Making Sense of Aircraft Endurance, Range, and Economy It isn t as simple as the textbook says it is! Photo: First aerial refueling, two DH- 4B aircraft, 27 June 1923, USAF Photo Most professional pilots earn their instrument- rating sitting behind one or more propellers, after having learned textbook definitions of maximum endurance and range speeds. We learn early on that maximum endurance speed uses less gas in a holding pattern and that gives us more time to wait for the weather, to sort out malfunctions, or to just make things right for an eventual landing. We also know that maximum range not only gives our craft its longest legs, but also gets us to our destination with the most fuel remaining. In each case, the correct answer to the question how fast? can give us needed flexibility. But how many of us realize that some of the rules changed when we graduated to jet engines? Or that in some cases the textbooks are just wrong? Maximum endurance speed carries more risk for a jet than a propeller- driven aircraft; the distinction can be lost on a novice jet pilot. While both aircraft types determine maximum range in the same way, adjustments for weight and altitude changes are different. In other ways the distinctions are disappearing. A turbofan engine gets most of its propulsive force from the fan, not the jet. A turboprop can get 15 percent or more of its forward force from the engine, not the prop. No matter what type of engine an airplane has, there are other factors that can completely outweigh engine economy. If you fly a turbofan or a turboprop and want to really maximize your fuel usage, you need to understand maximum endurance and range procedures for both.
Minimum Drag = Minimum Fuel Usage = Maximum Endurance A jet engine produces thrust by accelerating air and fuel aft. The fuel and air is the m (mass) of Sir Isaac Newton s first law of motion, F=ma. The a is the acceleration, and the resulting F is the force. Newton s third law tells us that for every action force (air and fuel going aft) there is an equal an opposite reaction force (aircraft going forward). An aeronautical engineer can diagram the thrust versus the velocity to look at the relationship of one to the other. In steady, unaccelerated flight, drag is equal to thrust so we can hypothesize the composition of induced and parasite drag. A jet aircraft requires high angles of attack at low speed and that spikes the induced drag. At high speeds the entire airplane becomes a speed brake, causing parasite drag to climb prohibitively. The result is the familiar u shape of the curve. Maximum endurance occurs where the total drag is at a minimum, and this point is also where the ratio of lift- to- drag is at its highest, the point of (L/D)MAX. A reciprocating engine connected to a propeller does not produce any thrust at all. If you disconnect the propeller, there will be no forward force. The engine imparts shaft horsepower to the propeller that develops an aerodynamic force that generates forward thrust. A turboprop may develop a small amount of thrust, but for the most part it too generates power imparted to the propeller. Unlike the turbojet, the propeller- driven airplane does not require large amounts of power to fly slowly and the minimum drag point is found at a point slower than (L/D)MAX. The shapes of these two curves are critically important. The propeller- driven aircraft s flatter curve means pilots can move from a little slow to a little fast with relative ease. Adding power makes you faster, subtracting power makes you slower. That is not always the case with the turbojet, where the minimum drag / minimum thrust required point
sharply divides two aircraft behaviors. When in the zone faster than the minimum thrust required point the thrust levers operate conventionally. If the pilot wants to fly faster, he or she simply adds thrust until at the desired speed, then reduces thrust to a point that is higher than the original thrust so as to maintain the newer speed. To fly slower, the pilot reduces thrust until the new speed is achieved, and then adds power to a point that is less than the original setting. This is fully expected. But when in the zone slower than the minimum thrust required point, things are not so straightforward. To fly slower, for example, reducing thrust will cause the speed to decrease. But to stabilize at the new, slower speed, more thrust is needed than the original setting. When attempting to accelerate, a large burst of thrust may suffice, but the only way to ensure an increase in speed is to sharply decrease the angle of attack. This region of reversed command is contrary to normal pilot reactions and is what many call, behind the power curve. Most aircraft manufacturers publish holding and endurance speeds that are well above the true maximum endurance speeds, as a safety factor. Turbojet pilots are well advised to treat even these published endurance speeds as absolute minimum speeds. Maximum V- to- F = Maximum Range A turbojet s thrust level is analogous to its fuel flow and a propeller- driven aircraft s power level is analogous to its fuel flow. The resulting charts offer a mathematical way to derive maximum range. Specific Range = distance fuel Algebra permits us to divide both sides by the same factor (time) without changing the result: Specific Range = distance/time fuel/time = V FF
We can then say that maximum range is obtained at (V/FF)MAX or (FF/V)MIN. That latter quantity can be found graphically by plotting the tangent to the curve and the intersection of the tangent and the curve determines where maximum range can be found. You can analyze the math or accept on faith that drawing a line from the origin to the curve will identify your maximum range speed. Why is this important? It will help you to understand how to set your power levers as you burn off fuel and why you should climb when you can. The Impact of Fuel Burn on Maximum Range Once you ve set your power levers to maintain maximum range speed your work is not done. As the aircraft burns fuel the weight decreases and the fuel flow required curves shift down and to the left. That means your speed is going to have to come back if you want to continue to make gas. This holds true for both aircraft types. When weight and ATC permit, your next step is to climb. The Impact of Altitude on Maximum Range Many pilot and aeronautical engineering texts are confused on the subject of climbing to achieve maximum range. Most acknowledge a jet engine performs best at higher RPMs and
that lower inlet temperatures reduce specific fuel consumption. But some claim all benefits end where the tropopause begins and fuel consumption may actually suffer at higher altitude. A propeller- driven aircraft, it is often said, is altitude ambivalent and does not need to climb to achieve maximum range. These claims ignore that high technology fuel control units and full authority digital engine (or electronics) control can extract performance gains at altitudes once thought impossible. A modern turbofan engine, moreover, can be said to be part propeller and a turboprop can be said to be part turbojet. Aircraft manufacturers should present accurate climb and cruise fuel numbers, pilots should refer to their manuals when venturing to beyond the limits of their textbooks. The Impact of Headwinds and Tailwinds on Maximum Range Many aeronautical engineering texts claim that flying slower with a tailwind and faster with a headwind will reduce overall fuel consumption, usually noting that the winds must be at least 25 percent of the true airspeed to yield benefits. The trigonometry of the chart seems to lend credence to this claim. Some manufacturers even give recommended speed adjustments. I ve run the numbers on a variety of aircraft, from the turboprop PC- 12 to the ultra- long range Gulfstream G650, and the results are the same. Making the recommended speed adjustments has a fifty- fifty chance of improving fuel burn, but only marginally. The adjustments will hurt fuel burn about as often, but again only marginally. My advice: don t bother adjusting your speed to account for a headwind or tailwind without doing the math first. One Final Myth: Flying Faster Means Less Flight Time and That Saves Money Consider a Gulfstream G450 cruising at 37,000 feet in a 100- knot headwind starting at 70,000 lbs. gross weight, and ISA conditions. The crew knows normal cruise speed will be M0.80 but are wondering if the owner will see an improved bottom line if they fly M0.03 slow, or even M0.03 faster. If you said, it depends, you are right. But it depends on more than just what the aeronautical engineer has to say; it depends on what the accountant is thinking. Are the pilots paid hourly or by salary? Are any of the maintenance programs billed by flight hour? Is the aircraft on a lease program, billed by flight time as opposed to calendar time? Each one of these variable costs may overwhelm the cost of fuel and make it financially advantageous to burn more fuel to reduce total flight time. You won t find the following equation in any aeronautical or pilot texts but it might provide you with the answer to the question, how fast do you want to fly?
where: Total Cost = D TAS WF FC FF FD + VA + VC + VE D Distance to cruise (since the climb and descent fuel will be about the same, we consider only the cruise portion) TAS True Air Speed during cruise WF Wind Factor (positive numbers for headwinds, negative for tailwinds) FF Fuel Flow (pounds per hour), average in cruise FC Fuel Cost ($ per gallon) FD Fuel Density (pounds per gallon) VA Variable Airframe costs ($ per hour) VC Variable Crew costs ($ per hour) VE Variable Engine costs ($ per hour) A salaried crewmember doesn t add to variable costs and does not lend to any incentive to fly faster. Some aircraft maintenance programs are fixed rate to a certain level of activity and then add per hour charges; while others count every hour from the first at an hourly rate. Variable costs can amount to $3,000 or more for a typical business jet. Fuel, on the other hand, will always be a factor. At $1.00 per gallon there are usually incentives to fly fast. But what about at $5.00 per gallon? Not so much! For the sake of our example, let s say it is an ISA day, the fuel costs $3.00 per gallon and has a density of 6.5 gallons per pound. The first hour fuel burn at M0.77 will be 2,996 lbs.; at M0.80 it will be 3,178 lbs.; and at M0.83 is will be 3,593 lbs. The speed up / slow down question depends entirely on those variable costs: Mach No Variable Costs $1,000 Variable Costs $3,000 Variable Costs M 0.77 $12,035 $20,828 $38,413 M 0.80 $12,278 $20,648 $37,389 M 0.83 $13,245 $21,233 $37,207 These numbers can be fine- tuned by adjusting fuel burn rates on an hourly basis but for demonstration purposes the conclusion in clear. In our example, it doesn t pay to fly faster until the variable costs overwhelm the cost of the increased fuel burn. As fuel costs go down, the incentive to increase speed go up. Similarly, as variable costs increase, speeding up becomes more attractive. The Answer to All Maximum Endurance and Range Questions Chances are you will not find the answers to all of your maximum endurance and range questions in a textbook because your aircraft is more advanced technologically than the textbook s author could have imagined. You are unlikely to get a complete answer from the airplane flight manual either because the accountants have just as much to say about costs
as do your computerized fuel control units. The correct answer depends on all of these variables and it is up to you, the pilot, to sort it all out. If you understand how turbojets and propeller- driven aircraft behave you will have a starting point when judging the performance of your turbofan or turboprop engines. You should also consider any variable costs in your operation. Only with these parts of the puzzle can you really have a well thought out answer to the question: how fast?