How could I determine the temperature at which an egg explodes A while back, some friends and I pondered "what will happen to an egg if placed very near a campfire".  So we placed an egg on a grill, approximately 12 inches above a real campfire. Something like 7-10 minutes later, thankfully right as we walked away, the egg exploded in every direction in a gooey, golden glory.
So I ask you, how can I determine the internal temperature of an egg that will cause it to explode?
 A: There can be few experimental scientists who haven't microwaved an egg to make it explode (if you have never done this shame on you :-). The reason it explodes is obvious, though I've never seen any detailed experiments measuring temperature and pressure inside the egg as it's heating. I can see that it would be hard to persuade the various research councils that this is worth funding.
The egg explodes because the water inside it boils and increases the pressure. Eventually the increased pressure ruptures the shell and the egg bursts. You specifically ask about the temperature. Obviously the temperature has to be above 100°C otherwise the water won't boil and the egg won't burst. However in the absence of any detailed experimental measurements it's hard to say any more.
Many, many years ago a group of we graduate students spent a Friday afternoon microwaving eggs (isn't it good to see your taxes are being well spent?) to see what happened. We found a lot of variation: some eggs just split while others burst with a real bang. I would guess that the best explosions come when you manage to superheat the water in the egg. Then when the steam bubbles nucleate you get a very rapid build up of pressure and a big bang. That would tally with your experience of bursting after relatively long slow heating, though in your case I'm a bit surprised you didn't end up with just a hard boiled egg.
Whilst looking for measurements of the egg temperature I found this article on the New Scientist web site. Since the egg exploded after removal from the microwave oven it supports the suggestion that the egg contents can be superheated. It also points out the dangers of this experiment as the young lady in question suffered substantial eye injuries.
A: Several narrow-angle pyrometers above and besides the egg, looking squarely at it and not seeing anything else (the firewood grass etc.) would be the simplest solution.

The temperature might be very uneven inside the egg because just one side is exposed to the fire. Several degrees of Celsius difference seems credible. The egg can be exploded by boiling just 1/10th part of it or even less. After 7-10 minutes convection inside the egg might be severely reduced due to proteins hardening.
Also, kitchen thermometers could be used to measure the fume temperature around the oil, both below and above the egg.

Both instruments should be calibrated to a boiling distilled water at 100°C. Then their precision will exceed that of gradients inside the egg. For even better measurements high speed infrared camera might be a slight improvement, but a visible light high-speed camera filming the moment of explosion would be even more gratifying and much cheaper, e.g. Panasonic HX-A100.
A: The explosion is caused by rapidly expanding steam, the critical temperature therefore is 100 degree Celsius (boiling of water), though the speed at which the water evaporates (proportional to outer temperature) will of course determine how violent the effect. ;)
