# Friction Factor Correlation for Regenerator Working
## 5 Results and Discussion
### Friction Factor Correlation  
In the porous medium theory, the friction factor of most porous media is presented by the permeability Reynolds number, \(Re_{K}=\rho uK^{1/2}/\mu\). The pressure drop characteristics of the porous medium are related to the permeability Reynolds number as given by [4]:  
\[-\
abla P=\frac{1}{2}\frac{f\rho u^{2}}{\sqrt{K}}. \tag{27}\]  
The pressure gradient can also be represented by the Darcy-Forchheimer model as in Equation (7). The following formula can be derived from Equations (7) and (27):  
\[f=\frac{2\sqrt{K}}{\rho u^{2}}\bigg{[}\frac{\mu}{K}u+\frac{F_{\rho}}{\sqrt{K}} u^{2}\bigg{]}. \tag{28}\]  
Figure 6: Validation of instantaneous pressures at locations P2 and P5: (**a,b**) for #180; and (**c,d**) for #94. Further modification of Equation (28) can be made to include the hydraulic diameter as follows:  
\[f=\frac{2\sqrt{K}}{\rho u^{2}}\cdot\frac{d_{h}}{d_{h}}\left[\frac{A_{1}\mu}{K}u+ \frac{A_{2}F\rho}{\sqrt{K}}u^{2}\right], \tag{29}\]  
\[f=\frac{2d_{h}}{\sqrt{K}}\cdot\frac{A_{1}}{Re_{h}}+2A_{2}F. \tag{30}\]  
The permeability coefficient, \(K\) has a unit (length)\({}^{2}\) and the Forchheimer inertial coefficient, \(F\), is dimensionless. Parameters \(A_{1}\) and \(A_{2}\) are constants introduced to correlate the equation to fit the experimental result. Simple dimensional analysis shows that Equation (30) is dimensionless. The derived correlation agrees with the well-known Ergun form presented in Equation (3). From this derived friction factor, the coefficients \(a_{1}\) and \(a_{2}\) in the Ergun equation are defined as:  
\[a_{1}=\frac{2A_{1}d_{h}}{\sqrt{K}}, \tag{31}\]  
\[a_{2}=2A_{2}F. \tag{32}\]  
In the simplest Darcy model, when the velocity is small, the inertia effect is neglected and the friction factor is left with only first term of right-hand side of Equation (30):  
\[f=\frac{2d_{h}}{\sqrt{K}}\cdot\frac{A_{1}}{Re_{d}}. \tag{33}\]  
Finally, with this correlation the porous coefficient predicted through CFD model is used to predict the friction factor of the porous medium and then compared to the experimentally calculated friction factor for the actual sample. The friction factor correlation presented in Equation (33) is derived from the Darcy correlation. Hence comparing the simulation results to experiment requires the friction factor predicted by simulation model to be divided by four. In this study, all comparisons are carried out in line with the Fanning-based correlation. The Fanning-based correlation for computational models is given as:  
\[f=\frac{d_{h}}{2\sqrt{K}}\cdot\frac{A_{1}}{Re_{d}}. \tag{34}\]  
Permeability is an empirical constant used in the porous medium community to characterize the ability to transmit fluid by the porous media. The permeability coefficients for the regenerator samples investigated, obtained from the CFD-assisted method, are tabulated in Table 3.  
A higher regenerator mesh number refers to a finer mesh. This is reflected by the value of the wire diameter of each sample as listed in \[f=\frac{2\sqrt{K}}{\rho u^{2}}\bigg{[}\frac{\mu}{K}u+\frac{F_{\rho}}{\sqrt{K}} u^{2}\bigg{]}. \tag{28}\]0 The permeability is high for a regenerator with a mesh screen number of #30 and decreases as the regenerator mesh screen number increases to #200 (the value for #180 being slightly outside this trend due to a lower porosity value--cf. Table 1). The permeability coefficient in Table 3 is used to calculate the friction factor using Equation (34). Several cases of measurement and numerical prediction were considered for each sample. These corresponded to different level of acoustic excitation produced by the linear motor, and result in different values of Reynolds number, \(Re_{d}\), as defined in Equation (6). The results are plotted in \[f=\frac{2\sqrt{K}}{\rho u^{2}}\bigg{[}\frac{\mu}{K}u+\frac{F_{\rho}}{\sqrt{K}} u^{2}\bigg{]}. \tag{28}\]1