The late B.H. Carson’s AIAA paper on “Fuel Efficiency of Small Aircraft” emerged unsought, like a dinosaur bone poking out of eroded Wyoming topsoil, from the scree of papers on my desk.
First published 45 years ago, Carson’s essay became a point of reference for the wonkier class of pilot, to whose vocabulary it contributed a couple of novel phrases: “Carson Speed” and “the least wasteful way of wasting.”
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Subscribe NowCarson’s argument began with the familiar observation that because airplanes need to climb, their engines are more powerful than they would be if they were sized solely for most efficient cruising. Since people love speed, they naturally make use of the surplus power to go faster, and as a result airplanes seldom realize in practice the efficiencies of which they are theoretically capable.
“Efficiency” here means specific range, which is measured in miles per gallon, as it is with cars. Operators of airplanes used for commercial transport of people or cargo prefer pound-miles per gallon, but for personal airplanes with randomly varying payloads mpg is a satisfactory yardstick for efficiency.
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Carson, a professor of aerospace engineering at the U.S. Naval Academy, approached the problem by setting lift-drag ratio, L/D, as his metric of efficiency. He noted that although most actual airplanes—he was talking about personal and business singles and twins—have maximum L/D ratios between 12 and 15, they typically cruise at L/Ds of 10 or less.
Given that pilots were always going to fly faster than the best L/D speed, Carson posed the problem of finding the cruising speed at which the most excess speed required the least excess fuel. A little algebra and voilà—out pops a number that would have pleased Pythagoras: 1.31607, aka the fourth root of 3.
Multiply the best L/D speed—which, to an approximation, is the gliding speed published in the POH—by 1.32, and you have the “Carson Speed”—the speed that yields the greatest saving in time for the least excess consumption of fuel. The best L/D speed minimizes fuel cost per unit of distance, while the Carson Speed minimizes fuel cost per unit of speed. The product of speed and L/D reaches a maximum at the Carson Speed.
To the extent that Carson’s paper continues to occupy a niche in an obscure corner of the chapel of aviation pop culture, it ends there. In fact, however, he was only at page 3 out of 8. He went on to discuss the efficiencies of vehicles in general—battleships, race horses, trolley cars—and finally presented a concise procedure, which as far as I know was original with him, by which a designer can quickly arrive at the optimum span, weight, and engine size for an airplane intended to cruise at a particular speed and altitude.
Carson’s approach was mathematically elegant but assumed that time and fuel were of equal value to a hypothetical pilot. That may not be the case—in fact, it often isn’t. But back on page 3, there was a proviso that glossed over another squishy spot in Carson’s argument. “By neglecting minor variations in propeller efficiency and specific fuel consumption that may exist at different airspeeds and power settings for any given aircraft…”
Actually, the minor variations may not be so minor, and as it happens they tend to reinforce rather than cancel one another.
The principal distortions come from the engine and the propeller. Engines are most efficient at high power settings. For instance, in the range of cruising power settings the specific fuel consumption of my Continental TSIO-360—the rate of fuel flow required to generate one horsepower—improves by about 1 percent with each increase in speed of three knots.
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Propellers, similarly, are usually pitched for cruise. Even variable-pitch propellers have a design speed for greatest efficiency, and manufacturers naturally place it where they think their customers will actually be flying. That design point is invariably above the Carson Speed.
The result is that the speed that yields the greatest product of speed and miles per gallon—call it the Real Carson Speed—is higher than the speed that yields the greatest product of speed and L/D ratio.
To see how the variations in engine and propeller efficiency affect the Real Carson Speeds of various airplanes, I ran some simulations of several types of single-engine propeller planes: a Cessna 172, a Mooney M20J, a Bonanza, and a Cirrus SR22. The simulations take variations in specific fuel consumption and propeller efficiency into account. I had the computer calculate the best L/D speed, multiplied it by 1.32 to get the theoretical Carson Speed, and compared the result with the calculated maximum product of speed and mpg.
The first thing that jumped out at me was that the published gliding speeds for my small fleet were in each case lower—typically, by 10 knots—than the calculated best L/D speeds. (These are indicated, not true, airspeeds.) I believe the reason is that flight testing reveals that with the added drag of a windmilling propeller, which my simulation ignores, a lower gliding speed is better. Published glide angles are also much steeper than the calculated L/Ds would suggest.
Using the calculated L/D speed as a baseline and multiplying by 1.32, I got Carson Speeds of 100 knots for the 172, 123 for the Mooney, 132 for the Bonanza, and 130 for the SR22. Again, these are indicated airspeeds.
Now, miles per gallon is a function of true airspeed, not indicated airspeed. So, since these are all naturally aspirated airplanes, I looked at their performance at 8,000 feet. I found that the highest products of true airspeed and mpg were achieved at indicated airspeeds of 106 knots for the C172, 134 knots for the M20J, and 151 knots for both the V35 and SR22. These speeds are in the range of 60 to 70 percent of power.
If you’ve stayed with me this far, you may be wondering how peaky the speed-mileage curve is. Does efficiency drop off sharply on either side of the best speed? What is the penalty for using 75 percent or even, as some operating handbooks permit, 80 percent of power?
It turns out that the Carson Speed is a very blunt instrument. In the Cirrus, for instance, going from 65 percent to 75 percent of power you exchange two extra gallons per hour of fuel flow for a speed gain of 9 knots; your mileage of 11.25 mpg drops to 10.5. That sounds huge—but the product of mpg and speed—the Real Carson Speed yardstick—loses only 1.5 percent.
So we were doing it right all along.
This column first appeared in the April Issue 957 of the FLYING print edition.