# Research on Regularities of Cyclic Air Motion through a Respiratory Filter
## 2 Materials and Methods
The total resistance to the air flow of the model is equal to the sum of the resistances of all cylinders (fibres), and then the resistance force, which is proportional to the speed, in the complex form, is written as follows:  
\[R_{0}=\gamma_{k}\cdot U, \tag{6}\]  
where \(\gamma_{k}=\gamma_{1}+i\gamma_{2}\) is the complex constant, (s/m), \(i\) is an imaginary unit. This expression can also be written as a sum of two components which are proportional, respectively, to the velocity \(U\)(m/s) and acceleration \(\dot{U}\) (m/s\({}^{2}\)) [39]. \[R_{0}=(\gamma_{1}+i\cdot\gamma_{2})\cdot U=\gamma_{1}\cdot U+\frac{\gamma_{2} }{\omega}\cdot\dot{U}, \tag{7}\]  
Having determined the acceleration as a derivative of the air velocity and rejecting the imaginary part, we obtain:  
\[R_{0}=U_{0}\cdot(\gamma_{1}\cos(\omega t)-\gamma_{2}\sin(\omega t)), \tag{8}\]  
Thus, the first part of the expression, which is related to the real part of the variable \(\gamma\), is proportional to the velocity and corresponds to the resistance force of friction on the cylinder surface. The other part can be called inertial; it depends on the frequency of respiratory movements. Variable \(\gamma_{1}\) depends on the density of fibres, which can be presented based on solving equations of stream function near the surface of an isolated cylinder, taking into account the influence of adjacent fibres by formula [39]\(\gamma_{1}=(a\cdot 4\cdot\pi)/(v\cdot(-0.5\cdot\ln\beta-\lambda))\), where \(v\) is the kinematic viscosity of the air, (m\({}^{2}\)/s). Variable \(\gamma_{2}\), according to the assumption of [PERSON] [40], is found by the formula: \(\gamma_{2}=\sqrt{2\cdot\rho_{p}/\omega\cdot\mu\cdot(i-1)}\), where \(\rho_{n}\) is the air flow density, (kg/m\({}^{3}\)). If we omit the imaginary part, the formula for calculating the value of non-dimensional resistance force of the filter fibres is written as follows:  
\[R_{0}=U_{0}\cdot\left[\frac{4\cdot a\cdot\pi\cdot v^{-1}\cdot\cos(\omega t)}{ -0.5\cdot\ln\beta-\lambda}+\sqrt{\frac{2\cdot\rho_{p}}{\omega\cdot\mu}}\cdot \sin(\omega t)\right], \tag{9}\]  
Meanwhile, the expression for estimating pressure drop (\(\Delta p\), Pa) is written as:  
\[\Delta p=\mu\cdot L^{-1}\cdot U_{0}^{2}\cdot\left[\frac{4\cdot a\cdot\pi \cdot v^{-1}\cdot\cos(\omega t)}{-0.5\cdot\ln\beta-\lambda}+\sqrt{\frac{2\cdot \rho_{p}}{\omega\cdot\mu}}\cdot\sin(\omega t)\right], \tag{10}\]  
As can be seen from formula (10), the increase in respiratory rate results in an increase in the amplitude of fluctuations in pressure drop. To determine a change in the pressure drop during the dust sediment accumulation on respirator filters, we use Krish's assumption that fibres with an asymmetric sediment can be approximated by a circular cylinder with a cross-sectional area equal to the midsection area of the dust-laden fibre (thickened fibre model) [41; 42]. It is considered that the length of the fibres per unit volume does not change, and the increase in the pressure drop is conditioned by the thickening of the sediment layer on the dusty fibres. This model, in contrast to the dendritic one [43; 44], allows consideration of the influence of the accumulated dust sediment on the hydrodynamics of the air flow movement. The pressure drop is determined by the formula [45]:  
Figure 1:0  
where \(R_{0}(a_{n}(x,t))\)--dimensionless force acting per unit length of dust-laden fibre with radius \(a_{n}(x,t)\), which is located at the depth of the filter \(x\) during the time t from the beginning of dust accumulation. To calculate the change in the fibre radius during the accumulation of dust sediment, we use the formula proposed by [PERSON] and [PERSON] [34]:  
Figure 1:1  
where \(h\) is the ratio of the height of the dust sediment on the filter fibre to its radius; \(k\), \(b\) are empirical coefficients depending on the physical-chemical properties of dust, air humidity, and particle shape; \(\xi\) is the angle formed by the dust sediment on the fibre.