Efficiencies of power plants, quasi-static models and the geometric-mean temperature

Observed efficiencies of industrial power plants are often approximated by the square-root formula: $1-\sqrt{T_-/T_+}$, where $T_+ (T_-)$ is the highest (lowest) temperature achieved in the plant. This expression can be derived within finite-time thermodynamics, or, by entropy generation minimization, based on finite rates of processes. A closely related quantity is the optimal value of the intermediate temperature for the hot stream, which is given by the geometric-mean value: $\sqrt{T_+ T_-}$. It is proposed to model the operation of plants by quasi-static work extraction models, with one reservoir (source/sink) as finite, while the other as practically infinite. No simplifying assumption is made on the nature of the finite system. This description is consistent with two model hypotheses, each yielding a specific value of the intermediate temperature. We show that the expected value of the intermediate temperature, defined as the arithmetic mean, is very closely given by the geometric-mean value. The definition is motivated as the use of inductive inference in the presence of limited information.