Starting with a pot of cold tap water, I want to cook a hard-boiled egg using the minimum amount of energy. Is it more energy efficient to bring a pot to boil first and then put the egg in it, or to put the egg in the pot of cold water first and let it heat up with the water?
I know the answer :).
The most energy efficient way to get a hard boiled egg is to have a pot with a good cover on an electric range.
1)cover the bottom of pot with 1mm water, put eggs in and pot on the range and and turn it on to the maximum.
2)when the cover starts popping, turn off heat completely, leave it on the range, for the residual heat.
3)wait 3 minutes for very soft, 4 to 6 for medium and 8 or more for hard.
Pot with good cover means the cover is not popping once the heat is off.
"Energy efficient" usually means "with minimal production of entropy". For that you need, 1) a well insulated (on the sides) pot, 2) minimal loss from heating element, 3) the smallest possible (and practical) amount of water amount that can boil an egg (i.e. conver its whole surface) and 4) put the egg at the beginning of the process (not drop it once the water boils because that generates extra entropy because of the temperature gradient).
Pressure cooking can be more efficient because it needs less time of the heating element. But since the temperature reaches more that ~100°C that can give a different cooking process, and not qualify as a "boiled egg" because of changes in taste and consistency. http://www.hippressurecooking.com/2011/04/hip-modernist-soft-medium-and-hard.html
My cookbook says an egg should simmer in water for about 10 minutes. If you wait till the water simmers and then add the egg, the water will start boiling again in a few seconds, then needs to simmer for the 10 minutes. The heat of vaporization loss takes place for about 10 minutes. If the egg is placed in cold water first, it'll cook in fewer than 10 minutes of simmering as the pot heats up to boiling, so the heat of vaporization loss will be less than 10 minutes.
A cooked egg has a lower free energy at room temperature and pressure than an uncooked egg (see this link, protein denaturation is complicated, but generally exothermic at room temperature). So boiling an egg does not require energy, it releases energy.
So it is most energy efficient to put the egg and water in a closed container, and use a reversible process to heat the water to boiling temperature. Then use the same pump in reverse to remove the heat. This cycle will net you a little bit of energy when all is said and done.
One simple way to do this is have the water touching a gas through a metal wall in an insulated container, so that the heat can flow, but the wall can sustain a large pressure difference. Then compress the gas slowly to raise the temperature of the gas and water to boiling temperature (this requires an enormous room-sized air-vapor container). If you do it slowly enough, the egg will be cooked when you reach 100 degrees, then you slowly relax the piston back to its starting position, recovering all the energy you put in, and a little more, due to the heat released by the egg as it was cooking.
This is a theoretical answer. The right answer for green purposes is that residental energy consumption is not a significant enough fraction of the total energy consumption for these measures to have any impact. So this type of conservation doesn't do anything, except keep yourself and others aware. To do something real, drive less and encourage energy efficient processes at work.