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I've been reading about extracting energy from heat, particularly Sterling engines. There's always a temperature gradient. Heat flows from the hot side to the cold side through the working fluid. As the temperature differential approaches zero the engine stops.

My question is that why do we even need a temperature differential? Even more broadly why do we use energy to cool things? Let's take a gallon of water at 50 deg C. Water has a specific heat of 4.18JdegC/g. Ignoring the heat of fusion and following Q = CMdT, just by virtue of it being "warm" (323 deg C above absolute zero) wouldn't the water "contain" 3780 * 4.18 * 323 = 5104 kJ of energy (3780 grams per gallon of water). Why couldn't you extract any of this energy as work and as a result of extracting the energy the water cools itself (it gives up energy, for every 3780*4.18 J extracted it should lower by 1 deg C). I'm invisioning the heated water (or anything heated above abs zero) as an energy sink or battery. Theoretically you should be able to extract 5104 kJ of energy from this 50 deg C gallon of water.

In the Sterling engine we could lower the cold side with liquid nitrogen but it takes energy to make this. In an AC system we spend energy to lower the temperature in the room. Since "cold" is the absence of "heat" or energy, shouldn't cooling things actually create energy for us (the hot things that earlier had energy imparted into them give up their energy to make work). It seems wasteful to just waste the heat on warming up something at a lower temperature differential such as the cool side of the Sterling engine or the ambient air by the condenser.

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The Ideal

The main problem is that all current technologies seem to waste a certain amount of energy on actually displacing some of the heat energy from A to B. If there were no such inefficiency, then we would in a sense obtain "free" energy as we could displace and focus an arbitrarily large amount of energy from a material into a finite space, and then use the laws of thermodynamics to power (e.g.) a stirling engine. If only there exists a structure that can naturally, without energy input, make one side cold and another warm. However, this gives us a contradiction.

The Contradiction

Due to the second law of thermodynamics and the conservation of energy, this becomes impossible. As having a perfect "structure" that can transfer heat from A to B would violate these laws. It is required that energy be put into the system to actually perform the displacement. Therefore, even if said system is 100% efficient, energy is still put into the system such that no "free" energy is ever gained.

For reference (2nd Law of Th.) (wikipedia)

When two initially isolated systems in separate but nearby regions of space, each in thermodynamic equilibrium with itself but not necessarily with each other, are then allowed to interact, they will eventually reach a mutual thermodynamic equilibrium. The sum of the entropies of the initially isolated systems is less than or equal to the total entropy of the final combination. Equality occurs just when the two original systems have all their respective intensive variables (temperature, pressure) equal; then the final system also has the same values.

A Law

Unfortunately, the question is hard to answer:

My question is that why do we even need a temperature differential?

The reason is that the contrary is made impossible by a law. A law in physics and mathematics doesn't necessarily equate a theorem or theory. It only states "such is so." whilst a theory would actually explain "why such is so".

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    $\begingroup$ As I stated in my answer, the second law of thermodynamics prevents us simply from doing so. Unfortunately there is no "why" to a law. If you are thinking about using the heat of a body to say propel an object, then the problem is the alignment of kinetic energy. You will be required to put energy into your system to align the molecules' kinetic energy such that it may purposeful. $\endgroup$ – Ultimate Hawk Jun 24 '14 at 18:47
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    $\begingroup$ There is a differential in the battery, the negative and positive pole. Unlike the water, most energy is stored as chemical potential. When the circuit is closed, the potential allows chemical reactions through oxidation and reduction to occur. The differential of the 2 poles causes the chemical reaction, and thus the amperage and power stream. $\endgroup$ – Ultimate Hawk Jun 24 '14 at 18:53
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    $\begingroup$ No energy is lost, there is simply no potential transfer from your tank to another system. Entropy therefore remains in a constant state. Therefore, energy in the system remains constant. However, you can not extract any energy using the thermoelectric effect or a stirling engine if you fail to find a suitable system to transfer between. $\endgroup$ – Ultimate Hawk Jun 24 '14 at 19:03
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    $\begingroup$ Thanks. I accepted your answer. I guess this is akin to dragging a jug of water up a mountain. It has a lot of energy (since it's high up), but relative to you you can only pour it onto the ground (small differential). If you can find a cliff to throw it off of you can extract the potential energy you imparted but other than that it's locked up. So you're telling me there is no way to extract the heat absent a suitable system to transfer to, $\endgroup$ – user52031 Jun 24 '14 at 19:17
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    $\begingroup$ Worth mentioning Maxwell's demon $\endgroup$ – WalyKu Jun 25 '14 at 12:49
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It may be possible using cold fraction from a vortex tube. Use heat source to boil water; use vortex tube to separate the steam into hotter steam and cooler steam; use hotter steam to replace original heat source. So the original body of water produces some kinetic energy and loses a lot of thermal energy. You would need a temporary external heat source but would gain that energy back and then some if the chain reaction consumed enough water.

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