The ice heat engine

No, the title’s not an oxymoron.

Water trapped in rocks is known to split them open upon freezing – this is in fact, one of the methods through which soil forms. The intense pressure it exerts suggests that the freezing of water may be utilized to do work for us. Here’s one way to go about it:

The simplest heat engine imaginable is a bucket of water at zero degrees celsius. If done very carefully, all the water can be made to solidify (thanks to the quintessential “infinitesimal” temperature difference) at the same temperature. Water expands when it solidifies, which means that with a piston like lid, the solidification of water at zero degrees can be used to raise a weight by a small height. Now the weight is removed, (moved horizontally, so we can neglect the energy used in doing so) and the ice is melted, again at zero degrees.

We have, in effect, created a heat engine that works at a single temperature.
And Viola! there we have it- the simplest perpetuum mobile ever built.

Of course, nature doesn’t work that way, and this is where the second law of thermodynamics steps in.

Clearly, one of the processes in the cycle must be impossible – but which one?
If you wish to stop and figure it out, do so before reading further.

1>With the weight on top, water at zero degrees solidifies to ice at zero degrees, and expands during the process.
2>With no weight on top, ice melts at zero degrees to give water at zero degrees, and contracts during the process.

This charming little thought experiment lends insight into the true utility of the second law of thermodynamics: By stating that at least two temperatures are required for cyclic operation, it tells us very little about heat engines – it does, however, tell us a lot about the properties of substances. In this case, it follows that water will not – it cannot – solidify at zero degrees with a weight on top (or equivalently, at a positive gage pressure). The result of this gendanken is information on the thermodynamic behaviour of water! At positive gage pressures, water must solidify at temperatures lower than zero degrees, and vice versa – a nice little snippet of information that comes right out of the second law.

The second law also provides an estimate of the pressure needed for water to solidify at, say, -5 degrees celcius. Assume that we run the heat engine reversibly (more on reversibility later. For now, let’s assume it runs very, very, slowly, with heat transferred across infinitesimal temperature differences), so that the efficiency (ratio of the work done to the heat absorbed at zero degrees) is :

1 – (low T)/(high T) ~ 0.018305

The net work done is : pressure X(volume change during solidification) – pressure X(volume change during liquification)

which nearly equals ~ (gage pressure during solidification) * (volume change)
= delta P * delta V

The volume change per kg of water during solidification, delta V is :

= (1/ice density – 1/water density)

= 0.00009051 cubic metres per kg

and the heat gained is: 334 kJ/kg (The latent heat of fusion for water)

Setting up the balance gives: delta P ~ 69260 KPa – Intense!
The gage pressure obtained here is within 15% of the actual gage pressure – quite acceptable, since we neglected changes in density and latent heat over a span of 5 degrees.

Incidentally, writing out the expression for the gage pressure (delta P) as obtained above gives:

(delta P) * (delta V) = (efficiency of “solidification engine”) * (latent heat)

Since the above analysis makes no use of the properties of water (in fact, it predicts them), it follows that this relation holds for any thermodynamically pure substance. We have derived what is called the Clausius -Clayperon equation, a very commonly used relation in thermodynamics.

The second law of thermodynamics is a statement that tells us little about heat engines, but a lot about matter!

PS: A few qualifiers: The perpetuum mobile explained above is actually a perpetuum mobile of the first kind – it violates the first and the second law of thermodynamics. Also, water doesn’t always expand on freezing – some phases of ice are denser than water! Finally, the second law states that at positive gage pressures, water must solidify at a different temperature – why is this temperature necessarily lower?(hint: look at the Clayperon equation) Next: A rubber band engine!