Low Reynolds number capture of small particles on a cylinder by diffusion, interception, and inertia at subcritical Stokes numbers: Numerical calculations, correlations, and small diffusivity asymptote

Abstract

When the particle-to-target radius ratio R and the inverse Peclet number 1/P are small, particle capture by interception and diffusion by cylinders at low Reynolds numbers may be described via Friedlander’s single similarity parameter Π ≡ R·P^1/3, in the full range 0 < Π < ∞. Particle inertia may substantially enhance this capture efficiency, even at subcritical Stokes numbers S < S*∼2. We have recently shown that this inertial enhancement is the product of an ‘outer’ function E(S) accounting for inertial particle concentration enrichment along the stagnation line, and an ‘inner’ function F(Π, S) describing particle transport near the target. F(Π,S) is first computed here over its full range, (0 < Π < ∞; 0 < S < S*) by numerically solving the corresponding diffusion equation, previously analyzed only in the limits Π = 0 and Π = ∞. Two PDE solvers used (pseudo-spectral orthogonal collocation method and Mathematica’s NDSolve statement) typically agree within 0.1%, with a 0.7% maximal difference. While the previously invoked “additive capture fraction” approximation is now seen to contain errors of up to –17%, new correlations are developed here with a maximum global error of ±0.7%. We also provide a detailed numerical and asymptotic description for the universal structure of the theoretically interesting limit Π → ∞, finding an unusual algebraic decay of the small diffusive contribution to the flux in the region upstream from the critical tangency angle. We use these results to compute the substantial inertial effects reducing the penetration of toxic aerosol fumes, or improving the recovery of valuable aerosol materials (e.g., noble metals and/or semiconductors) for submicron aerosol/carrier gas cases of: Pt(s)/N₂(g) and Ge(s)/He(g).

Copyright © 2019 American Association for Aerosol Research

Notes

1 If particle axial diffusion is negligible compared to axial convection and the overall pressure drop is small compared to the inlet pressure then the  relation between the overall filter capture fraction, η_{cap, SF}, and the single fiber capture fraction η_{cap, SF} will simply be: η_{cap, F} = 1 – exp[–(4/π) φ_f η_{cap, F} L_F/d_f], where  L_F  is the filter “depth” (see, e.g., Friedlander, 2000).

2 Note that for such low Re flows through fiber mattes the conventionally defined bed (matte) permeability “inherits” a weak dependence on both Re_f,0 and ϕ_f via the aforementioned Oseen-Stokes function C(Re).