Puzzle No 1: Cool Eggs

I like Puzzles, but its so hard to find good ones.

If an egg, initially at room temperature (20 degrees Celsius), is dropped into liquid nitrogen (-196 degrees Celsius), it takes approximately eight minutes to approach the temperature of the liquid nitrogen, if it is subsequently left out on a surface, it takes approximately 2 hours to return to room temperature. Why is the heating and cooling asymmetric?

So the essence of this problem is to identify, from among the various features, the one that dominates. Nearly everyone, when first given this question, comes up with the following answer:

The Air-Egg boundary transfers heat worse than the Air-liquid nitrogen boundary.

This is plausible, it is a well known fact that water conducts heat around twenty five times better than air. In practice, because the liquid nitrogen vaporises on contact, the egg is suspended in a bath of nitrogen vapour, which probably leaves the heat transfer properties similar to, or even worse than, that of an egg in air. It is very hard to know the exact consequences of this effect.

However, there is a second important effect:

Let us suppose that the coefficients of conduction are constant. In that case the rate of heat flow into the egg will be given, in both cases, by Newtons Law of cooling.

\dot{Q} = \kappa A (T_{outside} - T_{surface})

Obviously the area is also the same, for both eggs, so we can absorb the two constants into one. The dot stands for derivative, so the equation can be read: “The flow of heat is proportional to the temperature difference”. Now consider when the heat flows into the egg. The heat capacity of water is about four kilojoules per kilo per degree. The heat capacity of ice is about half that. However, the latent heat of freezing is about 330 kilojoules per kilo. Thus the total amount of heat flow needed is rougly

196\cdot 2 + 330 +20\cdot 4 = 802 KJ

Now consider, the act of melting/freezing. On the way up this happens with a temperature difference of 20 degrees, whereas on the way down this happened with a temperature difference of two hundred degrees. Thus it takes about ten times as long to melt as it did to freeze.

The solution to the law of cooling is

t = - C\cdot\ln\left(\frac{\Delta T_{end}}{\Delta T_{start}}\right)

, but this does not lend itself to an obvious solution, since it will never reach the temperature of the medium. This just shows the relative inaccuracy of this type of law. Suppose we measure the time taken to reach five degrees of the end points, then we can find that cooling takes about one third of the time to heating. So we conclude that the latent heat effect introduces a factor of about three, and transference effects introduce a factor of about six.

I find it interesting that even if the conduction coefficients were perfectly symmetrical, this process would still be asymettric.

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I know I live in my own world, but I like it: they know me here.

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