Analytical expression for the friction coefficient of DLCA aggregates based on extended Kirkwood–Riseman theory

ABSTRACT

We use a self-consistent field method, which we have previously validated, to calculate the translational friction coefficient of fractal aerosol particles formed by diffusion-limited cluster aggregation (DLCA). Our method involves solving the Bhatnagar–Gross–Krook model for the velocity around a sphere in the transition flow regime. The velocity and drag results are then used in an extension of Kirkwood–Riseman theory to obtain the drag on the aggregate. Our results span a range of primary sphere Knudsen numbers from 0.01 to 100 for clusters with up to N = 2000 primary spheres. Calculated friction coefficients are in good agreement with experimental data and approach the correct continuum and free molecule limits for small and large Knudsen numbers, respectively. Results show that particles exhibit more continuum-like behavior as the number of primary spheres increase, even when the primary particle is in the free molecule regime; as an illustrative example, the friction coefficient for aggregates with primary sphere Kn = 1 is approximately equal to the continuum friction coefficient for N > 500. We estimate that our calculations are within 10% of the true values of the friction coefficients for the range of Kn and N presented here. Finally, we use our results to develop an analytical expression (Equation (38[38] )) for the friction coefficient over a wide range of aggregate and primary particle sizes.

Copyright © 2017 American Association for Aerosol Research

EDITOR:

• Yannis Drossinos

Notes

¹ In this article, we define the viscosity by the relation , where is the gas density and is the mean thermal speed. This expression describes a hard sphere gas. Furthermore, we use Davies' coefficients (A = 1.257, B = 0.4, and C = 1.1) in the slip correction factor.

² If the flow is uniform, then , where is the uniform velocity.

³ This is because the friction coefficient obtained from our solution of the BGK equation is non-dimensionalized by the free molecule friction coefficient (Epstein's equation). As a result, the friction coefficient decays to zero for decreasing Knudsen number, meaning numerical errors are more prominent for the near-continuum regime.