Wednesday, November 5, 2014

Fundamentals of airliner performance, Part 1

By Bjorn Fehrm

As part of our premium content we provide a briefer form of our airliner performance analysis than we provide to our consulting clients. As we present this material, we presume a lot of knowledge on the part of the reader on the definitions we use and how these are employed. We thought it would be appropriate to give an easy-to-digest clinic on some of these definitions and concepts that we are using. Aired at the same time when we run our analysis series, we thereby present the background to our different analysis steps and some of the key parameters that influence these.

Leeham logo with Copyright message compactWe will provide these articles as free content to make them available to a broader audience. To make them more interesting and easy to digest we refrain from using formulas as much as possible, instead we illustrate our findings with real values from a modern aircraft , for that we have chosen the most common of them all, the Boeing 737.

We will fly this aircraft in the latest MAX 8 version on a typical short haul mission of 2.5 block hours covering a distance of 1,000 nautical miles. Starting from the cruise we will explain the factors that determine the performance of the aircraft and how we can estimate their influence. As we present the real values for the performance for the aircraft, we can also give the background to the different characteristics that contribute to the overall efficiency of the aircraft.

We don’t pretend to present a course in aeronautics or aerodynamics; there are many excellent books and online sites on the subject. We will merely highlight some areas which we know are important and where our experience says that the exact definition of how things interact can be a bit unclear. If we can contribute to making it easier to interpret the results of our analysis, our clinics have served their purpose. At the same time it can be interesting to understand how the OEMs go about improving their aircrafts by improving the key parameters that influence their performance.

The use of Boeing 737 MAX 8 as our example aircraft is well-timed as we this week start a detailed analysis of the 737 MAX 8 versus the Airbus A320neo as part of our premium content analysis series. This follows on our coverage of the 737 MAX 9 and A321neo/neoLR as part of our articles on a Boeing 757 replacement.

Cruise

We start the performance clinics with the aircraft’s cruise segment. This is relatively easy to describe and a modern airliner spends about 90% of its mission time in cruise mode, therefore it has a major influence of the aircraft’s design and performance.

Lift

Figure 1. Dominant forces on our airliner during cruise. Source: Leeham Co.

Figure 1 shows our airliner, the MAX 8 during cruise. We assume that we have climbed after take off to our initial cruise altitude of around Flight Level 350. The aircraft is relatively light at around 154,000 lb or 70 tonnes take-off weight. We have a combined passenger and cargo payload of 32,400 lb or 14.7 tonnes. In addition we have fuel for our 1,000nm sector plus reserves. We stopped at FL 350 as our wings would not carry us further (we will cover why in Part 2). As we consume fuel, our weight decreases and we can start a cruise climb to flight level 370, the next filed flight level on our flight plan. On FL 370 we look at our cruise data at a mid-mission weight of around 64 tonnes or 141,000 lb.

Steady state cruise is a relatively easy state to analyze as we can assume that a number of things are constant, simplifying the forces at play:

–        The aircraft is flying at a constant altitude, making the lift force generated by the wing equal to the aircrafts weight (in a first approximation, we handle the horizontal tails -influence below). So if our 737 is at mid-cruise weight 141,000 lb/64 tonnes, we know that the lift force is also 141,000 lb.

–        The steady state cruise makes life simple for the analysis as we have no acceleration or deceleration of the aircraft neither in the direction of flight nor in elevation. This also means that the aircraft’s engines are set to generate the thrust to maintain speed but not more, i.e. the thrust of the two engines are the same as the total drag of the aircraft.

Stable aircraft

Looking carefully in Figure 1 one can see that the lift has its center a little behind the center of weight. This is deliberate; otherwise the aircraft would not be controllable in pitch with a simple horizontal tail set at a fixed incidence. The nose down tipping moment generated by the difference in centers for lift and weight is called the aircraft’s (static) stability margin.

Lift with downforce

Figure 2. Airliner with down-force of tail added. Source: Leeham Co.

To counter this tendency to dip the nose, the horizontal tail is actually working as a small wing flying up-side down; it generates a small down-force, see Figure 2. It thereby bends the aircraft nose up by means of the lever caused by distance to the center of weight. This force is in the order of 3,000-5,000 lb force (we will denote the force expressed in pounds as lbf for force and the mass pound as just lb) for the 737 when evenly loaded at cruise and therefore is often ignored when calculating the lift the aircraft has to generate (to counter the aircraft weight plus this down-force, our model does consider it). 

Drag

In our first section we wrote, “This also means that the aircraft’s engines are set to generate the thrust to maintain speed but not more, i.e. the thrust of the two engines are the same as the total drag of the aircraft.” This means: if we can find the cruise drag of the aircraft we will automatically get the engines’ thrust setting. As we will explain in Part 2 this will then give us our cruise fuel consumption.

With the above we can see that there are some things we know or can deduce pretty easily (like the weight of the aircraft at different points during flight) and that the problem of understanding how much fuel our aircraft consumes per nautical mile (or any other measure) boils down to understanding the drag of the aircraft in steady state cruise (in all other flight states as well but as cruise constitutes 90% of the problem and is well defined this is also where analytical models work best).

It is because of the central importance drag has to the understanding of an aircraft’s performance that a lot of analytical work was done during the 1930s all the way to the 1980s to develop an analytical understanding of the drag of different aircraft parts and shapes. The analytical values were then checked and complemented by wind tunnel and full scale flight tests. In recent year computational fluid dynamics modeling has become a natural second step after an analytical solution is available, it can then bring further accuracy and understanding, especially of the cases which are a bit more complex aerodynamically.

Analytical models like the one Leeham Co has developed are using this research combined with the powerful computing resources available in today’s personal computers. The real computations are a bit more involved than what we present here but they follow the same principles and we describe the major contributors that make up the results. To understand the drag of an airliner one normally divides it into two major classes:

  1. Drag independent of the lift need to keep the aircraft flying level
  2. Drag due to the lift needed to keep the cruise level.

Drag independent of lift

We start with the drag which is independent of lift as this is the major cause of drag at cruise. There is one dominant cause of this type of drag, the air’s friction against the aircraft’s surface. When the aircraft flies through the air at high speed, in this case M 0.78 or 450 knots, it creates friction between the aircrafts skin and the air molecules which surrounds the aircraft. The level of friction is decided by the aircraft’s size and the smoothness of its skin. In the case of our 737 MAX8, the friction forces are in the order of 3.700 lbf out of a total value for drag independent of lift of 5,000 lbf. The main contributing areas of the MAX is shown in Figure 3.

Wetted areas

Figure 3. Surface areas contributing to friction drag. Source: Leeham Co.

The last 1,200 lbf is caused by (among others) drag due to increased local airspeed (the air has to flow around the bulbous aircraft) , interference drag (the air is squeezed in certain areas), drag caused by the strong up-sweep of the aircraft’s aft body and many more factors causing small increases of drag (gaps, inlets, outlets, antennae…). In our analysis articles we call the drag which is independent of lift for Drag due to size as this is the dominant cause for this kind of drag.

Drag dependent of lift

For the wings to generate lift they need to throw air downwards, behind the aircraft, see Figure 4.

Downwash

Figure 4. Flowfield behind our 737 MAX 8 aircraft showing the effect of the lift generating downwash and how it circles back up at the outer part of the downwash. Source: Boeing paper and Leeham Co.

Newton’s third law then says the lift force is equal to mass of air moved downwards times the acceleration which is induced on the air molecules by the wing. This air movement causes lower pressure above the wing and higher below, something that the air tries to equalize by flowing up on the sides. This whole process is continuing long since the aircraft has passed as shown in Figure 5.

Downwash live

Figure 5. Downwash vortices from Business-jet aircraft. Source: Innopedia.

The whole process of throwing the air sliding off the wings’ trailing edge down to generate lift costs energy. When separated from the friction drag and other effects that we described before the drag part which is due to lift is called induced drag. For our 737 MAX 8 at mid cruise weight it is around 2,900 lbf.

There is another common way to explain lift by using the law of energy conservation and Bernoulli’s law. This is just looking at the same phenomenon from another angle; the physical change in the airs properties is the same.

Induced drag, or drag due to lift as we will call it, is dependent on how long our wingspan is (real or virtually as extended by wingtip devices) and our weight. Luckily induced drag decreases not as a function of wingspan but as a function of wingspan taken twice, the square of the span. This means that if we could increase the wingspan (real or virtual) by a factor 1.2 the reduction in drag would be 1.2*1.2 = 1.44 (as a real world example Boeings extension of the wingspan of the 777X with 11% over the 777-300ER resulted in a decrease of the induced drag with 23%).

There is also another drag component that one can add to the category drag due to lift, transonic drag. This normally small component is created when the air speeds up on the top side of the wing and patch wise enters supersonic speed. The retardation down to subsonic speed on the aft part of the wing is critical. It is important that this happens with a weak transition region, called a shock, otherwise the attached flow of the wing can be disturbed and the drag caused by the supersonic regions rises abruptly. On a modern wing (a so called supercritical wing) the transonic drag is small. For our MAX 8, it is around 100 lbf at this stage of the flight.

Total drag

Figure 5 shows the total drag which is has to be compensated for by the engines at mid-cruise on our trip with the 737 MAX 8 flown at Flight level 370.

Drag components

Figure 5. The total drag and its components. Source: Leeham Co.

We see that our total drag is around 7900 lbf which shall be compared to generated lift of 141,000 lbf. One often compares these forces by dividing the lift with the drag which gives us the lift over drag value (L/D) of 17.9. This value is a little bit to low because we have not included the lift the wing has to generate to counter our tail down-force. When we include it we get 18.4 which is a good value for a short haul aircraft. In the next Part we will look at what this all means for the engines and their fuel consumption. We will also look at what effect the cruise altitude has on drag and therefore fuel consumption and what factors decides on what Flight level we start our cruise and how it is subsequently flown as we burn of fuel during the flight.

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