## 3 Results and Discussion
### Pressure-Drop and Velocity Phase Shift
If we relate the phase shift \(\phi_{i}\) of a regenerator of any structure (\(i\)) to the phase shift of the 30% porosity regenerator \(\phi_{30\%}\), the following relationship may be written:  
\[\frac{\phi_{i}}{\phi_{30\%}}=\frac{\alpha_{i}Re_{\omega,i}}{\alpha_{30\%}Re_{ \omega,30\%}}=\frac{\alpha_{i}}{\alpha_{30\%}}\frac{\rho_{i}\omega_{i}D_{hi} ^{2}}{\rho_{30\%}\omega_{30\%}D_{230\%}^{2}}=\frac{\alpha_{i}}{\alpha_{30 \%}}\frac{\rho_{i}\mu_{30\%}}{\rho_{30\%}\mu_{i}}\frac{D_{hi}^{2}}{D_{h30 \%}}. \tag{18}\]  
Furthermore, the ratio \(\rho/\mu\) is quite constant; it is then possible to write:  
\[\frac{\phi_{i}}{\phi_{30\%}}=\frac{\alpha_{i}}{\alpha_{30\%}}\frac{D_{hi}^{ 2}}{D_{h30\%}^{2}}. \tag{19}\]  
The calculations of those ratios for regenerators Str_35% and Str_40% are given in Table 4. It appears that they are the inverse of each other. \[\frac{\alpha_{i}}{\alpha_{30\%}}\cong\frac{D_{h30\%}^{2}}{D_{hi}^{2}}. \tag{20}\]  
Thus, Equation (20) leads to:  
\[\frac{\phi_{i}}{\phi_{30\%}}=\frac{\alpha_{i}}{\alpha_{30\%}}\frac{D_{hi}^{ 2}}{D_{h30\%}^{2}}=1, \tag{21}\]  
meaning:  
\[\phi_{i}\cong\phi_{30\%}. \tag{22}\]  
This result proves that the phase shift is only frequency-dependent in our experiments. When plotting the experimental phase shift measurements versus frequency altogether, Figure 17 is obtained. The arrangement of the experimental measurement points thus obtained seems to confirm the previous result. In order to obtain correlations depending on characteristics of the regenerators, a modified phase shift \(\phi^{*}\) as a new dimensionless parameter (as defined by [PERSON] et al. [20]) is introduced to obtain the proper following:  
Figure 13:0  
Thus, all our results are plotted for each regenerator for all the corresponding experiments independently of other parameters (length \(L\), stroke of the piston \(C\), and temperature gradient \(\Delta T\)). The proposed equation is also plotted on the graph for comparison in Figure 13:1 \begin{table}
\begin{tabular}{c c c} \hline \hline
**Structure of Regeneration** & \(\mathbf{\alpha_{i}/\alpha_{30}}\) & \((D_{h30\%}/D_{hi})^{2}\) \\ \hline Str\_35\% & 0.59 & 0.56 \\ Str\_40\% & 0.54 & 0.51 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Slope ratio and hydraulic diameter inversed ratio. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Structure of** & \(D_{h}\) & **Correlation** & \(\mathbf{\alpha}\) & **R-Square** \\
**Regenerator** & **(mm)** & **Model** & \(\mathbf{\alpha}\) & **R-Square** \\ \hline Str\_30\% & 0.178 & & 732.93 & 0.98 \\ Str\_35\% & 0.237 & \(\phi(\Delta P_{i}\mu)=\alpha Re_{\omega}\) & 432.91 & 0.99 \\ Str\_40\% & 0.250 & & 397.63 & 0.98 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Phase shift correlations between the pressure drop and speed for different regenerator structure. Therefore, the experimental modified phase shift \(\phi^{*}\) is well fitted by the following correlated equation, as shown in Figure 18. The linear regression plot definitively validates it, as the R-square of the fitting curves is 0.97 with an error of \(\pm 2.39\%\). Thus:  
\[\phi^{*}=2.12\times 10^{-4}Re_{\omega}=\phi(D_{h}/L)^{2.62}. \tag{24}\]  
Therefore, in our case, a correlation equation for phase shift \(\phi\) between the velocity and pressure drop is proposed for our experimental data:  
\[\phi=2.12\times 10^{-4}Re_{\omega}(D_{h}/L)^{-2.62}, \tag{25}\]  
which is valid in the range of \((0.0025\leq D_{h}/L\leq 0.0041;\ 0.0057\leq Re_{\omega}\leq 0.177)\). Equation (25) shows that the phase shift between the velocity and the pressure drop is governed by the hydraulic diameter to length ratio \(D_{h}/L\) and the kinetic Reynolds number \(Re_{\omega}\). It indicates that the effect of the kinetic Reynolds number \(Re_{\omega}\) on the phase shift is more dominant than that of the hydraulic diameter to length ratio \(D_{h}/L\). Figure 17: Phase shift between pressure drop and velocity for different regenerator structures, strokes, lengths, and temperature gradients.