## 4 Validation of the Numerical Model
### Previous Experimental Campaign by [PERSON] et al. [37]
#### 4.2.4 Results  
In this subsection, the results from the thermal nonlinear analysis of a calcium silicate substrate are presented, in which temperature values were measured at the closest integration Gauss point to the fire contact surface, which are points A, B, C, and D in Figure 17. Figure 18 provides the numerical results with the experimental values using three different approaches for the modelling of thermal properties: (i) constant at ambient temperature; (ii) temperature dependence with density ratio factors according to [56] and specific heat and thermal conductivity ratio factors according to [57]; (iii) temperature dependence with all ratio factors according to [56]. Figure 19 shows the temperature distribution that does not present discontinuities between layers as in the first validation model since in this case the substrate is completely continuous.  
Upon observing these results, the following observations can be made: (i) it was observed that in-depth temperature distribution over time follow the same patterns as the firebrand temperature during the analysed period for all cases; (ii) for the constant thermal properties (Figure 18a), it showed a good fit between results at the beginning of heating phase and then the values started to differ; (iii) the case with thermal properties with temperature dependence according to the reduction factors from [56; 57] exhibited the best behaviour, demonstrating a good agreement for the maximum temperature across all depths (Figure 18b); (iv) the case with thermal properties with temperature dependence according to the reduction factors from [56] (Figure 18c) showed similar results as using constant thermal properties at ambient temperature. This similarity can be attributed to the low variation in the ratio factors (\(k_{k}=1.24\) and \(k_{CP}=1.24\)), at the maximum measured temperature of 200 \({}^{\circ}\)C. In contrast, when utilising the thermal properties based on [54], these ratio factors were higher (\(k_{k}=0.48\) and \(k_{CP}=1.8\)), which can account for the significant difference in the obtained results; (v) in all cases the beginning of the heating phase is well modelled, but for lower temperatures the differences are higher; and (vi) finally, the cooling phase (>2500 s) could not have been accurately validated because the thermal properties are different from the heating phase. During the heat transfer, the material suffers internal damage, and the properties values change--the thermal property values from 20 \({}^{\circ}\)C to 200 \({}^{\circ}\)C are different from 200 \({}^{\circ}\)C to 20 \({}^{\circ}\)C--and these values could not be found in the literature.  
In view of this quite good correlation of both validation studies, it seems fair to argue that the finite element model employed in this work may be deemed satisfactorily validated.  
Figure 17: Boundary conditions adopted for the simulation of the previous experimental campaign by [PERSON] et al. [38].  
Figure 19: Temperature field distribution at 4000 s in Kelvin.  
Figure 18: Comparison between experimental and numerical values in-depth for (**a**) constant thermal properties at ambient temperature, (**b**) thermal properties with temperature dependence [56, 57], and (**c**) thermal properties with temperature dependence [56].