Mass transfer from small spheroids suspended in a turbulent fluid

Abstract

By coupling direct numerical simulation of homogeneous isotropic turbulence with a localised solution of the convection–diffusion equation, we model the rate of transfer of a solute (mass transfer) from the surface of small, neutrally buoyant, axisymmetric, ellipsoidal particles (spheroids) in dilute suspension within a turbulent fluid at large Péclet number, $\textit {Pe}$ . We observe that, at $\textit {Pe} = O(10)$ , the average transfer rate for prolate spheroids is larger than that of spheres with equivalent surface area, whereas oblate spheroids experience a lower average transfer rate. However, as the Péclet number is increased, oblate spheroids can experience an enhancement in mass transfer relative to spheres near an optimal aspect ratio $\lambda \approx 1/4$ . Furthermore, we observe that, for spherical particles, the Sherwood number $\textit {Sh}$ scales approximately as $\textit {Pe}^{0.26}$ over $\textit {Pe} = 1.4\times 10^{1}$ to $1.4\times 10^{4}$ , which is below the $\textit {Pe}^{1/3}$ scaling observed for inertial particles but consistent with available experimental data for tracer-like particles. The discrepancy is attributed to the diffusion-limited temporal response of the concentration boundary layer to turbulent strain fluctuations. A simple model, the quasi-steady flux model, captures both of these phenomena and shows good quantitative agreement with our numerical simulations.

Mechanisms of mass transfer to small spheres sinking in turbulence

“…Batchelor then argues that at large turbulent Péclet number Pe = a 2 τ η /D = Re 3/2 Sc, due to the convection-suppression effect (Batchelor 1979), particle motion due to slip has a negligible effect on the mass transfer rate because the particle rotates isotropically. This convection-suppression mechanism has recently been confirmed in turbulent flows to suppress convective mass transfer by velocity fluctuations occurring on a time scale more rapid than τ η Pe 1/3 (Lawson & Ganapathisubramani 2021). In the limit of Pe → ∞, Batchelor concluded that for small spherical particles, the only component of the relative flow field which is persistent upon averaging over time in the particle reference frame is an axisymmetric strain, whose magnitude scales with 1/τ η .…”

“…Therefore, in Batchelor's analysis, the sinking ratio does not appear as a control parameter. However, for neutrally buoyant spheroids, we have empirically shown shown that at large but finite Pe, the solution of (2.8) is best approximated by (2.12) when the time scale is chosen τ f ∼ Pe 1/3 (Lawson & Ganapathisubramani 2021). Under this hypothesis, it is therefore still possible for slip to contribute to the convective mass transfer, because v 0 does not vanish when averaging over a finite time interval.…”

“…The convection-diffusion equation can now be written in dimensionless form, normalising by the particle radius a and Kolmogorov timescale τ η (Leal 2012;Lawson & Ganapathisubramani 2021):…”

“…This phenomenon occurs in an abundance of engineered and natural processes: chemical products are produced by crystallisation, dissolution, fermentation and heterogeneous reactions in liquid-solid suspensions within stirred tanks and fluidised beds (Myerson 2002;Nienow 2006;Crowe et al 2012); planktonic osmotrophs absorb nutrients (Karp-Boss, Boss & Jumars 1996); pollutants leach from microplastics (Seidensticker et al 2017); bacteria encounter marine viruses (Guasto, Rusconi & Stocker 2012); and breakdown products of marine snow are swept away (Alcolombri et al 2021) as they sink through turbulent ocean waters. In these processes, turbulence, inertia and gravity may drive a convective flow around the particle which enhances mass transfer: gravity may cause particles to sink or float (Harriott 1962), inertia may cause particles to slip (Levins & Glastonbury 1972b) and turbulent eddies may convect material away by strain (Batchelor 1980;Lawson & Ganapathisubramani 2021). In this paper we ask: for small spherical particles sinking in a turbulent flow, what is the dominant mechanism of mass transfer?…”

“…Recently, we have examined Batchelor's scenario numerically for small, neutrally buoyant spheroids in isotropic turbulence at large but finite turbulent Péclet number (Lawson & Ganapathisubramani 2021). The motion of spherical and spheroidal tracer particles is simulated in homogeneous, isotropic turbulence and the relative velocity field, which is prescribed by the Lagrangian time history of the linear velocity gradient and Stokes perturbation sampled by the tracer, is used to force the numerical solution of the convection-diffusion equation in the vicinity of the particle.…”