Saturday, December 10, 2011

Jet Airliners, Tractor Pulls, and Math

Recently a friend of mine made a post about the AF-447 crash and some of the human factors involved. One of the things he noted was that the thrust levers were not designed to indicate, by their position, the current thrust setting of the engines they controlled. At first glance, this seems an odd decision, but there is indeed a method to the madness.

Before we dive into that method, we first need to take a little detour through the deliciously redneck world of competitive tractor pulling. Yes, you read that right, tractor pulling...

A number of years ago, apparently long before the invention of PVRs, I was surfing the channels and happened upon The Nashville Network showing a tractor pull event. Given there wasn't much else on at the time, I decided to watch it.

Now, for most people this would be an opportunity to take off their belt, crack open a bud lite, and kindly ask their lovely wife what's taking so long with making them a goddamn sandwich. For me, it was an opportunity for a little mathematical contemplation.

I never claimed I was normal.

You see, one of the drivers was a bit new to the game. It was only his second or third pull or something, I don't exactly remember the details. I do remember that it was a turbocharged diesel tractor, though. This is important.

Being somewhat inexperienced, this driver unfortunately had not yet mastered the art of the launch. As a result, instead of the usual high-pitched scream of the turbos winding up, followed by the roar of the engine spinning the tires with the might of two thousand or so horses, the audience was treated to a plume of black smoke followed by the tractor and sled edging forward about 10 feet before the driver bogged down and killed the run.

I should mention at this point that a diesel engine is an incredibly simple machine; there's very few things that could go wrong with it. There's no throttle plate, it essentially operates wide-open all the time, with the only meaningful control being the amount of fuel injected directly into the cylinder.

This, of course, leads us to the limiting factor in any diesel engine. Theoretically we could inject as much fuel as we like into the cylinder, but at a certain point we use up all the available oxygen and the excess fuel ends up only partially combusted, the remainder being ejected out the tail pipe as black smoke. The simple solution is to add more air, by way of forced injection either through a supercharger or, much more commonly, a turbocharger.

So what went wrong with this driver's run? Well, it all has to do with the other part of the turbo-diesel equation: the turbo charger. Now I don't off hand know how much power it takes to spin a turbo charger of that size to full operating pressure, but we can at least make a reasonable comparison: the supercharger on a drag racer, at full power, takes on the order of 400-500 horsepower to spin. We can use that as a rough estimate for how much power it takes to spin the turbo on this tractor.

Unfortunately, without the turbo spinning, the engine on the tractor has no way of generating the 400-500 horsepower needed to spin the turbo at full speed.

Thus we come to the chicken and egg problem: At idle, there's nowhere near enough power being generated by the engine for the turbo to spin up. Even revving up the engine to its redline (which would be on the order of 3000rpm) would barely be enough to put a noticeable dent in the turbo's speed. Hence the driver has to somehow come up with a way to make the engine generate a rather significant amount of power while the tractor itself is just sitting stationary on the starting line.

The way this is done is, in description, fairly simple. In practice, as the aforementioned unfortunate driver discovered, it's a bit tricky to do. If you just

Like any manual transmission vehicle, the first step is to apply a bit of throttle to bring the engine up off idle. Next, the clutch is partially engaged and more throttle is applied to keep the engine at its operating RPM. This balancing act is continued, with the clutch slowly being engaged and the throttle being increased until the engine is generating enough power for the turbo charger to spin up. Of course, at the same time, the clutch is soaking up massive amounts of heat, so you do want to be quick about it.

As the turbo starts to spin up, the power ceiling of the engine starts going up, and its the driver's job to follow this ceiling as quickly as possible, increasing the throttle while engaging the clutch to keep the engine at its peak power RPM. If done right, the tractor roars to life, the turbo screams at 50,000 RPM, and the massive rear wheels start spinning through the dirt, dragging the sled down the track.

If done wrong, well, you just end up with a bunch of black smoke and a stalled tractor about 10 feet from where you started.

Now at this point, the mathematically inclined among you might have recognized something. This system has all the hallmarks of being able to be represented as a system of differential equations, and you'd be right. Unfortunately, or perhaps fortunately depending on your point of view, I'm not that good with differential equations so I'll be skipping the rigorous details.

What's important is that we recognize the intuition behind the coupling of engine power and turbo speed. For the engine to produce more power, the turbo has to spin faster, and for the turbo to spin faster the engine must produce more power.

This sort of system has some desirable stable states. For example, the engine running at low power and the turbo running at low speed; a basic idling state. Alternately, the engine running at high power and the turbo running at high speed, a good state to be in when you want to pull a heavy sled down a track.

It's at this point where I'd like to bring jet engines back into the picture, because the differences between the two are actually quite subtle. A standard jet engine that you might find on any airliner has two sets of turbines operating coaxially: an inner high pressure turbine and an outer low pressure turbine, which surround the combustion chamber. The low pressure turbine is where power is extracted to drive the large fan at the front of the engine which provides the majority of forward thrust, and which provides a moderate amount of intake air compression for the engine, while the high pressure turbine is most analogous to the turbo charger in our turbo-diesel tractor: it provides the majority of the boost needed to compress enough air into the combustion chamber to burn the fuel. Additionally, in both cases the only meaningful input for the engine is the amount of fuel injected into the combustion chamber.

Much like our diesel tractor, throttling up a jet engine is a tricky affair. If we were to rapidly increase the amount of fuel injected into the engine while the high pressure turbine is operating at low speed, the increase in combustion chamber pressure would quickly overwhelm the intake compressor's ability to force air into the engine, resulting in an immediate reduction of fuel combustion. This reduction would allow the compressor to overcome the now reduced combustion chamber pressure and the engine would oscillate in this state until it either self destructs, or the throttle is reduced. This phenomenon is known as compressor surge, but in terms of differential equations it's merely an oscillating solution.

A similar sort of surging oscillation can be found in our diesel tractor if the turbo speed and engine RPM become mismatched while the engine is at a high power setting: Suppose that the engine is running at high speed, and the power is momentarily interrupted, perhaps due to a short blip of the throttle. This causes the turbo to start slowing down as the engine power output is increasing, which, a moment later, results in the turbo spinning up while the engine power is decreasing, a perfect oscillation that will continue until throttle input is reduced.

So how do we avoid these situations? Quite simply, by limiting the rate of throttle changes. In a traditional airliner, much like the turbo diesel tractor, this is left up to the skill and experience of the pilot. However, pilots may have other, more important things to worry about than manually keeping track of the throttle input relative to multiple engine performance parameters, so this leads us to looking for an automated solution.

Thus, along comes Airbus with fly-by-wire throttle levers. The pilot need only move the levers to the desired throttle level, and the computer takes care of monitoring the compressor RPM, combustion pressure and so on to ensure both a rapid and safe throttle response.

However, this leads us to an interface problem: If the pilot is setting the input, in this case the throttle levers, to the desired engine throttle level and not the actual engine throttle level, then where should the levers be when he removes his hand? On the one hand, one might argue that the levers should remain in the position the pilot set them, but this would not represent the actual throttle level input being delivered to the engine, especially once systems like autothrottle are taken into consideration. The other option is to have the throttle levers reset to a neutral position, so as not to present the pilots with a possibly misleading indication of actual engine throttle input. This of course has the downside that the levers no longer provide a glanceable indication of either current or desired engine performance.

I think it's quite likely that the throttle indication in the A330 and similar planes could be improved. However, I'm neither a pilot nor an aircraft engineer, so I'll leave it up to those more knowledgeable to determine what solution, if any, is needed.

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