Some PDAE aspects of the numerical simulation of a CO2 heat pump

We consider network modelling and numerical simulation of a simple CO2 heat pump consisting of a compressor, a valve and two heat exchangers. In a first step we investigate analytical and numerical properties of the heat exchanger model. The heat exchanger model is derived from the Euler equations under the assumption that the velocity of the refridgerant flow is small compared to the local speed of sound. While the Euler equations form a hyperbolic system, the character of the new system, called The zero Mach number limit of the Euler equations, is unclear. The lack of a time derivative in the momentum equation makes the heat exchanger model by itself a PDAE system. We analyse a frozen-coefficient linearisation of the heat exchanger model by transformation to a canonical form. The canonical form reveales that the system is equivalent to a hyperbolic equation and a parabolic block. The parabolic block is equivalent to a parabolic equation and an algebraic-differential relation, similar to the system that results when the heat equation ut = uxx + f is written as a first order system. We prove a stability estimate suggesting that the solution is more sensitive to perturbations, especially in time-dependent boundary conditions, than is indicated by previous results. Furthermore, we consider semidiscretisation of the linearised heat exchanger model. In a method of lines approach using collocation at the gridpoints, we suggest that it is possible to use a simple first order difference scheme taking into account the direction of the flow and the boundary conditions. We show that using this difference scheme, the solution to the semidiscrete equations satisfies a discrete analogue to the stability estimate in the continuous case. The results of the linear analysis is verified in numerical experiments with the nonlinear heat exchanger model.

QC 20101126