Solving Population Balance Equations Research Articles

Improved High-Resolution Algorithm for Solving Population Balance Equations

Improved High-Resolution Algorithm for Solving Population Balance Equations

Further developments of the extended quadrature method of moments to solve population balance equations

Developing numerical methods to solve polydispersed flows using a Population Balance Equation (PBE) is an active research topic with wide engineering applications. The Extended Quadrature Method of Moments (EQMOM) approximates the number density as a positive mixture of Kernel Density Functions (KDFs) that allows physical source terms in the PBEs to compute continuous or point-wise form according to the moments. The moment-inversion procedure used in EQMOM has limitations such as the inability to calculate certain roots even if it is defined, absence of consistent result when multiple roots exist or when the roots are nearly equal. To address these limitations, the study proposes a modification of the moment-inversion procedure to solve the PBE based on the proposed Halley-Ridder (H-R) method. Although there is no significant improvement in the extent of variability relative to the mean of the tested shape parameter σ values, an increase in the number of floating point operations (FLOPS) is observed which the proposed algorithm responds in limitations mentioned above. The total number of FLOPS for all the kernels used for the approximation increased by around 30%. This is an improvement towards the development of a more reliable and robust moment-inversion procedure.

Bubble size modeling approach for the simulation of bubble columns

The constant bubble size modeling approach (CBSM) and variable bubble size modeling approach (VBSM) are frequently employed in Eulerian–Eulerian simulation of bubble columns. However, the accuracy of CBSM is limited while the computational efficiency of VBSM needs to be improved. This work aims to develop method for bubble size modeling which has high computational efficiency and accuracy in the simulation of bubble columns. The distribution of bubble sizes is represented by a series of discrete points, and the percentage of bubbles with various sizes at gas inlet is determined by the results of computational fluid dynamics (CFD)–population balance model (PBM) simulations, whereas the influence of bubble coalescence and breakup is neglected. The simulated results of a 0.15 m diameter bubble column suggest that the developed method has high computational speed and can achieve similar accuracy as CFD–PBM modeling. Furthermore, the convergence issues caused by solving population balance equations are addressed.

Computationally Efficient Algorithm for Solving Population Balances with Size-Dependent Growth, Nucleation, and Growth-Dissolution Cycles

A computationally efficient algorithm for solving population balance equations (PBEs) for describing the evolution of crystal size distribution (CSD) in seeded batch crystallization processes is developed. The algorithm is particularly suitable for solving problems with size-dependent growth and dissolution kinetics where temperature cycling is applied to modify the CSD. It is much faster than the quadrature method of moments (QMOM) which has been widely applied to solve this type of problem and has the added benefit that it provides the complete crystal size distribution rather than only the moments. These features make it especially well suited for solving problems in optimization and determination of feasible regions, where many batch simulations are required. The algorithm is applied to determine attainable regions for seed-grown product crystals (characterized by the mean and variance of the seed-grown product CSD) and analyze the trade-off between the objectives of minimizing batch time and nucleated crystal volume in a batch crystallization process with temperature cycling. Two main conclusions are drawn from the results. First, the attainable region expands with an increasing number of growth-dissolution cycles as expected, but the extent of the increase or decrease in the attainable variance depends on the sensitivity of the growth and dissolution rates to the crystal size. Second, the trade-off between the abovementioned objectives subject to constraints on the seed-grown crystal mean size and variance is significant and an outcome representing a good compromise can be achieved by specifying proper supersaturation and undersaturation set points.

CFD-PBM simulation of bubble columns: Effect of parameters in the class method for solving PBEs

The class method (CM) has been frequently applied in solving population balance equations (PBEs) for the CFD-PBM simulation of bubble columns. There are four parameters in the CM: the volume ratio between successive bubble classes (rv), the minimum diameter (dmin), the maximum diameter (dmax) and the critical diameter (dc) that is used to distinguish small bubbles and large bubbles. These parameters have a significant impact on the computational precision and efficiency. In this work, numerical simulations are performed to investigate the effects of these parameters on the simulation of two typical bubble columns operated at heterogeneous regime. The simulation results quantitatively present the influences of these parameters on the simulated local gas holdup and global bubble size distribution. The simulated radial profiles of gas holdup at different heights are compared with experimental data. The optimized scheme for the values of these parameters in the CM is proposed.

A new method for solving population balance equations using a radial basis function network

This study presents a new method for solving the population balance equation (PBE) for particle coagulation. The method introduces a radial basis function network to approximate the number density ...

Application of extended method of classes for solving population balance equations and optimization study for crystallization process involving dissolution phenomena

Crystallization is a well-established chemical process for producing high quality of crystals. Several phenomena such as nucleation, crystal growth and dissolution are involved in producing good crystal’s quality. Thus modelling of crystallization process is essential for representing crystal quality as well as for process control and optimization purposes. One of the widely approach to represent crystal quality in terms of crystal size distribution (CSD) and to solve population balance equations (PBE) is by using method of classes (MOC). However, MOC is only applicable for the case of nucleation and crystal growth and thus needs to be extended for the case of dissolution. Therefore, the objective of this paper is to extend MOC for the case of dissolution and demonstrate crystallization process involving dissolution in order to achieve desired CSD with minimum fine crystals. The mathematical model of the process is developed and simulated in Matlab. The optimization algorithm is employed to generate set-point trajectory for Proportional-Integral (PI) controller in the case of with and without dissolution phenomena. Based on the simulations, the MOC have shown comparable results with published literature, which indicates MOC has been successfully extended to solve the PBE and representing CSD for size dependent growth rate system.

Population balance modeling: application in nanoparticle formation through rapid expansion of supercritical solution

The rapid expansion of supercritical solution (RESS) process is a novel method to produce free-solvent particles with narrow particle size distribution (PSD), and it is offered to replace conventional particle formation methods. The present study is aimed to investigate the numerical simulation of the RESS process comprising hydrodynamic and particle formation and solving population balance equation to compute PSD of solute-supercritical CO2 (SC-CO2) systems. Mass, momentum and energy balances in addition to an accurate equation of state are used to calculate hydrodynamic behavior of SC-CO2 at various conditions through a nozzle and supersonic free jet with considering heat exchange in the nozzle. The solubility of TBTPP (5, 10, 15, 20-tetrakis (3, 5-bis (trifluoromethyl) phenyl) porphyrin) in SC-CO2 is calculated using the Soave–Redlich–Kwong equation of state with van der Waals mixing rule. After that, supersaturation and nucleation of TBTPP and three other drugs, i.e., aspirin, salicylic acid and ibuprofen, are investigated. Eventually, the method of moments is used to solve the population balance equations to determine PSD of the solutes. The results are presented with and without coagulation. Furthermore, the effect of nozzle length and pre-expansion temperature on PSD is studied. Furthermore, to improve the results of coagulation model on PSD, collision frequency and lognormal function are modified. The modifications improve the coagulation results. The average absolute percent deviation of TBTPP solubility is 1.86. The CO2 hydrodynamic behavior shows the same trend as reported in the literature. Moreover, results of PSD calculation with coagulation indicated there is a good agreement with the experimental ones.

A hybrid particle-number and particle model for efficient solution of population balance equations

This work presents a hybrid particle-number and particle model to improve efficiency in solving population balance equations for type spaces spanning spherical and aggregate particles. The particle-number model tracks simpler, spherical particles cheaply by storing only the number of particles with a given one-dimensional internal coordinate, while the particle model allows resolution of the detailed aggregate structure that occurs due to collision and coagulation between particles by storing distinct computational entries for each particle. This approach is exact if primary particles are defined by their monomer count and the particle-number model increments in single monomers. A stochastic method is used to solve the population balance equations for the combined type space. The hybrid method works well for large ensembles (>212 particles) with a detailed particle model, where performing a finite number of particle-number updates is demonstrated to be 40–50% cheaper than updating an equivalent ensemble of discrete particles. These savings can be traded for a larger sample volume to increase the resolution in the particle size distribution or more repeat runs to reduce the total error. Run time improvements are curtailed at very high surface growth and coagulation rates due to the fixed cost of growth updates on the large aggregates formed; however, the hybrid method is still attractive in this case as its primary purpose is to reduce error by preventing saturation of the ensemble with simple particles at high inception rates.

Solution of population balance equations by logarithmic shape preserving interpolation on finite elements

A new numerical approach for solving population balance equations (PBE) is proposed and validated. The method employs a combination of basis functions, defined on finite elements, to approximate the sought distribution function. Similarly to other methods of the same family, the PBE are solved only in a finite number of values of the internal coordinate (grid points). The peculiarity of the method is the use of a logarithmic, shape-preserving interpolation (LSPI) procedure to estimate the values of the distribution in between grid points. The main advantages of the LSPI method compared to other approaches of the same category are: (i) the stability of the numerical approach (i.e., the absence of oscillations in the distribution function occurring when using “standard” cubic splines and a low number of elements), and (ii) the conceptual and implementation simplicity, as no mathematical manipulation of the PBE is required.

New developments of the Extended Quadrature Method of Moments to solve Population Balance Equations

Population Balance Models have a wide range of applications in many industrial fields as they allow accounting for heterogeneity among properties which are crucial for some system modelling. They actually describe the evolution of a Number Density Function (NDF) using a Population Balance Equation (PBE). For instance, they are applied to gas–liquid columns or stirred reactors, aerosol technology, crystallisation processes, fine particles or biological systems. There is a significant interest for fast, stable and accurate numerical methods in order to solve for PBEs, a class of such methods actually does not solve directly the NDF but resolves their moments. These methods of moments, and in particular quadrature-based methods of moments, have been successfully applied to a variety of systems. Point-wise values of the NDF are sometimes required but are not directly accessible from the moments. To address these issues, the Extended Quadrature Method of Moments (EQMOM) has been developed in the past few years and approximates the NDF, from its moments, as a convex mixture of Kernel Density Functions (KDFs) of the same parametric family. In the present work EQMOM is further developed on two aspects. The main one is a significant improvement of the core iterative procedure of that method, the corresponding reduction of its computational cost is estimated to range from 60% up to 95%. The second aspect is an extension of EQMOM to two new KDFs used for the approximation, the Weibull and the Laplace kernels. All MATLAB source codes used for this article are provided with this article.

A mathematical investigation of the Turkevich organizer theory in the citrate method for the synthesis of gold nanoparticles

Gold nanoparticles are commonly manufactured by the citrate reduction method, a synthesis method pioneered by Turkevich et al. (1951). Based on their experimental evidence, Turkevich et al. (1951) advanced the organizer theory, a nucleation-growth synthesis mechanism. Subsequently, Kumar et al. (2007) developed a mathematical model for the description of the synthesis, basing it on such a theory. However, this model has not been thoroughly tested. Recently, contrary to the evidence provided by Turkevich et al. (1951), other mechanistic descriptions of the synthesis, which emphasize the role of the pH of the solution, have been advanced in the literature. In this paper, we investigated the model of Kumar et al. (2007) for different conditions of pH, temperature and initial reactant concentrations. To solve the model, we used the numerical code Parsival, which is used for solving population balance equations. We tested the model for different synthesis conditions studied experimentally by various researchers, for which results are available in the literature. The model poorly predicted the experimental data because the Turkevich organizer theory does not account for the acid-base properties of chloroauric acid and sodium citrate. A new model, with a more accurate mechanistic description of the synthesis and of the chemistry involved, is therefore required.

Taylor series expansion scheme applied for solving population balance equation

AbstractPopulation balance equations (PBE) are widely applied to describe many physicochemical processes such as nanoparticle synthesis, chemical processes for particulates, colloid gel, aerosol dynamics, and disease progression. The numerical study for solving the PBE, i.e. population balance modeling, is undergoing rapid development. In this review, the application of the Taylor series expansion scheme in solving the PBE was discussed. The theories, implement criteria, and applications are presented here in a universal form for ease of use. The aforementioned method is mathematically economical and applicable to the combination of fine-particle physicochemical processes and can be used to numerically and pseudo-analytically describe the time evolution of statistical parameters governed by the PBE. This article summarizes the principal details of the method and discusses its application to engineering problems. Four key issues relevant to this method, namely, the optimization of type of moment sequence, selection of Taylor series expansion point, optimization of an order of Taylor series expansion, and selection of terms for Taylor series expansion, are emphasized. The possible direction for the development of this method and its advantages and shortcomings are also discussed.

New scheme for implementing the method of moments with interpolative closure

ABSTRACTThis article presents a new scheme for implementing the method of moments with interpolative closure (MOMIC), which was proposed by Frenklach and Harris in 1987. The new scheme can be implemented with arbitrary moment sequences. A new interpolation scheme, which is implemented among the square root of logarithms of newly defined normalized moments, was verified to be suitable for interpolation with only three known data points. The new scheme was applied to studies on Brownian coagulation in both free-molecular and continuum-slip regimes. The new MOMIC with integer moment sequence was verified to exhibit nearly identical accuracy to its original version as well as other state-of-the-art numerical methods, including QMOM, TEMOM, and log MM, and higher accuracy than these referenced numerical methods as the fractional moment sequence was employed. The study enhances the capability of the MOMIC for solving population balance equations.© 2017 American Association for Aerosol Research

A Facile pH Controlled Citrate-Based Reduction Method for Gold Nanoparticle Synthesis at Room Temperature.

The synthesis of gold nanoparticles using citrate reduction process has been revisited. A simplified room temperature approach to standard Turkevich synthesis is employed to obtain fairly monodisperse gold nanoparticles. The role of initial pH alongside the concentration ratio of reactants is explored for the size control of Au nanoparticles. The particle size distribution has been investigated using UV-vis spectroscopy and transmission electron microscope (TEM). At optimal pH of 5, gold nanoparticles obtained are highly monodisperse and spherical in shape and have narrower size distribution (sharp surface plasmon at 520 nm). For other pH conditions, particles are non-uniform and polydisperse, showing a red-shift in plasmon peak due to aggregation and large particle size distribution. The room temperature approach results in highly stable “colloidal” suspension of gold nanoparticles. The stability test through absorption spectroscopy indicates no sign of aggregation for a month. The rate of reduction of auric ionic species by citrate ions is determined via UV absorbance studies. The size of nanoparticles under various conditions is thus predicted using a theoretical model that incorporates nucleation, growth, and aggregation processes. The faster rate of reduction yields better size distribution for optimized pH and reactant concentrations. The model involves solving population balance equation for continuously evolving particle size distribution by discretization techniques. The particle sizes estimated from the simulations (13 to 25 nm) are close to the experimental ones (10 to 32 nm) and corroborate the similarity of reaction processes at 300 and 373 K (classical Turkevich reaction). Thus, substitution of experimentally measured rate of disappearance of auric ionic species into theoretical model enables us to capture the unusual experimental observations.Electronic supplementary materialThe online version of this article (doi:10.1186/s11671-016-1576-5) contains supplementary material, which is available to authorized users.

Application of N-phase algebraic slip model and direct quadrature method of moments to the simulation of air-water flow in vertical risers and bubble column reactor

In the present work the Direct Quadrature Method of Moments (DQMOM) has been implemented into the commercial CFD code TransAT. DQMOM has recently become a very attractive approach for solving population balance equation (PBE) due to its capability of representing the most interesting properties of the population, eg. Sauter mean diameter, void fraction, number of particles. The DQMOM technique was coupled with the turbulent N-phase Algebraic Slip Model (ASM) model in order to extend the model to handle dispersed phase populations such that each class has its own velocity field. The results compared to experimental data show that the developed numerical model accurately predicts void fraction profile in a long riser within a bubbly flow regime. Moreover the model is used for the simulation of bubble column, proving that it accurately predicts the gas hold up and the Sauter mean diameter.

CFD modelling and validation of upward bubbly flow in an adiabatic vertical pipe using the quadrature method of moments

An Eulerian–Eulerian approach was investigated to model adiabatic bubbly flow with CFD techniques. In the framework of the OpenFOAM® software, a two-fluid model solver was modified to include a population balance equation, solved with the quadrature method of moments approximation to predict upward bubbly flow in vertical pipes considering the polydisperse nature of two-phase flow. Some progress have been made recently solving population balance equations in OpenFOAM® and this research aims to extend its application to the case of vertical pipes under different conditions of liquid and gas velocities. In order to test the solver for nuclear applications, interfacial forces and bubble induced turbulence models were included to provide to this solver the capability to correctly predict the behavior of the continuous and disperse phases. Two-phase flow experiments with different superficial velocities of gas and liquid are used to validate the model and its implementation. Radial profiles of void fraction, gas and liquid velocities, Sauter mean diameter and turbulence intensity are compared to the computational results. These results are in satisfactory agreement with the experiments, showing the capability of the solver to predict two-phase flow characteristics.

Verification of Expansion Orders of the Taylor-Series Expansion Method of Moment Model for Solving Population Balance Equations

We verified the effect of the Taylor expansion order on the accuracy of the Taylor-series expansion method of moment (TEMOM) model in both the free molecular and continuum-slip regimes. The ordinary differential equations for moments with fourth-order Taylor expansion were first derived for fractal-like agglomerates, which were further compared to the existing TEMOM model with third-order Taylor expansion. We confirmed that the TEMOM model with a fourth-order Taylor expansion is less accurate than that with a third-order Taylor expansion. Moreover, the scope of application of the TEMOM model with a fourth-order Taylor expansion is limited. The existing TEMOM model with a third-order Taylor expansion was verified as the most reliable model for solving population balance equations for agglomerates undergoing Brownian coagulation.

Simulation and control of monomer conversion in a continuous emulsion polymerization reactor

A detailed pseudo-bulk model has been used for prediction of conversion and particle size distribution (PSD) of vinyl acetate in a continuous emulsion polymerization reactor. Finite volume (FV) and moment techniques are applied for solving population balance equation under continuous operation. It is found that both methods can predict sustained oscillations in the monomer conversion, however the FV method matches the experimental data better than the moment method. The monomer conversion and free surfactant concentration are controlled via two single control loops. In this work, a new control strategy for controlling monomer conversion has been proposed. It has been shown that monomer conversion and free surfactant concentration can be controlled simultaneously by manipulating initiator and surfactant feed rates.

A new analytical solution for solving the population balance equation in the continuum-slip regime

A new analytical solution is first proposed to solve the population balance equation due to Brownian coagulation in the continuum-slip regime. An assumption for a novel variable g (g=m0m2/m12, where m0, m1 and m2 are the first three moments, respectively) is successfully applied in executing a separate variable method for ordinary differential equations of the Taylor expansion method of moments. The sectional method is selected as a reference to verify the new solution. The accuracy between the new solution and Lee et al. analytical solution (Lee et al., 1997, Journal of Colloid and Interface Science, 188, 486–492) is mainly compared. The geometric standard deviation of number distribution for the new analytical solution is revealed to be limited to 1.6583. Within the valid range of the geometric standard deviation, the new analytical solution is confirmed to solve the population balance equation undergoing Brownian coagulation with the very nearly same accuracy as Lee et al. analytical solution. For the total particle number concentration, the new solution usually yields higher accuracy. The new solution and Lee et al. analytical solution approximately become one solution as the Knudsen number is smaller than 0.1000. The new solution has the potential to become a competitive analytical solution for solving population balance equation regarding its accuracy and very straightforward derivation.