From the ground...down (part 3)

Some people might think that we’ve already talked about this topic enough, but we know better. 

In the last two posts, we talked about how temperature fluctuations at the surface of the ground might affect the underground temperature, and specifically, how deeply those fluctuations might penetrate.

So, now imagine that there might be an application where it wasn’t the surface temperature nor the ground temperature that mattered, but the difference between the two.   (This actually is important for certain applications that propose to use that temperature difference to generate small amounts of electricity).

Looking at our temperature distribution equation:

you can see that there would be a depth where the ground temperature fluctuations are shifted 180 degrees out of phase from the surface fluctuations:

or

which, returning to a period of 24 hours and soil properties of α =0.1 mm^2/s results in a depth of about 16.5 cm where the temperature fluctuations stay 180 degrees out of phase with the surface temperature fluctuations.  Unfortunately, at that depth, the damping is so extreme (4% of the fluctuations are left at that depth) that the temperature is almost constant.  

So, the question naturally arises: Is there some intermediate depth where the phase shift isn’t so great, but the fluctuations are greater that would give us a maximum value for the temperature difference?  The answer is “yes”, and with a little calculus, we can find that the maximum temperature difference moves around with time according to this equation:

where the value of "n" can take on integral values like -1, 0, 1, etc.   

This figure shows the temperature distribution for three different times along with the point (red dot) where the difference between the surface temperature and the local ground temperature is greatest for those three times. (We’ve gone back to assuming an average temperature of 10 deg C, and fluctuations of +/- 5 deg C.)

It can be shown with a little more calculus that the square of the difference between the surface temperature and the underground temperature is greatest, on average, at

This point is important for some applications that are of interest for generating power from a ground-air temperature difference and is shown with a dashed blue line for the numbers used in the plot.

J.W. Stevens, 2003, "Optimal Placement Depth for Air-Ground Heat Transfer Systems," Applied Thermal Engineering, Vol. 24, pp. 149-157.