### 1. Introduction

_{c}) and Knudsen number (Kn) play an important role in estimating accurate filtration efficiency. In cases where the aerosol-particle size in air approaches the mean free path of gas molecules, the discontinuities of the medium should be considered. C

_{c}accounts for these discontinuities and can be useful for describing aerosol properties, such as sedimentation and diffusion [1, 8–10]. For the interaction of gases and particles, Kn, the ratio of mean free path (

*λ*) to particle diameter (d

_{p}), is widely used [11].

*C*

*was not considered because the derivation of an analytical expression considering this factor was complicated.*

_{c}*C*

*for different ranges of Kn, with three different formulas, which exhibited a good correlation with an original equation within specified ranges. However, these approximated equations are limited because they are valid only for a particular size range. However, in the diffusion-mechanism-dominant regime for small particle sizes, aerosol slip effects are non-negligible, and should be corrected using*

_{c}*C*

*.*

_{c}*C*

*is simplified and modified to accommodate all the particle-size ranges. The newly derived expression includes the slip correction, which is implicitly considered in the diffusion coefficient for aerosol particles with low Kn values.*

_{c}### 2. Theoretical Backgrounds

##### (1)

$${C}_{c}=1+\frac{\lambda}{{d}_{p}}\left(2.514+0.8\text{exp}\left(-0.55\frac{{d}_{p}}{\lambda}\right)\right)$$*λ*is the mean free path of air molecules as a constant, and

*d*

*is the particle diameter. However, this expression is complex, and requires simplification for parameterization studies [1, 12]. One of the simplified expressions for*

_{p}*C*

*, used in many studies, is as follows [2]:*

_{c}*C*

*(Eq. (2)) is simple and sufficiently accurate compared to original expression (Eq. (1)) [2]. For example, the diffusion coefficient (*

_{c}*D*) and Peclet number (

*Pe*), considering

*C*

*, can be expressed as follows:*

_{c}##### (3)

$$D=\frac{{kTC}_{c}}{3\pi \mu {d}_{p}}\cong \frac{kT}{3\pi \mu {d}_{p}}\left(1+3.34\frac{\lambda}{{d}_{p}}\right)$$##### (4)

$$\begin{array}{l}P{e}^{-2/3}={\left(\frac{2au}{D}\right)}^{-2/3}={\left(\frac{D}{{D}_{f}u}\right)}^{2/3}\\ ={\left(\frac{1}{{D}_{f}u}\right)}^{2/3}{\left(\frac{kT}{3\pi \mu {d}_{p}}\right)}^{2/3}{\left(1+3.34\frac{\lambda}{{d}_{p}}\right)}^{2/3}\end{array}$$*k*is the Boltzmann constant,

*T*is the absolute temperature, is the viscosity of air,

*D*

*is the diameter of the filter, and*

_{f}*u*is the flow velocity. Subsequently, the aerosol single fiber efficiency (

*η*

*) in association with the diffusion mechanism can be expressed as follows:*

_{D}##### (5)

$$\begin{array}{l}{\eta}_{D}=2.6{\left(\frac{1-\alpha}{K}\right)}^{1/3}P{e}^{-2/3}=2.6{\left(\frac{1-\alpha}{K}\right)}^{1/3}{\left(\frac{{kTC}_{c}}{3\pi \mu {d}_{p}u{D}_{f}}\right)}^{2/3}\\ =2.6{\left(\frac{1-\alpha}{K}\right)}^{1/3}{\left(\frac{kT}{3\pi \mu {d}_{p}u{D}_{f}}\right)}^{2/3}{\left(1+3.34\frac{\lambda}{{d}_{p}}\right)}^{2/3}\end{array}$$*K*is the hydrodynamic factor of the Kuwabara flow $\left(=-\frac{1}{2}ln\alpha -\frac{3}{4}+\alpha -\frac{{\alpha}^{2}}{4}\right)$, and

*α*is the volume fraction, solidity, or packing density of the filter [13].

*C*

*for three different size ranges, as follows:*

_{c}##### (6)

$$\begin{array}{l}{C}_{c}=1,\hspace{0.17em}\text{for\hspace{0.17em}large\hspace{0.17em}particles\hspace{0.17em}with\hspace{0.17em}a\hspace{0.17em}continuum-flow-regime\hspace{0.17em}assumption\hspace{0.17em}}\left(\frac{\lambda}{{d}_{p}}\ll 1\right);\\ {C}_{c}=3.33\frac{\lambda}{{d}_{p}},\hspace{0.17em}\text{for\hspace{0.17em}very\hspace{0.17em}small\hspace{0.17em}particles\hspace{0.17em}}\left(\frac{\lambda}{{d}_{p}}\gg 1\right);\\ {C}_{c}=3.69{\left(\frac{\lambda}{{d}_{p}}\right)}^{1/2},\hspace{0.17em}\text{for\hspace{0.17em}particles\hspace{0.17em}with\hspace{0.17em}intermediate\hspace{0.17em}sizes.}\end{array}$$*C*

*are considered, the filtration single fiber efficiency should be expressed by three different formulas. Moreover, if the minimum single fiber efficiency as well as the corresponding*

_{c}*d*

*varies for different formulas, the estimation of the filtration-mechanism characteristics considering*

_{p}*C*

*becomes confusing; therefore, this factor was neglected in several previous studies. Thus, for practical purposes, a more general approximation that covers all size ranges is required, considering the continuous changes in*

_{c}*C*

*.*

_{c}### 3. Results and Discussion

### 3.1. New Approximated Expression for the Cunningham Slip Correction Factor and Single Fiber Efficiency

*η*

*due to diffusion can be expressed in proportion to ${C}_{c}^{2/3}({\eta}_{D}\propto {C}_{c}^{2/3})$. Thus, for the further evolution of the analytical expression for the MPPS,*

_{D}*C*

*should be simplified. In this study, we evaluated ${C}_{c}^{2/3}$ using the 1*

_{c}^{st}order Taylor series expansion as a function of the

*d*

*:*

_{p}##### (7)

$${C}_{c}^{2/3}\cong {\left(1+3.34\frac{\lambda}{{d}_{p}}\right)}^{{\scriptstyle \frac{2}{3}}}\cong \zeta \left\{1+\frac{2}{3}\left(3.34\frac{\lambda}{{d}_{p}}\right)\right\}$$##### (8)

$${C}_{c}\cong {\zeta}^{3/2}\left\{1+\frac{2}{3}\left(3.34\frac{\lambda}{{d}_{p}}\right)\right\}=\vartheta {\left\{1+\frac{2}{3}\left(3.34\frac{\lambda}{{d}_{p}}\right)\right\}}^{{\scriptstyle \frac{3}{2}}}$$##### (10)

$$\begin{array}{l}D=\frac{{kTC}_{c}}{3\pi \mu {d}_{p}}\cong \frac{\vartheta kT}{3\pi \mu {d}_{p}}{\left\{1+\frac{2}{3}(3.34\frac{\lambda}{{d}_{p}})\right\}}^{{\scriptstyle \frac{3}{2}}}\\ =0.6{\left(\frac{{d}_{p}}{\lambda}\right)}^{0.1}\frac{kT}{3\pi \mu {d}_{p}}{\left\{1+\frac{2}{3}\left(3.34\frac{\lambda}{{d}_{p}}\right)\right\}}^{{\scriptstyle \frac{3}{2}}}\\ {D}^{2/3}={\left(\frac{{kTC}_{c}}{3\pi \mu {d}_{p}}\right)}^{2/3}\cong {\left\{0.6{\left(\frac{{d}_{p}}{\lambda}\right)}^{0.1}\right\}}^{2/3}{\left(\frac{kT}{3\pi \mu {d}_{p}}\right)}^{2/3}\left\{1+\frac{2}{3}\left(3.34\frac{\lambda}{{d}_{p}}\right)\right\}\\ P{e}^{-2/3}={\left(\frac{{D}_{f}u}{D}\right)}^{-2/3}={\left(\frac{D}{{D}_{f}u}\right)}^{2/3}={\left\{0.6{\left(\frac{{d}_{p}}{\lambda}\right)}^{0.1}\right\}}^{2/3}\left\{1+\frac{2}{3}\left(3.34\frac{\lambda}{{d}_{p}}\right)\right\}\\ ={(0.6)}^{2/3}{\left(\frac{{d}_{p}}{\lambda}\right)}^{1/15}{\left(\frac{kT}{3\pi \mu {d}_{p}{D}_{f}u}\right)}^{2/3}\left\{1+\frac{2}{3}\left(3.34\frac{\lambda}{{d}_{p}}\right)\right\}\end{array}$$*η*

*for diffusion can be expressed as a linear function of the*

_{D}*d*

*, as in Eq. (5) and (10).*

_{p}*C*

*,*

_{c}*D*, and

*Pe*, respectively; the results are in good agreement with each other. Here, the “exact”

*C*

*refers to values calculated from Eq. (1).*

_{c}*C*

*,*

_{c}*D*, and

*Pe*, respectively.

*η*

*, including that owing to interception and gravitational settling, is expressed as a function of the*

_{D}*d*

*:*

_{p}##### (11)

$$\begin{array}{l}{\eta}_{T}={\eta}_{D}+{\eta}_{R}+{\eta}_{G}\\ =2.6{(0.6)}^{2/3}{\left(\frac{1-\alpha}{K}\right)}^{1/3}{\left(\frac{1}{\lambda}\right)}^{1/15}{\left(\frac{kT}{3{D}_{f}u\pi \mu}\right)}^{2/3}{d}_{p}^{-3/5}\\ +\frac{5.2(3.34){(0.6)}^{2/3}}{3}{\left(\frac{1-\alpha}{K}\right)}^{1/3}{(\lambda )}^{14/15}\\ {\left(\frac{kT}{3{D}_{f}u\pi \mu}\right)}^{2/3}{d}_{p}^{-8/5}+\left(\frac{1-\alpha}{K}\right){\left(\frac{{d}_{p}}{{D}_{f}}\right)}^{2}+\frac{{\rho}_{p}{d}_{p}^{2}g}{18\mu u}\end{array}$$*η*

*and*

_{R}*η*

*are the single fiber efficiencies due to interception and gravitational settling, respectively.*

_{G}##### (12)

$$\begin{array}{l}{\eta}_{R}=\left(\frac{1-\alpha}{K}\right)\frac{{R}^{2}}{1+R}\cong \left(\frac{1-\alpha}{K}\right){R}^{2}=\left(\frac{1-\alpha}{K}\right){\left(\frac{{d}_{p}}{{D}_{f}}\right)}^{2},\\ {\eta}_{G}=\frac{{\rho}_{p}{d}_{p}^{2}g}{18\mu u}\end{array}$$*R*is the interception parameter $\left(=\frac{{d}_{p}}{{D}_{f}}\right)$ and

*g*is the gravitational coefficient. It should be noted that the newly approximated diffusion single fiber efficiency (

*η*

*) in Eq. (11) can now be expressed as a linear combination of the*

_{D}*d*

*.*

_{p}*C*

*estimates the exact*

_{c}*C*

*, and related parameters, such as*

_{c}*D*and

*Pe*, with good confidence. Fig. 2 shows

*C*

*,*

_{c}*D*, and

*Pe*as functions of the

*d*

*. The exact values of*

_{p}*C*

*which are calculated using Eq. (1) is compared with the approximated values calculated using Eq. (10). The filter diameter (*

_{c}*D*

*) is set at 0.02 mm, and the packing density and flow velocity are set at 0.08 and 1 cm/s, respectively. The results exhibit a good agreement (Fig. 2).*

_{f}*d*

*for different assumptions of*

_{p}*C*

*. For a filter diameter of 0.02 mm (*

_{c}*D*

*= 0.02 mm), and flow velocities of 1 cm/s and 10 cm/s, the values of the single fiber efficiencies for the exact*

_{f}*C*

*,*

_{c}*C*

*= 1, and the approximated*

_{c}*C*

*from this study are compared. As seen in Fig. 3, the approximated solutions of the single fiber efficiencies using the approximated*

_{c}*C*

*agree with those using the exact*

_{c}*C*

*. The single fiber efficiency with unity*

_{c}*C*

*exhibits a discrepancy compared to the exact values, increasing with the decreasing particle size.*

_{c}### 3.2. Analytic Expression for the MPPS of Fibrous Filters

*C*

*or with complex expressions [15–16]. Thus, the effects of*

_{c}*C*

*and the corresponding expression for the minimum single fiber efficiency, as well as the MPPS, must be improved in a way that considers*

_{c}*C*

*and simplifies those expressions. In this study, we approximated*

_{c}*C*

*to accurately estimate the single fiber efficiency and MPPS, as well as the minimum single fiber efficiency.*

_{c}*d*

*:*

_{p}##### (13)

$$\begin{array}{l}\frac{\partial {\eta}_{T}}{\partial {d}_{p}}=\frac{-3(2.6){(0.6)}^{2/3}}{3\pi \mu {d}_{p}}{\left(\frac{1-\alpha}{K}\right)}^{1/3}{\lambda}^{-1/15}{\left(\frac{kT}{3{D}_{f}u\pi \mu}\right)}^{2/3}{d}_{p}^{-\frac{8}{5}}\\ -\frac{8(5.2)(3.34){(0.6)}^{2/3}}{(3)(5)}{\left(\frac{1-\alpha}{K}\right)}^{1/3}{\lambda}^{14/15}{\left(\frac{kT}{3{D}_{f}u\pi \mu}\right)}^{2/3}{d}_{p}^{-13/5}\\ +\left\{2\left(\frac{1-\alpha}{K}\right){\left(\frac{1}{{D}_{f}}\right)}^{2}+\frac{{\rho}_{p}g}{9\mu u}\right\}{d}_{p}=0\end{array}$$##### (14)

$$\begin{array}{l}{\varphi}_{1}=-\frac{3(2.6){(0.6)}^{2/3}}{3\pi \mu {d}_{p}}{\left(\frac{1-\alpha}{K}\right)}^{1/3}{\lambda}^{-1/15}{\left(\frac{kT}{3{D}_{f}u\pi \mu}\right)}^{2/3}\\ {\varphi}_{2}=-\frac{8(5.2)(3.34){(0.6)}^{2/3}}{(3)(5)}{\left(\frac{1-\alpha}{K}\right)}^{1/3}{\lambda}^{14/15}{\left(\frac{kT}{3{D}_{f}u\pi \mu}\right)}^{2/3}\\ {\varphi}_{3}=\left\{2\left(\frac{1-\alpha}{K}\right){\left(\frac{1}{{D}_{f}}\right)}^{2}+\frac{{\rho}_{p}g}{9\mu u}\right\}\end{array}$$*ϕ*

_{1}and

*ϕ*

_{2}) refer to the diffusion, and the third term (

*ϕ*

_{3}) refers to the interception, and gravitational settling mechanisms in Eq. (14). The single fiber efficiency due to diffusion increases as the

*d*

*decreases, while that due to interception increases as the*

_{p}*d*

*increases. Thus, the leading terms can be estimated for each dominant particle or collector-size region. If the two dominant terms in Eq. (14) are determined, the resulting approximated equations can be generated as follows [6, 7, 16]:*

_{p}##### (15)

$$\begin{array}{l}{\varphi}_{1}{d}_{p}^{-8/5}+{\varphi}_{3}{d}_{p}=0\\ {\varphi}_{2}{d}_{p}^{-13/5}+{\varphi}_{3}{d}_{p}=0.\end{array}$$##### (16)

$${d}_{s1}={\left(\frac{-{\varphi}_{1}}{{\varphi}_{3}}\right)}^{13/5},\hspace{0.17em}{d}_{s2}={\left(\frac{-{\varphi}_{2}}{{\varphi}_{3}}\right)}^{5/18}$$##### (17)

$${d}_{p,min}=\overline{\omega}\sqrt{\frac{1}{\frac{1}{{d}_{s1}^{2}}+\frac{1}{{d}_{s2}^{2}}}}$$*ω̄*, can be adapted to minimize the errors between the exact and approximated MPPS. In this study, an

*ω̄*of 0.6 is suggested by comparing the exact and harmonic mean results.

*D*

*of 0.02 mm and 0.1 mm. The MPPS can be determined in accordance with a dominant mechanism, for any given condition. As seen in Fig. 4, the MPPS increases as the filtration velocity increases up to a point, and after the point, decreases with the increasing velocity. According to the previous study [5], the MPPS shifts to a larger size as the flow velocity increases for fiber diameter (*

_{f}*d*

*) of 0.02 mm, when Brownian diffusion and the gravitational force are dominant. The MPPS shifts to a smaller size as the flow velocity increases for*

_{f}*d*

*of 0.1 mm, when Brownian diffusion and interception are dominant. As shown in Fig. 4, the MPPS obtained using the analytical expression of this study agrees well with the exact MPPS. Fig. 5 depicts the MPPS as a function of the filter packing density;*

_{f}*d*

*of 0.02 and 0.2 mm are compared for filtration velocities of 1 and 10 cm/s, respectively. The actual MPPS and the one determined in this study show a good agreement. For a*

_{f}*D*

*of 0.2 mm, the MPPS increases with the increase in packing density up to a value of approximately 0.3, after which the MPPS starts to decrease. However, for a*

_{f}*D*

*of 0.02 mm, the MPPS decreases with the increase in packing density over the density range in Fig. 5.*

_{f}### 3.3. Minimum Single Fiber Efficiency

*η*

*) is obtained by substituting d*

_{min}_{p}in Eq. (11) with d

_{p,min}in Eq. (17). Fig. 6 shows the minimum single fiber efficiency as a function of the filtration velocity; the exact solution and approximated one from this study are compared. The minimum single fiber efficiency decreases with the increase in filtration velocity. The minimum single fiber efficiency is greater at smaller

*d*

*over the filtration velocities up to ~ 20 cm/s and for lower filtration velocities. Nevertheless, the difference in the minimum single fiber efficiency between different*

_{f}*d*

*decreases with the increase in filtration velocity.*

_{f}### 4. Conclusions

*is significant, for particle sizes below 2–3 μm, in air at ambient conditions. In filtration studies, the effects of*

_{c}*C*

*are critical and can be considered for the exact estimation of the collection efficiency and MPPS. For instance, in the regime where the diffusion mechanism is dominant, the aerosol slip effects cannot be neglected for small particle sizes, and the effects should be calibrated using*

_{c}*C*

*. However, the complex formula for*

_{c}*C*

*requires a simplified expression, for parameterization studies.*

_{c}*C*

*expression in association with the slip effects of small particles. Slip correction was implicitly embedded in the*

_{c}*D*for aerosol particles with low Kn. The comparison of the obtained analytical solution for the MPPS, which considers the particle slip of small particles, with the exact solution demonstrated a good agreement. This result indicates that the simplified expression for

*C*

*, demonstrated in this study, can be used for obtaining the analytical solution for the MPPS, including particles with low Knudsen numbers, with a high confidence level.*

_{c}