## 2 Materials and Methods  
For flowing through porous media with high enough velocity, the pressure gradient \(\mathrm{d}P/\mathrm{d}x\) versus flow velocity is described using the Forchheimer equation:  
\[-\frac{\mathrm{d}P}{\mathrm{d}x}=\frac{\mu v}{K}+\beta\frac{\rho v^{2}}{K^{1/2}}, \tag{1}\]  
where \(\mu\) is the air viscosity, \(v\) is the velocity, \(\rho\) is the density, \(K\) is the permeability and \(\beta\) is the inertial coefficient. The Forchheimer equation includes two terms denoting the viscous effect and the inertial effect. Removing the quadratic term yields the Darcy equation. The Forchheimer equation employs the square root of permeability as equivalent length characteristics in order to agree with the linear Darcy law at low velocities, and the dimensionless coefficient in the quadratic term is used to represent the inertial effect.  
Figure 1 schematically shows the physical model for air flow through a porous medium. The flow is assumed to be steady and fully developed, and only in the length direction (\(x\) coordinate). Actually, the distribution of pores with respect to shape and size is irregular in the porous medium. On the microscopic scale, the flow quantities (velocity, pressure, etc.) are also clearly irregular. The Forchheimer regime models the flow through a porous medium as a pipe flow, and therefore the quantities of interest are described using the space-averaged quantities for representative areas that cross many pores. That is, as shown in Figure 1, the pressure and velocity are considered to be uniformly distributed along a tiny length (\(\mathrm{d}x\)). Moreover, the porous medium is considered isotropic and homogeneous, and therefore the ratio of the circulation area and the total area equals the porosity.  
Accordingly, the mean flow velocity in each thin section (\(\mathrm{d}x\)) varies with the air density, and can be calculated by the following expression:  
\[v=\frac{G}{\rho A\varphi}, \tag{2}\]  
where \(G\) is the mass flow rate, \(A\) is the cross-sectional area, and \(\varphi\) is the porosity.  
According to the ideal gas state equation, the air density \(\rho\) can be expressed as below:  
\[\rho=\frac{P}{RT}, \tag{3}\]  
where \(R\) is the gas constant and \(T\) is the temperature.  
Substituting Equations (2) and (3) into Equation (1) and integrating the pressure with respect to length (\(x\)-coordinate), with the boundary conditions \(P=P_{1}\) (\(x=0\) [m]), \(P=P_{2}\) (\(x=L\) [m]), weobtain the following modified Forchheimer equation:  
\[\frac{\beta RT}{\sqrt{K}\varphi^{2}A^{2}}G^{2}+\frac{\mu RT}{K\varphi A}G+\frac {P_{2}^{2}-P_{1}^{2}}{2L}=0. \tag{4}\]  
Figure 1: Schematic diagram of airflow through a porous medium.  
The square root of the permeability is employed as a characteristic dimension and a Reynolds number is defined as  
\[Re=\frac{\rho vK^{1/2}}{\mu}. \tag{5}\]  
A dimensionless friction factor \(f\) is defined as  
\[f=\frac{(-\mathrm{d}P/\mathrm{d}x)\sqrt{K}}{\rho v^{2}}. \tag{6}\]  
Comparing Equation (6) with the Darcy-Weisbach equation for the case of flowing in a cylindrical pipe, it is observed that the square root of permeability is actually equivalent to two times the hydraulic diameter of the pipe. Substituting Equations (5) and (6) into Equation (1) obtains the following correlation for the friction factor as a function of the Reynolds number:  
\[f=\frac{1}{Re}+\beta. \tag{7}\]