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I don't know math or physics. If I misuse terms, just edit my post, thanks! Presuppose a normal uneventful flight, such that as the plane climbs and cruises, the pilot reduces engine thrust to 69%-97%. Presume the pilot never needs to increase thrust for long periods.

I don't understand the explanation in this YouTube video. Why does the graph have a local minimum at 2000-3000 nm. Why's it concave upward?

Why isn't the graph strictly decreasing like the blue or green segments, my first guesses? Unquestionably, idling and taxiing consume fuel. As soon as an airplane operates its engines, the amount of fuel strictly decreases. Thus the plane's mass strictly decreases. Thus, based on my pre supposition above on a constant thrust at cruising altitude, the fuel burn rate must strictly decrease.

enter image description here

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    $\begingroup$ That graph's labeling is terribly misleading. This is not "fuel burn rate". This is average fuel burn rate for entire flight. There's big difference between the two. They mean completely different things. What a snafu. $\endgroup$ Sep 24 '21 at 7:06
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    $\begingroup$ @Kuba, if labelled correctly, it is a useful graph - shows optimum flight distance for fuel efficiency. I do agree that as it is, it's very misleading. $\endgroup$ Sep 24 '21 at 7:56
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    $\begingroup$ In addition to the what's described in Bianfable's answer, I would also point out that just looking at the shape of this chart rather dramatically exaggerates this effect. The change from the minimum value on this chart to the maximum is only about 13%. Once the extra fuel spent to land, refuel, and climb back to cruise altitude and the additional crew time, ground service/airport costs, and inconvenience to passengers is taken into account, airlines prefer flying non-stop rather than adding a fuel stop with two shorter flights. Cargo flights do that, though, allowing them to haul more cargo. $\endgroup$
    – reirab
    Sep 25 '21 at 2:38
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    $\begingroup$ This is why you will see lots of 777s and 747s full of cargo from around the world if you visit the main Anchorage airport. While the passenger versions of those same planes would fly non-stop from Asia to the continental U.S., the freighters just carry enough fuel to get to Alaska and refuel. $\endgroup$
    – reirab
    Sep 25 '21 at 2:39
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What does the graph show?

The graph plots the average fuel required per distance as a function of the total flight distance. It does not show fuel flow rate as a function of time during the flight.

Why is it going up towards shorter flights?

For short flights (less than ~2500 NM for the 777), the takeoff and climb to cruise altitude are a larger fraction of the total fuel required. Since the aircraft has a higher fuel flow rate during the climb part compared to cruise, this will increase the average fuel per distance figure. The longer the flight gets, the less important is the climb. Therefore, the average fuel per distance is initially decreasing with flight distance.

Why is it going up again for longer flights?

The problem here is that you have to carry the fuel for the entire flight, which makes the aircraft significantly heavier. This is not an issue for short-haul regional airliners, but for long-haul flights the fuel weight can be enormous. E.g., the 777-200LR has a max. fuel capacity of 145 t, which is about equal to its OEW (Operating Empty Weight).

Since the aircraft is now significantly heavier during the initial stage of the flight, it will have a higher fuel flow rate due to the higher weight and due to the lower initial cruise altitude (e.g. optimum cruise altitude for a 777-200LR at a weight of 340 t is only FL285, but after burning 100 t of fuel it has increased to FL360). Long-haul flights will usually perform step climbs to increase their cruise altitude during the flight as their weight decreases. This further reduces the fuel flow rate during the flight, but the average fuel burn for the long-haul flight is still higher.

At what distance this effect starts to dominate over the first one (and average fuel per distance then increases again), depends on the aircraft (and actual payload). For the 777 it seems to be at about 2500 NM.

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    $\begingroup$ Boeing 747 400 can carry even more fuel: Up to 173t. It blew my mind when I found out that a Falcon 9 rocket first stage “only” contains 119t of kerosene, i.e. much less despite being huge and almost only fuel. $\endgroup$
    – Michael
    Sep 23 '21 at 19:20
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    $\begingroup$ @Michael well, the F9 isn't that huge (compared to e.g. Saturn V), and most of its “fuel mass” is in oxygen, not kerosene. A Falcon can send a handful of people around the globe, a 777 can carry more than 300, much more than even Starship, and the fuel that requires is something more like 1000 t of methane. $\endgroup$ Sep 23 '21 at 19:51
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    $\begingroup$ Does "t", as in 145 t, mean megagrams, as in 145,000kg? $\endgroup$ Sep 24 '21 at 14:24
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    $\begingroup$ huh, I never realized metric ton meant 1000kg...I guess that's self evident. TIL. $\endgroup$ Sep 24 '21 at 14:46
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    $\begingroup$ @USTopic I'm not sure if you're aware, but comments on Stack Exchange sites can only be edited for 5 minutes after posting them. They're not meant to be updated as responses to each other. $\endgroup$
    – reirab
    Sep 25 '21 at 2:32

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