Solution Of Population Balance Equation Research Articles

Exact Solutions of Population Balance Equation with Aggregation, Nucleation, Growth and Breakage Processes, Using Scaling Group Analysis

Population balance equations may be employed to handle a wide variety of particle processes has certainly received unprecedented attention, but the absence of explicit exact solutions necessitates the use of numerical approaches. In this paper, a (2 + 1) dimensional population balance equation with aggregation, nucleation, growth and breakage processes is solved analytically by use of the methods of scaling transformation group, observation and trial function. Symmetries, reduced equations, invariant solutions, exact solutions, existence of solutions, evolution analysis of dynamic behavior for solutions are presented. The exact solutions obtained can be compared with the numerical scheme. The obtained results also show that the method of scaling transformation group can be applied to study integro-partial differential equations.

Steady state population balance modelling of precipitation processes: Nucleation, growth and size-dependent agglomeration

In this work a numerical methodology to solve the steady state Population Balance Equation (PBE) is developed. Three crystallisation mechanisms are included, namely: nucleation, size-independent growth and size-dependent loose agglomeration. The numerical method is based on the discretisation of the crystal size as distributed variable. In order to describe the loose agglomeration, the numerical methodology solves two PBE: one including the nucleation and growth mechanisms and one accounting for the agglomeration process. From the first PBE, liquid phase composition, supersaturation, developed crystal surface and Crystallite Size Distribution (CSD) are obtained. Similarly, the second PBE leads to the Agglomerate Size Distribution (ASD). The study of the size-dependant agglomeration kernel induces an additional numerical difficulty due to the dependency of both PBE and agglomeration kernel on the particle size. An accelerated fixed point algorithm based on the crossed secant method is adapted to overcome the difficulty and accurately solve the agglomeration PBE. The oxalic precipitation of uranium is simulated using this numerical methodology. First, the experimental results of a reference case are compared with the numerical predictions in terms of particle size distribution, mean size, mass fraction and moments. Then, the operating conditions are varied in order to test the algorithm robustness and performances. In all cases, the crossed secant method ensures the size-dependent agglomeration PBE solution and properly predicts the ASD. The developed numerical methodology predicts the mean particle size under the experimental uncertainty in a reasonable computation time and number of iterations.

Self-similar Solution of Population Balance Equation for Aggregation with Constant Kernel

Self-similar solution of population balance equation includes the number density function which remains invariant or contains a part that is invariant. This paper describes self-similar solution of aggregation population balance model with constant kernel. Using this constant kernel, moment of the population balance system achieves the form μ_i (t)∝t^(i-1) and the aggregation population balance equation with constant kernel reduces to a first order linear ordinary integro-differential equation whose solution is exponential function with negative power.

An open-source computational framework for the solution of the bivariate population balance equation

The bivariate population balance equation (PBE) is a mathematical framework to explain the evolution of polydisperse multiphase systems. In this work, the direct quadrature method of moments (DQMOM) is implemented in an open-source CFD code, Fluidity, for the numerical solution of bivariate PBE. This efficient numerical framework is a highly-parallelised finite element (FE) CFD code that allows for the use of mesh adaptivity on fully-unstructured meshes. Various test cases to solve spatially homogeneous bivariate PBEs with aggregation, breakage, growth and dispersion were simulated and verified against analytical solutions, resulting in excellent agreement. Benchmarking, by comparison with the Monte Carlo method solutions from the literature, with realistic kernels in a gas–liquid system for simultaneous bivariate aggregation and breakage was also performed to show the feasibility of this implementation for realistic applications. This open-source framework demonstrates its impressive potential in the case of bivariate PBE and can be exploited for the simulation of complex polydisperse multiphase systems.

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

The present work focuses on the development of a performing numerical methodology to solve the steady state Population Balance Equation (PBE) including nucleation, independent size growth and loose agglomeration as crystallization mechanisms. The methodology is based on the solution of two PBEs: one for the isolated crystallites and one describing the loose agglomerates formation. Both are solved by a discretization method and only the last one is reformulated as a fixed point problem. The algorithm solving PBE for agglomeration includes the crossed-secant algorithm as a fixed point acceleration method. The numerical PBE solution method is first validated by comparison to analytical solutions and then applied to the neodymium oxalate precipitation in order to compare to experimental results in a wide range of operating conditions. The methodology is tested under highly restrictive numerical conditions: narrow tolerances, a large amount of points in the discretization scheme and a zero vector as initial condition. The crossed-secant method demonstrates to improve the robustness of the standard fixed point iterations by ensuring the convergence of the agglomerates PBE when penalizing conditions are applied and by reducing the number of iterations otherwise. In all cases, the developed methodology predicts accurately the crystal size distribution under the experimental uncertainty in a reasonable computation time and number of iterations.

Towards a CFD-PBE simulation of aerated stirred tanks at high gas hold ups and different flow regimes

In this work we analyze the challenges of the adoption of a population balance approach in aerated stirred tanks operating at high gas volume fractions, in different flow regimes and when segregation of the phases is present, i.e. due to the formation of aerated cavities behind flat blades. A decoupled solution of the equations governing the fluid flow and of the population balance equation (PBE) is adopted to obtain preliminary information on the bubble size distribution and to identify the specific issues that need to be addressed to extend the applicability of the population balance model to different regimes. A workflow is proposed for the simulation of aerated stirred tanks with geometries for which established correlations for the estimation of bubble size are not available and a simplified approach to determine whether a coupled CFD PBE solution may be required is presented.

Physics-informed deep learning for modelling particle aggregation and breakage processes

Particle aggregation and breakage phenomena are widely found in various industries such as chemical, agricultural and pharmaceutical processes. In this study, a physics-informed neural network is developed for solving both the forward and inverse problems of particle aggregation and breakage processes. In this method, the population balance equation is directly embedded in the loss function of a neural network so that the network can be trained efficiently and fulfil physical constraints. For the forward problems, solutions of population balance equations are obtained through the optimization of the neural network where the predictions well match the analytical solutions. In the inverse modelling, the data-driven discovery of model parameters of population balance equations is investigated. The sensitivity regarding the selection of different neural network structures is also investigated. The developed population balance equations embedded with neural network approach is promising for solving inverse problems of particle aggregation and breakage processes with noisy observation data.

Analytical Solution of Population Balance Equation Involving Growth, Nucleation and Aggregation in Terms of Auxiliary Equation Method

The Auxiliary Equation Method (AEM) has been modified to obtain the solutions of a Population Balance Equation (PBE) involving particulate growth, nucleation and aggregation phenomena. In all the cases examined, the volume density distributions are accurately predicted by the travelling wave solutions of the complementary equation of the nonlinear partial integro-differential equation with distinctly chosen parameters. Being a flexible techniq ue and a direct comparison for the existing analytical solut ions, this study proves the potential of the proposed methodology.

Numerical Methods for the Solution of Population Balance Equations Coupled with Computational Fluid Dynamics.

This review article discusses the solution of population balance equations, for the simulation of disperse multiphase systems, tightly coupled with computational fluid dynamics. Although several methods are discussed, the focus is on quadrature-based moment methods (QBMMs) with particular attention to the quadrature method of moments, the conditional quadrature method of moments, and the direct quadrature method of moments. The relationship between the population balance equation, in its generalized form, and the Euler-Euler multiphase flow models, notably the two-fluid model, is thoroughly discussed. Then the closure problem and the use of Gaussian quadratures to overcome it are analyzed. The review concludes with the presentation of numerical issues and guidelines for users of these modeling approaches.

An efficient methodology to use hydrolysate of phosphogypsum decomposition products for CO2 mineral sequestration and calcium carbonate production

There is an urgent need to find new, improved ways to reduce CO2 emissions and treat phosphogypsum (PG) waste. This research proposes an energy-efficient methodology for the production of pure calcium carbonate and CO2 mineral sequestration (CMS) by hydrolysate of PG decomposition products. The technical feasibility of this process and the influence of process variables on CMS were investigated using online experiments. Pure calcium carbonate was obtained by CO2 separation and mineralization using the hydrolysis solution of PG decomposition products. A carbonization rate greater than 90% was obtained when the liquid phase absorption reached saturation. In addition, lower temperatures and higher CO2 partial pressures favoured CMS. To confirm the operating strategy underlying calcium carbonate crystallization, real-time experiments data obtained from process analytical technologies (PATs) and solutions of population balance equations were used to determine parameters of crystallization kinetic models. The match between model simulation experiments and online experimental results throughout the crystallization process confirms the robustness of our findings.

Error analysis in stochastic solutions of population balance equations

Stochastic simulation of population balance equations (PBEs) is robust and flexible; however, it exhibits intrinsic stochastic errors which decreases at a very slow rate when increasing the computational resolution. Generally, these stochastic methods can be classified into two groups: (i) the classical Gillespie method and (ii) weighted flow algorithm. An analytical relationship is derived for the first time to connect the variances in these two groups. It also provides a detailed analysis of the resampling process, which has not been given appropriate attention previously. It is found that resampling has a profound effect on the numerical precision. Moreover, by comparing the time evolutions between systematic errors (i.e., errors in the mean value) and stochastic errors (i.e., variances), it is found that the former grows considerably faster than the latter; thus, systematic errors eventually dominate. The present findings facilitate the choice of the most suitable stochastic method for a specific PBE a priori in order to balance numerical precision and efficiency.

Exact solutions of population balance equations for breakage and growth processes, using group analysis

Population balance equations have been applied to model a wide range of processes including breakage, crystal nucleation, growth, agglomeration, comminution of ores and other solids, but the lack of exact solutions necessitates the use of numerical methods. A class of population balance equations with time independent but size dependent growth rate and breakage rate is investigated analytically. All invariant solutions, explicit exact solutions, reduced equations and optimal system of subalgebras for the corresponding system of equations are presented in this paper.

Comparison of a new mass-concentration, chain-reaction model with the population-balance model for early- and late-stage aggregation of shattered graphene oxide nanoparticles

Aggregation as an essential mechanism impacting nanoparticle (NP) functionality, fate, and transport in the environment is currently modelled using population-balance equation (PBE) models which are computationally expensive when combined with other continuum-scale reactive transport models. We propose a new simple mass-concentration-based, chain-reaction modelling (CRM) framework to alleviate computational expenses of PBE and potentially to facilitate combination with other fate, transport, and reaction models. Model performance is compared with analytical PBE solution and a standard numerical PBE technique (fixed pivot, FP) by fitting against experimental data (i.e., hydrodynamic diameter and derived count rate of dynamic light scattering used as a representative of mass concentration) for early- and late-stage, aggregation of shattered graphene oxide (SGO) NP across a broad range of solution chemistries. In general, the CRM approach demonstrates a better match with the experimental data with a mean Nash-Sutcliffe model efficiency (NSE) coefficient of 0.345 than the FP model with a mean NSE of 0.29. Comparing model parameters (aggregation rate constant and fractal dimension) obtained from fitting CRM and FP to the experimental data, similar trends or ranges are obtained between the two approaches. Computationally, the modified CRM is an order-of-magnitude faster than the FP technique, suggesting that it can be a promising modelling framework for efficient and accurate modelling of NP aggregation. However, in the scope of this study, reaction rate coefficients of the CRM have been linked to collision frequencies based on simplified and empirical relationships which need improvement in future studies.

Application of Transformation Matrices to the Solution of Population Balance Equations

The development of algorithms and methods for modelling flowsheets in the field of granular materials has a number of challenges. The difficulties are mainly related to the inhomogeneity of solid materials, requiring a description of granular materials using distributed parameters. To overcome some of these problems, an approach with transformation matrices can be used. This allows one to quantitatively describe the material transitions between different classes in a multidimensional distributed set of parameters, making it possible to properly handle dependent distributions. This contribution proposes a new method for formulating transformation matrices using population balance equations (PBE) for agglomeration and milling processes. The finite volume method for spatial discretization and the second-order Runge–Kutta method were used to obtain the complete discretized form of the PBE and to calculate the transformation matrices. The proposed method was implemented in the flowsheet modelling framework Dyssol to demonstrate and prove its applicability. Hence, it was revealed that this new approach allows the modelling of complex processes involving materials described by several interconnected distributed parameters, correctly taking into consideration their interdependency.

An Improved Analytical Solution of Population Balance Equation Involving Aggregation and Breakage via Fibonacci and Lucas Approximation Method

AbstractThis study presents a novel analytical solution for the Population Balance Equation (PBE) involving particulate aggregation and breakage by making use of the appropriate solution(s) of the associated complementary equation of a nonlinear PBE via Fibonacci and Lucas Approximation Method (FLAM). In a previously related study, travelling wave solutions of the complementary equation of the PBE using Auxiliary Equation Method (AEM) with sixth order nonlinearity was taken to be analogous to the description of the dynamic behavior of the particulate processes. However, in this study, the class of auxiliary equations is extended to Fibonacci and Lucas type equations with given transformations to solve the PBE. As a proof-of-concept for the novel approach, the general case when the number of particles varies with respect to time is chosen. Three cases i. e. balanced aggregation and breakage and when either aggregation or breakage can dominate are selected and solved for their corresponding analytical solution and compared with the available analytical approaches. The solution obtained using FLAM is found to be closer to the exact solution and requiring lesser parameters compared to the AEM and thereby being a more robust and reliable framework.

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.

Local Fixed Pivot Quadrature Method of Moments for Solution of Population Balance Equation

A local fixed pivot quadrature method of moments (LFPQMOM) is proposed for the solution of the population balance equation (PBE) for the aggregation and breakage process. First, the sectional representation for aggregation and breakage is presented. The continuous summation of the Dirac Delta function is adopted as the discrete form of the continuous particle size distribution in the local section as performed in short time Fourier transformation (STFT) and the moments in local sections are tracked successfully. Numerical simulation of benchmark test cases including aggregation, breakage, and aggregation breakage combined processes demonstrate that the new method could make good predictions for the moments along with particle size distribution without further assumption. The accuracy in the numerical results of the moments is comparable to or higher than the quadrature method of moment (QMOM) in most of the test cases. In theory, any number of moments can be tracked with the new method, but the computational expense can be relatively large due to many scalar equations that may be included.

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.

Fractional Monte Carlo time steps for the simulation of coagulation for parallelized flowsheet simulations

The event-driven acceptance rejection (AR) method is a computationally very advantageous Monte Carlo (MC) simulation technique for the solution of population balance equations (PBE) of coagulating systems. In the scope of the event-driven simulation approach, the simulation time is stepwise increased by a simulation time step τ, which is given be the simulated particle properties. Within this time step τ, exactly one coagulation event takes place. The method is therefore not applicable in situations, for which specific time points have to be reached by the simulation, or the time step has to be reset to a smaller value. We propose a methodology termed ‘fractional MC step’ which allows to reset the simulation time step of the AR method to any arbitrary smaller value than the one initially proposed. The proposed method is validated by simulations of coagulation for different initial conditions and comparison with results gained from the discrete sectional method. The potential increase of the stochastic noise is investigated by comparisons with the results from conventional MC simulation techniques. The advantages of a parallel implementation are briefly discussed.

Hybrid simulation algorithm for efficient numerical solution of population balance equations

Cell-to-cell variability of multicellular systems affects many processes from bioprocess engineering, systems biology and systems medicine. Multiscale modeling of these cellular heterogeneities with population balance equations is a valuable tool to understand the interplay of individual cell dynamics (intracellular level), interactions of cells with each other (population level) and interactions of cells with other molecular species (extracellular level). The resulting mathematical models represent high dimensional partial integro-differential equations which are rarely solvable analytically. As alternative to classical numerical solution methods which are not well suited for efficient solution, we propose a hybrid simulation algorithm in this contribution. This technique is based on a combination of the method of partial characteristics and approximate moment methods using monomial cubatures. The technique’s efficiency and accuracy is shown for a multicellular system describing cell-to-cell signaling in osteochondro-switch models.