Additional Math Pages & Resources

Thursday, January 6, 2011

Math Reasoning, Part II: Jaguar C-X75

Recently I saw and fell in love with the Jaguar C-X75 concept car. It's gorgeous. Take a look:

This high-performance hybrid vehicle uses one electric motor per wheel for propulsion. It has a pair of tiny gas turbines that turn generators to recharge the batteries or power the motors directly. The C-X75 accelerates from 0-62 mph in 3.4 seconds and can reach a top speed of 205 mph, says Jaguar.

What caught my eye was a number in the brochure: 35,000 liters per minute. That's right, the twin turbines consume 35,000 liters of air per minute at top output. Hence it has huge air intakes on its lower flanks, just in front of the rear wheels.

I wonder, Is that more or less air than a conventional high-performance gasoline-engined car consumes?

Let's see if elementary math can help us solve this (quite complicated) question. This is a multiple-step, higher-order, critical-thinking exercise and there are many ways to find a solution. Here's the process that makes sense to me.

I know a gasoline engine burns fuel most efficiently at a ratio of 14.7 parts of air to 1 part of fuel, by mass (weight). This is called the stoichiometric ratio. Look it up if you want to know more. I knew this from my previous career in automotive diagnostics.

There are the gas turbines!

Now we can start assembling our data.

FUEL weighs about 750 grams/liter  and  AIR weighs 1.3 grams/liter. These are standard numbers.

750 ÷ 1.3 = 576  which means FUEL is 576 times heavier than the same volume of AIR or to put it another way, we will need a LOT MORE AIR by volume than we do fuel (to have the same mass of both of them). Another way to say it is Air is less dense than gasoline which we all know. Now for some more math.

576 x 14.7 stoichiometric air/fuel ratio = 8467.2 or to simplify things, let's round this to 8500. A conventional car will burn 8500 liters of air per liter of fuel. Any car, at any speed.

So how much fuel will we burn in a minute, or to be more precise, how many liters of fuel do we consume at 200 mph?

Stay tuned; we'll finish this tomorrow. In the meantime, would you care to take the wheel?