Captain Astounding's NIGHTCLUB

x5 : What is it good for?
An example from our book The Power
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x5 might seem like a pretty arcane and unfathomable beastie, but it has got some perfectly down to earth applications. We even find a use for it down here in the club basement, which is about the most unsophisticated place I can think of. It's to do with the big fan that pumps fresh air through the ventilation system to all parts of the club. Now if you enlarge one of these fans by a certain scale factor x then what sort of an increase in your electricity bill would you expect? Well knock me down with a feather and call me Nancy if it isn't a factor of x5. That's right, a whopping x * x * x * x * x. This is assuming that you keep the basic shape of the fan the same and that you don't alter the speed of rotation.

This means that if you simply doubled the size of the fan then the power required to drive it would go up by a factor of 32. And if you were to triple the size of your fan it would increase by a factor of 243!! (That's because 35 equals 3 * 3 * 3 * 3 * 3 which equals 243). If that's not astounding then I don't know what is.

It's difficult to come up with a clear explanation of this enormous increase in power requirement, but try this one:
For a start, when we scale up the size of the fan we're making it x times taller and x times wider and x times deeper. So the volume of air held in the fan is going to go up by a factor of x cubed. So I can probably convince you that x cubed times as much air will get flung out of the fan on every revolution and therefore the power required will go up by at least x cubed.

But there's another completely separate effect of enlarging the fan. This effect is caused by the fact that the tips of the fan blades are now x times further away from the centre of the fan. That means that every litre of air that leaves the fan is going x times as fast as in the original fan. And that means that the kinetic energy in each litre of air whizzing out is x squared times what it was before (see discussion of x squared). We end up with x squared times as much energy in x cubed times as much air so we have to put x3 * x2 = x * x * x * x * x = x5 times as much expensive energy in every second.

It makes no difference if we slow down the leaving air to the same speed as before. If we did that then every litre of outgoing air would get squashed up against the slower moving air in front of it and it's kinetic energy would get converted into pressure energy. So we'd have x cubed times the volume or air at x squared times the pressure which still needs x5 times as much power. Actually, some of that energy would probably end up as heat but we still can't win.