Tunes of Glory

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Aeons ago, Flying had a regular column called Foreign Accent that reported on aeronautical doings abroad. I was writing it in 1976, and in the February issue I quoted a letter, originally published in the British magazine Flight, from one James Ferguson of Aberdeen, regarding the flight around England and Scotland of a World War II Avro Lancaster bomber, which was powered by four Rolls-Royce Merlins:

"... the attention of ... police ... was drawn to a badly parked car, and, breathalysers at the ready, the car's crew approached the driver, a well-dressed, middle-aged man. He appeared to be in some distress, and the crew called their sergeant, a much older officer, who eventually discovered that the driver had been making his way home when the Lancaster had passed overhead. An ex-Bomber Command air gunner, he had been overcome by the music of the Merlins and, blinded by tears, had stopped his car."

Ferguson closed with the hope that when the last Lanc is at the end of the road, someone "will have the wit to fly one around the UK so that we can all hear those glorious cadences for the last time."

The last Lanc is not yet at the end of the road, and will not be for the foreseeable future. Either 16 or 17 more or less complete airframes remain in the world, depending on whom you ask, and two of them, one in Canada and one in the UK, are airworthy. The UK one flew a 65th-anniversary reenactment of the famous Ruhr Valley dam buster raid last May; a couple of videos of that flight can be found on YouTube. So can lengthy excerpts from the pretty creditable 1955 movie about the attack, which involved using an ingenious "bouncing bomb" that, although it did succeed in destroying several dams, also had the unfortunate habit of occasionally bouncing back into the airplane that dropped it. The dam busting attack, Operation Chastise, is one of the great stories in the history of technical warfare; its real impact upon the course of events, like that of much strategic bombing, is a matter of dispute among historians.

I suspect that good quality recordings of the Lancaster's four Merlins in full song can be obtained somewhere. I'm checking. With that and an iPod, one could be overcome by emotion in the privacy of one's home.

Using Less Fuel, cont'd Two Duke University business school professors, Richard Larrick and Jack Soll, recently got their 15 minutes by publishing an article in which they suggested that using gallons per mile, rather than miles per gallon, as a measure of fuel consumption made it easier for people to assess the economics of auto purchases. The assumed criterion was fuel consumption, and the question was how best to reduce it. The gist of their argument was that replacing a 15 mpg pickup with a 20 mpg sedan saves more fuel than replacing a 35 mpg compact with a 45 mpg hybrid does, even though the difference in miles per gallon between the first two vehicles is only half that between the second two.

How does it come about? Suppose you drive 300 miles a week. The pickup will need 20 gallons, the sedan 15, the compact 8.6 and the hybrid 6.7. So the difference between the first two is five gallons, while that between the other two is only 1.9. If you think in terms of miles per gallon, however, the second difference looks twice as good as the first: An improvement of 10 mpg is obviously better -- right? -- than an improvement of five.

Now, suppose that instead of characterizing the fuel requirements of the cars in terms of miles per gallon, you expressed them in terms of gallons per 100 miles. The first thing you would need would be a convenient abbreviation to replace "mpg." I propose gpc -- gallons per centimile. (Centimile rhymes with "send a file," not with "facsimile.") In most of the world, fuel consumption is already expressed not, say, in kilometers per liter, but in liters per 100 kilometers. One gallon per centimile would equal 2.35 liters per 100 kilometers. (The reason for using hundreds of miles or kilometers is simply to avoid dealing with inconveniently small fractions.)

To convert mpg to gpc, all you have to do is divide the mpg figure into 100. The four cars consume, respectively, 6.7, 5.0, 2.9 and 2.2 gallons to go 100 miles. Now things fall into place, and it becomes clear that replacing the worst gas-guzzlers with somewhat more efficient vehicles -- not even very efficient ones -- achieves more than replacing very efficient ones with extremely efficient ones.

Gpc is not only intuitively clearer than mpg for the purpose of comparing different cars, as Larrick and Soll argued, but it is also more realistic. We don't use trips to get rid of gasoline; we use gasoline to get somewhere. The fixed item is the distance we need to go, not the amount of fuel in our tank, and so what should interest us is the amount of fuel a trip requires -- or, at the bottom line, what that fuel will cost.

Obviously, the same holds true for airplanes. An airplane that burns 7 gallons per hour while cruising at 115 knots is using 5.3 gpc (remember that the miles in gpc are statute, not nautical), and so it is in the same neighborhood as the 20 mpg sedan. In general, however, airplanes consume more gallons per mile than comparable cars do. The reason is not that they are inherently less efficient but simply that they are going faster and have to work harder in order to do so.

It's interesting to compare the fuel requirements of large jets with those of automobiles. A 747 uses something like 500 gpc. (Because we're talking in gallons rather than pounds, I should mention that jet fuel is somewhat more energetic than gasoline because it is denser; the difference is about 10 percent.) The 747 seems to be in an entirely different category from the family sedan until you reflect that the 747 carries about 100 times as many passengers, and carries them more of the time -- that is, the jet is nearly full far more often than the car is. So it turns out that the jet airliner is not that different from a car, despite its huge size and much higher speed.

To return to airplanes and engines of a slightly smaller scale, the power required, and hence the fuel consumption, of airplanes, over the range of normal cruising speeds, is approximately proportional to the cube of the speed. I don't know by what divine dispensation we began thinking of 75, 65 and 55 percent of power as the standard power settings for cruise, but as a matter of convenience we can stick with them for the sake of this discussion. If 75 percent of rated power is taken as 100 percent of cruising power, 65 percent becomes 87 percent (because 65 is 87 percent of 75), and 55 percent becomes 73 percent. Other things, like altitude and mixture setting, being equal, the corresponding speeds will be, respectively, around 95 percent and 90 percent of the maximum cruising speed (because .95 is the cube root of .87, and .90 is the cube root of .73 -- any errors you might spot are due to rounding). I know this sounds a bit cut-and-dried, and different airplanes will deviate a little from the rule because of different engine or propeller characteristics; but if you check a few performance charts you will find it to be quite close to the truth.

Readers more mathematically agile than I will already have figured out that while speed is proportional to the cube root of power, gpc is proportional to its square root. In practical terms, gpc at 65 and 55 percent of power will be around 93 and 86 percent of the 75 percent-power value. So we emerge from this thicket of exponents with a simple conclusion: reducing speed by 5 percent reduces fuel consumption by 7 percent, and reducing speed by 10 percent reduces fuel consumption by 14 percent.

What does this mean in practical terms? This is where the analogy to cars comes in. Comparatively slow airplanes using comparatively little fuel are analogous to the efficient car that gains little by becoming more efficient. They have comparatively little to gain by slowing down. A fast, powerful airplane with a larger fuel flow, on the other hand, is analogous to the high-gpc SUV or pickup, and it can potentially save a lot more.

A complicating factor for airplanes is trip length. Cars all travel at about the same speed; airplanes don't. For a trip of a given length, say 500 miles, the time lost by slowing down is less for a fast airplane than for a slow one, whereas more fuel is saved. There is, therefore, a greater incentive to slow a fast airplane down: You get back more dollars per extra minute spent en route.

A concrete example: You are in the habit of cruising your Baron at 75 percent power, burning 33 gph at 200 knots (that's 14.35 gpc -- big-rig territory). A 500 nm trip takes you 2.5 hours. Suppose that one day you start feeling a little nervous about fuel costs, and you throttle back to 55 percent of power. Your fuel flow is down to 25 gph, but because you've given up 20 knots your time en route will increase by 17 minutes. Your gpc drops to 12.1 -- not exactly green, but the trip that burned 72 gallons at 75 percent power -- I'm ignoring taxi and climb fuel -- burns 61 at 55 percent. So the extra 17 minutes you spent in the air saved you, at present fuel prices, around 60 dollars. That's more than $210 an hour -- fair pay, considering how easy the work is.

The counterexample is the 125-knot, 8-gallon-per-hour airplane. The trip, which lasts four hours at max cruise, will take an extra 24 minutes at 55 percent power and will save 4.5 gallons. The pilot is thus paid only $60/hr for the extra time and, what's more, the time is tacked onto the end of an already long flight.

So, mutatis mutandis, what Larrick and Soll said about cars applies to airplanes too: A small improvement among the big users of fuel counts for more -- and, incidentally, is less painful to make -- than a larger improvement among the small users.

Correction: In this column, I suggested that "gpc"-gallons per centimile-would be a more appropriate measure of the efficiency of airplanes than miles per gallon. I wish I had a C-note for each of the kiloreaders who took me to task for using the prefix centi-, which means "hundredth," instead of hecto-, which means "hundred." They were absolutely right. Unfortunately, gph, gallons per hectomile, won't work, because it already stands for gallons per hour. I think the best bet is ghm-gallons per hundred miles-not that anyone would have the faintest idea what you were talking about if you actually used it.