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This is just the curious cat in me. When heating water on a stove top in a covered pot, once the desired temperature is reached, does it take more energy to maintain the temp if it is higher (say 200F) vs if it is lower (say 150F)?

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I'm impressed if you're boiling water at 150-200F! –  Jefromi Mar 30 '13 at 23:25
    
@Jefromi I live in the international space station :) –  pixelfreak Mar 31 '13 at 21:17
    
I was trying to point out that your use of "boiling" was confusing. You've now gotten an answer assuming you're really talking about boiling, not varying cooking temperatures, so I went ahead and edited. –  Jefromi Apr 2 '13 at 19:27
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up vote 5 down vote accepted

The limiting case would be a pot that is only emitting energy via thermal radiation, because it is (in the imagination) perfectly insulated and so not losing heat into the air via escaping vapor through the lid, or conducting heat through the pot material into the air.

In that case, the energy loss from the pot is only that of the black body radiation, which is a function strictly of temperature. Energy radiates away at a faster rate at higher temperatures.

Even if you relax these stringent imaginary conditions, all heat engines are based on a temperature difference. All of the heat loss mechanisms on the pot (such as losing heat to vaporization, or conduction through the surface of the pot into the air) are also driven by temperature difference between the contents of the pot and the surrounding air. The larger the difference, the faster the transfer on an absolute basis.

So yes, maintaining a higher temperature in equilibrium requires more energy per unit time than a lower one. But you would need someone far better at physics than me to give you the absolute mathematical models.

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Generally the heat transfer that really matters is from pot to air; it's much less efficient than the conduction through the pot walls. But yes, roughly, you can approximate all the heat transfer (and therefore the input heat) as proportional to the temperature difference. –  Jefromi Mar 30 '13 at 23:28
    
Agreed. I was pretty much ignoring the pot due to its high conductivity. –  SAJ14SAJ Mar 30 '13 at 23:33
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