From the ground...down (part 2)

Last post we talked about the idea of a “penetration depth” to describe how deeply a fluctuating surface temperature might affect the temperature of the ground below.  

 Before we go on to talk more about the ground, we should point out that that kind of calculation isn’t limited to just temperature distribution in the ground.  It can be used to approximate the penetration depth of many situations where the surface temperature fluctuates periodically.  For example, if you wanted to estimate how deeply the temperature fluctuations penetrate into the wall of an engine cylinder, you might use properties for steel (α =17.7 mm^2/s) and the frequency of the periodic temperature variation of the cylinder (if the engine was running at 2000 rpm, you’d have 1000 temperature cycles per minute, or a frequency of 105 rad/s).  With those numbers, and using our definition of 2.2% fluctuation for the penetration from the last post, you could calculate a penetration depth of 2.2 mm. 

 It is important to remember that this is all based on a sinusoidal fluctuation in the surface temperature, which surely isn’t true in the case of the engine, but ignoring that allows you to make a ballpark estimate in a really simple way.  So, you shouldn’t get too fixated on the temperature fluctuations penetrating exactly 2.2 mm, but you can be pretty sure that they are penetrating more than, say, 1  mm, and they are probably pretty well damped out at say, 6 mm regardless of what the temperature profile at the surface looks like. 

Also, there is nothing magical about defining the penetration depth as the location where fluctuations are 2.2% of the total—we might have defined it as 1%, or 5%, or 10% depending on our mood, or what we were trying to accomplish with the definition.

So, now let’s go back to the frozen pipe problem from the previous post.  This figure shows the average monthly temperature of International Falls, Minnesota plotted with blue dots. 

I just took those temperatures from a random weather site on the internet.  Then, I made up the green line which is a sinusoid with an average value of 2.7 deg C, and fluctuations around that average of +/-17.6 deg C.  

 Notice a couple of things: first, you can see that for the  annual variation in the monthly average temperature,  the sinusoidal curve fits pretty well, so our assumption of that kind of variation at the ground surface probably isn’t too bad on a yearly basis.  Second,  International Falls gets REALLY cold.  I’m guessing that the people who live there must be tough.

However, what we’d like to know for the purposes of thinking about frozen pipes is where the temperature fluctuations are damped enough so that the lowest temperatures never dip below zero, as shown by the dotted purple line in this figure:

A little arithmetic shows that damping the full fluctuation to about 15% will work, and we can use this term again:

to determine that the depth should be about 6 feet.  This is assuming α =0.1 mm^2/s, and the frequency is based on a year.  Of course, given our discussion in the last post about the effects of variations in properties of the dirt, and thinking about cold snaps that get smoothed away by long-term monthly averages, we’d probably want to bury our pipe somewhat deeper than 6 feet to be safe, but this calculation gave us a starting point with hardly any work at all.