# 2D Conduction Simulation of a PCM Storage Coupled with a Heat Pump in a Ventilation System
## 4 Results
### Analysis of Temperature at the Centre of the Container  
The variation in temperature at the centre of the container is presented in Figures 15-18.  
For this measure, 1D and 2D models are found to offer similar predictions. For the 2D model, the reported temperature corresponds to the middle of the cavity, near the centre of PCM, and because of symmetry, no noticeable difference is observed with the 1D result.  
Figure 14: Heat flux density distribution on the surface of the container obtained with the 2D model. The heat flux is counted here positive when the heat goes to the inside of the material. No flux is exchanged at the start of the trial, or after 25 h.  
Figure 15: Temperature variation at the centre of the container. Cooling case with the heating \(c_{p}\) curve.  
In Figures 15-18 show, for experimental temperature evolution at the centre of the PCM steps for 35 \({}^{\circ}\)C and 32 \({}^{\circ}\)C for cooling and 34 \({}^{\circ}\)C and 40 \({}^{\circ}\)C for heating: this can be explained by the two peaks observed in the DSC curves. For the most part, the numerical results differ only slightly from  
Figure 16: Temperature variation at the centre of the container. Heating case with the heating \(c_{p}\) curve.  
Figure 17: Temperature variation at the centre of the container. Cooling case with the cooling \(c_{p}\) curve.  
Figure 18: Temperature variation at the centre of the container. Heating case with the cooling \(c_{p}\) curve.  
the observed behaviour, with marginal improvements depending on whether the cooling or heating curve of \(c_{p}\) has been used for the calculation. In Figure 19 the rate of temperature change is present at two points in the PCM: at the surface and at the centre of the container (for the two 2D results). The temperature of the PCM was assumed to lie this range.  
Intervals of \(\pm 0.15\)\({}^{\circ}\)C/minute shown by dotted lines in the figure indicate the rate used for the DSC characterisation of \(c_{p}\) for cooling and heating. Clearly a much large rate is observed at the start of surface temperature change and during the onset of phase change. These peaks last less than 30 min each and the model seems to give satisfactory results. At other times, a much lower rate than 0.15 \({}^{\circ}\)C/minute is observed, and for very low variations, the two DSC curves should coincide. The low rate indicates that the \(c_{p}\) value in the model is thus an underestimation during this period of low variation, and it explains that the solidification (and melting) is delayed.  
A solution to this discrepancy is thus to consider the mean result between the two predictions. As can be seen in Figure 20, this modification is found to improve the agreement with experimental data.  
Convection has been neglected in our model but, in fact, it seems to affect the behaviour in liquid state. The Rayleigh number for a fluid is a dimensionless number associated with buoyancy-driven  
Figure 19: Rate of temperature variation at the centre of the container. Heating case with cooling \(c_{p}\) curve.  
Figure 20: Comparison of measured and calculated temperature variation at the centre of the container comparison showing the mean of the modelled values and experiment results.  
flow, also known as free convection or natural convection. Under a value of \(2\times 10^{3}\), it can be considered only conduction transfer. In our case, the Rayleigh number has given a value of about \(5\times 10^{4}\) that can explain that convection is not completely negligible. Figure 20 shows effectively that the model gives temperatures lower than experimental one during heating. Convection enhances the heating speed. In references [21; 22], free convection is considered like an increase of thermal apparent conductivity depending on the Rayleigh number evolution. During cooling, the model gives higher results than experiments that can also be explained by the higher apparent conductivity of the PCM.