Béla G. Lakatos

Modelling and simulation of suspension polymerization of vinyl chloride via population balance model

Abstract A detailed population balance model is presented for suspension polymerization of vinyl chloride in an isothermal batch reactor perfectly mixed on macrolevel. Coalescence and breakage of monomer droplets, as well as mass exchange of species between the droplets induced by collisions, termed micromixing, are also included into the model forming a complex three-scale system. The resulted population balance equation is solved by coupling the deterministic continuous time computation of polymerization reactions inside the droplets with the random coalescence and breakage events of droplets using Monte Carlo simulation. The results obtained by simulation revealed that aggregation, breakage and micro-mixing of species induced by droplet collisions affect the process significantly.

Modeling and simulation of suspension polymerization of vinyl chloride via population balance model

Abstract A population balance model is presented for suspension polymerization of vinyl chloride (VC) in an isothermal batch reactor perfectly mixed on macrolevel. Coalescence and breakage of monomer drops, as well as mass exchange of species between the drops induced by collisions, termed micromixing, are also included into the model, forming a complex three-scale system. The resulted population balance equation is solved by coupling the deterministic polymerization reactions inside the drops with the discrete coalescence and breakage events of drops using Monte Carlo (MC) simulation. The results obtained by simulation revealed that the coalescence, breakage and micro-mixing of species induced by drops collisions affect the process significantly.

Axial dispersion/population balance model of heat transfer in gas-solid turbulent fluidized beds

Abstract An axial dispersion/population balance model is presented for describing heat transfer processes in gas–solid turbulent fluidized beds. In the model, the gas and particle transport is described by the axial dispersion model, while the particle–particle and particle–wall heat transfers are modeled as collisional random events, characterized by the collision frequencies and random variables with probability density functions determined on interval [0,1]. An infinite hierarchy of moment equations is derived from the population balance equation, which can be closed at any order of moments. The properties of the model and the effects of process parameters are examined by numerical experimentation.

Coupled Population Balance-CFD Modelling of a Continuous Precipitation Reactor

Abstract Precipitation of amorphous calcium phosphate in a Y-mixer-tubular reactor device is investigated using a population balance equation including nucleation, growth and agglomeration of particles and a three-dimensional CFD flow model. The population balance equation is reduced into a moment equations system the quadrature form of which is coupled with the detailed CFD model. Simulation results obtained by the detailed CFD flow model provided results comparable to those produced by assuming ideal mixing conditions. This comparison reveals that under the given process conditions of precipitation of amorphous calcium phosphate application of a simple ideal mixing model, for instance in controlling such processes can be justified.

Simulation and analysis of crystallization of high aspect ratio crystals with fragmentation

Abstract A two-dimensional population balance model of continuous cooling crystallization, involving nucleation, growth of the two characteristic crystal facets and random binary breakage along the length of high aspect ratio crystals is developed. The randomness of breakage is described by beta distribution of the broken fractions. The population balance model is reduced into a closed moment equation model for the joint moments of the two size variables of crystals by means of which the influence of parameters of breakage on the dynamic and steady state behavior of crystallizer is analyzed. It is shown that the mean value of width of crystals increases as the intensity of breakage along the crystal length resulted in increasing number of crystals, increases.

Axial dispersion/population balance model of heat transfer in turbulent fluidization

Abstract An axial dispersion/population balance model is presented for describing heat transfer processes in gas-solid turbulent fluidized beds. In the model, the gas and particle transport is described by the axial dispersion model, while the particle-particle and particle-wall heat transfers are modeled as collisional random events, characterized by the collision frequencies and random variables with probability density functions determined on interval [0,1]. An infinite hierarchy of moment equations is derived from the population balance equations, which can be closed from the first order of moments. The properties of the model and the effects of process parameters are examined by numerical experimentation.

Model-based analysis of stirred cooling crystallizer of high aspect ratio crystals with linear and nonlinear breakage

Abstract 2D population balance model is presented for stirred cooling crystallizer of high aspect ratio crystals including primary and secondary nucleation, size-dependent growth of the two characteristic crystal facets and linear and nonlinear breakage along the crystal length. Three breakage mechanisms are modelled: crystal-impeller and crystal-wall collisions linear and crystal–crystal collisions nonlinear breakage processes. The 2D population balance equation is reduced into a system of moment equations for the mixed moments of length and width variables which is closed applying the quadrature method of moments and solved by using a three point QMOM-ODE method. It was shown that strong interactions exist between the secondary nucleation, crystal growth and breakage processes connected by the CSD. The stirring rate has strong impact on crystallization of high aspect ratio crystals forming the crystalline product directly by breakage. The crystal-impeller breakage proved to be the dominant process but the crystal–crystal breakage also play significant role.

Multilevel Dynamical Models for Polydisperse Systems: a Volume Averaging Approach

Abstract The paper presents the fundamental elements of multilevel dynamical models of polydisperse systems of chemical engineering. A three-level model is derived and generalised by means of the volume averaging method, modified appropriately for dispersed systems. The model errors and computational aspects are analysed and discussed.

Three-scale Modeling and Simulation of a Batch Suspension Polymerization Vinyl Chloride Reactor

Abstract A three-scale mathematical model of batch suspension polymerization reactors of vinyl chloride is presented and analyzed by simulation. The multidimensional population balance equation which is a meso-scale model of the monomer-polymer droplets population is completed with the macro-scale heat balance equations for the continuous phase and the cooling medium. This hybrid model includes polymerization reactions inside the droplets, random breakage and aggregation of droplets as well as heat and mass exchange between droplets induced by binary collisions. The model equations are solved by a coupled continuous time-Monte Carlo method. The effects of constitutive and process parameters, and, because of polymerization is a strongly exothermic process, the influence of the cooling strategy are investigated by simulation. The simulation results demonstrated that the temperature rise in droplets causes differences in the polymer properties.