# Friction Factor Correlation for Regenerator Working
## 4 Computational Model  
The regenerator is modelled using a porous medium theory. Two-dimensional Darcy-Forchheimer porous medium equations are expressed as follows [14]:  
\[\phi\frac{\partial\rho}{\partial t}+\
abla\cdot(\rho\mathbf{v})=0, \tag{18}\]  
\[\frac{\partial(\rho\mathbf{v})}{\partial t}+\
abla\cdot\left(\frac{\rho\mathbf{ v}\cdot\mathbf{v}}{\phi}\right)+\
abla p-\
abla(\mathbf{\tau})=-\left(\frac{\mu \phi}{K}\mathbf{v}+\frac{F\phi^{2}\rho}{\sqrt{K}}\mathbf{v}|\mathbf{v}|\right). \tag{19}\]  
The terms \(\phi\), \(\rho\), \(t\), \(\mathbf{v}\), \(p\), \(\mathbf{\tau}\), \(\mu\), \(K\) and \(F\) represent the porosity, fluid density, time, velocity vector, porous pressure, stress tensor, fluid dynamic viscosity, permeability coefficient and Forchheimer inertial coefficient, respectively. For a homogeneous porous medium with local thermal equilibrium assumption, the one-dimensional energy equation for this porous region is given as [4]:  
\[ \begin{array}{l}\left[\left(\phi k_{f}+(1-\phi)k_{s}\right)\
abla^{2}T+(1- \phi)q_{s}^{\prime\prime\prime}+\mathbf{v}\left(-\frac{\partial p}{\partial x }+\rho_{f}g\right)\right]\\ =\left(\phi\rho_{f}c_{pf}+(1-\phi)\rho_{s}c_{s}\right)\frac{\partial T}{ \partial t}+\rho_{f}c_{pf}\mathbf{v}\frac{\partial T}{\partial x}\end{array}  \tag{20}\]  
where \(q_{s}^{\prime\prime\prime}\) refers to the heat generation rate and \(T\) is the temperature. The term \(c_{p}\) is the isobaric specific heat and \(c_{s}\) is the specific heat. The subscripts \(f\) and \(s\) refer to fluid and solid constituents, respectively. Assuming local thermal equilibrium, the temperature of the gas and solid structure is the same at any time and spatial location due to the condition of perfect thermal contact of the regenerator. In ANSYS-FLUENT [26], the porous medium is solved using equations given as:  
\[\frac{\partial(\phi\rho)}{\partial t}+\
abla\cdot(\phi\rho\mathbf{v})=0, \tag{21}\]  
\[\frac{\partial(\phi\rho\mathbf{v})}{\partial t}+\
abla\cdot(\phi\rho\mathbf{ v}\mathbf{v})+\phi\
abla\mathbf{p}+\
abla\cdot(\phi\tau)=\phi B_{f}-\left(D_{x} \mu+\frac{C_{x}\rho}{2}|\mathbf{v}|\right)\mathbf{v}, \tag{22}\]  
\[\frac{\partial}{\partial t}\left(\phi\rho_{f}E_{f}+(1-\phi)\rho_{s}E_{s} \right)+\
abla\cdot\left(\mathbf{v}\left(\rho_{f}E_{f}+p\right)\right)=\
abla \cdot\left[k_{eff}\
abla T-(\mathbf{\tau}\cdot\mathbf{v})\right]. \tag{23}\]  
The term \(E_{f}=c_{p}T-P/\rho+\mathbf{v}^{2}/2\) is the energy of the fluid and \(k_{eff}=\phi k_{f}+(1-\phi)k_{s}\) is the effective thermal conductivity within the porous regions. Note that the velocity vector \(\mathbf{v}\) used in ANSYS FLUENT [26] corresponds to the superficial velocity. The real value of velocity within the porous region is obtained by dividing the superficial velocity by the porosity of the region, \(\phi\). Relationships between coefficients \(C_{x}\) and \(D_{x}\) as used in [26] and the coefficients of the standard porous medium theory (permeability coefficient, \(K\), and Forchheimer inertial coefficient, \(F\)) may be obtained by comparing Equations (19) and (22). Note that the viscous dissipation represented by the stress tensor, \(\mathbf{\tau}\), and the body force, \(B_{f}\), in Equation (22) are neglected. The relationships obtained are shown as follows:  
\[K=\frac{\phi^{2}}{D_{x}}, \tag{24}\] \begin{table}
\begin{tabular}{l c c} \hline
**Properties** & **Helium** & **Stainless Steel** \\ \hline Ratio of heat coefficient, \(\gamma\) & 1.6667 & - \\ Speed of sound, \(a\) (m/s) & 1019.2 & - \\ Density, \(\rho\) (kg/m\({}^{3}\)) & 4.0115 & 7918.1 \\ Specific heat, \(c_{p}\) (J/kg\(\cdot\)K) & 5193.4 & 453.84 \\ Gas expansion, \(\beta\) (1/K) & 3.3333 \(\times\) 10\({}^{-3}\) & - \\ Thermal conductivity, \(k\) (W/m\(\cdot\)K) & 0.15243 & 14.388 \\ Dynamic viscosity, \(\mu\) (kg/m\(\cdot\)s) & 1.9938\(\times\)10\({}^{-5}\) & - \\ \hline \end{tabular}
\end{table} 
Table 2: Properties of helium gas and stainless steel solid matrix.