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.


β-PDF approach
The chemical reaction between magnesium and hydroxyl ions is instantaneous and irreversible and algebraic equations can be used to model it. Mixture fraction and its variance are enough to determine effective ion concentrations that can precipitate in crystal form. Being , the components concentrations depending on the local mixture fraction ( ), 0 , 0 the initial concentrations and the stoichiometric mixture fraction, the following equations can be written for + → reaction: As provided by these equations, since chemical reaction is instantaneous, reactants can't coexist when they are molecularly micro-mixed. Depending on fluid-dynamics conditions, though, reaction could be stunted. Based on the variance value, a probability density function is used to determine the probability that reactants are micro-mixed or micro-segregated in a volume of fluid. In other words, if variance is high enough (micro-segregated system), reaction doesn't begin depending on the lack of molecularly micro-mixed reactants and only reactants dilution arises.
For the 1D model, mixture fraction is kept constant and equal to the stoichiometric mixture fraction value in line with experiments. The stoichiometric mixture fraction value represents, in the mixture fraction space, the value at which reactants are stoichiometrically reacting. Variance time evolution and mixture fraction knowledge led to products concentrations calculation.
Full derivation is reported in sec. Analytical derivation. Authors aim is to give, for the first time concerning magnesium hydroxide precipitation, a comprehensive description of the fluiddynamics interference on the molecular precipitation phenomena (i.e., primary nucleation and growth). Assuming an initial concentration of magnesium chloride equal to 1 M (for the sake of clarity), equation 3-c can be properly written as: Figure S1. Micro-mixing effect on the molecular processes (i.e., primary nucleation and growth) quantified through the − approach Rearranging properly eq. (4), ( ̅, ̅ ′ 2 ) comes out and can be interpreted physically, having in mind the micro-mixing effect. At the very beginning of the simulation, variance is the highest possible; it means that ( ̅, ̅ ′ 2 ) → 0. At the beginning, due to the micro-segregation of the components, no molecular process can begin, which results in zero micro-mixing. On the other hand, as soon as the turbulence increases, variance is dissipated by turbulent eddies and reactants can interact within the Batchelor scale. The variance tends to zero, as described in Figure 5 (manuscript), and ( ̅, ̅ ′ 2 ) → 1 ( Figure S1). It means that, due to turbulence, the free-stream ions available can now integrally react.

Analytical derivation
These three integrals are closed analytically; all three derivations are reported in this Appendix. Not to weight down the notation, let's denote where Γ is the gamma function.
For B component: For the P component:

Bromley's activity coefficient
At the beginning of activity calculation for a solute in multi-component solution indexes must be fixed and used for related equation. In this regard, since Mg +2 and OHions are needed for supersaturation, following indexes were given: Thus, following the above-mentioned steps, multi-component solution parameters were calculated using these equations: where odd indexes refer to cations and even ones to anions, = 0.511 √ and is the solution ionic strength calculated as: where is the concentration of the i th ions in the solution. Moreover, parameters needed have this form: where is the i th ions molality and log 0 is the ten-based logarithm of activity coefficient of the pseudo-solution with the same ionic strength of the multi-component one but considering only the i-j ions pair. The governing equation is:

Micro-mixing modelling
As a test, we turned off the variance calculation in our model, effectively assuming instantaneous mixing and precipitation upon entering the T-mixers. However, the predictions for mean particle sizes based on this assumption were clearly off, as shown in Figure S2. The predictions failed to capture the correct values and trends observed in experimental data. We further understood the importance of the micro-mixing model by comparing the supersaturation evolution predicted by the model both with and without the micro-mixing, as shown in Figure  S3. Only when accounting for the micro-mixing ( Figure S3, top) the correct trend was observed, with supersaturation first generated by mixing and then consumed by precipitation. On the other hand, when micro-mixing was neglected ( Figure S3, bottom), precipitation started immediately, resulting in larger supersaturations. We also optimized the model parameters by fitting the experimental data, but this time without the micro-mixing model. Despite having the same number of parameters as the full model, the resulting parameter set was unable to reproduce the experimental trends shown in Figure S4 (cases #1-5, top; cases #6-8, bottom). We observed significant differences in all the parameters (Table S1), especially A 1 , which is involved in the homogeneous nucleation rate. The fact that A 1 decreased by three orders of magnitude, roughly corresponding to the increase in supersaturation shown in Figure S3, highlights the crucial role of micro-mixing in our description.  Table S1. The optimal set of parameters obtained from the comparison between the model without micromixing and experimental data (cases #1-5).

CFD component
CFD simulations might be used to estimate the mixing time and the Kolmogorov timescale, which are important parameters in predicting the particle size distribution.
The mixing time is the time required for two fluids to mix completely, and it is proportional to the ratio between the turbulent kinetic energy (k) and the turbulent dissipation rate (ε) (Eq. (15)).
The Kolmogorov timescale is the smallest timescale at which energy is dissipated in a turbulent flow, and it is proportional to the square root of the kinematic viscosity (ν) divided by the turbulent dissipation rate (Eq. (16)). These two parameters affect the degree of mixing, which in turn affects all the precipitation phenomena, and therefore, the resulting particle size distribution. While it is true that mixing time can be estimated using empirical correlations, such as those discussed in our previous work 2 , the accuracy of the estimates for the value may not always be high. For instance, it could be possible to estimate the value by passing through the pressure drops. Nevertheless, empirical correlations would not provide a detailed estimation of pressure drops, especially those due to the impingement between fluids. In this case an underestimate of the value, could lead to a quite important deviation of the model predictions as shown in Figure  S5. On the other hand, CFD simulations provide a more comprehensive understanding of the mixing process and can account for the effect of various parameters, such as fluid properties, mixer geometry, and flow rates, on the turbulence characteristics. Therefore, although other methods may be used to estimate the mixing time and turbulence parameters, CFD simulations remain a valuable tool for obtaining reliable and comprehensive information on the mixing process 3,4 .