How Much Suction in a Vacuum Cleaner?

I sometimes mention what a good resource Wikipedia is, especially in the sciences; in fact, I contributed $20 to their recent fundraising campaign because I have benefited so much from them over the years. But today I noticed a glaring error in their article about vacuum cleaners. Oh, I know I'm supposed to correct it myself...that's the whole idea of wikipedia. But I have no idea how to do that, so I'm just going to tell you about it here.

(EDIT: What follows is a very good calculation based on a fundamental misconception of the type of motor used in a vacuum cleaner. Eventually I figured out the mistake and posted a link to the corrected calculation at the bottom of the discussions. But since then, this article continues to get more hits than any other blog page of mine, while the correction gets little attention. So I'm posting the link to the revised calculation here. But you should still enjoy the discussion below.)

They say the suction of a typical household vacuum cleaner is around 20 kPa, or 80 inches of water. That's an outrageous amount of suction! It comes to around twelve pounds on a typical vacuum hose. Like you could pick a gallon jug of milk with a vacuum cleaner? That's a lot of suction.

Here's how you do the calculation. First of all, every vacuum cleaner I've seen uses a centrifugal (squirrel-cage) blower. I think that's correct. Now, the pressure you get from such a blower is limited by the maximum speed of the blade tip, and that in turn is limited by centrifugal forces. It is a fascinating and little-known fact of material science that a solid disc of steel has a maximum peripheral speed of around 400 feet per second, beyond which the internal stresses exceed the tensile strength of steel. I actually know enough about these things to do the calculation, but I'm not going to do it today. I'm just going to use that 400 feet per second for my blower calculation.

(First a short digression: at 3600 rpm, you hit 400 feet per second with a rotor diameter of around 2 feet. That's a fundamental size limitation for things like big electric motors or generators, and in fact that limit is observed in practise, with some margin to spare.)

How do you convert this to air pressure? The governing equation is I think due to Bernoulli, and it is

Pressure = 1/2*rho*v^2

where rho is the gas pressure. I normally like to derive these things but today I just want to get the answer. The density of air is 1.3 kg/meter^3, and using 120 meters/sec for velocity, I get a pressure of 10 000 Pascals, or 10 kPa, which is just about half of what Wikipedia gives for the "typical" home vacuum cleaner. So what's my problem...isn't that pretty much in the ballpark?

The problem is: do you have a 2-foot diameter blower in your vacuum cleaner? I don't think so. Do you have four six-inch diameter blowers in cascaded in series? I've never seen cascaded blowers in a home vacuum cleaner. Do you have a 4-inch blower and a 20 000 rpm motor? I don't think so.

With a six-inch diameter blower at 3600 rpm (that's a huge blower), I make it around 2500 pascals, or about 10 inches of water. You can do some pretty cool stuff with 10 inches of water pressure, but 80 inches would be downright scary.