Model geyser problem

A friend once asked me if I could solve the following problem he claimed it would be worth a lot of money, anyway I thought about it and came up with this which might help. It is given here in detail as a typical engineering problem.

The solution is actually relatively simple perhaps the reader would like to attempt it for himself first. As the problem description, I give a model for a physical hot water (household) geyser.

I formulate the problem. We wish to calculate the temperature of the water content as a function of time.
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Given an initial total internal heat energy (at start temperature), consider an insulated metal tank filled with heated water. The water contained is considered as of uniform temperature, a linear function of total internal heat energy.

We describe three types of feasible distinct events that change the geyser's internal temperature. I took the following in consideration:

Firstly, the water contents can be heated by a temperature controlled electric element, next, a temperature discontinuity as energy loss in tapping a volume of water and replacing of content at a given time. These two procedures could be controlled either manually or automatically.

Thirdly, heat is lost from the content due to heat leakage lost to the surroundings. The last two depend directly on ambient temperature as well. We want to find the water temperature with a given combination of these "events" and "procedures".
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Note that in applications a built in self-correction mechanism with time is essential. More details of the mathematics may be left to the reader if desired.

Two scans will accompany this forum entry and I plan to submit my own solution soon as an AbcTales story as very simplified analysis, the same ideas as in my own solution may be  refined and may provide more reliable and realistic results.

For more sophisticated models thus one would need more advanced methods to solve, typically the Laplace Transform.