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I got an egg boiler machine, on the instructions is stated:
My thermodynamics understanding cannot figure this out yet. Why would I need less water for more eggs?
The machine beeps when the water has evaporated, so the eggs are ready. In that case, why will less water cook more eggs?
Presumably, the rate of the steam escaping the cooker depends on the "resistance" of the steam path: from the opening in the bottom, where the steam enters the dome, to the opening on the side of the dome, from where the steam escapes.
The more eggs in the cooker, the narrower the path, the slower the flow. Also, as relatively slowly moving steam makes contact with more eggs, it is more likely to condense and make its way back to the water at the bottom of the cooker, which further reduces its escape rate.
So, with more eggs in the cooker, a smaller amount of water will last about as long as a greater amount of water with fewer eggs, resulting in a similar degree of cooking.
In boiling eggs, we note that each egg displaces a certain amount of water volume in the pot.
To cook eggs, we want them to be immersed in boiling water- immersed, but no more, because the excess water does not assist in the boiling process.
Why is this so? Because the water serves only as a heat transfer medium which completely surrounds the egg, and conveys thermal energy from the bottom of the pot to the eggs at a convenient temperature- that of boiling water.
This means that 1) we want the eggs to be just immersed in the water- but no more, 2) the more eggs we put in the pot, the less water is required to just immerse them- at least up to the limit of having the bottom of the pot completely populated with eggs.
The reason is, that your egg boiler recycles the water a couple of times.
You say that the process is finished when the water is gone, so you are talking about an egg boiler that cooks the eggs with steam. In these egg boilers you have a shallow, heated pan at the bottom into which you fill the water. Then the egg boiler boils this water at a constant rate (the heat input is constant, as the pan will remain at 100°C for the entire time and is electrically heated). So a constant amount of water is boiled off per second.
Now, above the water you have the eggs. And here comes the difference: If you put a single egg in, it has a certain surface on which the water condenses. This condensing is slower than the steam production, the additional steam just escapes through the hole in the top of the cover. The more eggs you put into the steam, the larger their surface, and the more water will condense on them. This condensed water then drips down back into the pan, and is evaporated another time.
As the steam production remains constant when you increase the number of eggs while the water condensation grows, less steam escapes through the hole in the cover. Thus, the water loss is slower with more eggs.
Now, to get well-cooked eggs, you need to heat them for a specific time. And because more eggs keep the water inside the boiler longer, you must reduce the starting amount of water to get the same cooking time. That's it: You reduce the amount of water to keep the cooking time constant.
I confess, I simplified a bit: I ignored the condensation on the inside of the cover itself. I did this, because that condensation is independent of the number of eggs. The condensation on the cover simply serves to quickly heat up the cover to some temperature slightly below 100°C, and then just supplies the heat necessary to keep it at that temperature.
Egg boiling machines are designed to cook eggs that have only a small part submerged in the water.
To ensure that the same volume is submerged regardless of the number of eggs, you need to add more water as you reduce the number of eggs.
This is because of displacement rather than thermodynamics. Fewer eggs displace less water, so more water is required.
I propose that the answer is found in the equation:
$$ PV=nRT $$
This sounds like an egg steamer, as many others have noted. As such, what's cooking the egg is the heat transfer from the steam under pressure. It's not apparent whether this is a sealed pressure cooker or not (probably not, given the lack of warnings); but it certainly creates some pressure.
Having more eggs in the container means that you have a smaller interior volume, and thus a higher pressure, or higher temperature for the same pressure, so less water is required.
