The fundamental optimal relation between heating load and coefficient of performance (COP) of a generalized irreversible Carnot heat pump is derived based on a new generalized heat transfer law, which includes the generalized convective heat transfer law and generalized radiative heat transfer law, $q \varpropto (\Delta T^{n})^{m}$. The generalized irreversible Carnot heat pump model incorporates several internal and external irreversibilities, such as heat resistance, bypass heat leakage, friction, turbulence and other undesirable irreversibility factors. The added irreversibilities besides heat resistance are characterized by a constant parameter and a constant coefficient. The effects of heat transfer laws and various loss terms are analysed. The heating load vs. COP characteristic of a generalized irreversible Carnot heat pump is a parabolic-like curve, which is consistent with the experimental result of thermoelectric heat pump. The obtained results include those obtained in many literatures and indicated that the analysis results of the generalized irreversible Carnot heat pump were more suitable for engineering practice than those of the endoreversible Carnot heat pump.

This paper deals with the performance analysis and optimization for irreversible heat pumps working on reversed Brayton cycle with constant-temperature heat reservoirs by taking exergetic efficiency as the optimization objective combining exergy concept with finite-time thermodynamics (FTT). Exergetic efficiency is defined as the ratio of rate of exergy output to rate of exergy input of the system. The irreversibilities considered in the system include heat resistance losses in the hot- and cold-side heat exchangers and non-isentropic losses in the compression and expansion processes. The analytical formulas of the heating load, coefficient of performance (COP) and exergetic efficiency for the heat pumps are derived. The results are compared with those obtained for the traditional heating load and coefficient of performance objectives. The influences of the pressure ratio of the compressor, the allocation of heat exchanger inventory, the temperature ratio of two reservoirs, the effectiveness of the hot- and cold-side heat exchangers and regenerator, the efficiencies of the compressor and expander, the ratio of hot-side heat reservoir temperature to ambient temperature, the total heat exchanger inventory, and the heat capacity rate of the working fluid on the exergetic efficiency of the heat pumps are analysed by numerical calculations. The results show that the exergetic efficiency optimization is an important and effective criterion for the evaluation of an irreversible heat pump working on reversed Brayton cycle.