# Numerical Study of the Filtration Performance for Electrospun Nanofiber Membranes
## 2 Numerical Simulation Method
### Theoretical Basis of Numerical Simulation
#### 2.2.3 Particle Control Equations  
When evaluating the initial filtration efficiency of electrospun nanofiber filter materials under steady-state conditions, the FilterDict module was used for numerical simulation. The simulation of filtration efficiency in the GeoDict software(GeoDict 2023.1) mainly involves different capture mechanisms when particles pass through the filter medium, such as interception and inertial collision, as well as the interactions between particles and fibers.  
The behavior of electrospun nanofiber filter materials capturing particulate matter from the airflow is actually a comprehensive problem involving gas-solid two-phase flow and dust particle adhesion. Therefore, when conducting numerical simulations to predict filtration efficiency, it is crucial to analyze the various forces acting on the particulate matter in the airflow passing through the fibers.  
Based on the forces acting on the particles in the flow field, the following equation representing particle motion can be established [23]. By solving the following equation, relevant information such as the particle position can be obtained.  
\[\overset{\rightarrow}{W}(t)=m\frac{d\overset{\rightarrow}{u}p}{dt}-6\pi\mu R \mathcal{C}_{c}(\overset{\rightarrow}{u}-\overset{\rightarrow}{u}_{p}) \tag{9}\]  
\[D=\frac{k_{B}T}{\gamma} \tag{10}\]In the equation, \(m\) is the mass of the particle, kg; \(\overset{\rightarrow}{v}\) is the velocity of the particle, m/s; \(\mu\) is the dynamic viscosity, Pa-s; R is the diameter of the particle, m; \(Cc\) is the Cunningham correction coefficient; \(\overset{\rightarrow}{u}\) is the gas velocity, m/s; and where W(t) is the Brownian force vector, which is a function of time t, representing the random force exerted by fluid molecules on particles due to Brownian motion. \(k_{B}\) is the Boltzmann constant, J/K; \(T\) is the temperature, K; and \(\gamma\) is the friction factor.  
When numerically simulating the motion of particles, the Lagrangian method is adopted. Meanwhile, to ensure consistency with the experimental conditions for the preparation of electrospun nanofiber membranes [24], the same flow field settings are maintained in the numerical simulation of the filtration efficiency in this work. Filtration efficiency is defined as the ratio of the number of particles captured by the filter medium to the total number of particles entering the filter medium, usually expressed as a percentage [25]. The consistent flow field settings include the following: face velocity = 6.89 cm/s; temperature = 20 \({}^{\circ}\)C; air density = 1.204 kg/m\({}^{3}\); dynamic viscosity = 1.834 \(\times\) 10\({}^{-5}\) kg/(m\(\cdot\)s); no-slip boundary at fiber surfaces; and periodic boundaries in X/Y directions. During the simulation, the distance from the centroid of the particle to the surface of the fiber is tracked in real time. If this distance is less than or equal to the radius of the particle, the particle is determined to be captured. This model is called the \"Caught on first touch\" model in the GeoDict software. The FilterDict module is used for numerical simulation. In the numerical calculation, the surface flow velocity is set to 6.89 cm/s, the particle diameter is 2.5 \(\upmu\)m, the density of the particles is 2165 kg/m\({}^{3}\), and the total number of particles tracked is 20,000 to ensure consistency with the experimental conditions for the preparation of electrospun nanofiber membranes [24]. The 20,000 particles (2.5 \(\upmu\)m diameter) are injected into the computational domain (2000 \(\times\) 2000 \(\times\) thickness voxels, 25 nm/voxel), corresponding to the experimental concentration (100 mg/m\({}^{3}\)) to ensure statistical significance in efficiency calculation.