In this paper, PhoSim V.0, a physics-based digital twin core of a wet-process phosphoric acid (WPPA) digestion reactor, is presented as an initial step toward a high-fidelity virtual representation of industrial WPPA plants. The simulator integrates dissolution, reaction, and crystallization phenomena under strongly non-ideal electrolyte conditions. The reactor is modeled as a batch reactor, where the dissolution of tricalcium phosphate is described using a shrinking-core approach, while gypsum precipitation is represented by a one-dimensional population balance equation capturing particle size distribution evolution. Supersaturation ratios governing nucleation and crystal growth are computed from non-ideal activities using a Pitzer-based thermodynamic model, ensuring consistent coupling between dissolution, reaction stoichiometry, and crystallization kinetics. The resulting stiff system of equations is solved to predict key process indicators such as speciation, supersaturation ratio, and characteristic crystal size. Implemented within a user-friendly simulation environment, PhoSim V.0 provides a modular and extensible foundation that will be progressively refined to mimic industrial WPPA digestion units and supports future monitoring, optimization, and control-oriented applications.

A first principles model is developed to describe the dissolution mechanism of phosphate ore particles in a solution of phosphoric acid. It is based on mass balance equations along with hydrodynamic and kinetic relations, combined with a shrinking core model with product removal. Furthermore, non-uniform size of ore particles is accounted for using particle size distributions (PSDs) (i.e., Rosin-Rammler-Bennett, Gamma, Gate-Gaudin-Schuhmann distributions). Moreover, a coefficient of variation (CV) is used to quantify the influence of the PSD shape on the model predictions at different temperatures, acid concentrations and stirring speeds. The resulting three-phase (i.e., solid particles, liquid film surrounding the particles and liquid bulk) involves several unknown parameters. A global estimability analysis is therefore carried out to determine the estimable parameters from the available experimental data. The most estimable parameters are then identified from measurements of the particle conversion fraction mainly collected from the literature. The optimal values of the estimable unknown parameters are very consistent with the values in the literature and the predictions show good agreement with the experimental data. Finally, the computed activation energy of the dissolution reaction is very low indicating that diffusion is the rate limiting step of the dissolution process.

Computational Fluid Dynamics (CFD) simulation of multiphase industrial flows is a significant research concern for studying the performance and efficiency of chemical processes. Within the last years, CFD had growth interest from researchers with the significant increase in computational resources capacities and high-performance calculations. The trend toward focusing on machine learning (ML) techniques to solve complex industrial problems is observed. In contrast, ML has found encouraging and promising applications in that research field by offering a wealth of techniques to extract unreachable knowledge from data that can improve the chemical processes understanding and allows efficient optimization and intensification. This study aims to present the different uses of ML in the chemical processes' field, particularly the CFD modeling, and highlight the main and variate use of ML for complex geometries design and mesh optimization. We have paid particular attention to the ML trend in turbulence modeling to generate data-driven physical models from CFD simulations at different fidelity levels. Then, the extended application of ML to accelerate CFD calculations, and the development of surrogate models are provided. This work reveals that the application of ML to chemical engineering is a promising way for developing this research area.

This paper deals with the development of accurate surrogate models for first-principles models constructed for the dissolution of phosphate ore in a phosphoric acid solution. The surrogate models are based on sparse multivariate polynomial interpolation. The main goal is to reduce the computational time of the first-principles model while preserving its properties, namely the outputs' monotonicity and positivity. The temporal profiles of the concentrations of the different components involved in the liquid phase and in the ore particles, the particle size, and the thickness of the liquid film surrounding the particles are approximated by the surrogate models. The inputs of the models are the particle size distribution, the initial acid concentration, and the hydrodynamic conditions. A design of experiments method is used to produce the required sampling points for the surrogate models and several simulations are conducted in the MATLAB environment. A final comparison of the proposed surrogate models' performance demonstrates high efficiency, both in terms of accuracy as well as computation time, and highlights the strength of the introduced surrogate modeling methodology.

In this work, a novel approach for the development of surrogate models is presented. It aims to combine traditional model-based design of experiments and global estimability analysis. The approach is based on the Fisher Information Matrix to set two objective functions to be maximized. The first is given by D-optimal design of experiments criterion that maximizes the determinant of FIM, thus minimizing the volume of the confidence region and resulting in more precise parameter estimate. The second is given by the sum of Euclidian norms of FIM’s columns to improve the estimability of all the unknown parameters. The resulting multi-objective optimization problem is solved to determine the Pareto front of the optimal solutions. The Multi-Attribute Utility Theory is used as a decision making aid method to select the best solution, needed for the development of the surrogate model. For demonstration purposes, a polynomial Response Surface Model (RSM) based surrogate is developed. It mimics a computationally expensive first-principles model that predicts the conversion rate of a phosphate ore digestion by phosphoric acid. The results are very promising, showing how the approach allows to select the most estimable model parameters from intelligently designed data points. High performance of the developed surrogate is demonstrated by comparing its predictions and computation time to those obtained by first-principles model.

In this paper, a benchmark study between chemical and biochemical reactors is carried out using Computational Fluid Dynamics (CFD) approach. The main motivation is to develop an optimized design of a stirred sparged gas liquid reactor for preneutralisation of phosphoric acid. A CFD multiphase model is tailored under ANSYS Fluent 2020 R1, to simulate the industrial preneutralizer. A new gas sparger is designed based on fermentation bioreactors ring spargers for axial gas dispersion. The Euler-Euler multiphase reacting flow model is used within the preneutralizer to examinate the impact of the spargers type on the hydrodynamics, species transport phenomena, and mass transfer between the different phases. K-Epsilon turbulence model is adopted to integrate the turbulence due to the Pitch Blade Turbine (PBT) agitator, and Multiple Reference Frame (MRF) approach allows to handle efficiently the rotational movement of the agitator. The simulations show that ring spargers enhance mass transfer by ensuring a high level of gas phase holdup with the axial dispersion compared to the pipe spargers. The numerical results revealed that gas-liquid flow hydrodynamics is extremely sensitive to the sparger design. The proposed new preneutralizer design is promising to increase the preneutralisation yield and allows to avoid several monitoring challenges.

In this paper, a surrogate model is developed from a first-principles model constructed for phosphate ore digestion with a phosphoric acid solution. This model is based on sparse multivariate polynomial interpolation. The main motivation is to reduce the computational time of the first principles model while preserving its properties, namely the monotonicity and positivity of the outputs. The temporal profiles of the concentrations of the different components involved in the liquid phase and in the ore particles, the particle radius and the thickness of the liquid film surrounding the particles are estimated by the developed surrogate model. The inputs to the model are the particle size distribution, the initial acid concentration and the hydrodynamic conditions. A design of experiments method is used to generate the sample points required for the surrogate model and several simulations are performed in the MATLAB environment. Comparison of the predictions of the surrogate model with those obtained by the first-principles model demonstrates the high performance (accuracy and computation time) of the developed surrogate model.

Computational fluid dynamics (CFD) have been extensively used to simulate the hydrodynamics of multiphase flows (MPFs) in rotating machinery. In the presence of a granular dense phase, the Kinetic Theory of Granular Flow (KTGF) is usually coupled to Eulerian multi-fluid models to obtain tractable computational fluid models. In the present work, the hydrodynamic behavior of a three dimensional, industrial scale, and rotating drum granulator with gas–solid flows is assessed using the Eulerian–Eulerian approach coupled with the k-ε standard turbulence model. A Eulerian–Eulerian Two-Fluid Model (TFM) is used with the KTGF model for the granular phase. The sensitivities to different operating parameters, including the rotational speed (8, 16, and 24 rpm), inclination degree (3.57∘, 5.57∘, and 7.57∘), and degree of filling (20%, 30%, and 40%) are studied. Moreover, the impact of the drag model on the simulation accuracy is investigated. The flow behavior, regime transitions, and particle distribution are numerically evaluated, while varying the operating conditions and the drag models. The rotational speed and filling degree appear to have greater influences on the granulation effectiveness than on the inclination degree. Three drag models are retained in our analysis. Both the Gidaspow and Wen and Yu models successfully predict the two-phase flow in comparison to the Syamlal and O’Brien model, which seems to underestimate the hydrodynamics of the flow in both its axial and radial distributions (a fill level less than 35%). The methodology followed in the current work lays the first stone for the optimization of the phosphates fertilizer wet-granulation process within an industrial installation.

This paper deals with the modeling and simulation of the acidulation (dissolution) of phosphate ore particles in a dilute phosphoric acid solution in a batch stirred tank reactor. A shrinking core model with elimination of the products is used to describe the reaction and mass transfer phenomena involved in the three phases considered, i.e., liquid bulk, liquid film surrounding the particles and solid phase. The model is based on mass balance equations in the three phases and the corresponding equations are ODEs in the liquid bulk and in the solid phase, and PDEs in the liquid film. An estimability analysis method based on global sensitivities is used to determine the most estimable unknown parameters involved in the model equations from the available experimental data. The experiments consist of measurements over time of the conversion rate of the ore particles obtained in operating conditions similar to those used in an industrial phosphoric acid plant. The values of the least estimable parameters are fixed from the literature, while gProms environment is used to implement and solve the equations and to identify the most estimable parameters. The results obtained show a good agreement between the model predictions and the measurements with coherent values of the identified parameters.

This paper deals with the modelling and simulation of the dissolution of phosphate ore particles in a dilute phosphoric acid solution in a batch stirred tank reactor. A shrinking core model with elimination of the products is used to describe the reaction and mass transfer phenomena involved in the three phases considered, i.e., liquid bulk, liquid film surrounding the particles and solid phase. The model is based on mass balance equations in the three phases and consist of algebraic equations in the liquid bulk, an ODE in the solid phase, and PDEs in the liquid film. An estimability analysis method based on global sensitivities is used to determine the most estimable unknown parameters involved in the model equations from the available experimental data. The values of the least estimable parameters are fixed from the literature, while gProms environment is used to implement and solve the equations and to identify the most estimable parameters. The results obtained exhibit a good agreement between the model predictions and the measurements with coherent values of the identified parameters. Furthermore, the model is tested on the measurements carried out by van der Sluis et al. (1987) and showed that the dissolution process is mainly controlled by the diffusion step.