## Wednesday, May 20, 2015

### hummingbird feeder physics

I couldn't figure out how the hummingbird feeder worked. Why didn't it overflow?

Forces need to balance, of course. Neglecting surface tension, there liquid level is higher in the inner reservoir than in the feeding chamber, so there must be a corresponding pressure difference. Suppose the pressure in the inner chamber were zero. Then the column height difference would need to be atmospheric pressure / (density of liquid × gravity). But this is over 9 meters! Obviously the height difference is only approximately 1% of this. So the pressure difference inside versus outside is only approximately 1%. The inside is only slightly below atmosphere.

So air is getting in. How? Does it diffuse through the liquid? If this were the dominant mechanism, it wouldn't take long for the pressure inside to go from 99% to 99.3%, for example, which should be plenty to push the column of liquid down in the inside chamber and thus push liquid out through the holes. It would overflow.

I tried to move the feeder to clean it (which I soon decided would be too messy a proposition without losing a large fraction of the contents) and saw the answer. When I moved it a bubble went up through the inner chamber, increasing the volume of air there. Correspondingly, the liquid level in the feeding chamber must have risen to make room for the air in the inner chamber. Ah! So that's how air got up there.

The answer was then easy. The liquid level outside is regulated not by the pressure difference directly, but rather when the liquid level drops outside too low, it provides a channel for a bubble to flow up into the chamber, which then decreases the pressure difference versus outside, and the liquid level inside then drops, pushing liquid up in the feeding chamber, and the path for any further bubbles is closed off. Eventually birds drink the liquid, or else it evaporates, and the level drops again, then another bubble flows, then the level outside increases again. It's a tight feedback loop. The key to the design is that the air channel is shut off sufficiently before the level reaches the top of the feeding chamber.

How far is sufficiently? During the day the volume of air in the inner chamber changes, as approximated by the ideal gas law: P V = N kB T, where T is the absolute temperature, kB is Boltzmann's constant, P is the pressure, and V is the volume. In this case I can divide both sides by the cross-sectional area A, yielding V/A = H, the height of the air in the chamber. N/A is assumed constant (gas doesn't enter or leave), so P = [ (N/A) / H ] kB T. As the temperature increases, the height of liquid in the chamber drops, and that increases the height of liquid in the feeding chamber. Daily variations in temperature in San Francisco right now are very small, on order 1%, but even 5% variations wouldn't be unusual elsewhere. This corresponds to a few mm shift in the liquid level outside (it's a bit complicated since as the liquid shifts in response to the pressure difference the pressure difference is canceled). But it's clear there's more buffer than that.

Anyway, the key to the problem was the bubbles. These are similar to the bubbles which flow up through water coolers when the liquid is drained. If such a channel wasn't available, you'd get a little water out, then no more. In the case of the feeder you'd barely be able to drain much liquid at all before the flow was suppressed.