Computational Modeling of Magnesium Hydroxide Precipitation and Kinetics Parameters Identification.

Magnesium is a critical raw material and its recovery as Mg(OH)2 from saltwork brines can be realized via precipitation. The effective design, optimization, and scale-up of such a process require the development of a computational model accounting for the effect of fluid dynamics, homogeneous and heterogeneous nucleation, molecular growth, and aggregation. The unknown kinetics parameters are inferred and validated in this work by using experimental data produced with a T2mm-mixer and a T3mm-mixer, guaranteeing fast and efficient mixing. The flow field in the T-mixers is fully characterized by using the k-ε turbulence model implemented in the computational fluid dynamics (CFD) code OpenFOAM. The model is based on a simplified plug flow reactor model, instructed by detailed CFD simulations. It incorporates Bromley's activity coefficient correction and a micro-mixing model for the calculation of the supersaturation ratio. The population balance equation is solved by exploiting the quadrature method of moments, and mass balances are used for updating the reactive ions concentrations, accounting for the precipitated solid. To avoid unphysical results, global constrained optimization is used for kinetics parameters identification, exploiting experimentally measured particle size distribution (PSD). The inferred kinetics set is validated by comparing PSDs at different operative conditions both in the T2mm-mixer and the T3mm-mixer. The developed computational model, including the kinetics parameters estimated for the first time in this work, will be used for the design of a prototype for the industrial precipitation of Mg(OH)2 from saltwork brines in an industrial environment.

Use of Microfluidic Capillary Electrophoresis for the Determination of Multi-Component Protein Adsorption Isotherms: Application to High-Throughput Analysis for Hydrophobic Interaction Chromatography.

Chromatography is a widely used separation process for purification of biopharmaceuticals that is able to obtain high purities and concentrations. The phenomena that occur during separation, mass transfer and adsorption are quite complex. To better understand these phenomena and their mechanisms, multi-component adsorption isotherms must be investigated. High-throughput methodologies are a very powerful tool to determine adsorption isotherms and they waste very small amounts of sample and chemicals, but the quantification of component concentrations is a real bottleneck in multi-component isotherm determination. The behavior of bovine serum albumin, Corynebacterium diphtheriae CRM197 protein and lysozyme, selected as model proteins in binary mixtures with hydrophobic resin, is investigated here. In this work we propose a new method for determining multi-component adsorption isotherms using high-throughput experiments with filter plates, by exploiting microfluidic capillary electrophoresis. The precision and accuracy of the microfluidic capillary electrophoresis platform were evaluated in order to assess the procedure; they were both found to be high and the procedure is thus reliable in determining adsorption isotherms for binary mixtures. Multi-component adsorption isotherms were determined with a totally high-throughput procedure that turned out to be a very fast and powerful tool. The same procedure can be applied to every kind of high-throughput screening.

Shear-Induced Heteroaggregation of Oppositely Charged Colloidal Particles.

This paper investigates numerically the shear-induced aggregation of mixed populations of colloidal particles leading to the formation of clusters. Suspensions with different amounts of positively and negatively charged colloidal particles are simulated. To resolve the aggregation kinetics and structural properties of the formed clusters, we resort to a mixed deterministic-stochastic simulation method. The method is built on a combination of a Monte Carlo algorithm to sample a statistically expected sequence of encounter events between the suspended particles and a discrete element method built in the framework of Stokesian dynamics to simulate the encounters in a fully predictive manner. Results reveal a strong influence of the composition of the population on both the aggregation kinetics and the aggregate structure. In particular, we observe a size-stabilization phenomenon taking place in the suspension when the relative concentration of the majority particles lies in the range 80-85%; i.e., starting from primary particles, after a short growth period, we observed a cessation of aggregation. Inspection of the aggregate morphology shows that the formed clusters are composed of few minority particles placed in the inner region, while the aggregate surface is covered by majority particles, acting to provide a shielding effect against further growth.

Numerical Methods for the Solution of Population Balance Equations Coupled with Computational Fluid Dynamics.

This review article discusses the solution of population balance equations, for the simulation of disperse multiphase systems, tightly coupled with computational fluid dynamics. Although several methods are discussed, the focus is on quadrature-based moment methods (QBMMs) with particular attention to the quadrature method of moments, the conditional quadrature method of moments, and the direct quadrature method of moments. The relationship between the population balance equation, in its generalized form, and the Euler-Euler multiphase flow models, notably the two-fluid model, is thoroughly discussed. Then the closure problem and the use of Gaussian quadratures to overcome it are analyzed. The review concludes with the presentation of numerical issues and guidelines for users of these modeling approaches.

The effect of flow configuration on hydrodynamic stresses and dispersion of low density rigid aggregates.

The effect of flow configuration on the distribution of internal stresses in rigid colloidal aggregates was investigated numerically for cluster-cluster aggregates with fractal dimensions ranging from 1.7 to 2.3. Stokesian dynamics was used to evaluate the hydrodynamic force on each monomer, while the internal intermonomer interactions were calculated by applying force and torque balances on each primary particle. The examined two-dimensional flows were characterised by a mixing index λ, that ranged from 0 (rotation) to 1 (elongation), with pure shear flow at λ=0.5. Two regimes of motion were identified: for λ > or approximately equal to 0.6 aggregates rotate continuously, whereas at higher values (λ or approximately equal to 0.7) they reach a stationary orientation with respect to the flow field. A transition region, in which only some of the aggregates reach the stationary condition, separates the two regimes. The stationary regime appeared more favourable to induce breakage or restructuring, because in this case the generated internal stresses are sustained in time, while in the rotational regime they vary cyclically. We showed that the greater effectiveness of elongation with respect to shear can be explained by relating the condition of breakage to the time-averaged value of the internal stress rather than to the instantaneous value. By exploiting this relationship, the information on the viscous stress required to break or restructure an aggregate in a particular flow configuration can be easily extended to a different flow type.

Hydrodynamic forces and critical stresses in low-density aggregates under shear flow.

The distribution of stresses in rigid colloidal aggregates under a shear flow was investigated numerically for particle-cluster and cluster-cluster aggregates with fractal dimensions ranging from 1.7 to 2.3. stokesian dynamics was used to calculate the hydrodynamic force on each monomer, while the internal intermonomer interactions were calculated by applying force and torque balances on each primary particle. Although the hydrodynamic forces act mainly on the periphery of the clusters, their filamentous structure propagates and accumulates internal stresses toward the inner region of the aggregates, where consequently the most loaded intermonomer bonds are located. The spatial stress distribution, when scaled by the proper power of the radius of gyration, is independent of aggregate size and fractal dimension. This feature has made it possible to identify the most probable locations of bond failure in the structure and to estimate the relationship between shear rate and particle size for the occurrence of restructuring and of breakage.

The distribution of stresses in rigid fractal-like aggregates in a uniform flow field.

The distribution of stresses in rigid fractal-like aggregates moving in a uniform flow field was investigated for particle-cluster and cluster-cluster aggregates with fractal dimensions ranging from 1.7 to 2.3. The method of reflections was used to calculate the drag force on each monomer, while the internal inter-monomer interactions were calculated by applying force and torque balances on each primary particle. The stress distribution was found to be very dissimilar from that of the applied external forces. Although the highest external forces act on the monomers located at the periphery of the aggregate where the drag is more intense, the most stressed inter-monomer bonds are always located in the internal part of the aggregate. This phenomenon is a consequence of the structure of the studied fractal aggregates, which are made mainly of filaments of monomers: the stress generated by the external forces is propagated and progressively accumulated by such filaments up to their roots, which are situated in the inner part of the cluster. Such a behaviour is different from that exhibited by highly connected structures, in which the loads are absorbed locally by the structure and the largest stresses are normally found in the proximity of the highest applied external forces.

Extension of the method of moments for population balances involving fractional moments and application to a typical agglomeration problem.

The method of moment (MOM) is a powerful tool for solving population balance. Nevertheless it cannot be used in every circumstance. Sometimes, in fact, it is not possible to write the governing equations in closed form. Higher moments, for instance, could appear in the evolution of the lower ones. This obstacle has often been resolved by prescribing some functional form for the particle size distribution. Another example is the occurrence of fractional moment, usually connected with the presence of fractal aggregates. For this case we propose a procedure that does not need any assumption on the form of the distribution but it is based on the "moments generating function" (that is the Laplace transform of the distribution). An important result of probability theory is that the kth derivative of the moments generating function represents the kth moment of the original distribution. This result concerns integer moments but, taking in account the Weyl fractional derivative, could be extended to fractional orders. Approximating fractional derivative makes it possible to express the fractional moments in terms of the integer ones and so to use regularly the method of moments.

Coagulation efficiency of colloidal particles in shear flow.

The mechanism of shear-induced coagulation of colloidal particles is reviewed, in order to define a method to evaluate the collision efficiency according to the present knowledge of the phenomenon. Therefore, a detailed description of the procedure for trajectory analysis and identification of collision is presented, as well as of the role and estimation of colloidal forces. Recent analytical expressions have been adopted for the Van der Waals interaction, that are based on the Lifshitz theory of dispersion forces and that are capable of considering ionic screening. The obtained results have been compared with those of older expressions for the dispersion force. The two formulations agree fairly well for the case of strongly destabilised systems; on the contrary, they may lead to significantly different results for slow coagulation. The proposed method results in a fully predictive procedure for estimating collision efficiency. The results have been compared favourably with experimental data concerning the system polystyrene-water for the regime of fast coagulation. Finally, the transition between primary and secondary coagulation has been analysed in detail and it has been shown that, in the fast coagulation regime, the type of coagulation depends only on surface potential and ionic strength, but not on particle size and shear rate.