One of the interesting things about learning about thrust for a rocket, is that we can use the same types of formulas for airplanes. So, let’s take a few minutes and figure out how much fuel an airplane uses to travel from one place to another. We can also look at why airplanes fly at the altitudes that they do, and how the wind affects the fuel used.
How to actually do this? Well, we need two equations that we have talked about on different posts: (1) the equation for thrust; and (2) the equation for the drag force. When a plane is traveling between two places at a constant altitude, we can ignore the forces in the vertical direction, since the gravity of Earth is balanced by the lift from the wings. In the horizontal direction, the forces are also balanced (since we are traveling at a constant velocity), namely the thrust of the airplane is balanced by the drag force from the airplane moving through the air.
So, what is the drag force on an airplane? Well, we can calculate it using the formula: F=0.5*Rho*Area*DragCoefficient*Speed², where Rho is the mass density of air, Area is the frontal area of the airplane, DragCoefficient is the Drag Coefficient of the airplane, and Speed is the speed of the airplane (with respect to the wind). Let’s do a simple example, and take the Boeing 747, like the plane shown above. Some assumptions about the 747:
Area = 158.3 m² (that is pretty big!)
Drag Coefficient = 0.05 (that is pretty small!)
Velocity = 562 miles per hour = 250 m/s
I basically found these by looking around on the web.
The surface mass density is 1.23 kg/m³. The density decreases pretty rapidly as you go up in the air. At 30,000 ft, the density is roughly 38% of the surface density (0.467 kg/m³). At 40,000 ft, the density is about 25% (0.308 kg/m³).
Ok, that was a lot of numbers. Sorry. What does this mean? Well, we could talk about the drag force that the 747 experiences. If we do all of the math, and we assume that the 747 is cruising at 40,000 ft, we get a force of 75,750 N, which is 17,030 lbs. If the airplane were to be flying just 10,000 ft lower, the force would be 25,885 lbs, which is much (50%) larger, showing that the altitude that the airplane flies is pretty important.
Now, let’s calculate how much fuel is used during a 6 hour flight (say New York to London). If we assume that it is all cruise (which is a bad assumption, since a lot of fuel is used to take off), how much fuel does the 747 use?
Well, we have to calculate how much fuel a 747 uses each second at cruise speed and at altitude. Remember that Thrust = MassFlowRate * ExhaustVelocity. For a rocket engine, the Exhaust Velocity is really the speed at which the gas comes out of the engine. For a jet engine, that is not really the case, and it is a bit more complicated. But, let’s skip over that and just take my word that the “ExhaustVelocity” of a jet engine is about 35,000 m/s. (If the exhaust velocity were really that large, it would be pretty dangerous to be around the backend of an airplane!)
To get the MassFlowRate, we can just divide the thrust by the ExhaustVelocity. At 40,000 ft altitude, the MassFlowRate would be 2.16 kg/s. A gallon of jet fuel is about 2.7 kg. So, a 747 uses just under (80%) a gallon of jet fuel every second. Depending on your point of view, this is either a lot (a car uses a gallon every few hours), or a tiny bit (a rocket uses a hundreds of gallons each second).
Over the course of a 6-hour (6*3600 seconds) flight, the airplane would use about 17,300 gallons of fuel (not counting takeoff and landing) if it flew at 40,000 ft.
If the airplane were to fly at 30,000 ft and keep the same exact speed (562 mph), the airplane would use 26,300 gallons! That is 9,000 gallons of jet fuel more, just for flying at 30,000 ft.
Hopefully this helps you understand why airplanes fly as high as they can. If you are on a very large airplane that is flying a long way, then the airplane may raise the altitude a couple of times as it uses fuel. A super heavy 747 can’t fly at 50,000 ft, since its wings can’t support the lift at 50,000 ft. As the 747 uses fuel and is less mass, it can fly at higher and higher altitudes. The best track would be to fly at the highest altitude all of the time, increasing altitude all of the time, but rules stop this – there are certain altitude “lanes” that planes can fly in.
Just for fun, if the airplane is at 40,000 ft, it gets about 0.195 miles per gallon. At 30,000 ft, it gets about 0.128 miles per gallon. If the flight had 400 people on the 747, and it flew at 40,000 ft, then each person would get the equivalent of about 78 MPG. Not really that bad! It would be hard to drive somewhere for this type of fuel economy!
Interestingly, if a 747 were to fly at ground level the entire flight, it would use 69,000 gallons of fuel to fly from New York to London, or would get about 0.05 miles per gallon. Yikes!
Finally, how does wind effect the amount of fuel used? Well, a 747 goes 562 MPH not with respect to the ground, but with respect to the background wind. So, if the 747 is flying in the jet stream, which can be about west-to-east at 100 MPH, then the ground speed of the 747 flying from New York to London would be 652 MPH, but coming back from London to New York, the ground speed would be 452 MPH. This doesn’t cause the amount of fuel used per second to change, but it changes the number of seconds the airplane is in the air. From New York to London, the flight would be shortened to 5:10, and back to New York, it would be lengthened to 7:30. The amount of fuel used would be 14,700 gallons (saving 2,600 gallons, NY to London) or 21,000 gallons (costing about 3,700 more gallons, London to New York).
Ah, physics. I love you.
Oh, on a side note, think about in the first Iron Man movie when he (Iron Man) flew from Los Angles to the Middle East in his suit. Obviously it must have been pressurized, since he would have to fly at incredibly high altitudes. Iron Man is quite a big smaller than a 747, but he probably was flying about twice as fast as a 747. So, if you do the calculations, he would have had to use about 147 kg of fuel. If this was jet fuel (which it was not, but that is a separate discussion) it would be about 55 gallons. Where did he put all of this fuel??? Marvel Fans want to know!