# Research on Regularities of Cyclic Air Motion through a Respiratory Filter
## 2 Materials and Methods
The values of the empirical coefficients are specified in the course of experimental research. Some data on their values can be found in a previous work [46]. The growth of the fibre radius depends on the number of settled particles, which can be represented as:  
Figure 1:2  
where \(\beta_{2}\) is density of sediment on the surface of the filter fibre, kg/m\({}^{3}\); \(M\)=\(\beta_{2}h_{1}\) is dust holding capacity, kg/m\({}^{2}\); \(h_{1}\) is dust sediment height, m; \(V_{p}\) is dust particle volume, m\({}^{3}\)  
The process of settling the aerosol particles on a cylinder can be expressed by the well-known equation:  
\[\left\{ \begin{array}{ll}\frac{\beta N_{0}}{\beta_{1}}=-\gamma(x,\ t)N_{0}\\ \frac{\beta N}{\beta_{1}}=U_{0}\gamma(x,\ t)N_{0}\end{array} \right., \tag{14}\]where \(N(x,t)\) is number of settled particles over time \(t\) per unit volume of the filter at depth \(x\); \(\gamma(x,t)\) is filtration coefficient, which is determined by the formula:  
\[\gamma=\frac{2\eta(a)\beta H}{\pi a^{2}}, \tag{15}\]  
where \(\eta(a)\) is the total coefficient of capturing the aerosol particles by the filter fibres. To introduce the dimensionless variables \(m=M/M_{0}\) into the second equation of the system (14), where \(M_{0}\) is the maximum dust holding capacity of the filter, when reaching the limiting pressure drop, kg/m\({}^{2}\); \(h=x/H\), \(n=N/N_{0}\), \(t^{\prime}=tLV_{p}U_{0}(t)\pi^{-1}\), then it can be written in a more general form:  
\[\frac{dm}{dt^{\prime}}=\frac{\eta(a_{n})n(t^{\prime})}{m}. \tag{16}\]  
From this equation, the time of the filtration phase end can be obtained:  
\[t^{\prime}=\int_{0}^{m_{1}}\frac{mdm}{\eta(a_{n})}, \tag{17}\]  
where \(m_{1}=m(0,\ t)\) is dimensionless dust holding capacity on the fibre surface, formed by a layer of aerosol particles, the thickness of which overlaps the distance between the fibres. Having determined \(n(t^{\prime})\) from Equation (15) and substituting it into the first equation of the system (14) after appropriate transformations, it is possible to obtain:  
\[\frac{\partial}{\partial h}\left(\frac{m}{\eta(a_{n})}\frac{dm}{dt^{\prime}} \right)+\frac{N_{0}a^{2}}{\beta}\frac{\partial}{\partial t^{\prime}}=0, \tag{18}\]  
with initial conditions \(m(h,\ 0)=1\), \(m(0,\ t^{\ })=m1\), \(c(0,\ t^{\ })=1\). Expression (17), taking into account the independence of the mixed derivative from the differentiation order, makes it possible to obtain an equation for calculating the amount of settled dust on the fibres, which changes with the dust sediment accumulation:  
\[\frac{dm}{dh}+\frac{\eta(a_{n})m}{m}\frac{N_{0}a^{2}}{\beta}=0. \tag{19}\]  
Thus, using expression (18) to replace the integration variable by \(m\) in (11), we obtain the formula for calculating the pressure drop during dust accumulation:  
\[\Delta p(t)=\mu\cdot L^{-1}\cdot U_{0}\cdot\int_{0}^{m_{2}}\frac{R_{0}(a_{n} )mdm}{\eta(a_{n})}. \tag{20}\]  
where \(m_{2}=m(h_{1},t)\) is dimensionless dust holding capacity on the fibre surface at the final moment with thickness \(h_{1}\). To verify the compliance of formulas (10) and (20), theoretical calculations have been compared with experimental data of determining the pressure drop on disposable filter respirators of the Standart brand with a surface area \(S=0.005\) m\({}^{2}\), manufactured commercially by SPE \"Standard\" (Dnipro, Ukraine) from \"Elephlen\" material, taking into consideration the dust sediment accumulation on the filter fibres. The parameters of the filtering polypropylene material are given in Table 1. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**The Average Radius of the Fibres,** & **Surface Density of Fibre Packing,** & **Filtering Layer Thickness,** & **Coefficient,** & **Fibre Length 1,** \\ \(a\), \(\mu\)m & _G_**; \(g/\)m\({}^{2}\) & _H_**, mm** & \(A\) & _L_**, **m** \\ \hline
2.5 & 45 & 3.2 & 0.8–1.3 & 7.96–\(10^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the filtering polypropylene material Elephlen.