## 3 Results
### Numerical Results  
#### 3.2.1 Design of Experiment Analyses (DOE)  
In order to understand the thermal problem and obtain the best possible fit between numerical and experimental results, two different optimizations were carried out that were based on the DOE methodology. First, the thermal properties of different layers were analyzed by applying the DOE methodology over a steady-state numerical wall FEM model. The input parameters used were the thermal conductivity and emissivity of the cavities in the block. The output parameter was the wall thermal transmittance. The sensitivity analysis and response surface results are presented in Figure 8.  
The sensitivity analysis shows that the two most influential materials were Layer 3 (LWMA) and Layer 2 (LWBA25), while the least influential were Layer 1 (GWG) and Layer 4 (GWP). Furthermore, the emissivity of the cavities in the block was the third most influential factor. The response surfaces in Figure 8 show that the highest material thermal conductivity also had the highest wall thermal transmittance, and this variation was more important for Layer 3 (LWMA) and Layer 2 (LWBA25). We found that the wall thermal transmittance was affected by the hygroscopic properties of the materials used. Small changes in the laboratory tests provide significant variation in the thermal transmittance of the wall. This fact was important for the materials in Layer 3 (LWMA) and Layer 2 (LWBA25). Accurate thermal conductivity values must be obtained, and the influence of the laboratory temperature and moisture conditions must be taken into account. In order to achieve this, a second optimization based on the DOE methodology was carried out to obtain a good agreement with real test conditions. Figure 7: Transient experimental results showing temperature and heat fluxes in thermal flux Direction 2. We also built a new DOE analysis over a thermal transient wall FEM model. In this case, the thermal conductivities of Layer 3 (LWMA) and Layer 2 (LWAR25), and the ambient laboratory temperature were used as input parameters. Initial values of these input parameters are shown in Table 2 and they varied by \(\pm\)20%. The maximum and minimum values, amplitude, and phase of the thermal flux were considered the output parameters for comparison with the experimental results. From this DOE analysis, the numerical transient thermal wall response was optimized using the following objective function (W/m\({}^{2}\)):  
\[\Psi=\sqrt{(A^{exp}-A^{num})^{2}+(\zeta^{exp}-\zeta^{num})^{2}}\ \ \sim 0 \tag{11}\]  
where \(A^{exp}=\frac{\hat{d}_{exp}^{max}-\hat{d}_{exp}^{min}}{2}\) is the experimental thermal flux amplitude (W/m\({}^{2}\)); \(A^{num}=\frac{\hat{d}_{max}^{max}-\hat{d}_{max}^{min}}{2}\) is the numerical thermal flux amplitude (W/m\({}^{2}\)); \(\zeta^{exp}=\hat{d}_{exp}^{max}-A^{exp}\) is the  
Figure 8: DOE results: (**a**) Sensitivity analysis of parameters and response surfaces of (**b**) LWAR25 and LWMA thermal conductivities vs. thermal transmittance and (**c**) GWG and GWP thermal conductivities vs. thermal transmittance. experimental thermal flux phase (W/m\({}^{2}\)); \(\hat{q}_{exp}^{max}=\hat{q}_{num}^{max}-A^{num}\), is the numerical thermal flux phase (W/m\({}^{2}\)); \(\hat{q}_{exp}^{max}\) is the maximum experimental thermal flux value (see Figure 7, W/m\({}^{2}\)); \(\hat{q}_{exp}^{min}\) is the minimum experimental thermal flux value (see Figure 7, W/m\({}^{2}\)); \(\hat{q}_{num}^{max}\) is the maximum numerical thermal flux value (W/m\({}^{2}\)); and \(\hat{q}_{num}^{min}\) is the minimum numerical thermal flux value (W/m\({}^{2}\)). The most relevant results from the multi-criteria optimization based on DOE are shown in Figure 9. This graph shows the input and output parameters, their range of variation, and the four candidate points, which meet the requirements of the objective function. Candidate point 4 was selected for the subsequent numerical simulations. It had a value of 0.54 W/m\({}^{2}\) for the objective function according to Equation (11).