(597a) A Multidimensional Population Balance Model for Predicting Crystal Size and Aspect Ratio in the Production of Phosphogypsum
A Multidimensional Population Balance Model for Predicting Crystal Size and Aspect Ratio in the Production of Phosphogypsum
Allan S. Myerson
Richard D. Braatz
Crystals of inappropriate crystal size distribution (CSD) and shape can cause problems in downstream processes . Many recent efforts have been directed towards modeling CSD and shape using a multidimensional population balance model (mD-PBM) that can be used to optimize crystal shape â??. The model equation is a hyperbolic integro-partial differential equation, which must be solved numerically except for some simple cases. Due to the advection term in the mD-PBM, most discretization methods experience numerical diffusion. Though more advanced schemes such as the high-resolution finite-volume method have demonstrated the ability to solve mD-PBM with much higher numerical accuracy , the computational cost of these methods is still not favorable for applications in process optimization and model predictive control. The method of characteristics (MOC) has demonstrated to be computationally efficient and accurate in handling the advection term in the 1D-PBM â??. This presentation extends the MOC for solving mD-PBM with both growth and nucleation for the reactive crystallization of phosphogypsum.
During the production of phosphoric acid by the Wet Process, gypsum crystal is produced as a byproduct in the crystallizer. The phosphoric acid is separated from the gypsum by passing the slurry to the filter, which is the bottleneck process due to the undesired gypsum CSD and shape. A two-dimensional PBM with simultaneous growth and secondary nucleation is developed to model the crystal size and its varying aspect ratio in a continuous crystallizer. The solution supersaturation is estimated using the state-of-art Mixed Solvent Electrolyte (MSE) thermodynamic model , which models both the solution nonideality and speciation. In the presence of metal impurities, the growth rate in each direction is inhibited differently and results in a change in aspect ratio. The relative change in growth rate due to the added impurity is modeled by growth inhibition model . The presented model is the first to combine the 2D-PBE, an impurity inhibition model, and the MSE model in the study of reactive crystallization under high ionic strength as found in the industrial process.
The proposed MOC method solves the 2D-PBM combined with the inhibition and the MSE model. Compared to existing PBM solvers, the MOC algorithm offers a computationally efficient and highly resolved method, especially in higher dimensions. The simulation results with experimentally measured crystal kinetic parameters show a good agreement with the measured CSD and mean aspect ratio. These simulation results also suggest that the presence of metal ion impurities cannot be ignored as they can modify crystal shape. This information would not be captured in a 1D-PBM for gypsum crystallization.
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