# Are Carnot engine efficieny and Fourier heat trasmission law related?

It just occured to me that the efficiency of Carnot cycles is $\eta= \frac{T_1 - T_2}{T_1}$, that is, the efficiency decreases as the difference between reservoir temperatures decreases. On the other side, Fourier's law states that the dissipation of heat is proportional to the temperature gradient, that is, to the temperature difference.

My question, then, is: are these two results related? Do they both have a common cause?

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2
It's a good question (+1), but to the best of my knowledge the answer is no. – Nathaniel Dec 21 '12 at 3:23
Fourier heat transmission law is like Hooke's law. They are empirical approximation. – C.R. Dec 21 '12 at 5:14

## 1 Answer

No, they are not directly related.

The law of heat conduction you cite can be thought of as one example of a general diffusion process, and involves retaining the first terms in a Taylor expansion of fluxes through zone boundaries. (see, for example, these MIT Introduction to Solid State Chemistry lecture notes on diffusion). On the other hand, the Carnot efficiency is an exact expression for the work that an engine undergoing a single Carnot cycle performs divided by the total energy flowing into the engine as heat during that cycle.

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I understand. However, they must both have a "cause" that explains them, don't them? Perhaps using statistical mechanics? My interest was if the causes or both phenomena could be traced back to a common one. – carllacan Dec 21 '12 at 14:50

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I do not think it is useful to look for a common cause connecting these formulae in this case. They are both expressed in terms of temperatures, and so they manifestly demonstrate their accordance with the second law of thermodynamics, but then again all laws of nature possess this feature. I do not think there is a connection that runs deeper than that. Something else to notice is that the formula for Carnot efficiency depends on the ratio of the two temperatures ($1 - T_2/T_1$), whereas a nondimensional formulation for heat conduction still depends on the temperature difference. – kleingordon Dec 22 '12 at 21:22
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