Why do we still interpolate in performance tables?

Question

I'm getting my Private Pilot License. In the coursework, we learn about flight planning, which requires looking at performance charts for your particular aircraft. These charts in my case are tables like Cruise Performance or Climb Performance, which, for a given Pressure Altitude and temperature, will give you RPM, Fuel Burn, and TAS.

The tables in my case (Cessna 162) have rows for Pressure Altitude equal to 0, 2000, 4000, etc. So if you're dealing with a Pressure Altitude of 3000 ft, you have to interpolate the values from the 2000 and 4000 rows.

My question is, these values are clearly coming from an equation of some kind that accounts for temperature, Pressure Altitude, maybe some other things. Why not give me, the operator of the plane, the actual equation so that I can plug in the exact values and get the exact answer, rather than having to pick the closest calculated value and then do a bunch of small calculations every time I want to plan a flight?

I'm a student, so I'm happy to be told I'm just supposed to learn it this way for historical sake in case I ever need to fly an old airplane, and that ForeFlight basically does what I want in real life. But I was curious if there was some benefit to keeping that information in tables that I'm just not thinking of.

Answer 1

I was in the technical publishing business (flight and maintenance manuals) in another life. Tables are used as an alternative to graphical plot presentations in flight manuals, and make it a little easier to access information, say, just sitting with the book in your lap. With the graphical plot, you have to unfold the sheet and have it laid out flat so you can work across the page, maybe needing things like straight edges to help locate data points relative to the margin scales on a graph. Mathematic formulae are not normally presented because then that would require pilots to do math, and most pilots go catatonic when forced to do math.

Breaking the information into tables shortens the process of sifting the data you need compared to a graphical plot, but in order to save publishing space, the increments of the data presentation are limited to fairly large increments. A table could be presented that included every unit of information, but to do so might require the table to extend across 20 pages instead of 2. Limiting the table to, say, 10 unit increments, cuts way back on the space it uses, but it requires you to interpolate, or make an educated guess, on the precise unit within that 10 unit block to use. When tables are used like that it's because the interpolated "guess" is accurate enough for the job.

The most extreme example of this might be found in Quick Reference Handbooks used by flight crews for procedures and performance data. The original information for some performance parameter will always be in the Airplane Flight Manual, in the form of a multi-element graphical chart (the root certification document basically), but you normally don't have that handy in the cockpit and if you did you couldn't really use it anyway; you have the QRH.

You are typically using the QRH sitting in the cockpit with a little map light to see, and a graphical plot format just won't work. The QRH condenses the data down to tables, making it much faster to work out the data you need, and when using the tables, interpolating values between listed increments is accurate enough to do the job.

So it's perfectly fine to see a table that gives a result of 10, 20, 30, and 40 say, and you make a guess that the value you need is between 10 and 20, and 17 "looks about right". The idea is that if the true value is 16 or 18, it makes little material difference in the real world in that instance.

It reminds me of when the use of electronic flight calculators to replace circular slide rules became popular in the 80s. Everybody I knew was buying those new fangled flight calculators, but I never bothered because I was cheap, and because the super precise result it gave was not something you could achieve in normal flying anyway. The Jeppesen circular slide rule may only be accurate to a knot or degree, whereas the electronic calculator goes down to decimal places, but nobody can actually fly with that precision and the slide rule is good enough if you're not trying to navigate to the Moon. Plus, if you get good with the Jepps circular slide rule, you can hold it and do solutions with one hand, getting a useable result in way less time than the calculator took with all the key punching.

Using a table in place of a precise formula or a graphical plot is kind of like that.

Answer 2

For most of the information in those charts there is not a single formula which would cover all aspects. The graphical way is the simplest and has other advantages, too:

1. With a formula it is easy to use a wrong number without noticing. On the graph you see where neighbouring values lead to so you can easily check whether your result is reasonably correct.
2. Some relations are non-linear, so you would need several formulae, each for its own range of valid input data. The chart simply adjusts the gradients of the curves - in most cases you will not even notice.
3. Aviation is extremely conservative. If a method has worked before, changing it means an additional risk for little reward. Reading performance values from charts works well enough that the gradual improvement of a formula (if it exists at all!) will not be worth it.

Answer 3

The equations, if they exist, are often rather non-linear and not enjoyable to calculate on the fly.

I do not have an equation for the RPM at a given pressure altitude for a Cessna 162, but if I may use a different equation as a concrete example: pressure correction for given altitude: $$P(h)=P_0 e^{-\frac{Mg}{RT}h}$$ That equation is the Barometic formula. It ignores all sorts of real effects, but its a pretty good little equation.$M$, $g$, $R$, and $T$ are constants, so we can reduce it to a more managable $P(h)=P_0 e^{-kh}$.

I don't know about you, but doing exponentials is tricky for me. I can't do them intuitively in my head, and its hard to confirm that I typed them correctly into the calculator. But I can do linear interpolation, or check my calculator if I have it do the linear interpolation.

And, frankly, that's one of the easier equations that show up. The equations you mention speak to the product of very complicated models (or just empirical data that has been sampled).

/Warning - calculus ahead!/

If you are interested, you can actually compute the maximum error caused by this linear interpolation. Mathematics has equations for this. Practically speaking, all of the equations you are interested have a 2nd derivative everywhere, so you can use Rolle's Theorem to find the error on this linear interpolation: $$|R_T| \le\frac{(x_1-x_0)^2}{8} max_{x_0\le x\le x_1}[F^{\prime\prime}(x)]$$ where you are looking from altitude $x_0$ to $x_1$ and $F^{\prime\prime}$ is the 2nd derivative of the function being interpolated or an upper bound on it. You don't have $F$ (which is the equation you are asking for in the question), but you can estimate it with finite differentiation, such as $F^{\prime\prime}(x_1)\approx \frac{F(x_0) + 2F(x_1) + F(x_2)}{x_1 - x_0}$ and then put a healthy margin on it.

This is a lot of work. I don't really expect you to do it. But it does provide a useful comparison. If one isn't willing to work out a set of error estimates while sitting on the ground, how willing is one to compute the actual equations in the air?

And if you do work out the error estimates, you can now peg them against uncertainties, such as how well can you actually calculate Pressure Altitude given the instruments you have on board?

Answer 4

As has already been pointed out, a lot of the data are not results of a (single) formula. Besides (non-linear) differential equations, even a lot of the algebraic correlations can't be solved analytically. These will need to be solved numerically (like x + ln(x) = 0 ..)

A lot of them are empirical equations themselves, meaning that the equations are not based on physical models, but merely an approximation trying to fit measured values, which is not very different to interpolation. e.g.. Darcy-Friction Factors (https://en.wikipedia.org/wiki/Darcy_friction_factor_formulae)

Some are even both - purely empirical and not analytically solvable: (for amusement.. scroll down all the way to "Table of Colebrook equation approximations")

Then there are mixed forms, like correlations derived by dimensional analysis ("pi theorem"), and the coefficients coming from empirical measurents.

Some examples:

Some data are likely the results of design process calculations and a lot of validation measurements:

• For fuel consumption at different altitudes and temperatures, I strongly assume that your Cessna 162 has been desiged for certain performance values, and then someone at Cessna has just flown at these conditions and actually measured, and that's the data points you have in the table (plus some data smoothing and adding a bit of margin)

Other data are purely empirical:

• which octane number do you need to avoid knock? How does fuel-air mixture ignition depend on octane number?
• what is the pressure in the tyre? (for reference - how did you determine the pressure in the tyre of your bike? car?)

Bottom line

Interpolating in a data table is good enough in accuracy (likely not worse than the other methods), but easier and less error-prone - hence it's done in the cockpit

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Additional note: The other methods lend themselves to predictions during the design phase, while the data tables are static and limited to known values within the operating boundaries once the design is fixed. I wouldn't use those data tables, if you intended to .. change the wingspan?