## 2 Methods
### Design of Experiments (DOE)
According to the literature, the relationship of these factors depends on the physical parameters of the thermoacoustic refrigerator; details are not given in Table 2 but in Table 3. These factors will be analyzed quantitatively and qualitatively in the DOE. Therefore, five parameters, each with two levels (low and high), are put forward as factors to create a full \(2^{5}\) factorial, using the full factorial design method defined in [33]. In a full factorial design, every possible combination of factor levels is tested. This means that for a system with \(n\) factors, each at \(k\) levels, the total number of experiments would be \(k^{n}=2^{5}\). Table 4 illustrates the matrix to be followed based on the \(2^{5}\) levels by the factors in Table 2. The low and high levels must remain orthogonal to each other, preserving the same proportion between the factors. These values depend on the nominal values chosen for the DOE. Multiplying the factors by this value to adjust for the lower and upper limits is referred to as orthogonality in the full factorial design, which helps to maintain the \"balance\" of the DOE. For instance, if the nominal value of a factor is 0.10, the low and high limits will be 0.05 and 0.15, respectively. The determination of the interaction signs is calculated by multiplying the signs of each interaction shown in Table 4 up to the third interaction for matrix size ratios. For example, the first value in column AB is \"+\", derived by multiplying the signs \(A-\) and \(B-\). For the higher interactions, the remaining signs are calculated similarly. Using these interaction signs and the \(COP\) response, the average \(COP\) values (low and high) are computed, to obtain the coefficients. These coefficients are calculated with MINITAB 19 [33] and are used to generate the main interaction effects plots. To verify that the chosen parameters in a DOE are orthogonal (independent of each other), the variance inflation factor (VIF) is calculated with MINITAB 19 from Table 5. The VIF is a measure of how much the variance of a regression coefficient is inflated due to multicollinearity between factors [33]. A VIF close to 1 for all factors suggests that the factors are independent; see Table 6. \begin{table}
\begin{tabular}{c l r r} \hline \hline
**Factor** & **Name** & \multicolumn{2}{c}{**Level**} & **Units** \\  & & **Low–High** & \\ \hline A & Drive ratio (\(D\)) & 1 & 3 & \% \\ B & Stack position (\(X_{s}\)) & 0.05 & 0.15 & m \\ C & Temperature difference (\(\Delta T_{m}\)) & 5 & 15 & \({}^{\circ}\)K \\ D & Blockage ratio (\(B\)) & 0.25 & 0.75 & N/A \\ E & Regenerator length (\(L_{s}\)) & 0.05 & 0.15 & m \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Level and interval values by factor used in the DOE. Several statistical decision approaches constitute the analysis phase: examining the significance of the distribution, performing analysis of variance (ANOVA), and evaluating the R-Sq coefficient (which represents the ratio of the variation explained by each variable). These methods enable the identification of significant coefficients and facilitate the construction of the transfer function in two stages. Firstly, the significance of the \(COP\) distribution is assessed, to establish a confidence level of 95%, which corresponds to a significance level of \(\alpha=0.05\) for a single proportion of the population. Alpha (\(\alpha\)) represents the maximum acceptable probability of incorrectly rejecting the null hypothesis when the alternative hypothesis is true.