(Barré) #1

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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