`http://www.saflyermag.com`

separation and so on. The longer an

aeroplane’s wingspan the lower its most

efficient speed and therefore the less its total

drag. This is why sailplanes have such long

wings, and why the two aeroplanes that have

circled the globe without refuelling, Voyager

and the late Steve Fossett’s Global Flyer, had

extremely long, sailplane-like wings and L/Ds

of 27 and 37 respectively.

The final component in the Breguet

Range Equation is the natural logarithm

of the ratio between the starting weight

and the ending weight. This weight ratio

is often expressed, indirectly, as the ‘fuel

fraction’ – the fraction of the take-off weight

that is fuel. Fuel fractions for conventional

aeroplanes range from 10 percent to 35

percent (the latter for long-range airliners).

Global Flyer had a fuel fraction of more than

80 percent – a very difficult thing to achieve,

structurally. The range of take-off-to-landing-

weight ratios is therefore from around 1.1 to

around 5, and the natural logarithms of these

numbers run from 0.1 (for 10 percent) to 1.

(for 80 percent), with .43 (for 35 percent)

representing the highest value found in

normal commercial aircraft.

There’s one more thing: To make the

result come out in nautical miles, we multiply

by 326.

Let’s try it out on some arbitrary numbers.

Combining the propeller efficiency, specific

fuel consumption (1.9 together, remember)

and the 326 factor gives us about 600. Say

we have an L/D of 10 and a fuel fraction of

15 percent. 600 x 10 x 0.1625 (the logarithm)

gives 1,000 nm range – sounds possible. Or

say that Voyager had an L/D of 27 and a fuel

fraction of 72 percent. The result is 20,

nm – again, sounds about right.

Now, DT complained that Breguet ranges

are always on the high side. In fact, since

he was quick to detect or imagine dishonest

intent in others, he concluded that Breguet

must have been either a bumbler or a liar.

Indeed, Breguet ranges do seem high.

But the range equation is an exact expression

of real physical processes and relationships,

and as such it is bound to be correct. Why

then does it appear to give inflated answers?

The reason is the usual one: faulty

inputs. Aeroplanes never fly to empty tanks

(though Voyager almost did). They taxi, and

use full power and a rich mixture for take-off

and initial climb. Pre-flight estimates of prop

efficiency, specific fuel consumption, L/D

and even fuel fraction tend to be optimistic,

and in any case may not be met under real

conditions. Propellers and engines are at their

best at only one speed and power output,

but a long range flight necessarily involves

`continually adjusting speed and power for`

diminishing weight. Generally speaking,

powerplant efficiency deteriorates as weight

and therefore power required diminish; but

since the aeroplane goes comparatively fast

and uses fuel rapidly early in the flight, but

both flies and burns fuel more slowly later

on, it spends more time operating at low

efficiency than at high.

Rather than concentrate on hypothetical

cases of extreme range, it perhaps makes

more sense to test the Breguet equation

against everyday flights. For example, I flew

from Santa Fe to Los Angeles in 3:45 and

observed a fuel burn en-route of 8.3 gph

(but a block burn of nine). I took off at 2,

pounds and landed at 1,825, so the logarithm

of the weight ratio is .0916. The distance

I covered, 625 nm, should be this number

multiplied by 326, and then multiplied by

another number which is the product of the

L/D and the prop efficiency divided by the

specific fuel consumption. That other number

has to be 20.93.

What combination of factors could

produce that number? One way to find good

candidates is to consult a performance

calculation program, which matches the

observed performance of my aeroplane pretty

well. (This is not a circular argument, since

the program uses methods quite unrelated

to Breguet’s.) For a cruising speed of 166

KTAS at 12,500 feet, it predicts an L/D of

10.7, a propeller efficiency of .85, and a

specific fuel consumption of .44, for a result

of 20.67. Pretty close, considering the many

uncertainties involved in the calculation.

Of course, I could be lying ...

`BELOW - Louis Breguet, a French`

mathematician and aeroplane designer,

reduced range prediction to a very simple

formula - the Bruguet Range Equation.

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