Local asymptotic stability of an irreversible heat pump subject to total thermal conductance constraint

The local asymptotic stability of an irreversible heat pump subject to the total thermal conductance constraint is investigated based on the stability criteria for almost linear systems. An irreversible heat pump system that is modeled by the differential equation expresses the rate of change of the temperatures of the working fluid in the heat addition and heat rejection processes. From the local stability analysis, we find that a critical point of an almost linear system is asymptotically stable. After an arbitrarily small perturbation, the system state exponentially decays to an asymptotically stable critical point with either of two characteristic relaxation times that are a function of the total thermal conductance UA, thermal conductance allocation ratio ϕ, heat capacity C, internal irreversible factor I, temperature of the heat reservoir TH and heating load Q. The behavior of solutions of the system is presented qualitatively by sketching its phase portrait. One eigendirection in a phase portrait is the nonzero constant vector, and the other is a function of UA, ϕ, I, QH, TH and TL. Finally, we discuss the local asymptotic stability and steady state energetic properties of the irreversible heat pump.