A model of a multielement thermoelectric refrigerator with another linear heat transfer law, the linear phenomenological heat transfer law $Q\propto \Delta (1/T)$, is established. The refrigerating capacity and coefficient of performance (COP) are analyzed and optimized. The junction temperature solution equations are derived. The optimum electrical currents and thermal conductance allocation are discussed. The influences of thermoelectric element quantity and refrigerating temperature difference on the optimum performances and optimum electrical currents are analyzed. The results show that different optimization objectives have different requirements for the distribution of electrical current and thermal conductance. The refrigeration capacity is not proportional to the number of thermoelectric elements. It is found that the refrigerating capacity can be achieved only when the number of thermoelectric elements is matched for fixed external heat exchangers. The input electrical current and the allocation of the thermal conductance between the two heat exchangers can be optimized synchronously to achieve maximum refrigerating capacity or maximum COP. Performance is compared with that with a Newtonian heat transfer law. The influences of the Thomson effect are also examined. Performance of the refrigerator with Newtonian heat transfer law is higher than that of the refrigerator with linear phenomenological heat transfer law. The Thomson effect can improve the performance of the refrigerator.

A model for the quantum Brayton refrigerator that takes the harmonic oscillator system as the working substance is established. Expressions of cooling load, COP and ecological function are derived. With numerical illustrations, the optimal ecological performance is investigated. At the same time, effects of heat leakage and quantum friction are also studied. For the case with the classical approximation, the optimal ecological performance, and effects of heat leakage and quantum friction are also investigated. For both general cases and the case with classical approximation, the results indicate that the ecological function has a maximum. The irreversible losses decrease the ecological performance, while have different effects on the optimal ecological performance. For the case with classical approximation, numerical calculation with friction coefficient and heat leakage coefficient shows that the cooling load (R_E) at the maximum ecological function is 6.23% smaller than the maximum cooling load (R_max). The COP is also increased by 12.1% and exergy loss rate is decreased by 27.6%. Compared with the maximum COP state, the COP (ε_E) at the maximum ecological function is 0.55% smaller than the maximum COP (ε_max) and that makes 7.63% increase in exergy loss rate, but also makes 6.17% increases in cooling load and 6.20% increases in exergy output rate.