A methodology to calculate the friction coefficient in the transition regime: Application to straight chains

Abstract

a b s t r a c tA methodology is introduced, the Collision Rate Method (CRM), to calculate the friction coefficient of power-law aggregates across the entire momentum-transfer regime. The friction coefficient is calculated via the ratio of two fictitious particle-aggregate collision rates evaluated in the continuum and slip-flow regimes. The effective collision rates are obtained from the numerical solution of the Laplace equation with Robin boundary condition. The methodology was justified by comparing the slip correction factor of straight-chains composed of up to 50 spherical monomers with literature results. We determined the validity of the CRM to lie in an extended slip-flow regime, the maximum monomer Knudsen number being 2. We calculated the adjusted-sphere radius, the radius of a sphere with the same slip correction factor as the chain, in slip flow. We found it to be weakly dependent on flow conditions, as specified by the carrier-gas mean free path, a dependence that leads to a weak effect on calculated slip correction factors. The CRM was combined with the Adjusted-Sphere Method to extend its validity to all Knudsen numbers. Excellent agreement of calculated, straight-chain, slip correction factors with literature values was obtained for monomer Knudsen numbers up to 100. Various characteristic length scales, geometric (equivalent volume radius, radius of gyration) and dynamic (equivalent hydrodynamic and mobility radii), were calculated.

The electrical mobilities and scalar friction factors of modest-to-high aspect ratio particles in the transition regime

“…(2a) should apply for non-spherical particles (i.e., an expression, which collapses to appropriate continuum and free molecular limits and varies with Knudsen number). By invoking a mass transfer-momentum transfer analogy (i.e., by using a distinct calculation approach from Zhang et al (2012)), Melas, Isella, Konstandopoulos, and Drossinos (2015) additionally have found excellent agreement between the adjusted sphere model and friction factor calculations in the transition regime. Equation (2a) has been recently tested experimentally for flame-synthesized TiO 2 aggregates (Thajudeen, Jeon, & Hogan, 2015), with reasonable agreement found between measurements and equation predictions.…”

Sensitivity analysis of aggregate morphology on mass-mobility relationship and improved parameterizations

“…In the transition regime, the accuracy of this approximation has also been repeatedly verified in various experimental studies (Rogak et al 1993;Chakrabarty et al 2007;Sorensen 2011). Very recently, Melas et al (2015) reported the range of validity of this approximation in the transition regime using the concept of monomer Knudsen number (Kn a D λ/a; not to be confused with Kn). They showed that for Kn a > 2, the projected area approximation held reasonably accurate.…”

Mobility of nanofiber, nanorod, and straight-chain nanoparticles in gases

“…A comparison of the predicted diffusion-equivalent spheres to the measured mobility equivalent diameters for a series of nanorods (Gopalakrishnan et al 2015b) showed excellent agreement (Tian et al 2016). Melas et al (2014Melas et al ( , 2015 investigated the friction coefficient and mobility radius of the fractal-like aggregates in the transition regime, and the application to straight chain agglomerates. These studies provided an opportunity to re-examine the high aspect ratio non-spherical particle dynamics, and the development of a convenient equivalent mobility diameter in high Knudsen number flows.…”

“…The "adjusted sphere" method has been supported by experiments (Cheng et al 1988;Rogak et al 1993), and most recently validated by Melas et al (2014Melas et al ( , 2015 For majority of the irregular-shaped particles, such as fractal-like agglomerates and straight chains, d adj is not available due to the lack of free molecular information. As a convenience, Equation (4) together with a volume equivalent diameter d ev, is frequently used to approximate the slip correction.…”

“…In Equation (37), Kn is defined as lpR s /A p , first presented by Rogak and Flagan (1992). Detailed analytical derivation can be found in the work of Zhang et al (2012) and Melas et al (2015).…”