Sunday, January 27, 2013

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.


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Marty Green said...

Thanks, Kiwi. Nice website you've got. You know, I started out in Window Cleaning...did it right through university, and still take the odd job even now.But your van is nicer than mine. And your helpers are better looking.

Mark Tannner said...

Thanks for the good article. I think there may be one type-o in the last example. With a six-inch diameter blower at 3600 rpm my calculations give 2.07 inches of water. To get a pressure of 10 inches of water you would need 7907 rpm. PS - I am also assuming a density of air of 1.25 kg/m3.

Marty Green said...

Well, let's see what happens. 3600 rpm is 377 radians per second. With a 15 cm diameter (I'm going metric in anticipation of the density of air etc.) that is close to....ouch! I should be taking the radius! That's where I went wrong.

Okay, it's 30 meters per second tip velocity, and using 1/2*rho*v-squared, it's around 500 Pascals, which is fifty kilograms per meter squared, which is like you say, about two inches of water. Good call on your part.

Marty Green said...

On re-reading my article and thinking it over, I realize I did something a little more stupid after all. I actually calculated the correct pressure for a two-foot-diameter blower, and then instead of taking the pressure as quadratic in the diameter, I took it as linear! I'm pretty sure that I was thinking by analogy with v-squared-over-r, the equation for centrifugal force, where the quadratic factor in velocity is partially compensated by a linear factor on radius on the bottom, making the overall function linear. So for constant angular velocity the force is linear with the radius. But that same reasoning doesn't work for pressure. So I kind of out-smarted myself.

Anonymous said...

HI Marty,

Thanks for the info.

But many companies (not only wikipedia) are claiming 20KPa for their products.
LOGIK L16VRP12 Cylinder Bagless Vacuum Cleaner claims 26 KPa. Are they lying?


Marty Green said...

Wow. I did not know that. Let's see if we can get to the bottom of this...I'm going to put up a new blogpost to flag this question. Thanks for the info. But I still say that's an insane amount of pressure.


Darrell Milton said...

I work in the industrial vacuum and compressor market (that's not vacuum cleaners, that's vacuum pumps for industrial suction applications).

A few months ago based on a discussion I was having with a customer to get them to understand flow versus pressure I put one of my vacuum gauges on a domestic vacuum cleaner and it pulled -22kPa. If a domestic vacuum cleaner had a hose with a greater seal then it might get closer to that -26kPa as previously mentioned, but with leakage, that might have to start with a -30kPa ability and lose some pressure through leakage to achieve that.

Marty Green said...

Okay, that's a lot more than I can justify by calculation. Is that a central vac system or just a regular vacuum cleaner? Because I'm assuming its a centrifugal blower of 6 inch diameter going at 3600 rpm, and I don't get anywhere close to that kind of pressure. What kind of blower system do you think they use?

Marty Green said...

OK everyone...I was wrong! It really is 20 kPa...Wikipedia was right. I've set the record straight in my latest blogpost, which you can read here:

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Andrew Judd said...

It is easy to test if the pressure created by a vacuum cleaner is 80" of water. Just get a thin clear plastic tube that goes from a bucket of water to the next floor up and seal it to the vacuum cleaner. Typical ceilings are 8 feet or 96" I would have though you would easy lift that height with a modern vacuum cleaner,

Andrew Judd said...

Actually the vacuum cleaner does not lift anything. The vacuum cleaner just lowers the pressure and then it is the higher pressure everywhere else that does the work - so you could lift humungous amounts of water with a single vacuum cleaner if you construct a very strong container over sufficient surface area of water

Anonymous said...

All cheap household vacuum cleaners use a universal motor, so lot more than 3600rpm.
But that change nothing in your calculation.
Thanks for the article.

Aneeq Raheem said...

Hi, I need to be able to lift a biscuit of size 5cm and 15g from a table. suction inlet would be 7cm in diameter.height of suction inlet would be 3cm high from the biscuit. I need to calculate the suction power to lift the biscuit
Aneeq Raheem

Aneeq Raheem said...

Hi, I need to be able to lift a biscuit of size 5cm and 15g from a table. suction inlet would be 7cm in diameter.height of suction inlet would be 3cm high from the biscuit. I need to calculate the suction power to lift the biscuit
Aneeq Raheem

larry finley said...

good article. I am planning to build a saltwater distillation unit, and since water boils at a lower temperature with lower pressure, it will be economical to lower the pressure. But since boiling water, or steam creates pressure, I will probably need more volume than suction to keep the process going. Can a suction hose pick up a gallon jug of water? How would I know. I sweep the floor before vacuuming.