## 3 Results and Discussion
### Friction Factor Evaluation  
As we have seen, in alternating flow, all the flow parameters (pressure, velocity, and temperature) vary over time. One can therefore determine the friction factor \(f\) at a particular time (\(\theta(0)\), \(\theta(u_{max})\),.......). The literature study showed that many authors,  
Figure 10: Effect of frequency on gas density. (**a**) Evolution of the gas density of regenerator Str_40% versus phase angle. (**b**) Density variation ratio versus the frequency of each regenerator (\(L_{1}\) = 60 mm, \(C_{1}\) = 64.4 mm, for isothermal flow). such as [PERSON] et al. [4], [PERSON] et al. [35], and [PERSON] and [PERSON] [27], determine the friction factor at the time of maximum velocity as:  
\[f(u_{max})=\frac{\Delta P(u_{max})D_{h}}{\frac{1}{2}\rho(u_{max})u(u_{max})^{2}L}, \tag{15}\]  
where the velocity \(u\) is obtained by dividing the frontal velocity by porosity \(\varepsilon\). We first determine the friction factor \(f\) on the basis of this relationship. In a second step, we discuss the suitability of this relationship for an alternating flow. In the formula for calculating \(f\), the flow parameters, such as density \(\rho\) and pressure drop \(\Delta P\), are determined at the position of maximum velocity \(\theta(u_{max})\) and denoted \(\rho(u_{max})\) and \(\Delta P(u_{max})\), respectively. These quantities are calculated during the compression phase (when the hot gas enters the regenerator) at the time when the fluid speed reaches its maximum value. Figure 11a-c show the evolution of the friction factor \(f(u_{max})\) as a function of the kinetic Reynolds number \(Re(u_{max})\) for different regenerator structures. Since the tested friction factor is a function of the velocity and pressure drop, the uncertainties of the tested friction factor, denoted by the error bars in Figure 11, are estimated by the law of propagation (Equation (11)). The largest uncertainty in the friction factor was computed to be approximately 5.50% for Str_30%, 7.60% for Str_35%, and 8.9% for Str_40%. We note that the uncertainty for the Str_30% regenerator is lower than that of Str_35% and Str_40% structures. In addition, the friction factor seems to decrease with the frequency of the Reynolds number for the regenerator of porosity 30%, despite some dispersion of values. This result is consistent with the works of [PERSON] et al. [36] on complex geometry regenerators. Nothing like this is clearly visible for other regenerators. Indeed, we observe that it is almost constant (less than 0.05% dispersion). Figure 11: Evolution of friction factor at maximum velocity of regenerator ((a): Str_30%, (b): Str_35%, (c): Str_40%. \(L_{1}=60\) mm, \(C_{1}=64.4\) mm) versus kinetic Reynolds number \(Re_{\omega}\). These dispersions are due to the velocity patterns being relatively flat around the maximum value (Figure 8). This leads to uncertainty in the determination of \(\theta(u_{max})\), which can have a significant impact. Indeed, the pressure drop \(\Delta P(u_{max})\) measured for maximum fluid velocity time can vary considerably over a small range of rotation angles \(\theta\) (Figure 12). This implies potential substantial dispersion in the friction factor \(f\) values. Although it is difficult to determine clear trends, there is a real quantitative difference in the friction factor depending on the type of regenerator. Indeed, if it seems logical that the porosity impacts the friction factor (\(\varepsilon\) decreases, \(f\) increases), it can be seen that for identical porosity variations of 5% between regenerators, the friction factor changes significantly in the first case (passage from 30% to 35%) and very little in the second (passage from 35% to 40%). This is due to the difference in structure geometry. The Str_30% regenerator is the only one with a pyramid structure that is much more disruptive than the channel structures of the Str_35% and Str_40% regenerators. Thus, the 30% to 35% evolution is more relevant than the 35% to 40% one.