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. 

From the ground...down

If you’ve ever lived in a house with a basement, or spent time in a cave, you are probably aware that the temperature underground remains fairly constant regardless of what the air temperature is doing.  Many times the basement is the warmest part of the house in the winter, and the coolest in the summer.  Old time root cellars took advantage of this uniform temperature before air conditioners and central heat were common. 

  

Obviously, fluctuations in the air temperature will penetrate some distance into the ground.  How deep will they go?  While this question is fascinating all by itself, it has some practical implications, too.

Thermal Explosions

While the term thermal explosion sounds very dramatic, it really just refers to a thermal event where the heat release (or absorption) occurs over a time period that is very small relative to the time scale of interest, and in a volume that is negligibly small compared to the surroundings in which the temperature distribution is to be calculated.  So, for example, the heat released by the combustion of  blasting powder in a small hole (time scale of milli-seconds) could be analyzed as a thermal explosion if the time scale of interest was on the order of seconds.  Also the decay heat generated by a pocket of radioactive rock (say, over 100 years) could be analyzed as a thermal explosion if the period of interest were, say, 10,000 years.

Hot (or Cold) to the Touch

At one time or another we’ve all experienced the sensation of feeling a different temperature depending on the material that we are touching.  Try it now—touch a nearby piece of metal with your fingertip, then, with another finger, touch a piece of wood or cloth.  Most likely the metal felt cooler to the touch than the wood did.  

How could this be if both objects have been sitting in the room long enough to come to thermal equilibrium with the surroundings? 

Approximate vs. Exact Solutions in Heat Transfer (part 2)

In the last post, we showed an approximate integral solution and an exact analytical solution to the same heat transfer problem: an isothermal semi-infinite solid subjected to a step change in temperature at the surface.

Exact solutions are very useful to have because they provide the true solution to the given problem.  Unfortunately, out of all possible problems, exact solutions tend to be relatively rare and also tend be limited to fairly simple situations, like this one.

Approximate vs. Exact Solutions in Heat Transfer

In analytical heat transfer, there are a variety of techniques to determine both exact and approximate solutions to problems.  To one way of thinking, all numerical solutions are approximate, since the exact differential equations have been discretized to enable an approximate solution via a system of algebraic equations.

Both exact and approximate solutions can be useful.  Let’s look at an example, then talk about some useful features of both types of solutions.

Liquid-piston dynamometers

A dynamometer is an important tool for studying liquid piston engines (or any kind of engine, for that matter) .  A dynamometer allows us to run the engine under varying external loads and also to measure the power produced by the engine.  All kinds of dynamometers have been developed over the years suited to many different engines and purposes.  While the testing of the solar-powered engine discussed in my last post provided the proof-of-concept, and yielded some useful operational information, the fact that the engine ran completely unloaded (only operating against internal friction) imposed a pretty severe limitation on the amount and usefulness of the information that could be learned.  Essentially, it would be like evaluating a car engine if the car were never taken out of neutral.  Actually, it is even more limiting than that because at least you could rev up a car engine whereas liquid piston engines operate in a resonance, and hence single speed, mode.