Solution Of Population Balance Equation Research Articles

An open-source quadrature-based population balance solver for OpenFOAM

The extended quadrature method of moments (EQMOM) for the solution of population balance equations (PBE) is implemented in the open-source computational fluid dynamic (CFD) toolbox OpenFOAM as part of the OpenQBMM project. The moment inversion procedure was designed (Nguyen et al., 2016) to maximize the number of conserved moments in the transported moment set. The algorithm is implemented in a general structure to allow the addition of other kernel density functions defined on R+, and arbitrary kernels to describe physical phenomena involved in the evolution of the number density function. The implementation is verified with a set of zero-dimensional cases involving aggregation and breakage problems. Comparison to the rigorous solution of the PBE provides validation for these cases. The coupling of the EQMOM procedure with a CFD solver to address aggregation and breakage problems of non-inertial particles is validated against experimental measurements in a Taylor-Couette reactor from the literature.

PUFoam : A novel open-source CFD solver for the simulation of polyurethane foams

In this work a transient three-dimensional mathematical model is formulated and validated for the simulation of polyurethane (PU) foams. The model is based on computational fluid dynamics (CFD) and is coupled with a population balance equation (PBE) to describe the evolution of the gas bubbles/cells within the PU foam. The front face of the expanding foam is monitored on the basis of the volume-of-fluid (VOF) method using a compressible solver available in OpenFOAM version 3.0.1. The solver is additionally supplemented to include the PBE, solved with the quadrature method of moments (QMOM), the polymerization kinetics, an adequate rheological model and a simple model for the foam thermal conductivity. The new solver is labelled as PUFoam and is, for the first time in this work, validated for 12 different mixing-cup experiments. Comparison of the time evolution of the predicted and experimentally measured density and temperature of the PU foam shows the potentials and limitations of the approach.Program summaryProgram Title: PUFoamProgram Files doi:http://dx.doi.org/10.17632/62ggzx623g.1Licensing provisions: GNU General Public License 3 (GPL)Programming language: C++.Supplementary material: In order to test the main solver all the required input files have been provided within the testCase directory. This could be either directly used or modified according to user’s needs.Nature of problem: The CFD solver developed through this research work provides a numerical mean for the simulation of polyurethane foam. The problem includes a reacting multi-phase system in which the liquid mixture expands due to the polymerization phenomenon and the presence of different additives. In that, the gas bubbles nuclei within the reacting liquid mixture start to grow owing to the diffusion of gases produced due to the chemical reactions. Further, industrial applications such as mould-filling seek for capturing the foam front face and the evolution of its physical and thermal properties during the foaming process. Thus, the solver shall facilitate the evolution of gas bubbles via a population balance equation, capturing the foam interface using a volume-of-fluid method, and eventually predicts the foam characteristics.Solution method: Conservation equations for mass and momentum as well as phase fraction equations are solved using the standard finite volume method. The indicator function (i.e., the phase fraction equation in this case) is solved for the primary phase taking into account the density variation of the foam which is implemented as the compressibility effect. Additional transport equations are also solved to yield the progress of the polymerization process through the conversions of water, isocyanate, and the amount of gases produced. Finally, the evolution of the cell size distribution inside the foam phase is evaluated by using a population balance equation. The solution of PBE inside the CFD code is performed by transforming the problem into a set of transport equations for the moments of bubble/cell size distribution. The quadrature method of moments (QMOM) is applied to approximate the right hand side of moments equations which represent the growth and coalescence of bubbles.External routines/libraries: OpenFOAM® (version 3.0.1) (http://www.openfoam.org)Restrictions: Due to lack of available experimentally-based or analytical models, the current version of the solver supports two different blowing agents including n-pentane and R-11. Further, the growth rate of the bubbles is assumed to be diffusion-controlled and its rate associated with the concentration gradient around the bubbles. This restriction will be relaxed in near future by adopting a detailed model for the bubble growth rate. It must be also reminded that the solver is compiled with OpenFOAM version 3.0.1 and compiling it with other versions might require additional efforts.

Solution of population balance equations in applications with fine particles: Mathematical modeling and numerical schemes

The accurate description and robust simulation, at relatively low cost, of global quantities (e.g. number density or volume fraction) as well as the size distribution of a population of fine particles in a carrier fluid is still a major challenge for many applications. For this purpose, two types of methods are investigated for solving the population balance equation with aggregation, continuous particle size change (growth and size reduction), and nucleation: the extended quadrature method of moments (EQMOM) based on the work of Yuan et al. [52] and a hybrid method (TSM) between the sectional and moment methods, considering two moments per section based on the work of Laurent et al. [30]. For both methods, the closure employs a continuous reconstruction of the number density function of the particles from its moments, thus allowing evaluation of all the unclosed terms in the moment equations, including the negative flux due to the disappearance of particles. Here, new robust and efficient algorithms are developed for this reconstruction step and two kinds of reconstruction are tested for each method. Moreover, robust and accurate numerical methods are developed, ensuring the realizability of the moments. The robustness is ensured with efficient and tractable algorithms despite the numerous couplings and various algebraic constraints thanks to a tailored overall strategy. EQMOM and TSM are compared to a sectional method for various simple but relevant test cases, showing their ability to describe accurately the fine-particle population with a much lower number of variables. These results demonstrate the efficiency of the modeling and numerical choices, and their potential for the simulation of real-world applications.

Algorithms for the Haar wavelet based fast evaluation of aggregation integrals in population balance equations

In several production processes, the distribution of particles dispersed in an environmental phase may be mathematically described by the solution of population balance equations. We are concerned with the development of efficient numerical techniques for the aggregation process: It invokes an integral term that is usually numerically expensive to evaluate and often dominates the total simulation cost.We describe an approach on locally refined nested grids to evaluate both the source and the sink terms in almost linear complexity (instead of quadratic complexity resulting from a direct approach). The key is to switch from a nodal to a wavelet basis representation of the density function. We illustrate the numerical performance of this approach, both in comparison to a discretization of piecewise constant functions on a uniform grid as well as to the fixed pivot method on a geometric grid.

Multiscale Modeling of Expanding Polyurethane Foams via Computational Fluid Dynamics and Population Balance Equation

SummaryThis study is aimed to formulate a numerical modeling recipe for polyurethane foams. The model is capable of simulating the foam principal characteristics during mold filling. The model is formulated upon coupling of Computational Fluid Dynamics (CFD) and Population Balance Equation (PBE) to predict and simulate the evolution of foam features including apparent density and viscosity, bubble (or cell) size distribution (BSD) during the polymerization, as well as its kinetics. The solution of PBE inside the CFD code is performed with Quadrature Method of Moments (QMOM). The foam, constituted by a liquid polymer and gas bubbles, is simulated as a pseudo‐single‐phase system, while the interface between the foam and the surrounding air is tracked by a Volume‐of‐Fluid (VOF) solver within the open‐source CFD code OpenFOAM. The modeling is applied for a simple foaming experiment and attention is paid to the effect of the rheological model on the predictions.

Exact solutions of population balance equation

Population balance equations have been used to model a wide range of processes including polymerization, crystallization, cloud formation, and cell dynamics, but the lack of analytical solutions necessitates the use of numerical techniques. The one-dimensional homogeneous population balance equation with time dependent but size independent growth rate and time dependent nucleation rate is investigated. The corresponding system of equations is solved analytically in this paper.

Effect of Triplet T 1-State Population on Fluorescence Polarization Kinetics of Organic Fluorophores

The fluorescence polarization kinetics of fluorescent organic molecules and molecules with two interconverting photoisomeric forms and fixed molecular skeletons were investigated by numerical methods based on the solution of population balance equations. The depolarizing effect of the lowest triplet T 1 state and the polarization time dependence for quasi-continuous-wave (cw) and pulsed photoexcitation were studied using model compounds. The fluorescence depolarization with cw excitation was characterized by times that were shorter than the T 1-state lifetime; for the isomers, also by times that correlated with the photoisomerization times. Intense pulsed photoexcitation distorted significantly the initial oscillator orientation distribution in the singlet S 0 and S 1 states and could lead to sign reversal of the polarization.

Application of the Lagrangian meshfree approach to modelling of batch crystallisation: Part I—Modelling of stirred tank hydrodynamics

Crystallization phenomena in stirred reactors are influenced by local hydrodynamic conditions and these must be taken into account for successful process scale-up and optimization. This article is the first of two presenting the application of the Smoothed Particle Hydrodynamics (SPH) method to modelling of batch crystallisation in stirred tanks. The benefits of the Lagrangian meshfree methods were discussed and the SPH method was proposed as an efficient method for the rapid prediction of the global mean flow in stirred reactors. Various aspects of the simulation results were discussed such as quality of the fluid prediction, computational requirements and availability of the crystal size distribution without reconstruction. It has been shown that the computational requirements and accuracy of the fluid flow prediction can be controlled through the particle size and that the particles with a radius of 1.50–1.75mm in the reactor of 2.65l provide a good balance between the quality of the prediction and the computational requirements.The developed Smoothed Particle Hydrodynamics model was applied to a numerical solution of coupled computational fluid dynamics and discretised population balance equations to model a batch crystallization process. Due to the specific formulation of the SPH equations the resulting ODE system is solved using the weighted contributions rather than numerically by solving a linear system of equations. Therefore, a large number of additional transport equations resulting from the discretisation of the population balance leads to only a minor increase in computational requirements (around 60% for 200 equations). The non-idealities in the reactor result in non-uniform mixing and contribute to the dispersion of the CSD. The effect of the hydrodynamics on the local temperature/supersaturation and the resulting crystal size distribution was captured and compared to the ideal mixing case.The developed SPH method served as a basis for the Part II of the series where a methodology for solution of population balance equations using high-resolution finite volume schemes or method of characteristics computed in parallel and independently from the Navier–Stokes equations is presented.

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

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

Population balance discretization for growth, attrition, aggregation, breakage and nucleation

This paper presents a new discretization method to solve one-dimensional population balance equations (PBE) for batch and unsteady/steady-state continuous perfectly mixed systems. The numerical technique is valid for any size change mechanism (i.e., growth, aggregation, attrition, breakage and nucleation occurring alone or in combination) and different discretization grids.The developed strategy is based on the moving pivot technique of Kumar and Ramkrishna and the cell-average method of Kumar et al. A novel contribution is proposed to numerically handle the growth and attrition terms, for which a new representation of the number density function within each size class is developed. This method allows describing the number particle fluxes through the class interfaces accurately by preserving two sectional population moments.By comparing the numerical particle size distributions with analytical solutions of one-dimensional PBEs (including different size change mechanisms and particle-size dependent kinetics), the accuracy of the proposed numerical method was proved.

Two Analytical Approaches for Solution of Population Balance Equations: Particle Breakage Process

AbstractVarious particulate systems were modeled by the population balance equation (PBE). However, only few cases of analytical solutions for the breakage process do exist, with most solutions being valid for the batch stirred vessel. The analytical solutions of the PBE for particulate processes under the influence of particle breakage in batch and continuous processes were investigated. Such solutions are obtained from the integro‐differential PBE governing the particle size distribution density function by two analytical approaches: the Adomian decomposition method (ADM) and the homotopy perturbation method (HPM). ADM generates an infinite series which converges uniformly to the exact solution of the problem, while HPM transforms a difficult problem into a simple one which can be easily handled. The results indicate that the two methods can avoid numerical stability problems which often characterize general numerical techniques in this area.

A modified cell average technique for the solution of population balance equation

A new generalized discretization technique, based on the cell average technique (CAT), for solving the population balance equation and predicting the zeroth and first moments has been developed. The proposed algorithm can be easily cast into computer programs, is conceptually straightforward and demonstrated to be computationally economic. Additionally, a new type of mesh, for breakage, aggregation and combined processes is introduced. A new solver, compatible with aforementioned processes, for improving the accuracy of technique has been also proposed. A number of solvable processes were simulated by this technique and compared with the analytical solution and the CAT method. The results were very close to the analytical solution and much more accurate than CAT results.

Accuracy and optimal sampling in Monte Carlo solution of population balance equations

Implementation of a Monte Carlo simulation for the solution of population balance equations (PBEs) requires choice of initial sample number (N0), number of replicates (M), and number of bins for probability distribution reconstruction (n). It is found that Squared Hellinger Distance, H2, is a useful measurement of the accuracy of Monte Carlo (MC) simulation, and can be related directly to N0, M, and n. Asymptotic approximations of H2 are deduced and tested for both one‐dimensional (1‐D) and 2‐D PBEs with coalescence. The central processing unit (CPU) cost, C, is found in a power‐law relationship, , with the CPU cost index, b, indicating the weighting of N0 in the total CPU cost. n must be chosen to balance accuracy and resolution. For fixed n, M × N0 determines the accuracy of MC prediction; if b > 1, then the optimal solution strategy uses multiple replications and small sample size. Conversely, if 0 < b < 1, one replicate and a large initial sample size is preferred. © 2015 American Institute of Chemical Engineers AIChE J, 61: 2394–2402, 2015

High dimensional population balances for the growth of faceted crystals: Combining Monte Carlo integral estimates and the method of characteristics

A numerical scheme is proposed for the solution of population balance equations that describe the growth of faceted crystals. Therefore, two issues are discussed. First, a numerical scheme for solution of high dimensional population balance equations is proposed. The scheme is based on Monte Carlo integration for the integral distribution properties (number, area, volume) combined with the method of characteristics. Samples’ points are initially obtained from an arbitrary probability distribution via the Metropolis–Hastings algorithm. The evolution of these sample points is subsequently computed using the method of characteristics, while the sample points themselves are the basis of the Monte Carlo integration of the population to obtain e.g. total crystal volume to close the mass balance. The theoretical background of the Monte Carlo integration allows us to estimate the error introduced in the mass balance depending on a particular choice of sampling points. The effectiveness of the scheme is verified for an analytically solvable 3-dimensional population. That case study is also employed to demonstrate that an adequate choice of the probability distribution for the generation of the initial sample points can improve the accuracy without increasing the computational effort. The second issue that must simultaneously be considered comprises the disappearance of faces during growth. Such conditions create a constraint for each face distance, measured from a chosen origin, which, prior to the disappearance of that face, had freely evolved according to its growth rate. This constraint is defined for general faceted crystals, and a procedure is proposed to correct the growth rate so that the constraint can be maintained in dynamic simulations. The application of this procedure is demonstrated in a second case study that simulates a 7-dimensional population balance equation.

Fast evaluation of univariate aggregation integrals on equidistant grids

A variety of production processes in chemistry and biotechnology are concerned with particles dispersed in an environmental phase. The particle distribution is mathematically described by the solution of population balance equations of integro-differential type. We are concerned with the aggregation process: it invokes an integral term that is usually numerically expensive to evaluate and often dominates the total simulation cost. We will expose the algorithmic details of an efficient approach based on a separable approximation of the aggregation kernel and a subsequent fast Fourier transformation. This approach reduces the originally quadratic complexity to an almost optimal complexity O(nlogn) in the dimension of the approximation space. We include numerical tests illustrating its application to representative aggregation kernels from the literature. While originally developed in the context of a discretization with piecewise constant functions, we illustrate how these ideas can be applied in the setting of the popular sectional methods.

Analytical solution of population balance equation involving aggregation and breakage in terms of auxiliary equation method

This paper presents an effective analytical simulation to solve population balance equation (PBE), involving particulate aggregation and breakage, by making use of appropriate solution(s) of associated complementary equation via auxiliary equation method (AEM). Travelling wave solutions of the complementary equation of a nonlinear PBE with appropriately chosen parameters is taken to be analogous to the description of the dynamic behaviour of the particulate processes. For an initial proof-of-concept, a general case when the number of particles varies with respect to time is chosen. Three cases, i.e. (1) balanced aggregation and breakage, (2) when aggregation can dominate and (3) breakage can dominate, are selected and solved for their corresponding analytical solutions. The results are then compared with the available analytical solution, based on Laplace transform obtained from literature. In this communication, it is shown that the solution approach proposed via AEM is flexible and thereby more efficient than the analytical approach used in the literature.

Adomian decomposition method solution of population balance equations for aggregation, nucleation, growth and breakup processes

The dynamic behavior of particulate processes under the influence of the nonlinear aggregation term, nucleation, growth and breakup is studied. Analytic solutions are obtained from the integro-differential population balance equation governing the particle size distribution density function for special cases by the Adomian decomposition method (ADM) and are compared with other analytical solutions available in the literature. It avoids the difficulty numerical stability that often characterizes general numerical techniques in this area. It generates an infinite series which converges uniformly to the exact solution of the problem. For the case where there are no previous results a comparison between the present method and projection method which include collocation techniques is made.The results obtained indicate that the Adomian decomposition method is highly accurate, efficient and are useful for further work.

Ultrasound‐facilitated particle breakage: Estimation of kinetic parameters using population balance modelling

AbstractUltrasound‐facilitated particle breakage has been proven to be a good technique for size reduction. However, because of the difficulties associated with the solution of population balance equation (PBE), there has been little attention on the estimation of kinetics of this process. In this paper an attempt has been made to develop a generic approach for solving PBE with a given particle size distribution (PSD) data to estimate the breakage kinetics. For this purpose, discretization technique along with gPROMS parameter estimation tool was used. This method was found to be a comprehensive method for the estimation of breakage kinetics in general and was subsequently used for the estimation of ultrasound‐facilitated breakage kinetics, using a simple model system. Potassium nitrate (KNO3) particles suspended in toluene was used as model system, and the effect of induced power on breakage strength was studied. The data obtained were used for the estimation of specific breakage rate with different particle size distribution function. The distribution function, for which simulated PSD was found to match with the experimental PSD, was considered for deriving energy‐size reduction law.

System model of a tablet manufacturing process

A novel process system model of a roll compaction tablet manufacturing process has been developed. The model couples together multiple unit operations, namely roller compaction, milling and tablet press, providing a framework to simulate cause and effect relationships between process parameters and product performance. Models for tablet tensile strength and dissolution are implemented to evaluate tablet performance. The models for the milling and the dissolution units are based on the solution of population balance equations. The model is validated by comparison with experimental data. This work demonstrates that it is possible to develop a model for tablet manufacturing that is based on fundamental process understanding and that the resulting simulation tool, requiring limited experimental calibration, can be used easily and effectively in the workflow of pharmaceutical product development.

A computationally efficient technique for the solution of multi-dimensional PBMs of granulation via tensor decomposition

Multi-dimensional population balance equations (PBEs) are commonly used to describe the dynamics of particulate processes such as granulation. Such a class of equations are numerically complex and computationally intensive to solve due to the multiple internal coordinates involved. A computationally efficient model reduction technique would overcome the computational overheads associated with the solution of multi-dimensional PBEs. Moreover, this enables the process model to be used efficiently in process control and optimization. This study is concerned with the development of a novel reduced order model for a three-dimensional population balance model (PBM) for granulation, using a tensor decomposition technique in combination with separation of variables and singular value decomposition. These techniques were used to decompose the complex aggregation and breakage integrals. The developed model is faster by two orders of magnitude, requires less memory allocation for the storage of variables and results in negligible error when compared with the full model.