## 4 Discussion
### Results of the Discharge Method  
In the steady-state case, the permeability is first determined in the Darcy regime, and then the inertial coefficient is determined in the Forchheimer regime with the resulting permeability. This treatment is laborious since at least two types of flow sensors and pressure sensors with different ranges and accuracies are needed to satisfy the requirements of measurements across different flow ranges. During the discharge process, the instantaneous flow rate can be indirectly obtained thanks to the isothermal chamber, and the permeability and inertial coefficient can be simultaneously determined by Equation (4). If this method works, it will undoubtedly shortens the measuring time and greatly reduce the air consumption. Figure 9 shows the pressure variation in the chamber during the discharge process for the samples of group 2. The initial data immediately after the opening of the valve are left because those data might include pressure disturbance. The acquired pressure data were first smoothed by the moving average method using 10 points, and then differentiated numerically with eight points to calculate the flow rate. Figure 10 shows the relation of chamber pressure versus the calculated mass flow rate. A subscript \(i\), as the sequence number, is used to refer to any data point (e.g., chamber pressure \(P_{i}\), mass flow rate \(G_{i}\)) in the dataset, and the number of the total data points is denoted as \(n\). To facilitate iterative computation, Equation (4) is written in the following form:  
\[G=-\frac{\mu A\varphi}{2\sqrt{K}\beta}+\sqrt{\left(\frac{\mu A\varphi}{2\sqrt{ K}\beta}\right)^{2}+\frac{\sqrt{K}A^{2}\varphi^{2}(P_{1}^{2}-P_{a}^{2})}{2 \beta R\theta L}}=f(P,K,\beta), \tag{10}\]  
where \(P_{a}\) is the atmospheric pressure. Then, the permeability and inertial coefficient can be determined by the Gauss-Newton fitting method using the following procedure:  
1. Initial value \(K_{0}=1\times 10^{-12}\) and \(\beta_{0}=1\) are given for the Forchheimer flow rate equation, Equation (10): \(G=f(P,K,\beta)\). 2. The square residual error \(S_{\mathrm{pre}}\) is calculated by: \(\sum\left(G_{i}-f(P_{i},K,\beta)\right)^{2}\). Here, The \(G_{i}\) values are obtained by differentiating the measured pressure in the chamber. 3. The elements of the first derivative \(\frac{\partial f}{\partial K}\), \(\frac{\partial f}{\partial\beta}\), and the residuals \(G_{i}-f(P_{i},K,\beta)\) are calculated for \(i\) =1\(\
eg n\). 4. Solve the following two linear equations to obtain \(\Delta K\) and \(\Delta\beta\). \[\sum\left(\frac{\partial f_{i}}{\partial K}\right)^{2}\Delta K+\sum\frac{ \partial f_{i}}{\partial K}\frac{\partial f_{i}}{\partial\beta}\Delta\beta= \sum\frac{\partial f_{i}}{\partial K}(G_{i}-f(P_{i},K,\beta)),\] \[\sum\frac{\partial f_{i}}{\partial K}\frac{\partial f_{i}}{\partial\beta} \Delta K+\sum\left(\frac{\partial f_{i}}{\partial\beta}\right)^{2}\Delta\beta =\sum\frac{\partial f_{i}}{\partial\beta}(G_{i}-f(P_{i},K,\beta)),\] (5) Update \(K\) and \(\beta\) by \(K=K_{0}+\Delta K\) and \(\beta=\beta_{0}+\Delta\beta\). 6. Calculate \(S=\sum\left(G_{i}-f(P_{i},K,\beta)\right)^{2}\), if this value satisfies the convergence condition, i.e., \(\mid S-S_{\mathrm{pre}}\mid\)\(<\varepsilon\), the determination process is ended. If it does not satisfy the condition, the aforementioned process is repeated with \(K_{0}\) and \(\beta_{0}\) replaced by \(K\) and \(\beta\). In Figure 10, five pressure subsections over the whole range, denoted as I, II, III, IV, and V, are uniformly divided from 100 kPa to 600 kPa to calculate the permeability. The measuring accuracy changes at different stages of the discharge process since the rate of pressure drop is always changing. The pressure bands are uniformly determined in order to find a satisfactory section to measure the permeability accurately. Figure 11 shows the permeability \(K\) determined by the discharge method in different pressure subsections for group 2.