# The Very Best Speed to Fly

###### Peter Garrison

I've noticed lately that I fly more and more slowly and pay more and more attention to fuel flow. Clearly this behavior is related to the rising cost of fuel; as it goes up, the best speed to fly goes down. But that relationship - speed divided by fuel flow - must be corrected by a scaling effect related to one's impatience, anxiety regarding disposable income, age, number of children in college and so on. No doubt one could construct a long equation, loaded with fractional exponents whose values would vary from day to day, and whose solution is the reading on my airspeed indicator.

At a first glance you might suppose that by slowing down I was doing nothing more complicated than trying to maximize my airplane's specific range, or what in common parlance is called its gas mileage. If that is the case I'm not doing a very good job of it, because that number, which is about 25 nautical miles per gallon for Melmoth 2 at 12,000 feet - I usually cruise as high as I can without needing oxygen - is achieved at an indicated airspeed of 100 knots. I'm going slower, but I'm not going that slow.

What I'm actually doing is indicating around 135 knots on shorter flights and 145 knots on longer ones. At 12,000 feet those speeds correspond to between 45 and 55 percent power, fuel flows of 7.1 to 8.5 gph, and mileages of around 22 and 20 nmpg respectively.

Something is raising my speed well above the "ideal" of 100 kias. That thing is obviously speed itself. I want to go reasonably fast. That's the point of an airplane. And so speed per se must factor into the equation as something other than just a measure of distance traveled. To supplement speed-as-distance with speed-as-time-saved, I need to inject speed into the equation once again. The figure of merit, previously speed divided by fuel flow, becomes speed squared divided by fuel flow, or in other terms, speed times mpg.

It so happens that this was the basis of the scoring formula for the Comparative Aircraft Fuel Efficiency (CAFE) races, which were held in California between 1981 and 1990. Cabin payload was considered as well-the actual formula defining efficiency was speed times payload times mpg-and the results pretty well coincided with one's intuitive sense of which airplanes were the "best" at doing what it is that airplanes do.

The speed to fly for the CAFE races was the speed at which the product of groundspeed and miles per gallon was greatest. It happens that that speed coincides with the "Carson speed," named after a former Naval Academy professor of aerospace engineering, Bernard H. Carson. In a 1980 paper entitled "Fuel Efficiency of Small Aircraft," which is in the public domain and ought to be available on the internet for download but isn't, Carson observed that, because they need to climb, airplanes are provided with much more power than they need to cruise at their most efficient speed, the "best L/D speed" or "best range speed." Since this additional power allows airplanes to go much faster, the question naturally arises of how to get the most extra speed for the least extra cost - or, as Carson put it, of finding "the least wasteful way of wasting." The speed in question turned out, theoretically at least, to be the one at which the product of speed and L/D, or lift-drag ratio, was greatest. This was 1.32 times the speed for most miles per gallon (that is, the best L/D speed).

In 1993, Russ Erb of EAA Chapter 1000 (Muroc, California) published a paper, which can be found online (search for "Carson speed"), in which he identified the Carson speed as the speed for the highest ratio of speed to thrust (which is equal in magnitude to drag). Erb pointed out, again, that this was 1.32 (the fourth root of three) times the best L/D speed. (The best L/D speed, by the way, is in turn 1.32 times the best endurance speed or least-power speed. Who would have guessed that airplanes had such a profound affinity for the fourth root of three?) As a practical matter, however, the real Carson/CAFE speed is higher than calculation would suggest, because both propeller efficiency and specific fuel consumption (the amount of fuel required per horsepower per hour) improve, up to a point, as the airplane goes faster, and as a result the actual best range speed (in Melmoth's case, 100 kias) is higher than the best L/D speed (87 kias). By the same token, the product of speed and miles per gallon turns out to reach its maximum at a speed quite a bit higher than the theoretical one. For Melmoth 2, this speed is around 135 kias, whereas 1.32 times the best-L/D speed would be only around 115 kias.

Having identified the peak of the curve, one also wants to know how rapidly it falls off on either side. It turns out that it falls off more rapidly on the low side than on the high, and the figure of merit is less than five percent lower at 75 percent of power than at 50 percent. This observation seems to provide a handy mathematical justification for the cruising speeds that we use in real life.

If two other fundamental speeds are called the best-endurance and best-range speeds, the Carson speed could perhaps be called the best-speed speed. Its underlying assumption is that speed per se and economy are of equal value. Actually, one could generate an infinity of different best-speed curves by assigning different weights to the two components, speed and gas mileage. For instance, if you considered speed to be less important than cost per mile, you might give it a fractional exponent, such as three-quarters. In that case, the "best" speed for Melmoth 2 would drop to 130 kias, or about 45 percent power. If you used one-half as your exponent, diminishing the weight of speed still more, it would fall to almost 120 kias and 35 percent power. These fractional exponents reflect the variable premium put on speed in different circumstances. For instance, if you're just taking a friend sightseeing you might throttle way back and coast along, since you're really in no hurry to get anywhere. On the other hand, if time is of the essence and cost is no object, speed would receive an exponent greater than one, and the "best" cruising speed would be the highest possible.

Having derived a "best" speed for a particular aircraft (which you can do by multiplying some representative speeds and fuel flows taken from the performance charts), you naturally wonder about a way to compare different airplanes in order to see which is "most efficient" in the dividend of speed that it yields for incremental fuel flow. At the end of his paper Carson suggests a way of comparing all airplanes and, incidentally, comparing them with submarines, horses, bicycles and trains-in short, with any kind of conveyance whatever.

This seems to be a tall order, but in the great tradition of scholarly knight-moves, Carson recalls a "celebrated" 1950 paper by Giuseppe Gabrielli and Theodore von Kármán, entitled "What Price Speed?", which regards an apparent natural limit upon the speed that may be attained by any vehicle. In an attempt at a sort of Grand Unified Theory of locomotion, the authors constructed a graph having speed as one axis and, as the other, a dimensionless parameter to which they assigned the Greek letter epsilon, e, which is the letter that "efficiency" would start with if we used the Greek alphabet. This efficiency was the quotient of the power and what could be called the mass-power, defined as the weight multiplied by the speed. In airplane terms, this amounts to the L/D ratio; for other kinds of conveyances, it is the weight multiplied by the velocity squared, and divided by the power. Mapping all sorts of moving things, from pedestrians to merchant ships, on this graph, they found that all lay to the left of a diagonal line that appeared to represent an absolute limit to vehicle efficiency. "It should be noted," Carson wrote, "that this limit was not derived, but rather was postulated according to the evidence developed in their exhaustive study." In other words, no physical reason for the limit was found; it just seemed to be there.

Carson did some mathematical manipulation to convert this diagonal barrier into an upper limit, which has a value of 4,962 knots. This is not a real speed, but the product of speed and efficiency, where efficiency is defined as using a minimum of power to move a maximum of weight as fast as possible. It implies, for example, that the highest speed at which an L/D ratio of 10 is physically possible is 496.2 knots. Carson then proposed that all conveyances could be compared by dividing their actual performance by this limiting value; this procedure, called normalizing, has the effect of fitting all scores into a range of zero to one. (If you liked, you could multiply by 100 and give the rankings as percentages, like scores in school.)

Obviously, if you're comparing wheelbarrows and submarines, a very high level of generality is required. Carson's plot seems to have it. The performance of each conveyance maps onto it as a curve having the shape of an inverted J. The "best" score for each, which coincides with the Carson speed, is the highest point on the curve, that is, the point at which it most closely approaches the upper limit. This method of scoring, applied to airplanes, is sensitive to altitude, because the true speed at which a given L/D ratio can be attained increases with altitude, and so, therefore, does the Carson score. But it does not reflect the vagaries of propulsive efficiency. When you get down to comparing specific airplanes, however, you can refine the procedure by taking into account the variations in propulsive efficiency with speed, and creating, in effect, a sort of corrected L/D curve for each airplane.

At this point in his paper Carson, his ingenuity not yet exhausted, proposes reversing his procedure in order to use it as a preliminary design tool. Suppose, he says, we wish to design an airplane that is to have a Carson efficiency of 0.6 (which is pretty high) while cruising at 250 knots at 25,000 feet on 390 brake horsepower. We deftly toss our salad of numbers and presto! it emerges that our design should weigh 4,900 pounds and have a wingspan of 32.25 feet and an equivalent flat plate area of 3.43 square feet-"which should be well within present design capabilities," he somewhat casually adds.

The original plot accompanying the Gabrielli-von Kármán paper showed a few vehicles closely approaching the theoretical limit; they included, in order of increasing speed, a merchant ship, an autorail (a French one-car train), a commercial airplane (this would have been something like a DC-7 or Constellation) and Boeing B-47 jet bomber. Among the not very efficient conveyances were a race car, a destroyer and an ordinary horse (not a race horse). The least efficient of all, curiously-because neither heavy nor fast-is a walking man.

Peter Garrison taught himself to use a slide rule and tin snips, built an airplane in his backyard, and flew it to Japan. He began contributing to FLYING in 1968, and he continues to share his columns, "Technicalities" and "Aftermath," with FLYING readers.

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