<< Page 13 Page 14: Page 15 >>   Cruise Speed & Max Range Just using the things you learned about Drag back in pages 5 through 9, it is worthwhile to realize you can already figure out how a given airplane can fly at a speed that is optimized for range (i.e. miles per gallon). This is the cruise speed. You already know that induced drag (wingtip vortex drag) is high at lower speeds, while viscous drag and compressibility drag are higher at high speeds. At first, you might think that the way to squeeze out the most miles per gallon is to fly at the “middle” speed where the sum of these drags is lowest. That is not quite true. Imagine a graph of the drag of an airplane vs the speed at which it flies (*1). Now, let’s assume that the thrust produced by the engine is proportional to the fuel it takes in; in other words, to get twice the thrust out, you need to feed the engine twice as much fuel per unit time (*2). Drag and thrust are equal in steady level flight, so the drag is roughly proportional to the fuel consumption (in, say, gallons per hour). Any point on that graph means a certain drag (and thus a certain rate of fuel consumption) at a certain speed. The inverse of the slope between that point and the origin is the miles per gallon – it’s the speed divided by rate of fuel consumption. That slope is gallons per hour divided by miles per hour, which gives you gallons per mile – so a point that is “high”, on a steep line (points that make up the red line on the graph) has more gallons per mile than a “low” point, with fewer gallons per mile and thus more miles per gallon (points on the green line on the graph). But at what points CAN the airplane fly? Well, let’s look at how the components of drag change with speed: Induced, viscous, and sonic. Add them all together, and you have how much drag is seen by the airplane at different speeds, the total drag: The optimal or cruise speed is not the one at the very bottom of the graph (lowest drag, or lowest thrust). It’s the one where the point has the lowest slope relative to the origin. An easy way to find this is the line tangent to our graph-curve that goes through the origin (because the line that goes through the origin and through any other point in our curve will be higher, and thus correspond to more drag-per-speed (and thus less speed-per-drag, less gallons-per-hour per miles-per-gallon) and thus more miles per gallon): In fact, notice how flying just a little slower than cruise (like at the minimum-drag speed), or just a little faster, gives you worse mileage (orange), and flying quite a bit slower gives you much worse mileage. Perhaps surprisingly, flying very slow gives you the worst mileage possible (purple), even worse than at a very draggy maximum speed (red). This is why airplanes are at their most inefficient during takeoff and landing, using up a disproportionately high amount of fuel during takeoff/climb and during approach. (Well, also because an airplane is flying lower at those times, where higher densities mean even higher drag at any speed compared to that same speed at altitude). It is also worthwhile noting that the minimum-drag speed is not useless (despite it giving worse gas mileage than the cruise speed). The minimum-drag speed maximizes endurance (time in the air) by minimizing the rate of fuel consumption. This is useful for aircraft performing patrol missions, search and rescue, traffic reporting (or other aerial coverage of newsworthy events), communications relaying, command and control, etc, as well as for low-energy flight (human-powered, electric or solar-powered, or gliding) where power consumption must be minimized. Additionally, the minimum-drag speed is also the speed at which the lift-to-drag ratio is minimized, so it is the speed at which the glide slope is least steep. Pilots know this speed as the "Best Glide" speed, since, in case of an engine failure, it is the speed that will allow you to fly the furthest (and thus be more likely to reach a landing strip). *1: This is an oversimplification, as drag at a given speed is different at different altitudes, temperatures, weights, etc. If you want to make this really accurate, let’s say the graph is drag “divided by air density” vs speed when the airplane is at a certain weight – say, maximum take-off weight. *2: This too is an oversimplification. Thrust is roughly proportional to fuel consumption squared (to get twice the thrust, the engine must double the increase in speed of the air it affects, which means giving that air four times as much kinetic energy (half of m times v squared), which means using up four times as much fuel). However, the conclusions on this page still hold, as the only consequence of incorporating the better model would be to make fuel consumption proportional to drag squared, which would make the colored “mileage lines” parabolas (of the form x=ky2) rather than straight lines, with the parabolas nearer the drag axis being bad mileage and the parabolas near the speed axis being good mileage. Again, all these parabolas go through the origin, and the optimal one is the one tangent to the drag/speed curve, which touches it not at the very bottom (lowest drag) but on the lower right (cruise), like with the straight-line approximation above. Therefore, if we took the “square” effect into consideration, mileage would vary with speed in much the same way as in the simpler approximation of the graphs above. *PS: For a more in-depth analysis of how different speeds and altitudes affect the range, efficiency and endurance of different kinds of aircraft, I highly recommend checking out http://www.professionalpilot.ca/aerodynamics/performance/range_jet.htm