# Calculations of heat transfer in a tank system

1. ### petterg

117
Hi guys

This is my first post here. I studied physics about 15 years ago and some weeks ago I realized that I'd forgotten most of it. So these days I'm playing around with physics just for the fun of it. So, I got to the subject of heat transfer, and though I would like to learn some more than what's in my old books. So I thought of the following problem:

Imagine you have a tank of hot water and a tank of cold water.
There is a tube going into the bottom of the hot tank, coming out on the top of the tank, then it goes into the top of the cold water tank, coming out at the bottom of the tank and then back to the bottom of the hot water tank. In this tube water is circulating.

The water circulating in the tube is moving heat to the top of the cold water tank, while it's also "moving cold" from the bottom of the cold water tank to the bottom of the hot water tank.

Now, what I'm curious about is: How can the temperature at any given point at any given time in the tube (or the tanks) be calculated?

2. ### xts

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–Poiseuille_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.

Last edited: Sep 20, 2011
3. ### petterg

117
Thank you, XTS

I can see how this formula works. But I would guess that the temperature at the top and bottom of the tanks will be very different. So the approach with assuming uniform temperature seems a bit too optimistic. How would this look if it also should count for the not uniform temperatures?

4. ### xts

882
Frankly - I am too lazy to make calculations for asymmetric setup.
But if:
- both tanks have identical dimensions;
- both tanks have uniform width (vertical cyllinder, cuboid, rather than cone);
- the mixing in both are the same;
the results are independent of temperature distribution in any of them - what matters is only average temperature in the tank. You'll get the same results for all spectrum of models: from uniform temperature to sharp thermocline (hot water above, cold below the thermocline). Of course, the reality is somewhere in between, but the rate of heat transfer is not dependent on actual distribution of the temperature, unless it is symmetric between tanks.

5. ### Curl

757
This problem is impossible since convection occurs in water. Such turbulent effect cannot be calculated (not even numerically, let alone analytically).

6. ### xts

882
As long as the heat transfer is slow (and water flow in the pipe is also slow) - and the tanks are symmetric, convection do not interfere much.

Although I made no exact calculations, I am brave enough to bet a bottle of good wine that predictions of such model will be acurate to 5% if you connect two 1m^3 cubic tanks one containing 80C, other 5C water with 0.5" copper pipe.