How automatic water feeder works

Few days ago, I was thinking how the automatic water feeder (dispenser) of pets and birds works. I searched over the internet and the answers seemed to be very traditional "air has to replace the water going out"!!!!.

The automatic water feeder is a simple mechanical device that is intended to continuously and automatically feed water for pets and birds without the need to refill it frequently. The innovation in this device that water feeding is done without flooding or over flow.

Simply, the automatic water feeder consists mainly of two elements: tank and sump. Referring to the schematic below, the tank has two properly sized holes at two different heights (extremely small holes will not allow air flow or water flow because of surface tension force).  The bottom hole is the water feeding (bleeding) hole and the top one is the air hole. The ratio between the volume of the tank and the volume of the sump decides how frequent the user has to refill the tank. The top hole level also decides the maximum water level in the sump. The lower hole level decides the wasted water volume inside the tank that can not be fed to the sump.

The same concept also exist in the domestic cold and hot water dispenser.

The tank used in most common automatic water feeder for birds is a lid which is filled with water, the sump is placed over it, and then the whole setup is flipped vertically to start working.

So, how this automatic water feeder works?

To answer this question I will explain this on the general physical model of this automatic water feeder as shown in the image above.

After filling the tank and flipping it over, air pressure inside the tank is atmospheric pressure. Water pressure at top and bottom holes is higher than the atmospheric pressure outside, so water flows from both holes but with higher rate from the lower hole (assuming that both holes have the same diameter). As a result, air vacuum increases inside the tank.

Modelling water as a rigid body, there will be a force acting on it because of pressure difference between its sides: a vacuum pressure from one side and atmospheric pressure from the other side. Outside air will push water surface (at hole 1)  inside till the applied force can overcome the surface tension of water, penetrate water, trapped in, and then move up because of buoyancy force until it joins the trapped air at the top of the tank resulting in decreasing the vacuum effect.

This surface tension describes why air is getting inside in an intermittent way in a form of large bubbles. As a result, water will flow out and so on. This will repeat over and over until water level is above the top hole center where a tiny area of the hole is not covered with water. At this instant, air pressure can not exert the enough force on this very tiny area to overcome surface tension and penetrate water and air and water flow will stop.

When outside water level is just covering the top hole, and although the top layer of the outside water still has atmospheric pressure, air can not get inside the tank because air pressure is applied on the whole water surface and water is an incompressible fluid. 

By consuming outside water by pets or birds or due to evaporation, and as long as the water level inside the tank is going down, the flow rate of water from the bottom hole is getting slower because the pressure difference across its sides is reduced.

In this post I tried to proof why water stops at the level of the top hole using mathematical method by studying different general water levels in the sump and assuming a static steady state where the water level is static and this -as a result- means that there is no water flow through both holes. By concept, water will not flow through hole when the pressures on its sides are the same (ignoring pressure drop across the hole). In this model, vapor pressure of water will be ignored.

First case: Outside water level below the bottom hole:
By applying the following equations and solving them simultaneously it has two physical solutions:

[A] Top and bottom holes are coincident (one hole). This is not our case because we have two holes.
[B] Impossible to happen if there are two holes at different levels.

Second case: Outside water level above the bottom hole:

By applying the following equations and solving them simultaneously it has two physical solutions:

[C] Water is at the level of the top hole.
[D] Impossible to happen

based on solutions [B] and [C], water will flow as long as it is below the top hole and it will stop flowing only when the water level reaches the top hole.

Third case: Outside water level above the top hole:

This case will only happen when extra water is added by the user above the top hole level. By applying the following equations and solving them it has only one physical solution:

[E] It is possible for all water levels above the top hole level as long as the water level is lower than the tip of the sump.

Fourth case: Inside water level below the top hole:

In this case water levels inside and outside will be same.

This theory of operation is only valid for relatively short tanks which are used in domestic and household applications and is impossible to apply for any tank with water level higher than approximately 10 meters (assuming zero absolute pressure above water inside the tank).

References and useful links:

https://physics.stackexchange.com/questions/88669/physics-of-the-inverted-bottle-dispenser

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