Analyzing trade-offs in electron heat engine optimization using the ecological-efficient function

Abstract

We analyze evolutionary transitions between performance criteria for electron heat engines. The transition from power P to efficient power \(P_{\eta }\) improves efficiency and also reduces entropy generation, while the transition from ecological function \(\Omega\) to efficient-ecological function \(\Omega _{\eta }\) achieves a more balanced multi-objective compromise. Through multi-objective optimization using the NSGA-II algorithm, we obtain the four-dimensional Pareto front encompassing P, \(P_{\eta }\), \(\Omega\), and \(\Omega _{\eta }\). The Pareto analysis reveals that the \(\Omega _{\eta }\)-optimal region concentrates along a central ridge of the frontier where ecological and efficient-power objectives naturally align, confirming its Pareto-efficient nature. Quantitative comparison shows that the \(\Omega \rightarrow \Omega _{\eta }\) transition requires a substantial sacrifice of \(58.71\%\) in the ecological function but achieves a significant reduction in entropy generation of \(25.10\%\) and a modest efficiency gain of \(3.35\%\). In contrast, the \(P \rightarrow P_{\eta }\) transition sacrifices only \(7.91\%\) of power for a \(21.92\%\) efficiency gain while achieving a \(48.55\%\) reduction in entropy generation. Thus, both pathways reduce entropy, but with different trade-offs. Our results do not establish an absolute superiority of either criterion. Instead, they reveal a design-dependent trade-off: the \(P \rightarrow P_{\eta }\) pathway is preferable when power output and efficiency are the primary concerns and a narrow operational window is acceptable, as it achieves large efficiency gains (\(21.92\%\)) and entropy reductions (\(48.55\%\)) with minimal power sacrifice (\(7.91\%\)). Conversely, the \(\Omega \rightarrow \Omega _{\eta }\) pathway is preferable when operational robustness, design flexibility, and a balanced multi-objective compromise between power, efficiency, and entropy generation are prioritized, despite requiring a much larger sacrifice in its primary objective (\(58.71\%\)). The choice between these pathways thus depends on the specific design priorities and operational constraints of the target application.