# Heat Transfer problem

## Main Question or Discussion Point

Hi,
I am currently faced with a problem. I am trying to design a cooling system that uses Geo-Thermal energy to cool water. From my research I have found that 12 feet below the earth in the month of August the temperature is maintained at approximately 62-65 degrees Fahrenheit.

I would like to utilize this natural energy to cool water. Right now I am trying to determine the best type of material that can be placed 12 feet below the surface of the earth to maximize heat dissipation from the water into the soil.

My goal is to come up with the right material to achieve timely cooling (I am thinking of using PVC its thermal conduction is low but on the upside it does not rot or corrode).

Secondly I would like to come up with a mathematical equation for the system, so that I can tell exactly what the temperature of the water would be at a given time (here is where I am having difficulties)

At the moment I am aware of 2 formulas

1) Newtons law of cooling: T(t) = Ta + (To - Ta)*e^(-kt)

2) Conduction Law: Q = [(K)(A)(Tw-Tc)(t)]/L

Where K = thermal conductance of material; units : J/(s m degree Celsius)
A = Cross sectional Area; units : m^2
Tw = Warm temperature
Tc = Cool temperature
t = time in seconds
L = length; units: m

The first formula will give me the temperature at a given time however I would need to determine the time constant "k" for the material that I am using (not sure how to to that unless I do a physical test first to find values for T(t), Ta, To and t then solve for k).

The seconds formula gives me the amount of heat transferred in Joules (how does heat transferred in joules correlate to temperature transferred?)

I would appreciate some advice because I feel I am not quite understanding the dynamics of how to approach this problem or even if I am using the right formulas to do so.

Thanks
J