The Single-Fiber Collision Rate and Filtration Efficiency for Nanoparticles I: The First-Passage Time Calculation Approach

Figures & data

FIG. 1. Schematic describing the MFPT approach to determine dimensionless collision kernels. Relevant dimensionless length scales for calculation are labeled, and a depiction of particle motion is provided.

FIG. 2. The dimensionless collision kernel as function of the diffusive Knudsen number for a cylindrical cell model in the absence of fluid flow (χ_f = 0), as determined by MFPT calculations. Circles (red): V_f = 6.83 × 10⁻⁴; squares (blue): V_f = 3.42 × 10⁻³; long dashed line (red): Equation (5b)[5b] calculated values with V_f = 6.83 × 10⁻⁴; short dashed line (blue): Equation (5b)[5b] calculated values with V_f = 3.42x 10⁻³; solid line (black): Equation (6b) calculated values.

FIG. 3. Values of the parameter (H/H_C – 1) with H determined from MFPT calculations and H_C calculated with Equation (5b)[5b] for (a) V_f = 0.01 and (b) V_f = 0.07. Circles (red): R = 0; squares (blue): R = 0.03; triangles (yellow): R = 0.10. Plots such as those displayed are used to determine the coefficients C₁ and C₂ as functions of V_f and R.

TABLE 1. Values of the coefficients (a) C₁ and (b) C₂ determined from regressions, as functions of R and V_f

	Vf
(a) C1 values	0.005	0.007	0.01	0.03	0.05	0.07	0.1
R  0.00	0.605	0.626	0.596	0.526	0.504	0.491	0.508
0.01	0.580	0.610	0.569	0.520	0.502	0.490	0.492
0.03	0.548	0.550	0.549	0.492	0.441	0.442	0.443
0.07	0.494	0.472	0.466	0.411	0.383	0.371	0.354
0.10	0.459	0.378	0.408	0.383	0.351	0.338	0.335
0.20	0.311	0.333	0.346	0.282	0.241	0.239	0.277
(b) C2 values
	0.005	0.007	0.010	0.030	0.050	0.070	0.100
R  0.00	0.394	0.382	0.383	0.388	0.386	0.387	0.380
0.01	0.406	0.406	0.400	0.396	0.397	0.396	0.393
0.03	0.430	0.422	0.442	0.421	0.434	0.432	0.431
0.07	0.471	0.477	0.475	0.485	0.495	0.497	0.509
0.10	0.501	0.540	0.518	0.519	0.532	0.542	0.548
0.20	0.628	0.615	0.602	0.641	0.671	0.677	0.660

FIG. 4. A comparison of the dimensionless collision kernels inferred from MFPT calculations to those calculated from Equation (12a)[11b] (Modeled H). Circles (red): C₁ and C₂ determined Equations (13a) and (13b). Squares (blue): tabulated C₁ and C₂ values used for calculations. The central dashed line (black) denotes 1:1 values, whereas the outer dashed lines (gray) denote 1:1.25 and 1:0.75 values.

FIG. 5. Depth filtration predicted single-fiber efficiencies as compared to those inferred from MFPT calculations. (a) Calculated with R ≤ 0.03; (b) calculations with R ≥ 0.07. Circles (red): diffusion collection only (Equation (14a)); squares (blue): combined diffusion and interception collection (Equation (14a) + Equation (14c)[14b] ); triangles (yellow): combined diffusion and interception collection with the correction term (Equation (14d)[14c] ) included.

FIG. 6. Comparison of MFPT results to the experimental measurements provided by Lee and Liu (1982a,b). The parameters noted on the abscissa and ordinate are equivalent to those used by Lee and Liu (1982b). Circles (red): experimental results from Lee and Liu (1982); squares (blue): MFPT calculations.

FIG. 7. A comparison of H calculated from equation (15c) for staggered (circles, red) and aligned (squares, blue) cylinders to results of MFPT calculations under all conditions where R = 0. Dashed curve (green): a 1:1 guideline.

Lee, K.W. and Liu, B.Y. H. (1982b). Theoretical-Study of Aerosol Filtration by Fibrous Filters. Aerosol Sci. Technol. 1:147–161.

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