Is it possible for a pan to reach a higher temperature than its heat source? There is a claim in this cookware infomercial that a "traditional" (aluminum) non-stick pan, placed on a burner set to 400 degrees F, will reach a surface temperature of about 700 degrees F (while, by contrast, the pan being advertised remains around 400 degrees F). "A pan that is made mostly of aluminum is just going to absorb the heat and keep climbing[...] There is no excuse for your pan being at nearly 700 degrees minutes into being set at 400 degrees"
My intuition is that the pan would reach an equilibrium point at a temperature somewhat below that of the heat source, where it is shedding heat to the environment at the same rate as it is absorbing energy from the source. That being said, I'm far from an expert in thermodynamics, so very possible that my intuition is wrong.
Is it possible (and -- as suggested in the ad -- common) for a pan to reach a higher temperature than the heat source?
 A: Your intuition is right. I believe when they say that they are setting the burner to 400 degrees, they mean that they are setting the burner to a level of heat output that would allow the pan to reach an equilibrium temperature of 400 degrees. The pan itself is gaining energy from the burner, but it is also losing energy to the environment, so the pan ends up at a different, lower equilibrium temperature than the burner does. Modern electric burners can generally get as hot as 1500 degrees.
A: Something is definitely amiss here. There is no way the aluminum pan can reach 700 degrees if the heat source never exceeds 400 degrees. That would require heat to naturally flow from a lower temperature object to a higher temperature object in violation of the second law of thermodynamics. I would bet that the aluminum pan reached 700 degrees only initially because of an initial overshoot of the regulation of the temperature of the burner combined with the faster rate of temperature rise of the aluminum pan.
So, let's consider in more detail what might be actually happening here.
First the commercial shows the temperature of the aluminum pan rising at a much faster rate than the combination aluminum/stainless steel pan. That part appears to be correct because the volumetric heat capacity of the combination stainless steel/aluminum (Ninja pan being promoted) would be greater than the volumetric heat capacity of the aluminum pan. The volumetric heat capacity is the amount of heat needed to to be added in order to raise one unit volume of the object  one unit in temperature. So for the same dimensions pan it takes more heat to raise the promoted pan per degree than the aluminum pan. In addition, the thermal conductivity of the aluminum pan will be much greater than the stainless steel/aluminum pan, so heat will transfer more quickly to the aluminum pan.
Next we need to look at the heat source. The promo did not describe what the actual product was and, perhaps most importantly, what it means by "setting the burner to 400 degrees" and how the stove actually performs the function. Is it a fixed power setting that makes the burner surface 400 degrees? Is it a sensor (e.g., thermostat) that regulates the power source to produce a steady burner surface temperature of 400 degrees? I believe it is the latter.
If the burner temperature is regulated by a thermal device (e.g., thermostat), then there is the possibility of an initial temperature overshoot prior to reaching the design equilibrium temperature on the surface. Then, if that is the case, the maximum initial temperature on the aluminum pan would likely be higher than the Ninja pan since the temperature rises faster on the aluminum pan and any overshoot will be greater for the aluminum pan. But after a while, when the cycling device settles down, both pans should be at around 400 degrees.
The most important thing is that at the end of the promo they actually say that both pans are at 400 degrees. That would support my suggestion that the 700 degree temperature on the aluminum pan is due to an overshoot of the cooking device. The promoted pan may not do much better than the aluminum pan if the cooking product (the stove), which is a variable in the demo, had better thermal regulation.
At the very end, when cooking pancakes, only part of the pancake is burned at the edges of the aluminum pan where, in all likelihood, the temperatures did not have time to go down.
Hope this helps.
A: The pan can be at a higher temperature than the heat source only if: work is also done on the pan and/or is there mass transfer of hot material into the pan.
