We investigated the dynamics of a transcritical CO2 heat pump system including hot and cold thermal storages, which makes up the concept “thermal battery”. The analytical model is used for the study of the dynamics of the system involving simultaneous supply of heating and cooling for buildings. The model includes the dynamics of the gas cooler, evaporator and the thermal storages, while the compressor and the expansion valve are considered quasi-static. The heat transfer in the dynamically modeled components is described by partial differential equations (PDEs) consisting of heat conduction, convection, and source terms. Each component is divided into a number of volumes adjusted according to the required precision and reasonable computational time. We applied two discretization schemes in order to find a numerical solution to the PDEs. The spatial discretization for the heat exchangers is performed by using the upwind scheme, where the fluid properties are individually calculated within each volume. Due to the discrete events in form of tapping and loading (or charging and discharging) of the heat storages, the discretization approach takes into account the sharp spatial transitions within the thermal storages. Therefore, the method of lines in combination with the Superbee slope-limiter was applied for the spatial discretization for high resolution calculation. The modeling approach results in a set of algebraic and ordinary differential equations (ODEs), hence the model becomes an algebraic differential equation problem, which we solved by using MATLAB solver ODE15s.
This extended model was used to simulate a dynamic response of the case with varying heating and cooling consumption over a period of 24 hours in a building. The heating and cooling energy consumption follow a sinusoidal and continuous pattern. The results include the effect on both the outlet temperatures and the system coefficient-of-performance (COP). The outlet energy from the hot storage and the cold storage is used for heating tap water and a chilled water space cooling application subject to temperature requirements. Dimensioning of both storages is crucial for obtaining the required temperatures. The model identifies the critical storage levels required to satisfy the periodic but out-of-phase combination of heating and cooling demands. The volume of the cold storage will have to be considerably larger than the hot storage due to the lower temperature difference.
On spatial discretization of evolutionary differential equations on the periodic domain with a mixed derivative
