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        riddles >> medium >> Bottle + Faucet =
        
(Message started by: James Fingas on Apr 15^th, 2003, 6:02am)

Title: Bottle + Faucet =
Post by James Fingas on Apr 15^th, 2003, 6:02am

I performed an interesting experiment this morning. I was washing out my lunch bottle (it's one of those plastic spring water bottles that I'm reusing to bring drinks to work) with hot water. When I press the bottle up against the faucet, I get this interesting sound: ffffft .... fffft ... ffft .. fft .. fft. The bottle pressurizes a bit and then the excess pressure escapes. As the bottle fills, the escaping air bursts get shorter and closer together. Today, I decided to see how the spacing between bursts of air was dependent on how much force I apply holding the bottle on the faucet.

So I emptied the bottle, and pushed it hard against the faucet. It filled up about half way, and then let out a single burst of air. However, I noticed that as the bottle got more full, it had filled more slowly. I realized that I was pressurizing the bottle to near the pressure of the water tank in the basement (we have well water). So I emptied the bottle again, and held it tightly agains the faucet. It turns out that the bottle neck is just large enough to completely cover the faucet end if you force it on. Now I turned on the water.

Safety note: DON'T try this if you have a town water supply, or have your pressure turned way up. The bottle might just explode!

The bottom portion of the bottle filled normally, but as the bottle neared half-full, I could see a noticeable slowdown in the fill rate. The water flow slowed to a trickle at about three-quarters full, and then a final few drops of hot water dripped into the bottle. I turned off the kitchen tap. The bottle was now pressurized to the same pressure as the water tank. My puzzle for you is: what happened when I pulled the bottle off, any why?

Title: Re: Bottle + Faucet =
Post by aero_guy on Apr 15^th, 2003, 3:30pm

It seems that something interesting must have happened, but I don't see it. Seems like you should get both the faucet and the bottle depressurizing by a ffft as soon as you release the pressure holding it on. My guess is that the water being hot had something to do with the actual answer though.

Maybe it wouldn't come off because you had expended the faucet due to the heat of the water?

Title: Re: Bottle + Faucet =
Post by cho on Apr 15^th, 2003, 3:40pm

If the pressure was high enough, I suppose the air suddenly coming out might result in enough turbulence to stir up the water and send a whole lot of water in your face. Were you hoping we'd try it out ourselves and get the same result?

Title: Re: Bottle + Faucet =
Post by SWF on Apr 15^th, 2003, 5:55pm

The gas pressure forced the water out and the resulting thrust propelled it like a water rocket. It bounced around the kitchen, unfortunately destroying many "Formulas of Ramanujan" commemorative plates which were on display. Or that would be my guess if you had held the bottle upside down before releasing pressure.

Due to the internal pressure, the bottle probably increased in volume--maybe even enough that there was more water inside than it holds with no pressure, so it overflowed?

Title: Re: Bottle + Faucet =
Post by LZJ on Apr 17^th, 2003, 1:21am

Hot water...you got scalded?

Assuming that the bottle was the right side up when you pulled it out, I should think that the pressurized water spurted out of the bottle, but after that initial spurt, the bottle was still full.

Title: Re: Bottle + Faucet =
Post by aero_guy on Apr 17^th, 2003, 7:58am

It was only 3/4 full with water when he had finished filling it, anyways it shouldn't spurt much as water is generally incompressible, unless the bottle had been stretched by the pressure.

Title: Re: Bottle + Faucet =
Post by James Fingas on Apr 17^th, 2003, 1:56pm

I'm sure the bottle did get a little stretched being under the pressure that it was, but I don't think that has much to do with the answer. A lot more energy was stored in the gases under pressure than was stored in the bottle's elastic (or not-so-elastic) expansion.

The result was both visible and audible, but I (luckily) didn't get hurt, and neither did my Ming dynasty vases ...

Title: Re: Bottle + Faucet =
Post by LZJ on Apr 17^th, 2003, 5:40pm

3/4 full only? I guess I missed that part. :-[

Title: Re: Bottle + Faucet =
Post by maryl on Apr 18^th, 2003, 2:32pm

My guess is it boomeranged around the room, maybe even keeping the water in it until it dropped.

Title: Re: Bottle + Faucet =
Post by James Fingas on Apr 29^th, 2003, 1:55pm

It looks like nobody is going to guess what I saw. So I'll tell you what I saw and see if you can explain it. But first, for the people who think the bottle rocketed around the room:

Consider the speed that the bottle could obtain if all stored energy (pressure*volume) was converted to kinetic energy. First, how much energy is stored? We consider the force of the air coming out acting through a distance. How much force is there?

Force = Pressure*Area = 3 atm (roughly) * 2 cm² (roughly)
= 303 kPa * 2 * 10^-4m²
= 60 N

Over what distance is this force exerted? A total of roughly 375 mL of gas escapes (it's a 500 mL bottle), so we get a distance of 375 cm³/2 cm² = 188 cm = 1.9 m. Now we multiply, remembering that the force decreases linearly from 60 N to zero with distance:

Energy = 1/2 * Force * distance
= 0.5 * 60 N * 1.9 m
= 57 J

Now how fast could this energy make the bottle go? There is about 375 mL of water in the bottle.

Energy = 1/2 * mass * velocity²
velocity = sqrt( 2 * energy / mass )
= sqrt( 2 * 57 J / 0.375 kg )
= sqrt( 400 m² / s² )
= 17 m/s (faster than you're supposed to drive your car in town)

This is pretty darn fast. But before you say "haha! I told you so", think about Newton's second law. Where is the reaction mass? For the first 1cm or so, the bottle is still stuck on the faucet, but after that, the only reaction mass is the expelled air. Air weighs about 1g/L, so we have about a 0.375g reaction mass we're pushing in the other direction. By conservation of momentum, the speed of the air is therefore 375g/0.375g = 1000 times as large as the speed of the bottle of water. Our new energy balance is therefore:

Energy = 1/2 * mass of water * (velocity of water)² + 1/2 * mass of air * (velocity of air)²

Writing in terms of the water's mass and velocity:

Energy = 1/2 * mass * velocity² + 1/2 * 1/1000 * mass * (1000 * velocity)²
= 1/2 * (1 + 1000) * mass * velocity²

velocity = sqrt( 2 * Energy / mass / 1001 )
= 0.6 m/s (a slow walking pace)

The energy it acquires before it gets off the faucet end is about 60 N * 1 cm = 0.6 J, for a velocity of about 1.8 m/s. This is more significant. The total velocity is therefore at most 2.4 m/s (assuming no friction coming off the end of the faucet). This is a moderate jogging pace.

So it doesn't really rocket around the room. It has somewhat of a kick to it, but nothing uncontrollable. This analysis predicts that there will be a very fast jet of escaping air. I did observe a a short powerful spray of air and small water droplets (and no, it didn't make a mess). Unlike the prediction, there was no sonic boom, indicating some fluid flow restriction making the escaping air speed less than the predicted 600 m/s.

But what I observed after that was a very fine mist inside the bottle. As I moved the bottle around, the mist would leak out the top somewhat, but no new mist was being generated. How, when, and why did this mist form?

Title: Re: Bottle + Faucet =
Post by aero_guy on Apr 30^th, 2003, 1:25pm

HaHa, I was right! A little pfft and a small kick! As to the mist, you hac high pressure air with hot water below and falling through it. The humidity probably reached near 100%. When the pressure was released there is a decrease in temperature (without any checking on this I believe this is how it works) and as lower temperature air can hold less moisture than higher temperature air, some of it condensed in the form of mist. I could be wrong on one or two of these points, but don't really feel like checking.

Title: Re: Bottle + Faucet =
Post by James Fingas on Apr 30^th, 2003, 2:31pm

aero_guy,

The concept of air "holding" water can be misleading. Consider two same-sized containers at the same temperature, one at atmospheric pressure (zero percent humidity), and one with a complete vacuum. Put a glass of water in them, and wait. After equilibrium has been reached, the same amount of water will have evaporated.

A better concept is "equilibrium vapour pressure", in which we simply state that at a given temperature, liquid water is in equilibrium with a given partial pressure of water vapour. The prescence of air doesn't affect this one way or another.

And I'm disappointed that you don't have any math ... that's no fun ;)

Title: Re: Bottle + Faucet =
Post by Nick on Jul 17^th, 2003, 4:42pm

About your warning, I've hydrostatically tested 2 liter bottles. They rupture at an average at 150psi. The smaller 1 liter bottles are higher. So, I doubt that you would have to worry about a bottle rupturing. Plus, being how you can't pressurize water, if the bottle were to rupture, only a small amount of water would leak out before the pressure is equalized. Now, If you pressurize it with air only, there's a very impressive bang. We do that at home, around holidays. It's comparable to a 10 Guage shotgun. What's even more fun is cutting a hole inside of a watermellon, and putting a 1 liter bottle inside. (We made a custom adapter just for this reason.) When the bottle explodes, it sends watermellon rind over 100 feet away. (The insides kinda just liquify) I wouldn't really worry about the bottle on the faucet though.

Title: Re: Bottle + Faucet =
Post by aero_guy on Jul 22^nd, 2003, 7:09am

"...Plus, being how you can't pressurize water..."

Actually water is (for all intents and purposes) incompressible, but you can easily pressurize water. Think about the change in pressure as you descend in the ocean, not to mention that this is the reason hydraulics work. You are right that if you use only water there will be a very depressing little pop when you depressurize it, not because there was no change in pressure, but because there was very little change in volume.

Title: Re: Bottle + Faucet =
Post by Nick on Jul 22^nd, 2003, 5:15pm

Ah, so I never took physics in high school. I still atleast had a concept of what was going on. And I looked into it more, Ideal water pressure for your house is around 80psi. But generally is around 40 to 60. This is to avoid bursting the compression fittings.