How is heat and temperature related? How will a thermometer react in the following conditions? Consider the cooking stove and LPG fuel for example, all my questions will be related to it.
Assume I have a frying pan made of steel which is kept on the the stove with the knob set to moderate.
Questions : 


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*Will the pan ever melt if I never took it off the stove with an unlimited supply of fuel? (How will a thermometer react int this case?)

*If not, will the pan melt if I increase the burner's radius so that more flame can hit the pan? (How will a thermometer react int this case?)

*If not, will the pan melt if I set the gas knob to max, or will it only take less time for the pan to melt by increasing the knob (if melting is possible)? (How will a thermometer react int this case?)



What will the factor be to melt the steel pan?
 A: A flame requires an oxidizer. For the cooking stove, that means it would have to be in the air. The melting point for steel is around 1300-1500C (depending on grade and quality). Most LPG fuels don't reach a burn temperature above 2000C (note that's an ideal case, but you won't reach that with a commercially available cooking stove). You have the knob set to moderate, so the flame probably won't get anywhere near 2000C. Then, once you account for radiative cooling, conductive cooling with whatever the pan is resting on, and the increased convective cooling from all the air motion that such high temperatures would cause, it's safe to say that in standard conditions, this stove will never be able to melt the steel pan. It may soften the steel, but not melt it.
But you asked what happens to the thermometer when the steel pan melts. Okay, to melt the steel pan the flame needs to be hot enough. The fuel does allow for this, so let's say this is a custom designed stove that can get a 1900C flame. Then we should cover the pan with a reflective dome (or similar cover). Wide enough so as not to touch the pan and reflective to keep most of the radiation inside it. That would also keep the hot air around the pan, but we need to make sure not to smother the flame. There has to be a vent to allow oxygenated air into the burner. If we do this, it's possible to get everything hot enough to melt the steel pan.
What happens to the thermometer? That really depends on the thermometer. Is it a glass-alcohol or glass-mercury thermometer? Is it a digital thermometer? Is it rated for these high temperatures? Is it an IR thermometer? An IR thermometer will be fine. It will record the temperature of whatever it is pointed at. A digital thermometer could melt. But if it's rated for those temperatures, it will be fine. An analog thermometer (one with a bulb and liquid inside) is not a smart thing to use. If the casing doesn't melt, you could get boiling of the liquid and it could explode. But as usual, if the thermometer is rated for that temperature, it will be fine.
