A Dynamic Analysis of Refrigerant Mass in Vapor Compression Cycles

Abstract

Numerical simulation of a thermofluid vapor compressioncycle (VCC) model in Modelica, for example,can exhibit a variation in the total fluid (refrigerant)mass. This paper provides a dynamicanalysis of a commonly used VCC model, identifies andanalyzes the root causeof this variation, and proposes a number of remedies. Thecause lies within the dynamicequations that result from application of the principle ofmass conservation.In many common formulations, these equations express theconservation of mass asone or more differential equations that equate the timederivative of mass to zero.The resulting set of n ordinary differential equations (anda number of auxiliaryalgebraic equations) include the time derivative of a massconstraint function,but not the actual mass constraint function itself. As aresult, this modelingformulation has the following properties: (1) equilibriumsolutions of the systemare neither isolated, nor exponentially stable; (2) alinearization about any equilibriumsolution has at least one eigenvalue equal to zero, makingan equilibrium solutionstable, but not exponentially stable; (3) for a VCC modelformulated using two fluidstates per control volume, a one-dimensional equilibriummanifold exists containingall of the equilibrium solutions, and is parameterized bythe total fluid mass;(4) an (n-1) dimensional, stable, invariant manifold existstransverse to theequilibrium manifold, defined by the mass constraintfunction, and on which analyticsolutions to the model evolve and the total fluid massremains constant; and(5) numerical solutions may drift off of this manifold,resulting in an observeddrift of fluid mass. These properties have consequences forsimulation,control design, numerical model reduction, and stateestimation.A number of methods to stabilize the mass constraint areproposed and anumber of examples that illustrate the behavior, analysisand remedies are provided.