Crystal Population Balance Formulation and Solution Methods: A Review

Kinetics of fluidised bed melt granulationIV. Selecting the breakage model

Predictive simulation of nanoparticle precipitation based on the population balance equation

Particle size distribution of nanoparticles plays an important role in modelling many scientific and engineering problems. In this article, we proposed a Finite Volume Method (FVM) to model TiO2 nanoparticles formation using population balance equations (PBEs) by incorporating the simultaneous agglomeration and disintegration processes. The superposition of the PBEs for agglomeration and disintegration with different kernels leads to a system of partial-integro differential equations, which are numerically solved by using FVM. The precipitation of TiO2 nanoparticles in the batch reactor is studied experimentally as well as by numerical simulations based on Austin and Diemer disintegration kernels and Shear agglomeration kernel. Finally, the capability of the precipitation model is evaluated and the experimental results on particle sizes are compared with the numerical results.

Numerical analysis and implementation of finite domain complete trial functions method of moments (FCMOM) in CFD codes

A comparison of model order reduction techniques for a four-dimensional population balance model describing multi-component wet granulation processes

High shear breakage of compact polyelectrolyte-bridged flocs: A method for obtaining model-independent breakage rate function data

Kinetics and population balance modeling of antisolvent crystallization of polymorphic indomethacin

Attrition and fragmentation of limestone during FB combustion of sulphur-bearing fuels have a profound influence on sorbent inventory and particle size distribution establishing at steady state in the bed. A population balance model is presented aiming at the prediction of the inventory and of the particle size distribution of sorbent particles establishing at steady state in the bed of an air-blown CFBC. The core of the model is represented by a population balance equation on sorbent particles which embodies terms expressing the extent/rate of each attrition/fragmentation process. The effect of the progress of sulphation on attrition/fragmentation is also taken into account. Constitutive equations needed to quantify attrition/fragmentation are developed on the basis of published data. Model results are presented and discussed with the aim of clarifying the influence of particle attrition/fragmentation on sorbent inventory and particle size distribution in a CFBC and on the closely related variables. Research needs and priorities within the specific field of investigation are also discussed.

A crystal is defined as a solid material that is inclusive of the molecules, atoms, and ions that are arranged in an orderly fashion and a repetitive pattern in a spatial arrangement. Crystal growth is a major step in the process of crystallization. The process of crystal growth and kinetics is a dynamic process that is impacted by several factors. Crystals of a compound are a form of solid state that is one the purest form of the solid. The crystals are produced through the process of crystallization that is inclusive of two primary processes. The first process is nucleation, and the second process is crystal growth. In low supersaturation or desupersaturation, crystals are grown and accelerated with nucleation. This results in larger crystal size and distribution. A supersaturated solution is present in thermodynamic equilibrium with the solid phase at a specified temperature. The state of supersaturation is critical for the operation of the crystallization experiment. The crystallization process in the pharmaceutical sciences also helps in improved downstream processes that include filtration, milling and drying. These applications demand a crystallization of even more complicated molecules that can be used as drugs and duplication at a large scale. This paper discusses the process of crystallization of paracetamol in the solvent, isopropanol. The developed crystals have been studied in this thesis by application of the process supersaturation. The developed crystals have also been analyzed by using the PAT tools. The bioavailability of the compounds is essentially dependent on three factors that include solubility, permeability, and dosage. Based on the solubility of the compounds, the physiological pH value and permeability parameters of drugs are classified into several categories. These properties are altered and are impacted by the process of crystallization and crystal growth in the chemical setting. Paracetamol has three polymorphs with the form I being most stable and form II being the least stable. The process of crystal growth and crystallization focuses on the inclusion of the development of the most stable crystal polymorph to be used as a drug and to be mass-produced and applied in the pharmaceutical industry. The crystal data of the molecule reveals that the crystal growth of the molecule is subject to change with alterations in temperature. The bond length and bond angles for the crystal polymorphs of the paracetamol remain unchanged. The optimized geometry for a free molecule crystal is planar, where the phenyl rings and the acetamide lie in different planes. Bulk crystallization of paracetamol has been done as per the controlled conditions in clinical conditions for saturation experiments. The induction and the nucleation time in the process are determined by the changes in the concentration of the solute in the process of crystal growth. The products of the process are microcrystals and are characterized for the strain, shape, and defect in the crystals using diffraction and microscopic studies. Through this thesis, the researchers aim to establish a method of crystal formation for paracetamol through the process of supersaturation and using isopropanol as a solvent with growth trajectory by second-order growth kinetics. This will help in the development of paracetamol crystals that are highly pure and can be used for replication in the medical industry for drug production and development.

A numerically robust method of moments with number density function reconstruction and its application to soot formation, growth and oxidation

The stability condition for solving the population balance equation (PBE) involving bubble growth and shrinkage within the Eulerian framework is proposed. The particle flux weighted average Courant number, , which reflects the propagation of particle state of the entire size classes on internal coordinates is first derived. Stability conditions of internal convection for PISO and PIMPLE algorithms are obtained by evaluating a series of tests concerning the constant bubble growth. The proposed stability conditions are then applied to simulate two kinds of laboratory experiments, that is, the bubble growth in stagnant liquid and condensing steam‐water pipe flows. The results show that the stable PBE solutions hold for the cases using either PISO or PIMPLE algorithms combined with the corresponding stability condition. Meanwhile, the conservation of the zeroth and first moments of the number density function can be guaranteed when the stability condition is satisfied. In addition, the effect of heat transfer coefficient correlations is also discussed.

Mathematical modeling of emulsion copolymerization regarding particle size distribution and average molecular weights

The paper presents a study of formation of the primary particle size distribution in suspension “powder” polymerization of vinyl chloride. The process is modelled by means of a population balance model, and the primary particle size distribution inside the polymerizing monomer droplets is determined by analysing the population balance equation, governing the nucleation, growth, and aggregation of the primary particles, using the moment method. The infinite set of moment equations obtained by moment transformation was closed using a sum aggregation kernel, and for numerical experimentation a second order moment equation model was used. The results show how important are to choose the correct parameters in production of poly(vinyl chloride) by suspension polymerization. Changing the parameters a bit the quality of product may change significantly. The results presented in the paper illustrate well that the population balance model can be used for describing the process and a number its properties with sufficient accuracy.

Algorithm for the accurate numerical solution of PBE for drop breakup and coalescence under high shear rates

Fixed point convergence and acceleration for steady state population balance modelling of precipitation processes: Application to neodymium oxalate