Population Balance Models for Particulate Flows in Porous Media: Breakage and Shear-Induced Events

Abstract

Transport and particulate processes are ubiquitous in environmental, industrial and biological applications, often involving complex geometries and porous media. In this work we present a general population balance model for particle transport at the pore-scale, including aggregation, breakage and surface deposition. The various terms in the equations are analysed with a dimensional analysis, including a novel collision-induced breakage mechanism, and split into one- and two-particles processes. While the first are linear processes, they might both depend on local flow properties (e.g. shear). This means that the upscaling (via volume averaging and homogenisation) to a macroscopic (Darcy-scale) description requires closures assumptions. We discuss this problem and derive an effective macroscopic term for the shear-induced events, such as breakage caused by shear forces on the transported particles. We focus on breakage events as prototype for linear shear-induced events and derive upscaled breakage frequencies in periodic geometries, starting from nonlinear power-law dependence on the local fluid shear rate. Results are presented for a two-dimensional channel flow and a three dimensional regular arrangement of spheres, for arbitrarily fast (mixing-limited) events. Implications for linearised shear-induced collisions are also discussed. This work lays the foundations of a new general framework for multiscale modelling of particulate flows.

Aggregation of amphiphilic nanocubes in equilibrium and under shear

“…Shear can detach pieces from an isolated aggregate, if the hydrodynamic drag force exerted on the cubes exposed to the flow exceeds their characteristic binding force F bind . We describe this event through the kernel 56 with shear rate .…”

Numerical treatments for large eddy simulations of liquid–liquid dispersions via population balance equation

“…In order to obtain the solution of the PBE, different methods can be employed for different industrial applications. [1][2][3] The quadrature-based moments method (QBMM) accurately captures the dynamics of the particle size distribution while reducing computational complexity compared to other methods, attracting more attention. 4 The most popular QBMM is the quadrature method of moments (QMOM), [5][6][7][8] whereas it is worth mentioning that other moment methods exist beyond QBMM, as, for example, the Taylorseries expansion method of moment (TEMOM).…”