Calculations of heat transfer in a tank system

In summary, the conversation discussed a problem involving heat transfer between two tanks of water connected by a circulating tube. The formula for calculating the hydrostatic pressure difference causing circulation was mentioned, as well as the assumption of laminar flow and the use of Poiseille's law to determine friction. The characteristic time for temperature change was also mentioned, and it was noted that the approach of assuming uniform temperature would still result in accurate predictions for the problem. However, it was pointed out that in reality, the temperature distribution in the tanks would not be uniform and would affect the rate of heat transfer. The challenge of accurately modeling this problem due to the turbulent effects of convection was also brought up.
  • #1
petterg
162
7
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?
 
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  • #2
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 [itex](\rho_c-\rho_h)gh[/itex]

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 [itex]\tau[/itex] of the temperature change: [tex]T_h(t)=\frac{T_{h0}+T_{c0}}{2}+\frac{T_{h0}-T_{c0}}{2}e^{-t/\tau}[/tex]

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.
 
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  • #3
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
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
This problem is impossible since convection occurs in water. Such turbulent effect cannot be calculated (not even numerically, let alone analytically).
 
  • #6
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.
 

1. What is heat transfer and why is it important in a tank system?

Heat transfer refers to the movement of thermal energy from one object to another. In a tank system, it is important to understand the rate and amount of heat transfer in order to maintain the desired temperature and ensure efficient operation.

2. How do you calculate the heat transfer coefficient in a tank system?

The heat transfer coefficient can be calculated using the equation: h = Q/(A * ΔT), where h is the heat transfer coefficient, Q is the heat transferred, A is the surface area, and ΔT is the temperature difference between the two objects.

3. What factors can affect heat transfer in a tank system?

Some factors that can affect heat transfer in a tank system are the temperature difference between the two objects, the material and thickness of the tank walls, the type of fluid being heated or cooled, and the flow rate of the fluid.

4. How do you determine the amount of heat transferred in a tank system?

The amount of heat transferred in a tank system can be determined using the equation: Q = m * Cp * ΔT, where Q is the heat transferred, m is the mass of the fluid, Cp is the specific heat capacity of the fluid, and ΔT is the change in temperature.

5. What are some methods to improve heat transfer in a tank system?

Some methods to improve heat transfer in a tank system include increasing the surface area of the tank, using a more conductive material for the tank walls, and increasing the flow rate of the fluid. Additionally, properly insulating the tank can help to minimize heat loss and maximize heat transfer efficiency.

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