# 2D Conduction Simulation of a PCM Storage Coupled with a Heat Pump in a Ventilation System
## 3 Modelling  
Simulations of PCM heat storage solutions have previously been proposed using analytical [11], semi-analytical [12] and empirical [13] models. Such simplified descriptions of a system are useful at an early stage of design. In order to address technical design considerations however, more detailed numerical models are required to predict the behaviour of the storage unit. In [13], a simplified heat storage model is proposed neglecting conduction in the PCM slab. A finite difference approach was adopted to discretize the conservation laws, and the governing differential equations were solved using an explicit scheme. In the model introduced in [14], conduction heat transfer was considered along the airflow direction for a PCM slab. For small values of the Biot number (3), these simplified models are adequate because temperature gradients within the storage material can be ignored. For configurations implying a large Biot number, as would typically be encountered for a storage material with a low thermal conductivity, an accurate prediction of the transient response of the unit can only be obtained by modelling conduction in the PCM slab itself. \[Bi\ =\ h_{e}\times e/2 \tag{3}\]  
Several experimental studies [13; 15; 16] have considered systems with a Biot number near to or less than unity. Experimental data from higher Biot number configurations are needed, firstly to understand the influence of the temperature gradient in the plate on the energy storage and release, and secondly to validate numerical models taking into account such non-negligible temperature gradients in the slab. The configuration described in Section 2 represents a system with an associated value over \(Bi\) = 4. Hence, a detailed calculation of conduction inside the PCM is mandatory. In [17], 2D conduction in a PCM slab was solved via an implicit scheme, for finite volume discretization and enthalpy formulation. Only advection and convection terms were taken into account in the energy balance of the flowing air. In [18], the 2D conduction problem was solved using the equivalent heat capacity method. The variation of thermal conductivity in the PCM slab was not modelled. In [19], another study also took into account the variation in conductivity and was validated with experimental data. In that study, an encapsulated PCM device was considered without a metallic container. Consequently, qualitative differences to transversal heat transfer should be expected with the configuration studied in the present work. For the present study we have constructed 2D and 1D models of the heat storage container responsible for the complex heat transfer phenomena, using COMSOL Multiphysics simulation platform [20]. It is a powerful code for modelling and solving various sciences and engineering problems based on the Finite Element (FE) method. The software is particularly well suited for the treatment of multiphysical and non-linear problems where several phenomena are involved, giving rise to non-linear behaviour, as in the case for the present study. The apparent heat capacity of the PCM was modelled according to Figure 5 and its thermal conductivity is illustrated in Figure 6. For the aluminium components of the system the model assumes constant thermophysical properties: \(\varrho_{\mathrm{alu}}=2700\) kg m\({}^{-3}\) and \(c_{p\,alu}=897\) J kg\({}^{-1}\). K\({}^{-1}\). Figure 5: Equivalent heat capacity of the Microtek 37 from DSC (Differential Scanning Calorimetry) measurement. Figure 6: Thermal conductivity of PCM in the model. For the 2D model, the mesh generated by COMSOL to discretise the heat storage system consists mainly of triangular elements for the PCM domain and edge elements for the aluminium container. Complete mesh consists of more than 1700 elements with a minimum quality of 0.7 and an average quality: 0.9. The time discretization was realized according to the backward differentiation formula with a maximum time step equal to 1 min.