# Friction Factor Correlation for Regenerator Working
## 2 Literature Review  
The friction factor is a dimensionless parameter used to represent the hydrodynamic condition of a flow across a structure. In oscillatory flow the friction factor is defined as [5]:  
\[f_{osc}=\frac{X_{\Delta P}d_{h}}{\frac{1}{2}\rho X_{u_{m}}^{2}L_{r}}. \tag{1}\]  
The terms \(X_{\Delta P}\), \(d_{h}\), \(\rho\), \(X_{u_{m}}\), and \(L_{r}\) represent the amplitude of pressure drop, hydraulic diameter, gas density, amplitude of velocity and length of the regenerator, respectively. The subscript \(u_{m}\) relates to the mean value taken between two measured points. The hydraulic diameter, \(d_{h}\), is defined as [5]:  
\[d_{h}=\frac{\phi}{1-\phi}d_{w}. \tag{2}\]The value of porosity, \(\phi\), and the wire diameter, \(d_{w}\), are usually supplied by the manufacturer of mesh screens. The experimentally measured pressure drop and velocity are used to calculate the friction factor. The results calculated through Equation (1) may be fitted with suitable correlation to represent the hydrodynamic condition of the flow investigated. The basic form of friction factor correlation is given by the standard two-parameter Ergun equation [6]:  
\[f=\frac{a_{1}}{Re}+a_{2}, \tag{3}\]  
where \(a_{1}\) and \(a_{2}\) are constants that fit the condition tested. [PERSON] and [PERSON] [7] showed that the two constants of Equation (3) can be represented by an equation for porosity defined as:  
\[a_{1}=1268-3545\phi+2544\phi^{2}, \tag{4}\]  
\[a_{2}=-2.82+10.7\phi-8.6\phi^{2}. \tag{5}\]  
In their correlation, [PERSON] and [PERSON] [7] defined the Reynolds number by taking into consideration the simple harmonic oscillation in regenerator and expressed it as the first order amplitude of complex Reynolds number, \(Re_{d}\), given by:  
\[Re_{d}=\frac{4<u_{1}>r_{h}\rho_{m}}{\mu}, \tag{6}\]  
where \(<u_{1}>\), \(r_{h}\), \(\rho_{m}\) and \(\mu\) are the first order spatial average velocity, hydraulic radius, mean density and viscosity, respectively. Assuming a simple harmonic oscillation in the regenerator, their correlation based on Equations (3)-(5) is reported to fit well the steady experimental data of [PERSON] and [PERSON] [8]. This equation had been widely used in thermoacoustic community in conjunction with DeltaEC, design software for thermoacoustic devices [9]. The suitability of [PERSON] and [PERSON]'s equation for use in an oscillatory flow is however questionable because it is fitted to steady flow data. A number of research works considered various specific formulations of equations for friction factor, including [PERSON] and [PERSON] [6], [PERSON] et al. [10], [PERSON] and [PERSON] [11] or [PERSON] and [PERSON] [12]. These formulations are generally of the same form as the standard Ergun Equation (3), but are typically multiplied by a certain dimensionless modification factor. This is typically obtained through dimensional analysis, experiments, numerical simulations or their useful combination, and its presence highlights the important differences between oscillatory and steady flow conditions. Another important parameter in an oscillatory flow is referred to as (acoustic) impedance. It is defined as a ratio between pressure and velocity of a flow. However, impedance is a complex number and so in addition to their amplitudes it also includes the phase difference between them [3]. In travelling wave systems, the phase difference between pressure and velocity can be controlled by an RLC network which is also known in some papers as a \"phase shifter\". [PERSON] et al. [13] reported that the phase shifter reduces losses within the regenerator and improves system performance. This indicates that the pressure drop characteristics within the regenerator depend on the phase shift between pressure and velocity. In most of the papers discussed above, the phase difference between pressure and velocity is not set experimentally to a specific value. This may introduce additional unknown effects. There is yet no clear explanation as to how this affects the friction losses of the whole system.