Heat transfer rate between an underground concrete cistern and the soi 2

[Jerre]

I am building an underground water heat transfer system. In order to calculate the heat capacity of the system, I need the heat exchange rate (BTU's per hour) between the red clay soil(typical of North Georgia), which is at 56 degree F and the water in the concrete cistern. Help, please! Typical equation would be nice.

 

[Gianfranco]

Hi Jerre,

We assuming no ground water movements in soil (no seepage) and that the cistern is an horizontal cylinder of diameter Do and leght L, buried ad a depth = X + Do/2.

The heat flow from the cistern is:

Q = ( tw - ta ) / Rt

Where:

tw = temperature of water (average along the predefined time)
ta = temperature of surface (average along the time as tw). In first approssimation, if the time = 1 year, ta = temperature of freatic layer.
Rt = Rc + Rs = total equivalent thermal resistance water-surface
Rc = 1 / (hw * Ai) + (Do - Di)/(2 * kc * Amc)
Rs = 1/(2 * ks * L * 3.14)* ln (4 * X / Do) = calculated by a conduction shape factor you'll find out on Rohsenow-Hartnett, Handbook of Heat transfer or on others reference textbooks on heat transfer.
hw = f(Reynolds, Pr) or f(Grashof) depending on the water movement (natural o forced)
Ai = inner surface of the cistern
Ao = outer surface of the cistern
Amc = (Ao - Ai) / ln (Ao/Ai) = mean surface of the cistern
kc = thermal conductivity of the concrete of the cistern
ks = thermal conductivity of the soil depends on % water
L = lenght of the cistern
X = distance centre of the cistern and surface

The evaluation of hw could be easy if you know the Nusselt correlations.
Otherwise take into account the following values hw = 500 W/m2K (natural) hw = 1200 W/m2K (forced).

For ks search on the Web with the key words: thermal conductivity soil.
I've found out several useful sources of data, for instance:

http://www.geo.umnw.ethz.ch/staff/homepages/fuhrer/dipl/node10.html

and

http://newton.dep.anl.gov/askasci/env99/env140.htm

from which I excerped this interesting text:

<< The thermal conductivity of soils is mostly controlled by water content, although, obviously, the type of soil is also important. For an average clay, the thermal conductivity in W/(m K) is 0.25 for no soil moisture,
about 1.0 for 10% soil moisture (% by volume), 1.25 at 14% (a bend in the thermal conductivity curve occurs here), 1.67 at 30%, and about 2.0 at 50%. That will give you a rough idea - David R. Cook - Atmospheric Section - Environmental Research Division - Argonne National Laboratory>>

Hoping this is useful to you for a ... good calculation

Gianfranco.

 

[Jerre]

I'm sorry I didn't get back to you earlier, but I'm working with rectangular cisterns that are 3 1/2&quot; thick concrete and buried with 8 feet of soil. Your equation were helpful and made me research further to find the proper solution, which I am still working on.

Thanks Jerre