# Calculations of heat transfer in a tank system

by petterg
Tags: heat equation
 P: 882 As the simplest approach you may assume that water in your pipe heats instantly after entering hot tank and cools instantly after entering cold tank, and both cold and hot tanks has uniform temperatures. So now you may calculate the hydrostatic pressure difference causing circulation: it is $(\rho_c-\rho_h)gh$ Then (if your pipes are not too wide) you may assume that the flow is laminar - so you may determine the friction using Poiseille' law: http://en.wikipedia.org/wiki/Hagen%E...uille_equation. So now you have flow of water, thus amount of heat transferred in a unit of time from hot to cold tank, thus change of temperature over time, thus, finally, characteristic time $\tau$ of the temperature change: $$T_h(t)=\frac{T_{h0}+T_{c0}}{2}+\frac{T_{h0}-T_{c0}}{2}e^{-t/\tau}$$ Actually, if you assume that in both tanks you have some cold water at the bottom and some hot at the top rather than uniform temperature - the results will be the same - just ending with two tanks half filled with cold water and half with hot (I am not 100% sure, my intuition says me so - but you may want to check it making calculations only a bit more complicated than those sketched above, assuming that the water in pipe changes temperature immediately after passing the boundary between cold and hot water in a tank...) In reality you'll have something in between of those models. But I have no idea if it is possible (I rather doubt) to model the temperature distribution inside each tank without going to deep details about its geometry - and then numerically (I can't believe it may be possible to do analytically) modelling internal circulation and heat diffusion.