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I was considering some theoretical limits on transferring heat from one system at to another to another. It is well known that if you have two systems at temperatures $T_1$ and $T_2$, respectively, with $T_2> T_1$ then you need to provide work to transfer heat to the system at temperature $T_2$. Now, that work is then coming from some different thermodynamic process that we choose not to describe. But leaving that out is actually compromising the ideal solution as extracting work first and then having to dump that at the system at $T_2$ is not necessarily optimal.

This led me to consider three systems at temperatures $T_1$, $T_2$, and $T_3$, respectively, with $T_1<T_2<T_3$ where we want to pump heat to the system at temperature $T_3$ but now there is no external source of work. In the limit of infinite heat capacities, we then find that ratio between the heat supplied to the system at $T_3$ and the heat extracted from the system at $T_2$ in the optimal reversible case will be:

$$\eta = \frac{T_3}{T_2}\frac{T_2 - T_1}{T_3 - T_1}$$

See below for the derivation. So, in principle at least, one can construct a heat pump that heats a house using outside environments at two different temperatures without the need for work to be provided. E.g. in winter the outside temperature near the ground could be -10 °C while at some depth below the ground it could be 10 °C.

The question is then how to realize this in practice. Presumably one would need some sort of a Stirling machine. What can one expect about the amount of heat that can be extracted from the environment per unit time in some realistic setting?


To derive the efficiency formula, one can consider a Carnot process that extracts work from the systems at temperature $T_1$ and $T_2$ and another Carnot process that acts as a heat pump between the systems at temperatures $T_2$ and $T_3$ using the work provided for by the first Carnot process. However, a simpler derivation is to consider any arbitrary process that adds an amount of heat of $q_i$ to system $i$, the First law then implies that:

$$\sum_{i=1}^3 q_i = 0$$

Reversibility implies:

$$\sum_{i=1}^3 \frac{q_i}{T_i} = 0$$

Solving for the $\eta \equiv -\frac{q_3}{q_2}$ then leads to the formula given above.

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  • $\begingroup$ Can you add more details on the efficiency relationship you obtained? $\endgroup$ – Algo Apr 27 '16 at 7:48
  • $\begingroup$ You could be a bit more rigorous in defining the system boundaries for this analysis; the work done by the heat engine is external to the system of the heat pump. There is still no such thing as a free lunch. $\endgroup$ – Air Apr 27 '16 at 23:43
  • $\begingroup$ The Stirling cycle isn't the only way to harness a thermal gradient to do useful work; there's also the Seebeck effect, which would be convenient in that residential heat pumps tend to be powered by electricity but not very promising in terms of efficiency. $\endgroup$ – Air Apr 27 '16 at 23:59
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There are currently houses that are built with heat pumps to heat or cool the house using the ground a thermal reservoir. It's usually much more efficient than using the outside air as their thermal reservoir. However, as far as I know none of the heat pumps used approach the theoretical carnot efficiency as the heat transfer would be too slow without massive heat exchangers. Instead, they relinquish some of their thermodynamic efficiency to make their cold side colder than their cold reservoir, and their hot side hotter than their hot reservoir. this allows for the heat transfer rates to be greatly increased, making the heat exchanger sizes reasonable.

Unfortunately, trying the same tactic won't work when trying to increase the heat transfer rate of a stirling machine. Increasing, the difference between the cold side of the engine and the cold reservoir just cuts into energy available. To be effective, you'd have to have giant heat exchangers. Turns out it would be way cheaper to install solar panels, or just buy the electricity.

Unless...

you have a really large temperature difference. For example if you live near yellowstone national park, the temperature deep underground can be hundreds of degrees farenheit. Then it's "easy" to build a geothermal generator.

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