Flying Frugally

Route Kansas City to Dallas with Winds Aloft at 12,000 ft

There are several definitions of efficient flight. For example, a flight could be considered most efficient if (1) the least amount of fuel is used to cover a given distance; or if (2) the least amount of time is spent in the air for the given distance; or if (3) the least amount of fuel per knot of speed is used.

In the landmark technical paper Fuel Efficiency of Small Aircraft (AIAA-80-1847, 1980), B. H. Carson argues that the third criterion, which minimizes expenditure per knot of airspeed, is a good compromise between the first two criteria, which minimize fuel used or time in the air. The airspeed producing the least expenditure per knot is now known as the Carson speed. The optimal speed for the first criterion is the best-glide speed.

Carson's paper establishes a neat formula that links Carson speed and best-glide speed. It says that the Carson speed is the best-glide speed times sqrt(sqrt(3)) = 1.31607, where sqrt denotes square root. Let's round this factor to 1.3 for the discussion to follow.

Best-glide speed is not a constant for a given airplane, since it goes up with payload. Thus, Carson speed also goes up with payload. This complicates the use of Carson's formula.

So far, we have assumed no-wind conditions. But typically there is some wind aloft, and we are actually interested in efficient groundspeed, where it is not quite clear how we would want to define efficiency for that case. The definition should somehow take advantage of the push of tailwinds, and should judiciously deal with headwinds.

Here we suggest a simple rule that accommodates these goals while using both Carson speed and best-glide speed.

Recall that best-glide speed depends on payload, indeed, increases with payload. For small two-seaters, we ignore that aspect and use the best-glide speed when the plane is operated single-pilot and with some reasonable amount of baggage, say 50 lbs.

Exact measurement of that best-glide speed is not easy. The excellent paper Maximum Endurance, Maximum Range, and Optimum Cruise Speeds by R. Erb explains some ways for doing so. A simple method whose precision suffices for our purposes first estimates the airspeed with minimum sink rate, which is called the max-endurance airspeed. Carson's paper establishes that best-glide speed is max-endurance speed times sqrt(sqrt(3)).

To estimate max-endurance airspeed with single pilot and 50 lbs of baggage, go to sufficient altitude, like 3,000 ft AGL. Pull power, let the airplane slow down and adjust trim gradually until the airplane has minimum sink rate. Now increase engine power just enough that the propeller turns in a high idle. Ideally, the rpm should be such that the propeller neither pulls the plane nor creates a drag. Then trim again to get minimum sink rate. The airspeed accompanying that sink rate is an approximation of the max-endurance speed.

The difficult part of the estimation process is done. Indeed, using Carson's formulas, we estimate the best-glide speed by max-endurance speed times 1.3, and then the Carson speed by best-glide speed times 1.3.

For my Zenith 601HDS, this approach produces a best-glide speed estimate of 70 kts and a Carson speed estimate of 90 kts.

We now use the estimates for best-glide speed and Carson speed in an easily applied rule for efficient flight along a given route.

First, determine the altitude with most favorable wind. Of course, you can only consider altitudes where you can legally fly. For us non-oxygen and non-pressurized folks, the limit is 12,500 ft MSL, with exception up to 14,000 MSL for up to 30 min. By the way, this restriction continues to amaze me, considering that any person of any age with a valid driver's license can drive up 14,000 ft mountains in Colorado without any time restriction. Oh well.

Suppose that the best wind is a tailwind, as shown in the above snapshot of the Garmin Pilot on my iPad for the route Kansas City to Dallas. For that route, the best wind is at 12,000 ft MSL. Since this is a westerly flight, 12,500 ft is the correct altitude. 

Go to that altitude, set the autopilot for the course, and monitor the ground speed. If the groundspeed is higher than the Carson speed, pull power until one of two things happens: The groundspeed becomes the Carson speed, or the indicated airspeed drops to the best-glide speed.

Climbing to altitude: For comparatively low fuel consumption and good forward visibility over the nose, climb at best-glide speed and a modest rate of climb, say 300 ft/min.

Suppose you have a headwind. Then gradually increase power and stop whichever of two events occurs first: Ground speed increases to Carson speed, or 75% of rated engine power is used.

What about crosswind components accompanying head- or tailwinds, or differing payloads? Ignore all those aspects. They are implicitly accounted for in the following Frugal Rule:

(1) Go to altitude at best-glide speed with modest climb rate, say 300 ft/min. 

(2) Let the autopilot take care of crosswind corrections, and reduce or increase power depending on whether groundspeed is above or below Carson speed. 

(3) Change power as described to achieve groundspeed = Carson speed. But stop decreasing power when indicated airspeed reaches best-glide speed, and stop increasing power when 75% of rated power is reached.

We used the rule on a recent flight from Ankeny, Iowa, to Dallas, a distance of 530 nm = 610 statute miles. Flying mostly at 10,500 ft, a tailwind with significant crosswind component was so strong that we reduced power until indicated airspeed = best-glide speed. The crosswind component required a significant heading correction by the autopilot, which was of no concern to us. 

At 10,500 ft and indicated airspeed = best-glide speed, fuel consumption dropped to a miserly 2.1 gal/hr. The entire trip took 6 hrs, including time for a refueling stop near Kansas City, and required a total of 14.2 gal of fuel. In automotive terms, we achieved 43 mpg and traveled at an average speed of 100 mph, including the time for the stopover.

If you are in a hurry, then you may argue that the frugal rule leads to airspeeds that are too low and flying times that are too long. The argument is correct. As for us, we fly our plane not just to get to a destination, but to spend time in the air and experience our world in three dimensions instead of just two as done by car travel. The frugal rule lowers the cost of such flying.