Fundamentals of airliner performance, Part 2.

By Bjorn Fehrm

In our first article on how to understand the fundamentals that make up airliner performance we defined the main forces acting on an aircraft flying in steady state cruise. We used the ubiquitous Boeing 737 in its latest form, the 737 MAX 8, to illustrate the size of these forces. Leeham logo with Copyright message compact

Here a short recap of what we found and then some more fundamentals on aircraft’s performance, this time around the engines:

When flying steady state (Figure 1) we only need to find the aircraft’s drag force to have all important forces defined.

Lift with downforce

Figure 1. Elementary forces acting on an aircraft at cruise. Source: Leeham Co.

The lift force is given as equal to and opposite to the aircraft’s weight and the tail downforce that we need to add to this was small. We also presented the two classes of drag that we will talk about:

  1. Drag independent of lift or as we often call it drag due to size as almost all drag components here scale with the aircraft’s size.
  2. Drag due to lift or drag due to weight as we call it as this drag scales with weight when one flies in steady state conditions.

We could see that the aircraft’s flight through the air created a total drag force of 7900 lbf, Figure 2 ( lb with an f added as we prefer to write it as this is a force and not a measure of mass. Mass we denote with just lb or the metric units kg or tonne = 2205 lb).

Drag components

Figure 2. Drag of our 737 MAX 8 and how it divides between lift and non lift drag. Source: Leeham Co.

We also learned that if the drag is 7900 lbf then the engine thrust is opposite and equal. It is then 3950 lbf per engine when cruising at our mean cruise weight of 65 tonnes or 143.000 lb on our 1000 nm mission. Drag due to size consumes 63% of our thrust and drag due to weight 37%.

Drag reduction

Drag is what stands in the way of efficiency for an aircraft therefore the quest for drag reduction is never ending. Here in short what is being done:

Drag due to size is the dominant drag force and there are several technologies being employed to try and reduce it. One can influence this drag with good packaging of the aircraft so that payload and necessary system components creates an aircraft of minimum wetted area. Further one can employ advanced control methods so that the size of empennage needed to keep the aircraft stable can be reduced (full time Fly-By-Wire enables this).

A wing that is not wasting area to produce a wide span (see below why we want wide span) is also helpful, this is called a high aspect ratio wing. One can also employ techniques to keep a smooth airflow over a larger part of the aircraft (laminar flow technology, minimize disturbances like antennas etc.). By and large this drag type scales pretty monotonously with aircraft size however until we change configuration principle to e.g. blended wing body aircraft. Then a step change in drag due to size can be achieved.

The component that can be influenced more readily is drag due to weight or induced drag. This drag is reduced with a wider span (real or virtually extended by wingtip devices). Induced drag reduces with the span counted twice or its square, so extending the span is a worthwhile exercise. This is also where modern wing technology uses advanced materials like Carbon Fiber Reinforced Plastic (CFRP) to make large gains. The limiting factors for the extension of span is space on ground (the gate classification where the MAX 8 fits in class C for up to 36m span) and the weight consequences of a long and slender wing with the necessary thin wing profile (thin due to our high cruise speed).

With advanced transonic design techniques one can keep the wing profile thicker and by using the tailored strength of CFRP the wing can be longer without the weight becoming prohibitive. Clever folding wingtips get wings past airport class limits (the Boeing 777X folds its wingtips when on the ground to fit in class E for max. 65m span aircraft). Our MAX wing has an aspect ratio of 11 (a high and therefore good value) so it is a long and slender wing despite the use of classical materials. It has gained this good value in part from being on a second round of virtual span extension with new split winglets adding to its effective span. Reducing induced drag helps in all aspect of flight ( take off, climb, cruise) so it has a high priority with aircraft designers.

Engine thrust

We now know our two engines shall deliver 3950 lbf thrust each at our mid cruise point. The 737 MAX 8 take-off thrust specification calls for 26-28000 lbf of thrust so where are the other 22,000 lbf? sitting idle? i.e. are the engines working at 15% of their capacity during cruise? In fact no, they are working at around 90% of their capacity. To understand why requires us to look at how thrust from a turbofan is generated and how atmosphere and aircraft speed affects thrust.

A turbofan engine works in exactly the same way as a piston engine with propeller on a private plane to generate thrust, it is pushing air backwards. Air is not that light; it weighs 1.2 kg per cubic meter or 0.081 lb per cubic feet at ground level. The generated thrust is then dependent on how much air is pushed back and at what speed. The speed we talk about here is always the over-speed relative to the aircraft, therefore:

Engine thrust = air mass moved * air mass over-speed

Both these factors are affected by our aircraft humming along with M 0.8 or 450kts at a very high cruise altitude. We start with the first term, air mass moved. As said air weighs 1.2 kg per m3 at ground level. It weighs less then a third of a kilo at our cruise altitude of FL 390 (39.000 feet) however, Figure 3.

Std athmosphere

Figure 3. ISA Standard atmosphere for a standard day. Source: Leeham Co

This means our engine only pumps through around 1/3 of air mass at cruise with the engines at the same rpm as at take-off. But it does not stop there; thrust is generated from air which is travelling faster out the back of the engine then surrounding air.

If we assume that our take off thrust of 28,000 lbf was generated by 900lb (400kg) of air being accelerated to 1000 feet per second (300 m/s or 590 kts) we have an over-speed of 590kts versus the surrounding air when lining up for take-off. At M 0.8 at 39.000 feet the aircraft is travelling at 450kts so we only have an over-speed of 140kts if no other measures were taken. This only gives us 2,200 lbf when we combine the mass and over-speed loss and we need 3950 lbf minimum. In practice the engines use a narrower output area, the nozzle, to accelerate the air to the level needed, in this case to around 400 ft/s or 240 kts over-speed so we can get our 4000lbf of thrust.

The loss of thrust with speed is called Lapse and it has a significant influence on aircraft performance not only at cruise or climb but also take-off. We will cover this in our next fundamentals article.

Fuel consumption

Aircraft turbofan engines are rated efficiency wise with how much fuel they consume per generated thrust. In our case we have a MAX 8 aircraft with a LEAP-1B engine representing the state of the art for aircraft engines when entering service in 2017. Such engines are consuming around 0.53 lb of fuel per generated lbf of thrust and hour. This means we can now estimate our fuel consumption at cruise, we have 7900 lbf * 0.53 lb/lbf/hr which equates to 4200 lb fuel per hour, a very good figure and about 15% lower than today’s 737-800.

The low fuel consumption from modern turbofans comes from a high efficiency. This can be divided in the efficiency to generate shaft horse power to drive the fan from the stored energy in the fuel, thermal efficiency, and the engines efficiency to transfer that horse power to effective thrust driving the aircraft forward, propulsive efficiency. Thermal efficiency comes from burning the fuel under very high pressure (compression ratio in a car, pressure ratio in turbofans) and turbines that are effective in converting the gas energy into shaft energy. Propulsive efficiency comes from moving a large air mass with low overspeed, the lower the overspeed the better. This is why high bypass ratios are good, they create a larger air mass travelling at a lower overspeed for a given amount of thrust. The negative of high by-pass ratios are larger engines which becomes heavy and their nacelles and pylons also have larger wetted areas.

Modern engine design

CFM LEAP-1 engine

Figure 4. CFM LEAP 1 engine with markings for different areas. Source: Leeham Co from CFM brochure.

Figure 4 show a cut through of our engine for the MAX 8 (it is a generic LEAP picture which has 7 low pressure turbine stages. The MAX 8 variant has 5, it is therefore a cut through of the larger LEAP-1A or C. Principal components and technologies are the same for all variants however so it serves our purpose of understanding the engine technologies for the LEAP. We could not find a suitable picture of the MAX 8 LEAP-1B variant).

Using the figure markings we will now describe the used technologies enabling the MAX engines high efficiency:

  1. shows the SAFRAN developed Carbon Fiber Reinforced Plastic (CFRP) fan and fan case. This produces a large lightweight fan which also adjusts the blades angles dependent on fan load (by virtue of the directional strength capabilities 3D woven CFRP fan blades enables).
  2. denotes the debris rejection that the bleed ports after the engines booster compressor enables.
  3. shows the fan case mounted accessory gearbox which makes servicing of the engines accessories more accessible and therefore easier.
  4. marks the start of the General Electric (GE) developed core which is responsible for the thermal efficiency.
  5. the advanced ten stage high pressure compressor which gives a 22 to 1 pressure ratio when spinning at full rpm. Together with the booster it generates the 50 to 1 pressure ratio that gives an efficient combustion in
  6. the GE TAPSII combustor which combines pilot and main zone technology to give low emissions at those high combustion pressures.
  7. twin high pressure turbine which uses regulated cooling and un-cooled ceramic matrix stators (outer turbine housings) to increase efficiency. The advanced cooling and metal coating techniques used shall keep the metal temperatures on the same levels as today’s CFM 56 engines despite having considerably higher gas temperatures according to CFM, thereby laying the foundation for a similar reliability.
  8. the large low pressure turbine area where all the power is generated to drive the fan and the booster.

Optimal cruise altitude

With today’s example we have learned there is a loss of engine thrust with height. So why do we then want to fly so high? The main reason is the reduction in drag we can achieve as the dominant skin friction drag reduces due to the thinner air.  At the same time the wings are countering the weight of the aircraft with down-wash air which is thinner, thus generating less of an opposing lift force. This puts a limit to how high we can go. There is also a limit posed by the air having to speed up more over the wing to create the stronger downwash. This causes transonic buffeting (pre stall shaking) if we go to high. The engines loss of power also puts a limit on high we can go, we need to get to our flight level with acceptable climb performance left.

This all means there is an optimal cruise altitude for the aircraft where the combined effects of the atmospheres influence on the engines, the reduced friction forces, increased induced drag and the transonic effects all balance to an optimum. This is the ideal cruise altitude for a given aircraft weight. As we burn off fuel our weight reduces and we seek higher cruise flight levels (the induced drag and transonic problems reduces with lower weight). For our 737 MAX 8 flying a relatively short mission (and therefore being relatively light as we don’t need to carry so much fuel) a start of cruise at around FL 350 and a finish around 400 before descending to destination would be normal.

Summary

Today’s fundamentals has been about the engines and how they counter our drag. There is a gain in going high with the cruise as the dominant drag reduces, the drag due to skin friction. Transonic effects, induced drag and weaker engines limits how high we can go however.

In our next part we will look at take off and landing. These are to a large part influenced by the fundamentals we covered today.

25 Comments on “Fundamentals of airliner performance, Part 2.

  1. Excellent reading. Thanks to Bjorn. I have a question which relates more to part I. Engines are located below the center line of aircraft. Not much in 737, more on A320 and other aircraft. Since much of drag is generated by fuselage in center line and also by wings a little bit lower, I think this to forces(drag and thrust) does not meet exactly. Does this generate twisting torque, which tends to put nose up? If it does, wouldn’t over-wing engine be better solution? I know under wing has its advantages like easier maintenance, help of gravity in case of fuel pump failure…

    • It does generate a small nose up force, the lenght of the moment arm is short however, something like 1.5m / 5 ft for 737 and about 2m or 6.5 ft on an A320. The horizontal tailplane which is sitting on on a moment arm of 17-19m has an easy job of correcting that during cruise when the engine thrust is less then 10klbf. At low speed, like a go-around from landing speed, one feels this change in pitch moment and needs to compensate going from the finals low throttle to full go around thrust. But once again the HTP can correct this quite easily, it has such an advantage in moment arm.

      Re overwing engines, these flied on the VFW 614 which was in operational service for many years and is making their comeback on the Honda bizjet. The engines are mounted on the sensitive top side of the wings rather then the more un-sensitive underside, it should require quite some research and testing to get right for marginal gains and would bring engine daily maintenance headaches especially in winter on slippery wings.

  2. Thank you for this terrific multi-part explanatory text. By your insight and analysis, are the OEMs looking to start a migration toward a “flying wing” planform, possibly with root extensions and/or a more articulated wing-box as a first step? Or are there other reduced drag planforms currently being studied?

    • LERXs primarily provide value in high angle of attack situations, i.e. dogfighting. They could improve takeoff and landing performance, but their complete absence on modern commercial and B-Jet designs (for which short/low speed takeoff/landing has a much higher functional value than for a 737) would indicate that the benefits do not exceed the costs in this use case.

      BWBs suffer from two main problems within the commercial aircraft industry: Fear and Manufacturability.

      The Fear component is based on perceived technical and certification risk combined with fear of customer acceptance issues (as I have stated before if people will put up with a 17*28 middle seat in a 3-5-3 747 for 8 hour flights, they will quickly accept a video screen with a selection of exterior camera views in lieu of windows)

      The Manufacturability component is tied to both the difficulty of making a structurally efficient non-cylindrical pressure vessel (i.e. potentially a lot of extra weight) and the fact that you can’t just do a simple stretch to add capacity, there is a much more involved in adding another 10 rows of seats.

      however, we have largely squeezed all the juice in terms of structural and aero efficiency out of Tube and Wing designs that can be got with the latest gen of 787/a350, yes, the next gen of CF tech might get us another 5% in weight, but in order to get the kind of efficiency step change that will be needed going forwards, new forms will need to be explored.

      The logical place for this to start is the B-Jet segment. This is a market where the Prestige value of new and different designs as well as the functional value of a dramatically different passenger space configuration could really drive the value proposition.

      Imagine a BWB G7…

    • Root extensions are actually drag hogs, they are only used on military aircraft to generate a last gasp of lift and controlability at very high angles of attack in exchange for a lot of drag. So not something that you would like to use on a civil airliner, there you try to suppress as many vortices as you can. The intensive work right now is around more laminar flow. Today you have laminar flow (which reduces skin friction drag significantly) for the first 5% or max 10%of a wing or engine nacelle cord.

      Where you can ie for non lifting surfaces like nacelle lips or winglets (737 MAX) you give it a shape that delays the transition from laminar to turbulent flow. There are also passive (787-9, -10) or active (Airbus research projects) boundary layer suction schemes (which once again causes a delay of laminar to turbulent flow) that are being explored. There is a lot to do here before using more exotic schemes.

  3. Exotic geometries (whereof eg the naturally unstable ones) were long considered unfit for commercial passenger transport … that may still prevail as ‘prudent’ today … yet, there are a number of naturally stable yet ‘unusual’ geometries which have been tested in military applications and which possibly could show benefits if re-applied to future commercial newbuilds, whereof eg to airfreighters, specially with ultrahigh MTOW and PAYLOAD in the mire, such as eg Prandtl’s BWB (best wingbox double-decker). Any chance Bjorn that your lectures may dwell somewhat incidently upon similar fancy designs ?

    • I doubt that, I am staying as mainstream as possible without fancy nomenclature and formulas to make it fun and easy to digest, there is quite enough to cover for just a vanilla airliner like the MAX 8 (its wingtips are not so vanilla, quite nice actually). Such discussions will have to be left for the many forums available.

  4. ” ( lb with an f added as we prefer to write it as this is a force and not a measure of mass. Mass we denote with just lb or the metric units kg or tonne = 2205 lb).”

    So…. many years ago in Physics class, our teacher, wishing to impress upon us jacket-and-tie-clad prep-schoolers the difference between “mass” and “weight” got all of us to stand-up, start jumping up and down, and shout:

    Mass is NOT Weight!
    Mass is NOT Weight!
    Mass is NOT Weight!

    Thankfully, the Physics classroom was down in the basement so our efforts did not disturb the rest of the school.

    With that intro, pounds are a unit of weight, not mass 🙂

    • However, the standard aircraft term for its maximum mass on take-off is MTOW, not MTOM. One of those many annoying times when everybody persists in being wrong just because it would be too hard to get people to change (much like using imperial/ british units – Newtons are the unit of force :-p)

      • Quite silly how in usual aviation speak, some things are measured in imperial units – e.g. thrust, altitude, distance, while others are measured in metric – e.g. aircraft weights, dimensions, fuel quantities.

    • Rick, I am indebted! Of course we should have used slug instead 🙂 . You are right, but even Wiki now says that lb stands for both mass and force:

      The slug (sl) is an Imperial unit of mass, (about 14.6 kg) similar to the kilogram.
      The pound (lb) is a unit of both mass and force, used mainly in the United States. (about 0.45 kg or 4.5 N) In scientific contexts where pound (force) and pound (mass) need to be distinguished, SI units are usually used instead.

      The imperial units are practical (everyone knows what a lb is) but the definition is crappy. As our most readers are from the US we will continue to use Imperial units mixed with SI units, everyone to their taste 🙂 .

      • I would love to see this whole article redone using nothing but archaic units of measure.

        having velocity represented in Furlongs per Fortnight
        fuel volume measured in Hogsheads
        aircraft dimensions in Smoots and Ears (or rods, chains, cubits and digits)
        weights in stones or slugs…

      • Slugs are too slimy 🙂 I say go with stone. Or we could get really arcane and go with curling stone 🙂 As an added bonus, curling stones come in a range rather than a single weight.

  5. Nice writeup, Bjorn. Very clear explanations, even though it skips some second and third order details. Linking up with you was a good find for Scott.
    – Retired Airplane Performance specialist, private pilot, and Managing Director of Boeing Product Development.

  6. Bjorn Fehrm can u explain how much heavier 777x can be as ecnomical as lighter a350-1000?..
    it dkes have a bigget wing i know…but same fomula failed for airbus…a350-800 vs 787s

    • The 777X is relying on its 6m extra wingspan to keep the drag to a level so that its more efficient engines (the GE9X is a 5 year later design then the TXWB 97k) and larger cabin can give it about the same fuel burn per seat as a lighter A350-1000. It only works per seat, on an aircraft level it consumes considerably more fuel. Better fill those seats as well.

  7. If the tube is flying at a slight upward angle, it would think air migrates up the side as it moves aft. How about some chines running under the doors to trap the air under the fuselage at least until the wings, and use the energy of pushing the fuselage through the air to help create more lift, similar to a downward winglet to more efficiently ride the high pressure cushion.

  8. Air is like water, it finds its ways around things. Any chine having any angle towards the air stream would generate a vortex and any vortex steals energy from the aircraft, i.e. increases fuel consumption. Civil airliners are trying to fly vortex free in contrast to fighters who are more interested in ultimate manouverabililty. The air streaming up the side of the aircraft means the fuselage is generating lift. This is not good as it is the worst possible lift generator on the aircraft, it has an extremly low aspect ratio ie a lot of induced drag.

    Variable camber wings are there to keep the fuselage from acting as a lifting device, i.e. being canted upwards to much when the aircraft needs a lot of lift early in the cruise (a lot of fuel makes it heavy then). Heavier aircraft means the wing needs more canting (higher alfa angle) which you only can do by canting the whole aircraft. A variable camber wing aircraft like 787 or A350 can produce the extra lift needed without canting the fuselage more (increase fuselage alfa angle), that is the point of the variable camber wings, to optimise the total aircraft’s lift versus drag.

    • Avoiding vortices makes sense. So the optimum angle for the fuselage is level, or a certain degree up? Seems like if it was level, the nose should be above center, with the windscreen below the nose. And the taper at the tail below center, but I guess that would interfere with rotation.

  9. Thanks for this post! I am wondering about the effect of cruise altitude on performance generally. I understand that parasitic drag decreases with altitude, but so does the speed of sound therefore ground speed at constant mach number. And yet airlines clearly value flying higher, are eager to climb on even short hops. Is there a shorthand reference for quantifying the relationship between flight level and trip fuel efficiency?

  10. Hi Bjorn,

    I have a question regarding your variable camber wing explanation. Does it really have to do with fuselage drag at higher AoA for higher weights, or is is it more to keep optimum cruise AoA of the wing itself for a range of weights? Or perhaps, both?

    Another thing – I have no experience with the 737, but I fly an A320, whose cruise performance is (I guess) very similar to the 737.

    To me, the weights you assume for a 1000 Nm mission seem slightly conservative.

    Of the top of my head – in my Bus, for payload of 14.7 tonnes and 2.5 hour flight I’d expect block fuel of ~10 tonnes and initial cruise weight of ~65 tonnes (assumuing ~2000kg burnoff during taxi/takeoff/climb). Also, I’d expect the initial cruise level at this weight to be FL360-370…

    I guess the MAX (and the NEO) should have markedly better figures – ie. less fuel required for the same mission and higher initial cruising level – so perhaps your calculations are slightly conservative?

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