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An immersion heater works by supplying heat at a constant rate Q to a tank of...

An immersion heater works by supplying heat at a constant rate Q to a tank of water. At the same time, heat is lost to the surrouding medium at a rate proportional to the temperature difference between the water (assumed the same throughout the tank) and the ambient temperature (this is Newton's law of cooling). These assumptions imply:

du/dx = -ku + {Q, when the heater is on... 0, when the heater is off

where u(t) is the temperature of the water over ambient at time t. We want to know if it is cheaper to leave the heater on all the time or only heat the water when it is needed. Assume U is the desired water temperature (over ambient) and that the heater is off for a fixed time T and then switched back on until the water reheats to U.

Starting with temperature U and the heater off, find u(T).

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genius_generous answered 2 years ago
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