A simple model for Carnot heat engines
Résumé
Contrary to what some texts are saying, the theory of heat engines and heat pumps presented by Carnot around 1824 is entirely accurate. We offer in this paper a simple mechanical model consisting of reservoirs at altitude $\epsilon$ containing $N$ ball locations and $n$ weight-1 balls. The reservoir energy is $Q=n\epsilon$. We consider particularly two such reservoirs, with the label $l$ referring to the lower reservoir and the label $h$ to the higher reservoir. A cycle consists of exchanging balls between the reservoirs. It is straightforward to show that the efficiency, defined as the ratio of the work produced to the energy lost by the higher reservoir is $\eta=1-\epsilon_l/\epsilon_h$. To relate this energy-generating device to a \emph{heat} engine, we introduce the entropy, defined as the logarithm of the number of ball configurations in a reservoir, $S(n)=\ln[N!/n!(N-n)!]$. The absolute temperature is then defined as $T=[(Q(n+1)-Q(n))/[(S(n+1)-S(n))]$, and the large $n$ limit is assumed. It follows that when $n_l\approx n_h$, the system efficiency $\eta=1-\epsilon_l/\epsilon_h=1-T_l/T_h$ is the Carnot efficiency. Because the concept of time is not involved, a treatment such as the one presented here may be given early in Physics courses.
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