Maximizing Air Miles

Right after I set up the background aerodynamics to talk about jet performance, I saw Dave's latest post on a real life situation involving setting maximum range speed to offset unforecast headwinds. This topic corresponds to two of my best aha! moments while reading this textbook, so I'm going to seize Dave's story and festoon it with trigonometry in order to cement my knowledge and share it with you.

Dole and Lewis treat jet performance before propeller aircraft performance, which confused me at first, but now I realize it's the logical progression. A jet is actually simpler than a prop plane when it comes to performance. Jet fuel is converted directly into thrust. Thrust is a force, and the thrust produced by a jet depends on three things: the speed of the intake air (V₁), the speed of the exiting exhaust (V₂), and the mass flow of air through the engine (Q). If you want an equation, you can use this one: Thrust = Q(V₂ - V₁). The faster the jet is going, the more air will go through the engine, and the greater the intake air velocity, but the exit speed remains pretty much constant at different airspeeds. That decrease in acceleration through the engine almost exactly cancels out the increase in airflow, so that at a given RPM, a jet can be considered to produce the same amount of thrust regardless of airspeed.

Remember the graph of total drag versus airspeed described a few days ago. Here's a graphic of the curve from wikipedia. It doesn't show the high end compressibility, but you'll get the idea. Because thrust counteracts drag, the y-axis can be relabelled in thrust units, making the same total drag curve into thrust required for level flight (T_r) versus airspeed. We can add another line to the graph representing thrust available (T_a) at max power. It's a straight, horizontal line, because jet thrust available does not vary with airspeed. The line runs across the graph and intersects the thrust required curve somewhere on the high airspeed end of the upward curve. When thrust required equals thrust available, you have just enough thrust to sustain level flight. The line and the curve intersect at the highest airspeed available for level flight. At lower airspeeds, there is excess power, so the aircraft can accelerate or climb. At higher airspeeds there is not enough thrust to overcome the drag, so the airplane would have to be descending to sustain that airspeed. At lower rpm a smaller amount of thrust is available, but it's still a straight line to determine the maximum cruising speed. Set the thrust low enough and there are two points of intersection with the T_r curve: one at the high end for maximum cruise at that thrust setting and one at the low end, representing the lowest speed you can fly with that thrust. Fly any slower and you need more thrust to overcome induced drag. There is one thrust setting that is so low that when you plot its horizontal line on the graph, the (L/D)_max point at the base of the U-shape curve rests on the line. Set T_a any lower and it wouldn't intersect the T_r curve at all. Where T_a intersects T_r at the T_r minimum, that's the minimum fuel flow required to sustain flight, and the corresponding speed is called best endurance speed. Fly at that speed to stay aloft for the maximum possible time before running out of fuel.

That's not the fuel flow Dave set. He wasn't trying to stay in the air long enough to finish showing the onboard movie. He was trying to get to Anchorage before his fuel reserves dropped low enough to require paperwork. Note also that he wasn't trying to get there before the bars closed: he had to reduce fuel flow and slow down to set best range speed. As Dave put it, groundspeed/fuel flow x fuel remaining = range. If that is unfamiliar, realize that when you divide nautical miles per hour by pounds of fuel per hour, the per hours cancel and you get nautical miles per pound of fuel. Nautical miles per pound times the number of pounds you have left tells you how many nautical miles you have left.

Another way of putting it is that to get the furthest distance possible using the available fuel, you want to burn the minimum number of pounds per nautical mile. You want the smallest possible value of fuel flow divided by groundspeed. That's flipping it over from the way Dave did it, but that's how it works best with the graph. If you can increase the airspeed without increasing the fuel flow by a larger factor, you win. And if you can decrease the fuel flow while keeping the airspeed from reducing as much, you win, too.

The same y-axis that yesterday time I called drag, and two paragraphs ago I called thrust, could just as soon be labelled fuel flow. Thrust overcomes drag, and fuel flow determines thrust. So we have a graph of fuel flow versus airspeed. Note that this is airspeed not ground speed, the difference being the considerable headwind Dave was fighting. As it stands, the graph is concerned only with the speed through the air, the result of the fuel flow. I'll get to the headwind after solving the zero wind case.

For any point on the curve, its distance right of the y-axis represents the airspeed and its distance up from the x-axis represents the fuel flow required to sustain that airspeed. That is a somewhat duh statement, as it simply restates what is plotted on the graph, but I'm setting up my first aha. Consider any point on the curve and draw a vertical line from it to the x-axis. The length of that line is fuel flow. Draw another line from the origin (zero-zero point on the graph) to the point under consideration. That line is the hypotenuse of a right triangle. The base of the triangle runs along the x-axis and its length represents airspeed. Fuel flow divided by airspeed equals fuel per distance, but now you can see that it's also the perpendicular divided by the base of a right triangle. If you know trigonometry you see what is going on, and if you don't, this should be enough to make you rush out and learn, because the perpendicular of a right triangle divided by the base is equal to the tangent of the angle between the base and the hypotenuse. The tangent of an angle decreases with the angle. Now remember that we're looking for the lowest possible value for fuel per distance. So now all we have to do is find the line that goes from the origin to the curve, making the smallest possible angle with the x-axis. Inspection quickly reveals this to be a line tangent to (i.e. barely touching) the curve.

There's a way to realize this without the trigonometry. Lets say we start at the (L/D)_max point at the base of the curve, with the lowest possible fuel flow and a low airspeed. If we add a little more thrust, we get an increase in airspeed. Add a little more, a little more airspeed. The goal is to stop adding more thrust at the point when the airspeed increases less than the fuel flow does. That's the point where the curve bends away, curving more up than forward, the point described in the final two sentences of the previous paragraph.

I had known the handwaving argument that the best range speed could be determined from a tangent drawn from the origin to the thrust required curve, but I hadn't noticed how to mathematically prove of it. I like it.

The speed and fuel flow represented this way corresponds to the maximum distance the aircraft can travel through the air with a set amount of fuel: its air range. My other aha! moment in this chapter was an even simpler demonstration of how to adjust the best range speed to compensate for a head or tailwind: best ground range. I'll write about that soon.