## 3 Model Validation
### Discharged Thermal Energy  
The nodal temperature equations for each \(\Delta t\) compute the transient energy transfer process into the external medium. The model's energy conservation has been validated by analyzing the released heat through the outer boundary. All released energy must go through all control volumes, meaning that the last passing segment \(V1\), relates directly to the boundary point 1. The nodal discharged energy is given by:  
\[\Delta E=\rho_{1}.V_{1}.C_{1}(T^{n+1}-T^{n}) \tag{6}\]  
Two test cases were simulated to validate the model, comparing the slab's thermal response for the constant and temperature dependent scenarios, during 100 s. Both cases start with the same energy storage capacity. The initial specific heat values were set to \(C_{p}=C_{p}(320\,\mathrm{K})\) and the thermal conductivity was fixed at \(10.5\,\mathrm{W}\,\mathrm{m}^{-1}\,\mathrm{K}^{-1}\). The total accumulated thermal energy has a value of \(105.6\,\mathrm{mJ}\) and it is fully discharged in 15 s (\(\Delta E\)) into the exterior, allowing the heat transfer system to reach (and remain) at steady-state conditions. Figure 3: The heat capacity of gadolinium as a function of temperature in zero magnetic field. Adapted from [PERSON] et al. (2008). Figure 5: Nodal temperature response (**top**), discharged energy (**middle**) and \(k(T)\) behaviour at the considered nodal points and at the control volumes interfaces (**bottom**). Figure insets highlight the effect of \(k\) temperature dependency. Figure 6: Nodal temperature response (**top**), discharged energy (**middle**) and \(k(T)\) behaviour at the considered nodal points and at the control volumes interfaces (**bottom**). Figure insets highlight the effect of \(k\) temperature dependency. Tables 2-4 highlight the difference in thermal response between the considered cases, analysing the temperature difference between control and temperature dependent models. The temperature response for the first second in a cool-down process is of significant relevance as current magnetocaloric cycle simulations consider frequencies around 1 Hz [40,41]. \(\Delta T_{t=1s}\) and \(\Delta T_{Max}\) represent the temperature difference for \(t=1s\) and the instant where this difference reaches its peak value, respectively. Another essential aspect when characterizing the system's heat transfer process is the time that a given scenario takes to reach a complete temperature response (CTR). The time difference \(\Delta t_{CTR}\) provides us with the time between the two studied models when the system settles close to the steady-state temperature. The negative values for this time  
Figure 8: Nodal temperature response (**top**) and discharged energy (**middle1**). \(C_{p}(T)\) (**middle2**) and \(k(T)\) (**bottom**) behaviours at the considered nodal points and at the control volumes interfaces. Figure insets highlight the combined effects of \(k\) and \(C_{p}\) temperature dependencies. Figure 7: Nodal temperature response (**top**) and discharged energy (**middle1**). \(C_{p}(T)\) (**middle2**) and \(k(T)\) (**bottom**) behaviours at the considered nodal points and at the control volumes interfaces. Figure insets highlight the combined effects of \(k\) and \(C_{p}\) temperature dependencies. difference indicate that the temperature dependent process is slower for fixed \(C_{p}\) and \(k\) values. Figure 9 shows the normalized temperature difference for the boundary point 1. \[\frac{T(t)-T_{\infty}}{T_{i}-T_{\infty}}=\frac{\Delta T(t)}{\Delta T_{Max}} \tag{7}\]  
Equation (7) represents the ratio between the temperature change from the beginning for each instant \(\mathbf{f}\) and the maximum temperature change that the system can undergo. The normalized temperature change gives a general overlook about which thermal property affects the temperature response the most.