# Friction Factor Correlation for Regenerator Working
## 2 Literature Review
It is found, however, that several situations have been reported in the literature where the meaning of friction factor is unclear when the phase difference is present. [PERSON] [14] showed that when a phase difference exists, setting the amplitude alone is not sufficient to predict the hydrodynamics of flow in the porous medium. The experimental detail is reported by [PERSON] et al. [15]. The coefficients \(a_{1}\) and \(a_{2}\) of the friction correlation (Equation (3)) are presented by [PERSON] et al. [15] as Darcy and Forchheimer coefficients. The friction factor gained from experimental results is discussed from a theoretical point of view. The theoretical explanation involves Stokes drag force, \(C_{s}\), frictional force due to boundary layer, \(C_{B}\), and inviscid form drag, \(C_{I}\), first defined by [PERSON] and [PERSON] [16] and further developed theoretically by [PERSON] [14]. The theoretical prediction in that study is in good agreement with experimental results when the effect of phase difference is considered. In this paper, the phase difference between pressure and velocity will be referred to as \"phase shifting\" for brevity. The findings of [PERSON] (1984) may be related to the \"phase shifting\" phenomena reported by [PERSON] et al. [PERSON] et al. (2005) and the breathing factor, \(B\), introduced by [PERSON] and [PERSON] and [PERSON] (2006). The phase shift effect is also observed in the experimental study of [PERSON] and [PERSON] [PERSON] and [PERSON] (2006) when the kinetic Reynolds number increases with frequency. Clearly the phase shifting between pressure and velocity has an effect on friction losses reported in many investigations of oscillatory flows. There is a possibility that these phenomena occur as a result of the experimental conditions not being set to a particular time-phasing between pressure and velocity. However, no attention has been given to the possibility of controlling the time-phase between pressure and velocity in the experimentation for determination of the hydrodynamic condition of the oscillating flow through a regenerator. This is crucial in thermoacoustic applications because the regenerator should work with a well-controlled travelling-wave time-phasing. The porous and tortuous structure of the regenerator is difficult and computationally very expensive to model using CFD if the full original structure is to be replicated. An economical way of modelling such a structure is through the use of the porous medium theory. The porous medium theory models a porous structure through the use of additional parameters in the transport equation that represent the flow resistivity. Flow resistance is modelled through the knowledge of pressure drop caused by the structure. According to the Darcy-Forchheimer model, the pressure gradient, \(\
abla p\), is defined as [PERSON] (1961):  
\[\
abla p=-\frac{\mu}{K}\mathbf{v}-FK^{-1/2}\rho|\mathbf{v}|\mathbf{v}, \tag{7}\]  
where \(\mu,\rho\), and \(\mathbf{v}\) are the gas dynamic viscosity, density and velocity, respectively. The terms \(K\) and \(F\) represent the additional parameters representing form drag to the flow and the terms are known as permeability coefficient and Forchheimer inertial coefficient, respectively. In the simplest case, the flow across the porous medium can be represented by [PERSON]'s law. [PERSON]'s law simply neglects the second term on the right hand side of Equation (7) leaving only the term with permeability coefficient, \(K\), to represent the porosity experienced by the flow. This law is valid only when the Reynolds number, defined by the average flow velocity, is an order of magnitude smaller than one. When the velocity is high, the Forchheimer modified equation should be considered to account for the inertial losses in the porous medium. In this condition, Equation (7) should be applied in full. In a highly viscous flow, a further modification is suggested where the second term on the right hand side of Equation (7) is replaced with the [PERSON]'s term, \(K\
abla^{2}\mathbf{v}\).