Efficiency of Non-Ideal Engines

Collective Working Regimes for Coupled Heat Engines

B. Jimenez de Cisneros and A. Calvo Hernandez

PRL 98, 130602 (2007)

URL: http://link.aps.org/abstract/PRL/v98/e130602

The authors consider the efficiency of an array of coupled heat engines between two reservoirs at temperature t and T, with T > t. Long ago, Carnot showed that the maximum efficiency is

e = 1 - (t/T).

This efficiency can only be realized in an adiabatic, reversible process.

In the late 50s and mid 70s, physicists extended Carnot's analysis to finite-time endoreversible processes and found

e = 1 - sqrt(t/T).

This is called the Curzon-Ahlborn efficiency. (For T = 4t, this implies a reduction in efficiency from 50% to 25% -- significant!) The authors claim this efficiency provides a good approximation to the observed efficiency of several power plants, which suggests they are closer to maximum theoretical efficiency than one might have thought. If you want your power in finite time, you might have to settle for significantly less efficiency.

The authors analyze an array of coupled heat engines between two reservoirs. First, they show that the efficiency only depends on the endpoints and not the intermediate mechanisms. They derive an efficiency that depends on the heat fluxes at the ends, not the temperatures:

e = 1 - j/J.

They also calculate the rate of entropy production, which leads to an analysis of thermodynamic forces and Onsager coefficients, which I am not familiar with. The authors show that the Carnot efficiency is realized when the rate of entropy production is zero.

The rest of the paper is devoted to solutions of a Ricatti differential equation the authors derive for the Onsager coefficients. They show that the Carnot and Curzan-Ahlborn efficiencies are specific cases of their more general theory.

A surprising result is that global optimization of the total power of the system does not require that every element perform at its individual maximum power. I also infer from this analysis that the key to improving efficiency is reducing entropy production, or isolating the system from the environment.

A final note: I am nearly certain that Eq. (20) or (21) is incorrect. I can solve the differential equation, and the solution to Eq. (20) is not Eq. (21). I'm not sure where the error is, but something is amiss.