# Numerical Study of the Filtration Performance for Electrospun Nanofiber Membranes
## 2 Numerical Simulation Method
### Theoretical Basis of Numerical Simulation
#### 2.2.2 Pressure Drop Analysis Model  
Darcy's law describes the pressure drop \(\triangle\)P along the thickness of \(\delta\) as an incompressible fluid passing through a porous medium under static and laminar flow conditions, as shown in Equation (4):  
\[\Delta P=\frac{1}{k}\mu v\delta \tag{4}\]In the equation, \(k\) is the fiber permeability coefficient (m\({}^{2}\)); \(\mu\) is the dynamic viscosity, Pa\(\cdot\)s; and \(v\) is the fluid surface velocity, defined as the volumetric flow rate per unit cross-sectional area of the filter, m/s. For fibrous media, the fiber permeability coefficient is affected by the solid volume fraction, fiber size, shape, and their arrangement. When the fibers exhibit a unimodal distribution, it can be assumed that the fiber permeability coefficient k is a function of the solid volume fraction s and the fiber radius \(\beta\) in the medium, as shown in Equation (5):  
\[k=\beta^{2}\cdot f(s) \tag{5}\]  
For fibrous media with high porosity, [PERSON] proposed an empirical expression for f(s) [19], which is the dimensionless permeability, as shown in Equation (6). The empirical expression of f(s) is based on experimental data of gas flow (specifically air) in high-porosity fiber media. The experiments focused on measuring pressure drops across various fibrous filters under different flow rates, which were then correlated with the solid volume fraction to develop the dimensionless permeability function. The original experiments were conducted with airflows at low to moderate velocities (0.1-10 cm/s) through glass fiber and cotton fiber media with porosities ranging from 0.85 to 0.99. The derived expression has been widely validated for gas filtration applications and is particularly suitable for the nanofiber membranes in this study (porosity > 0.95). \[f(s)=\frac{1}{16s^{\frac{3}{2}}(1+56s^{3})} \tag{6}\]  
In the Happel model [20] and the Kuwabara model [21], the calculation formulas for the fiber permeability coefficient k are as follows: Equation (7), Happel model; Equation (8), Kuwabara model. \[k=\frac{\beta^{2}(-0.5-\frac{1}{2}\ln s)}{4s}\text{ Happel model} \tag{7}\]  
\[k=\frac{\beta^{2}(-0.75-\frac{1}{2}\ln s)}{4s}\text{ Kuwabara model} \tag{8}\]  
The three permeability models have specific applicability ranges and characteristics. [PERSON] model [19]: Applicable for solid volume fraction (s) < 0.2, derived from airflow experiments in high-porosity fibrous media. It assumes random fiber distribution and ignores fiber interactions. Happel model [20]: Valid for s < 0.1. Theoretically, permeability becomes zero at s = 0.367, but practical application is limited to s < 0.1 to avoid significant errors. Kuwabara model [21]: Suitable for s < 0.15. It exhibits zero permeability at s = 0.223 but is only reliable for dilute fiber systems (s < 0.15). All samples in this study have s = 11.3-17.7% (Table 1), approaching the upper limits of the models, leading to deviations between calculations and experiments. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sample** & **SVF/\%** & **Sample** & **SVF/\%** \\ \hline
7\% & 17.7 & 3.0 h & 11.3 \\
9\% & 11.3 & 3.5 h & 11.6 \\
11\% & 12.0 & 4.0 h & 12.7 \\
13\% & 15.3 & 4.5 h & 13.3 \\
15\% & 13.5 & 5.0 h & 13.5 \\ \hline \hline \end{tabular} Note: For PAN concentration samples (7–15%), the electrospinning time = 3 h; for electrospinning time samples (3.0 h–5.0 h), the PAN concentration = 9%.  
\end{table} 
Table 1: Solid volume fraction of fiber filter media model. When using the GeoDict software to solve the fluid flow control equations, the built-in LIR (Left-Identify-Right) solver is adopted. The numerical calculation convergence is evaluated based on the residual. A residual less than 0.001 is set as the standard for the convergence of the simulation calculation. The calculation direction of the flow field is set along the Z-axis, that is, along the thickness direction of the electrospun nanofiber filter material.