PROJECT 1: How fast can my car go (theoretically)?

The object of this project is to “predict” the top speed of a vehicle constructed according to a certain design, the design in fact being the typical design of a modern automobile. However I want you to be in the position of an inventor proposing a new concept (as the automobile was 120 years ago), who has not built any prototypes or anything, and is trying to get someone to invest in his crazy idea. In order to be convincing, you have to be able to answer some questions about how your new invention will perform.

As I described in class, the essential part of the design is an engine consisting of cylinders and pistons, which execute a 4-stroke cycle (sometimes called `Otto’ in honor of that pioneer of internal combustion engines). Without going into detail again, the 4 strokes are the “intake” when fuel-air mix is sucked into the cylinder (that’s the one whose name I forgot in class), “compression” when the piston compresses the mixture, “power” when the fuel is ignited and the hot burned gases push the piston down, and “exhaust” when the piston pushes the cold burned gases out of the cylinder in preparation for another intake.

The key to the model as I presented it is to relate both the power dissipated in air friction and the power produced in the engine to the velocity of the car. The power dissipated in air friction is pretty easy to write down a formula for, i.e. . (You can assume the drag coefficient equals 1 for your car.) The power produced by an engine is a little more complicated, because the process of converting it from chemical to mechanical is less direct. But it follows the argument I outlined on 20 January, which gives the power as 2 . The number of cylinders is easy to find, and the engine revolution rate can be related to the velocity through the size of the wheels and the transmission ratio. But the energy release still requires some thinking.

The way to approach the energy release from combustion is to figure out how many moles of pentane (the main ingredient of gasoline) can react with a cylinderful of air, or more specifically with the oxygen within a cylinderful of air. It’s pretty easy to find the volume of a single cylinder; it’s the total engine volume (in cubic inches or centimeters, or liters) divided by the number of cylinders. The mass of air in the cylinder is the volume times the density of air (which we’ll assume is at atmospheric conditions when it is taken into the cylinder). The mass of oxygen is approximately 30% of the air mass; I would suggest you invent a symbol for this percentage because it might be useful for investigating the effect of oxygen-enriched air mixtures for specialized engines. Then the number of moles of oxygen in the form of molecules is the mass divided by the molecular weight of , which is 32. This is not really a variable, but I suggest you use a symbol for it so that we remember where it comes from in the final formulas.

So you have now figured out how many moles of are in a single cylinder. But the energy release is known in terms of moles of . So we need to figure out how many moles of react with a mole of , which is a problem from chemistry. Actually it’s a problem in Chemistry 101A which you have all done before, though it might be some time since you last set up chemical balance reactions. You start by recalling that when you burn a compound consisting of carbon and hydrogen in oxygen, you produce mostly carbon dioxide and water . Our imaginary car produces only these combustion products, and a little bit of thinking indicates that the reaction balance is . Therefore the number of moles of is 1/8 the number of moles of . FINALLY, then, we can figure the energy release in Joules by multiplying the number of moles of by the magic number 3509 kilojoules per mole.

YIKES! The calculation of the energy released in one cylinder combustion seems incredibly complicated. But it’s really just multiplying a bunch of numerical factors like density of air, 30%, 1/8, 1/32, 3509, and so on. All you have to do is remember them all! You can make this easier by giving them units such as grams per mole of , grams of per gram of air, and so on. You know you have the right combination when all the units cancel and you’re left with a quantity in units of joules (energy) only.

Ultimately you’ll find that the power produced by the engine is proportional to the velocity of the car, with a complicated but constant factor in front of it. Then it’s basically simple to set this equal to the power dissipated by air friction and solve for the velocity. One solution is V=0, but that probably does not represent the car’s maximum speed!

I’d like you to express this calculation in terms of symbols representing the various numerical quantities, so that you can identify the effect of possibly changing some of the parameters. In particular I’d like you to notice where the factor of the transmission ratio is. I suspect it suggests that the car actually goes faster in a lower gear, which is somewhat non-intuitive. This might suggest at least one shortcoming of this model as a representation of a real car, though it is probably not the easiest to fix.

When you come to your second model of the car, one way to make it more realistic is to include tire friction, with a coefficient of friction that you can measure indirectly. This can be done by getting your car to a certain speed and then cutting the engine and coasting to a stop. The distance coasted is related to the coefficient of friction.

Another improvement of the model is to realize that not all the heat produced by combustion is converted into mechanical work during the power stroke of the engine. If you assume the hot gas expands adiabatically (and remember your Ideal Gas Laws from physics) you can relate the fraction of thermal energy converted to work to the compression ratio of the piston in the cylinder, which is another design parameter of the engine. If you can’t find it specifically for your car, it’s probably 9 to 1 or thereabouts.

These improvements of your first model are also based on properties of the imaginary car that are only approximations to a real vehicle. But they do reflect increasing similarity between the real and imaginary situations.

By the way, some of you have already completed a very simple model of the car, in which your imaginary car has an engine that produces a given amount of power. This is indeed a legitimate mathematical model, and has the significant advantage that if you already know how much power an engine will produce you can come up with a very good estimate of a vehicle’s top speed. (If you did this, please include it as Model 0, then go on to models 1 and 2).

However such a model is unable to say anything about the effects of design parameters (number and sizes of cylinders, type of fuel, transmission ratio, wheel diameter) on the top speed, which is some of the most valuable information a company that actually makes cars could get from a mathematical modeler. In fact, if you think about it, the “power” of an engine is not a precisely-defined quantity in the same way that a cylinder volume or a heat of combustion is. One of the advantages of a good model is that the parameters that specify the properties of the model correspond to well-defined quantities in the real system.

Although our simplistic models are unlikely to give a very precise estimate, the kind of thinking we went through to get them and analyze them is the kind of thinking this course is intended to get you into.