will be

W,= [ (SFC)(kW) dt

Jo

This can be written as

Wf = J‘(SFC^kW)

Since the shaft power equals the thrust power divided by the propeller efficiency,

1 f (SFC)D,

ioooj.——* where D is the drag. The constant represents the fact that lkW equals 1000 mN/s.

Given the velocity and weight, Equation 7.31 can be integrated numeric­ally. One of the difficulties in evaluating Equation 7.31 rests with the weight, which! is continually decreasing as fuel is burnt off.

A closed-form solution can be obtained for Equation 7.31 by assuming that the SFC and tj are constant and that the airplane is flown at a constant CL. With these assumptions, the fuel flow rate, with respect to distance, becomes

dW, _ (SFC)e ds 1000 tj

where e is the drag-to-lift ratio, which is a function of CL, and W is the airplane weight. dWflds is the negative of dW/ds. Thus,

dW (SFC)e

w 1000 tj dS

W, (SFC)e W 1000 tj

where Wі is the initial weight of the airplane. If WF is the total fuel weight, the distance, or range, R, that the airplane can fly on this fuel is finally, in meters,

WE denotes “weight empty,” meaning “empty of fuel.” Normally, weight empty refers to the airplane weight without any fuel or payload. This equation, which holds only for propeller-driven aircraft, is a classical one known as the Breguet range equation.

In the case of a turbojet-propelled airplane, the fuel flow becomes

Wf = (TSFC)D

so that

dW (TSFC)e ds W V

In order to integrate this relationship, we must assume that the airplane operates at a constant e/V and that TSFC is constant. When this is done, the modified Breguet range equation for jet-propelled aircraft is obtained.

R (TSFC)e ln 0 + wO (7’33)

Thus, for maximum range, e should be minimized for propeller-driven

airplanes and e/V should be minimized for turbojets. In the case of turbojets, this can lead to the airplane cruising slightly into the drag rise region that results from transonic flow.

Referring to Equation 7.18, e will be a minimum when

/С ^4

Vop, = (^j (propeller-driven airplane) (7.34)

e/V will have a minimum at

(

ip l/4

-jjf) (turbojet) (7.35)

For propeller-driven airplanes this leads to a minimum e value of

e ■ = 2(ЩШ
mm rreA)

r 4 (wis)2y14

opt

For turbojet-propelled airplanes, e/V has a minimum value of

The optimum V for the above is equal to that given by Equation 7.37 multiplied by 31/4.

Some interesting observations can be made based on Equations 7.36, 7.37, and 7.38. For either propeller or turbojet airplanes, the indicated air-

speed for maximum range is a constant independent of altitude. However, for the same wing loading, elfective aspect ratio, and parasite drag coefficient, the optimum cruising speed for the turbojet airplane is higher than that for the propeller-driven case by a factor of 1.316. The optimum range for a propeller – driven airplane is independent of density ratio and hence altitude. However, with the indicated airspeed being constant, the trip time will be shorter at a higher altitude.

The optimum range for a turbojet is seen to increase with altitude being inversely proportional to the square root of the density ratio. This fact, together with the increase in true airspeed with altitude, results in appreciably higher cruising speeds for jet transports when compared with a propeller – driven airplane. As an example in the use of Equations 7.38 and 7.37 (multiplied by 1.316), consider once again the 747-100 at a gross weight of ^2700 kN. In this case,

Y = 5284 N/m2

fls = 0.0182 Ae – 4.9

Vopt = 167/Vo – m/s

Dividing this velocity by the speed of sound, the optimum Mach number as a function of altitude shown in Figure 7.17 can be obtained. Above a Mach number of approximately 0.8, this curve cannot be expected to hold, since drag divergence will occur.

The second curve shown in Figure 7.17 presents the optimum range divided by the sea level value of this quantity. This curve is calculated on the basis of Equation 7.38 and the TSFC values for the JT9D-7A engine presented in Figure 6.38 as a function of altitude and Mach number. This curve is reasonably valid up to an altitude of 7500 m. Above this, because of Mach number limitations, the range ratio will level off. However, despite the Mach number limitations, the gains to be realized in the range by flying at the higher altitudes фе appreciable, of the order of 30% or more.

In the case of propeller-driven airplanes, the optimum cruising velocity given by Equation 7.37 does not reflect practice. To see why, consider the Cherokee Arrow. In this case, at a gross weight of 26501b, the optimum velocity is calculated to equal 87.8 kt. This velocity is appreciably slower than the speeds at which the airplane is capable of flying. It is generally true of a piston engine airplane that the installed power needed to provide adequate climb performance is capable of providing an airspeed appreciably higher than the speed for optimum range. Therefore, ranges of such aircraft are quoted at some percentage of rated power, usually 65 or 75%.

йор t Ropt @ sea level Figure 7.17 Effect of altitude on optimum range and cruising Mach number for 747-100.

The cruising speed at some specified percentage of the rated power can be found from the power curves, such as those presented in Figure 7.15 for the Cherokee Arrow. For example, 75% of the rated power corresponds to approximately 81% of the available power shown in Figure 7.15. This increase^ results from the rating of 200 bhp at 2700 rp. m as compared to only 185 bhp output at 2500 rpm for which the figurefwas prepared. A line that is 81% of the available power crosses the power-required curve at a speed of 223 fps or 132 kt. This speed is therefore estimated to be the cruising speed at 75% of lirated power at this particular rpm.

The penalty in the range incurred by cruising at other than the optimum speed can be found approximately from Equations 7.18 and 7.34. The ratio of the drag at any speed to the minimum drag can be expressed as a function of the ratio of the speed to the optimum speed. The result is

This relationship is presented graphically in Figure 7.18. In the preceding example of the Cherokee, this figure shows a loss of approximately 25% in the range by cruising at 75% power instead of the optimum. Of course, the time required (to get to your destination is 33% less by cruising at 75% power.

The effect of wind on range is pronounced. To take an extreme, suppose you were cruising at the optimum airspeed for no wind into a headwind of equal magnitude. Your ground speed would be zero. Obviously, your airspeed is no longer optimum, and it would behoove you to increase your airspeed. Thus, without going through any derivations, we conclude that the optimum airspeed increases with headwind.

Correcting Equation 7.32 for headwind is left to you. If V* denotes the headwind, this equation becomes (now expressed in the English system),

The effect of headwind on the optimum cruising airspeed can be obtained by minimizing e/(l – VJ V). Without going into the details, this leads to the following polynomial

v

+ —= 0

v.

* opt

Here V is the optimum cruising velocity for a given headwind and Fopt is the value of V for a Vw of zero.

Vw

v;pt

т^= 1 + AVIVopt

‘ Opt

A VIVopt is presented as a function of VJVapt in Figure 7.19. This figure shows, for example, that if one has a headwind equal to 50% of the optimum velocity for no wind, he or she should cruise a. t an airspeed 20% higher than the optimum, no-wind velocity.