For the given initial finite high-temperature heat reservoir temperature, continuous Hamilton–Jacobi–Bellman equations are established to obtain optimal finite high-temperature heat reservoir temperature for minimum power consumption of multistage Carnot heat pumping system with generalized convective heat transfer law [q ∝ (ΔT) m ]. Analytical expression of optimal heat reservoir temperature with Newtonian heat transfer law (m = 1) is obtained based on generalized optimization results for minimum power consumption. For other heat transfer laws (m ≠ 1), numerical solutions for minimum power consumption are provided. Optimization results for multistage Carnot heat pumps are compared with maximum power output solutions of multistage irreversible Carnot heat engines.

中文翻译：

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q ∝ (ΔT) m 传热定律的多级不可逆卡诺热泵的最小功耗

对于给定的初始有限高温热库温度，建立连续的Hamilton-Jacobi-Bellman方程以获得具有广义对流传热定律的多级卡诺热泵系统功耗最小的最优有限高温热库温度[q∝ (Δ吨) 米 ]. 用牛顿传热定律解析最佳储热温度的表达式（米= 1) 是基于最小功耗的广义优化结果获得的。对于其他传热定律 (米≠ 1)，提供了最小功耗的数值解。将多级卡诺热泵的优化结果与多级不可逆卡诺热机的最大功率输出解决方案进行了比较。