A moment ago, I was emptying bottles filled with water (2 liters) that are on the terrace of my house. As I did so I remembered something I saw on TV a some time ago (I don't remember when or where or exactly what) but I made a move on the horizontal plane as if the bottle drew a circle several times.
To my surprise, after several laps, a vortex was formed and the bottle did empty in half the time than those that are simply put upside down.
I repeated the experiment 20 times to be sure, and always got the same results.
- Bottle upside down: 20 seconds. (approx)
- Bottle upside down and rotation to form a vortex: 10 seconds (approx)
Clearly the vortex allows air to enter the bottle faster than in the other case.
I would like to understand what happens in physical terms.
EDIT:
Another interesting fact is that in the first 6/7 seconds just the first liter is gone. And in just two seconds (after the vortex is formed) the remaining water goes fast, very fast.
Answer
When water leaves the bottle, the pressure above it drops. This reduces the net force pushing the water out of the opening, until it stops and a bubble can rise up. When the bubble has left the mouth of the bottle, the water can start flowing again. The stop-start of the water, and the reduced pressure inside the bottle, contribute to the lower flow rate in the bubbling case. We can actually estimate the difference in efficiency.
Case 1: vortex.
Simplifying assumptions: - the water comes down half the aperture and the air comes up the other half. - Air pressure is maintained inside the bottle at atmospheric pressure. - The tangential water flow velocity was generated by swirling the bottle initially and we only concern ourselves with vertical velocity.
For water height in the bottle of $h$, the vertical velocity is given (from conservation of energy) by
$$v = \sqrt{2gh}$$
And the mass flow rate $M = \rho v \frac{A}{2}$ where $\rho$ is the density and $A$ the full area of the bottle opening.
Case 2: bubbles.
When the bottle is bubbling the water will keep stopping and starting - but when it flows it has the whole aperture available. But since the water needs to accelerate, then decelerate (as the pressure above the water drops, the velocity goes back down to zero) we can see that the mean velocity will be quite a bit lower. In fact, if the time between bubbles is $T$ and we assume that the pressure above the water is lowered by the weight of the column of water below it when the next bubble is formed, we can write an approximate expression for the net pressure:
$$P_{net} = \rho g h (1 - \frac{t}{T})$$
On average, this is half the pressure experienced by the free flowing water. And using this mean pressure, the water needs to first accelerate, then decelerate. This means that the average velocity of the water will be half what it would otherwise be (if it didn't keep stopping and starting), and half again since the mean pressure difference is halved. This gets a mean flow velocity that is $\frac14$ of the free flowing value - but we have twice the aperture. The net result is that the flow with bubbles is about half as fast as with the vortex. Which, coincidentally, is exactly what you observed.
Note that the above uses many simplifying assumptions - but the basic mechanics I described is plausible. If anyone has a more complete mathematical description I invite them to offer it up.
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