A heat treatment plant needs hot water at temperature (T_{c} pm Delta T_{c}) for the heat treatment

Question:

A heat treatment plant needs hot water at temperature \(T_{c} \pm \Delta T_{c}\) for the heat treatment process. A storage tank of volume \(V\) and surface area \(A\) is employed for this purpose. Whenever hot water is withdrawn from the tank, cold water at temperature \(T_{c}\) flows into the tank. A heater supplying energy at the rate of \(Q\) turns on whenever the temperature reaches \(T_{c}-\Delta T_{c}\) and turns off when it reaches \(T_{c}+\Delta T_{c}\). The heater is submerged inside water contained in the tank. Assume uniform temperature in the tank and a convective heat loss to the environment at the surface, with a heat transfer coefficient \(h\). Obtain a mathematical model for this system.

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