Numerical Simulation of Open-Circuit Continuous Mills Using a Non-Linear Population Balance Framework: Incorporation of Non-First-Order Effects

Abstract

Population balance modeling has been used as a tool for simulating, optimizing, and designing various particulate processes, including milling. A fundamental tenet of the traditional models for milling processes is the first-order breakage kinetics. Ample data obtained from batch milling studies show that this assumption is not necessarily valid for certain milling systems. In the present theoretical investigation, an attempt has been made to incorporate these experimentally observed non-first-order effects into continuous mill models within the context of a novel non-linear population balance framework. In view of two idealized flow regimes, i.e., perfect mixing and plug-flow, continuous mills operating in the open-circuit mode are numerically simulated. The simulations indicate that not only does the product size distribution depend on the degree of mixedness in a continuous mill, but also on the non-first-order effects arising from multi-particle interactions.

A Novel PBM for Nanomilling of Drugs in a Recirculating Wet Stirred Media Mill: Impacts of Batch Size, Flow Rate, and Back-Mixing

“…Space time, which equals the average residence time if there are no dead zones and bypass in the milling chamber, can be calculated using τ s = V m (1 − c )/ Q . For a single-pass continuous mill, an increase in τ s results in a finer PSD; a plug-flow regime yields a narrower and somewhat finer PSD in the mill than the perfect-mixing regime [ 53 , 54 ]. It should be noted that the average residence time in the mill for a recirculation operation τ c can be regarded as the effective milling time and is described by τ c = N to τ s , where N to is the number of turnovers defined by N to = Qt / V s .…”

“…Space time, which equals the average residence time if there are no dead zones and bypass in the milling chamber, can be calculated using τs = Vm(1 − c)/Q. For a single-pass continuous mill, an increase in τs results in a finer PSD; a plug-flow regime yields a narrower and somewhat finer PSD in the mill than the perfect-mixing regime [53,54]. It should be noted The volume-based PSD of the drug suspensions at different milling times was determined using laser diffraction with an LS 13-320 Beckman Coulter instrument (Brea, CA, USA).…”

A study of the collisional fragmentation problem using the Gamma distribution approximation

“…As it was shown, only in case p(x, y; z) = p(x, y) Eq. (4) can be considered as a subcase of the general nonlinear fragmentation equation developed in [6].…”

“…One possible extension is the introduction of the nonlinearity in the breakage process, i.e., to take into account that the rate and the way in which a particle breaks depends on the other particles of the population. An extensive discussion regarding the need of the nonlinear fragmentation equation to model some grinding processes can be found in [6]. A particular type of the nonlinear fragmentation is the collisional fragmentation, where the particle breaks upon collision with other particles.…”