## 4 Discussion  
### Results of the Steady-State Method  
Figure 6 plots the pressure difference as a function of the mass flow rate for all the test media. The pressure difference increases with the flow rate and appears slightly nonlinear as the flow rate becomes adequately large. The viscous effect and the inertial effect can be viewed by permeability \(K\) and inertial coefficient \(\beta\), which can be determined according to the modified Forchheimer equation. Observing these coefficients provides more insight than observing the pressure drop. The viscous effect dominates the flow pattern at a sufficiently small flow velocity, and in this case the Darcy equation can be used to characterize the flow behavior. However, how small flow velocity can be used as the boundary for the transition from viscous flow to inertial flow is unknown. To the best knowledge of the authors, it is almost impossible to define a universal boundary for the transition of regimes applicable to all kinds of porous media. Although some researchers presented a definite boundary of the Reynolds number for the regime transition, it depends on the properties of the porous  
Figure 5: Schematic of the apparatus for discharge method. Figure 6: Variation of measured pressure difference with mass flow rate. material and the working fluid they used. For example, [PERSON] and [PERSON] (2016) reported that the boundary for viscous flow regime and inertial flow regime is \(Re=1\); however, it cannot be used in this work. For this problem, firstly, experiments are conducted under a slight pressure drop less than 3 kPa, where the air compressibility can be neglected and the pressure difference should exhibit a linear relationship with the flow rate. Figure 7 shows the measuring results in the form of pressure difference versus the mass flow rate. The data points are fitted using a least squares straight line, and the permeability can be calculated by the slope. The selected flow region proved appropriate since the \(R^{2}\) value of the fitting line exceeds 0.98. Then, experiments are conducted with large pressure drops. With the resulting permeability \(K\), the inertial coefficient \(\beta\) is determined by the data points close to the maximum flow rate, where the inertial effect is regarded as completely dominating the flow pattern. Table 3 lists the determined coefficients \(K\) and \(\beta\). In Figure 8, the experimental data for the samples of group 2 are presented in the form of friction factor versus Reynolds number, and the axes are plotted on a log-log scale. The Darcy equation in the form of \(f\)-Re is represented by a linear dashed line. Clearly, the data points significantly depart from the Darcy regime when \(Re>0.1\). Therefore, \(Re=0.1\) can be considered as the boundary separating the viscous flow from the inertial flow. Actually, there exists a transitional region between the viscous and inertial regimes; however, the region is uncertain and thus neglected in order to facilitate analysis. The maximum flow rate used for determination of the permeability for the media 7-11 mm (group 2) corresponds to the Reynolds number of 0.019, 0.024, 0.033, 0.039, and 0.052, respectively, and thus the reliability of the obtained permeability can be confirmed. Moreover, the friction factor converges to the inertial coefficient as the Reynolds number is adequately large. Taking group 2 as an example, the data points for the sample of 11 mm, 10 mm, 9 mm, 8 mm, and 7 mm can be well represented by analytical expressions \(f=1/Re+0.623,f=1/Re+0.578,f=1/Re+0.613,f=1/Re+0.585,\) and \(f=1/Re+0.522\), respectively.