Waxing Rhapsodic

I've been at Swarthmore enjoying Senior Week activities and my graduation yesterday morning.  One night last week, the dining hall served us an elegant, candlelit dinner.  I mention the candlelit part because looking at this

how could I avoid thinking about the physics of dripping candle wax?

As the wax melted, it would run down the column of solidified drips, until it reached the bottom.  It would sit there until it cooled and solidified, unless a second drop reached it first, in which case the combined weight would be too much, and they would both drip off.

To figure out how quickly the wax was cooling, we can use Newton's Law of Cooling.  Unfortunately, I couldn't find anything that explicitly gave the cooling rate for wax, but I did find some example data from a cooling candle.  Fitting the exponential form given by Newton's law to this gives

I'd estimate it took a little under a minute for each drop to solidify.  Supposing the wax solidifies at about 60°C, an initial temperature of 61°C would say that it remained molten for about 45 seconds.

We can figure out how much power the candle was putting out by looking at how much energy is required to raise the wax from room temperature to 61°C.  Using the equation

where ρ is the density of the drop, V its volume, and c the heat capacity of the wax, we find that it takes 0.08 Joules of energy to raise the temperature of a cubic millimeter of wax.  Since this involves melting the wax, we also need to take into account the latent heat of the wax.  This is the amount of energy required to induce the actual phase transition.  It's actually a pretty significant amount, and brings the total energy to 0.23 J.  If the candle is producing this much energy every 45 seconds, then its power output is about 5 milliwatts.  According to this page, candles put out something like 100 watts of power, so clearly not all of it is going toward melting the wax.  I would guess also that for such a small bit of wax, Newton's Law of Cooling may be a poor approximation.  This is a case where empirical data may have been helpful, but sadly I did not have my temperature probes and precision heater with me at dinner.