Endo-reversible heat engines coupled to finite thermal reservoirs: A rigorous treatment

We consider two specific thermodynamic cycles of engine operating in a finite time coupled to two thermal reservoirs with a finite heat capacity: The Carnot-type cycle and the Lorenz-type cycle. By means of the endo-reversible thermodynamics, we then discuss the power output of engine and its optimization. In doing so, we treat the temporal duration of a single cycle rigorously, i.e., without neglecting the duration of its adiabatic parts. Then we find that the maximally obtainable power output P_m and the engine efficiency \eta_m at the point of P_m explicitly depend on the heat conductance and the compression ratio. From this, it is immediate to observe that the well-known results available in many references, in particular the (compression-ratio-independent) Curzon-Ahlborn-Novikov expressions such as \eta_m --> \eta_{CAN} = 1 - (T_L/T_H)^(1/2) with the temperatures (T_H, T_L) of hot and cold reservoirs only, can be recovered, but, significantly enough, in the limit of a vanishingly small heat conductance and an infinitely large compression ratio only. Our result implies that the endo-reversible model of a thermal machine operating in a finite time and so producing a finite power output with the Curzon-Ahlborn-Novikov results should be limited in its own validity regime.