Power-consumption minimization for multistage sequential endoreversible heat-pump systems with complex heat-resistance model

Model of multistage sequential endoreversible heat-pump (EHP) system with a finite-sink and an infinite-environment and with complex heat-resistance model of [\(q \propto (\Delta (T^{\text{n}} ))^{\text{m}}\)] is established and investigated. The \(q \propto (\Delta (T^{\text{n}} ))^{\text{m}}\) model includes many cases, such as linear-phenomenological model [\({\text{n}} = - 1\), \(m = 1\), \(q \propto \Delta (T^{ - 1} )\)], linear model [\({\text{n}} = 1\), \(m = 1\), \(q \propto \Delta (T)\)], Dulong-Petit model [\({\text{n}} = 1\), \(m = 1.25\), \(q \propto \Delta (T)^{1.25}\)], radiative model [\({\text{n}} = 4\), \(m = 1\), \(q \propto \Delta (T^{4} )\)], generalized convection model [\(q \propto (\Delta T)^{\text{m}}\)], generalized radiative model [\(q \propto \Delta (T^{\text{n}} )\)], special model [\({\text{n}} = 4\), \(m = 1.25\), \(q \propto (\Delta (T^{4} ))^{1.25}\)], etc.. Continuous Hamilton–Jacobi–Bellman (HJB) equations for optimal-configurations of sink-temperature with power-consumption minimization objective (PCMO) are obtained. General results are provided, and analytical solution with linear heat-resistance model is further obtained. Discrete HJB equations are obtained, and dynamic program method is utilized to obtain numerical-solutions of optimal-configurations with non-linear heat-resistance models. Optimization results are compared with those obtained for multistage discrete sequential endoreversible heat-engine systems with five different heat-resistance models. For some fixed parameters, PCMO of multistage discrete sequential EHP system for linear model is \(\dot{W}_{{\min }} = 8.29 \times 10^{4} {\text{W}}\); for Dulong-Petit model, it is \(\dot{W}_{\min } = 8.41 \times 10^{4} {\text{W}}\); for linear-phenomenological model, it is \(\dot{W}_{\min } = 8.59 \times 10^{4} {\text{W}}\); and for radiative model, it is \(\dot{W}_{\min } = 8.{4}9 \times 10^{4} {\text{W}}\); for [\(q \propto (\Delta (T^{4} ))^{1.25}\)] model, it is \(\dot{W}_{\min } = 8.21 \times 10^{4} {\text{W}}\). Only if cycle-period tends to infinite-long, \(\dot{W}_{\min } = \dot{W}_{\text{rev}}\).