Bivariate Population Balance Equation Research Articles

Mathematical modeling of the bivariate molecular weight—Long chain branching distribution of highly branched polymers.: A population balance approach

In the present study a population balance approach is described to follow the time evolution of molecular polymer properties in free-radical polymerizations. The model formulation is based on the fixed pivot technique (FPT) which was properly adapted in two dimensions in order to calculate the bivariate molecular weight–long chain branching distribution (MW–LCBD) for highly branched polymers like poly(vinyl acetate) (PVAc) and low-density poly(ethylene) (LDPE) produced in chemically initiated free-radical polymerization systems. The method assumes that the overall polymer population can be assigned to selected discrete points which are also called ‘grid points’. Accordingly, dynamic bivariate population balance equations (PBEs) are derived for the ‘live’ and ‘dead’ polymer chains, which are solved at the specified grid points. Thus, the infinite size of differential molar balance equations is reduced to a smaller and feasible system of differential equations. The validity of the numerical calculations is examined via a direct comparison of simulation results obtained by the 2-D FPT with available experimental data of the molecular weight distribution (MWD), number and weight average molecular weights ( M n , M w ) and number average degree of branching ( B n ) of highly branched polymerization systems (i.e., Vac and LDPE). In general, the 2-D FPT can provide very accurate predictions of the molecular and branching characteristics of highly branched polymers in a relatively short time. It is important to point out that, to our knowledge, this is the first time that the joint MW–LCBD for branched polymers has been calculated by a sectional grid method via the direct solution of the governing PBEs for both ‘live’ and ‘dead’ polymer chains.

Solution of the bivariate dynamic population balance equation in batch particulate systems: Combined aggregation and breakage

This paper presents a study on the application of an extended sectional grid technique for the solution of the dynamic bivariate population balance equation (PBE) in batch particulate systems. Specifically, particulate processes operating under the action of constant particle aggregation as well as the combined action of constant particle aggregation and an idealized linear breakage mechanism are examined. In order to solve the bivariate PBE an extended sectional grid technique is developed with improved computational efficiency. The performance of the extended sectional grid technique in terms of numerical accuracy and stability is assessed by a direct comparison of the calculated bivariate particle size distributions and/or distribution moments to available analytical solutions. For combined aggregation and breakage processes it is observed that the time required for the computed bivariate PSDs to evolve towards a steady-state distribution is considerably longer than the time required for the univariate number distributions and volume moments to reach steady-state.

Population Balance Models and Monte Carlo Simulation for Nanoparticle Formation in Water-in-Oil Microemulsions: Implications for CdS Synthesis

We address controlled CdS nanoparticle formation by tuning experimental synthesis conditions. To this end, a bivariate population balance equation (PBE) model has been developed based on time scale analysis, to explain the mechanism of nanoparticle formation in self-assembled templates. It addresses the process of mixing two water-in-oil (w/o) microemulsions, each containing a pre-dissolved reactant in the microemulsion drops. Brownian collision and coalescence of two water drops of nanometer size results in mixing and exchange of reactant molecules, leading to chemical reaction. The water insoluble reaction product nucleates to form a nanoparticle in an individual drop, which subsequently grows internally by consuming the excess product and by coalescence-exchange with other drops. Finite rates of nucleation and coalescence-exchange are accounted for in the PBE, while the rates of reaction and internal growth of nanoparticles are found to be instantaneous. Experimentally proven binomial redistribution of reactant and product molecules upon drop coalescence is implemented in the present work. This results in a very good prediction of experimental data of the mean aggregate number (MAN) and hence size of CdS nanoparticles. Both our model and Monte Carlo (MC) simulation quantitatively capture the reported variation of MAN with molar excess of Cd2+ concentration and microemulsion drop size. Our results together with previous experimental data establish that usage of stoichiometrically five times or more of excess Cd2+ concentration can cause surface adsorption and desirable enhanced emission intensity of CdS nanoparticles, without altering particle size. We also propose a simplified and computationally efficient univariate PBE model. The univariate model gives very fast (in minutes) and accurate estimates (for low reactant concentrations) of the number and mean size of CdS nanoparticles. Time-scale analysis offers a good a priori choice of the appropriate model based on range of reactant concentrations.

Monte Carlo simulation for the solution of the bi-variate dynamic population balance equation in batch particulate systems

This paper presents a study on the solution of the bi-variate population balance equation (PBE) in batch particulate processes using stochastic Monte Carlo (MC) simulations. Numerical simulations were carried out over a wide range of variation of particle aggregation and growth rate models. The PBE was numerically solved in terms of the number density function, n ( V , x , t ) , for the prediction of the dynamic evolution of the two-dimensional (with respect to particle volume, V , and a second internal property, x ) particle size distribution (PSD), as well as for the calculation of the leading moments of the distribution. The performance of the method was assessed in terms of accuracy via a direct comparison of the calculated PSDs and the respective leading moments to available analytical solutions. The comparisons showed that the MC algorithm was capable of predicting the dynamic evolution of the bi-variate distribution with high accuracy, while its computational requirements were relatively low.

Bivariate direct quadrature method of moments for coagulation and sintering of particle populations

The direct quadrature method of moments (DQMOM) is applied to a bivariate population balance equation governing the joint number density function n ( v , a ) of particle volume and area. Using a test case proposed by Wright et al. [2001. Bivariate extension of the quadrature method of moments for modeling simultaneous coagulation and sintering of particle populations. Journal of Colloid and Interface Science, 236, 242–251], it is shown that DQMOM reproduces the results found by using bivariate extensions of QMOM. The effects of the choice of bivariate moments and the order ( N) of the quadrature approximation are also discussed. In general, it is found that for fixed N all non-singular sets of bivariate moments yield nearly identical results as reported by Wright et al. However, particular sets leading to lower condition numbers for the linear system used to compute the source terms in DQMOM are preferable for numerical reasons. For any value of N ⩾ 3 , the bivariate moment predictions are very similar. Overall, it is found that the coagulation and sintering kernels used in this test case are easily handled by lower-order moment methods, and are relatively insensitive to the choice of bivariate moments used for DQMOM.

Numerical solution of the bivariate population balance equation for the interacting hydrodynamics and mass transfer in liquid–liquid extraction columns

A comprehensive model for predicting the interacting hydrodynamics and mass transfer is formulated on the basis of a spatially distributed population balance equation in terms of the bivariate number density function with respect to droplet diameter and solute concentration. The two macro- (droplet breakage and coalescence) and micro- (interphase mass transfer) droplet phenomena are allowed to interact through the dispersion interfacial tension. The resulting model equations are composed of a system of partial and algebraic equations that are dominated by convection, and hence it calls for a specialized discretization approach. The model equations are applied to a laboratory segment of an RDC column using an experimentally validated droplet transport and interaction functions. Aside from the model spatial discretization, two methods for the discretization of the droplet diameter are extended to include the droplet solute concentration. These methods are the generalized fixed-pivot technique (GFP) and the quadrature method of moments (QMOM). The numerical results obtained from the two extended methods are almost identical, and the CPU time of both methods is found acceptable so that the two methods are being extended to simulate a full-scale liquid–liquid extraction column.

Direct Simulation and Mass Flow Stochastic Algorithms to Solve a Sintering-Coagulation Equation

In this paper we introduce an efficient stochastic method to solve the time evolution of a bivariate population balance equation which has been developed for modelling nano-particle dynamics. We have adapted the existing stochastic models used in the study of coagulation dynamics to solve a variant of the sintering-coagulation equation proposed by Xiong & Pratsinis. Hitherto stochastic models based on Markov jump processes have not taken into account the surface area evolution. We produce numerical results efficiently with the direct simulation and mass flow algorithms and study the convergence behaviour as the number of stochastic particles increases. We find a marked preference for using the mass flow algorithm to determine the higher order volume and area moments of the particle size distribution function. The computational efficiency of these algorithms is remarkable when compared to the sectional method that has been used previously to study this equation.

Direct Simulation and Mass Flow Stochastic Algorithms to Solve a Sintering-Coagulation Equation

In this paper we introduce an efficient stochastic method to solve the time evolution of a bivariate population balance equation which has been developed for modelling nano-particle dynamics. We have adapted the existing stochastic models used in the study of coagulation dynamics to solve a variant of the sintering-coagulation equation proposed by Xiong & Pratsinis. Hitherto stochastic models based on Markov jump processes have not taken into account the surface area evolution. We produce numerical results efficiently with the direct simulation and mass flow algorithms and study the convergence behaviour as the number of stochastic particles increases. We find a marked preference for using the mass flow algorithm to determine the higher order volume and area moments of the particle size distribution function. The computational efficiency of these algorithms is remarkable when compared to the sectional method that has been used previously to study this equation.