Bivariate Population Balance Equation Research Articles

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.

Numerical study on fractal-like soot aggregate dynamics of turbulent ethylene-oxygen flame

The soot aggregate dynamics of turbulent ethylene-oxygen flame is numerically studied for different equivalence ratios and jet Reynolds numbers (Rej). Our developed Taylor-series expansion method of moments (TEMOM) model in our previous research studies is further extended to solve the bivariate population balance equation (PBE) and formulate the novel Bivariate TEMOM model scheme. Full numerical validations are performed with the stochastically weighted operator splitting Monte Carlo method and moving sectional method, the Bivariate TEMOM model scheme coupled with large eddy simulation (LES) method and soot formation model is used to simulate fractal-like soot aggregate dynamics in turbulent ethylene-oxygen flame. The results show that soot nucleation and surface growth processes are enhanced with increasing equivalence ratio while the coagulation rate is hardly varied as the total soot volume fraction is quite low. The increasingly uniform distributions of fractal dimension and particle size can also be observed. As Rej increases from 14,400 to 36,000, both the mean diameter and mean fractal dimension of soot aggregates gradually decrease as the surface growth rates of soot aggregates decrease significantly. However, the effect of increasing Rej on coagulation and nucleation rates are slight due to the decreasing residence time of soot particles in the combustor.

Analytical solution for a three-dimensional non-homogeneous bivariate population balance equation—a special case

There has been a dramatic increase in the number of research publications using the population balance equation (PBE). The PBE allows the prediction of the spatial distribution of the dispersed phase size for an accurate estimation of the flow fields in multiphase flows. A few recent studies have proposed new efficient numerical methods to solve non-homogeneous multivariate PBE and implemented the same in computational fluid dynamics (CFD) codes. However, these codes are generally benchmarked against other numerical methods and applied without verification. To address this gap, an analytical solution for a three-dimensional non-homogeneous bivariate PBE is presented here for the first time. The method of manufactured solutions (MMS) has been used to construct a solution of the non-homogeneous PBE containing breakage and coalescence terms, and an additional source term appearing as a result of this method. The analytical solution presented in this work can be used for the rigorous verification of computer codes written to solve the non-homogeneous bivariate PBE. Quantification of the errors due to different numerical schemes will also become possible with the availability of this analytical solution for the PBE.

Steady state modeling of Kühni liquid extraction column using the Spatially Mixed Sectional Quadrature Method of Moments (SM-SQMOM)

We present a new steady state algorithm for modeling the hydrodynamics and mass transfer behavior of liquid extraction columns based on the SQMOM in a one dimensional domain. The SQMOM is extended to solve the spatially distributed bivariate population balance equation (w.r.t. droplet diameter and the solute concentration) at steady state. The integral spatial numerical flux is closed using the Multi Primary one Secondary Particle Method, while the hydrodynamics integral source terms are closed using the analytical Two-Equal Weight Quadrature formula. To facilitate the source terms implementation, an analytical solution based on the algebraic velocity model is derived to calculate the required dispersed phase mean droplet velocity. In addition to this, the hydrodynamics moment transport equations are coupled with the One Primary One Secondary Particle Method to close the mass transport equations. The resulting system of ordinary differential equations is solved using a fifth order numerical scheme in space. As a case study, the model prediction is validated using published experimental data for DN80 Kühni extraction column. By using SQMOM, fifteen sections (w.r.t. the droplet diameter as an internal coordinate) are found enough to predict the droplet volume distribution and the column hydrodynamics as compared to the experimental data. On the other hand, one section (w.r.t. the droplet solute concentration) along droplet size distribution is found enough to predict the mass transfer profiles along the column height.

Improved numerical inversion methods for the recovery of bivariate distributions of polymer properties from 2D probability generating function domains

Abstract The 2D probability-generating function technique is a powerful method for modeling bivariate distributions of polymer properties. It is based on the transformation of bivariate population balance equations using 2D probability generating functions (pgf) followed by a recovery of the distributions from the transform domain by numerical inversion. A key step of this method is the inversion of the pgf transforms. Available numerical inversion methods yield excellent results for pgf transforms of distributions with independent dimensions with similar orders of magnitude, for example bivariate molecular weight distributions in copolymerization systems. However, numerical problems are found for 2D distributions in which the independent dimensions have very different ranges of values, such as the molecular weight distribution-branching distribution in branched polymers. In this work, two new 2D pgf inversion methods are developed, which regard the pgf as a complex variable. The superior accuracy of these innovative methods makes them suitable for recovering any type of bivariate distribution. This enhances the capabilities of the 2D pgf modeling technique for simulation and optimization of polymer processes. An application example of the technique in a polymeric system of industrial interest is presented.

Discussion on DQMOM to solve a bivariate population balance equation applied to a grinding process

A bivariate population balance equation applied to a grinding process is implemented in a model (PBM). The particles are simultaneously characterized by their size and their mechanical strength, expressed here by the minimum energy needed to break them. PBM is solved by the Direct Quadrature Method of Moments (DQMOM). The mixed moments of the distribution are expressed by the quadrature form of the population density defined for one order (N) and incorporating the weights and the abscissas defined for the two properties. The effect of the quadrature order (N=2,3,4) and the selected set of the 3N moments needed to solve the system on the accuracy of the results is discussed. For a given order of the quadrature, the selected set of the initial mixed moments slightly affects first the weights and abscissas derived from the initial particle distribution. The set of moments also affects the precision of the moments calculated versus time but only those having high orders in relation with the respective range of the solid properties considered. Problems of convergence and significant differences in the predicted mixed moments are also observed when the order of the quadrature is equal to 2. However, the changes of a bivariate distribution versus time applied to a grinding process are well predicted using the DQMOM approach, choosing a number of nodes equal to 3, associated with a smart selection of the moment set, incorporating all the moments of interest.

Solution of bivariate population balance equations with high-order moment-conserving method of classes

In this work the high-order moment-conserving method of classes (HMMC) (Alopaeus et al., 2006) is extended to solve the bivariate Population Balance Equation (PBE). The method is capable of guaranteeing the internal consistency of the discretized equations for a generic moment set, including mixed-order moments of the distribution. The construction of the product tables in the case of aggregation, breakage and convection in internal coordinate space are discussed. Eventually, several test cases are considered to assess the accuracy of the method. The application to a realistic mass transfer problems in a liquid–liquid system is preliminarily discussed. The comparison with analytical solutions of pure aggregation problems shows that the proposed method is accurate with only a limited number of categories.

Comminution process modeling based on the monovariate and bivariate direct quadrature method of moments

Population balance equations (PBE) applied to comminution processes are commonly based on selection and breakage functions, which allow the description of changes of particle‐size distributions vs. time. However, other properties, such as the particle strength, also influence the grinding kinetics. A bivariate PBE was developed and resolved by the direct quadrature method of moments. This equation includes both the particle size and strength, the latter of which is defined as the minimum energy required for breakage. The monovariate case was first validated by comparing the predicted moments with those calculated from the size distributions given by an analytical solution of the PBE derived for specific selection and breakage functions. The bivariate model was then compared with a discretized model to evaluate its validity. Finally, the benefit of the bivariate model was proven by analyzing the sensitivity of some parameters and comparing the results of the monovariate and bivariate cases. © 2014 American Institute of Chemical Engineers AIChE J, 60: 1621–1631, 2014

Modelling of strongly coupled particle growth and aggregation

The mathematical modelling of the dynamics of particle suspension is based on the population balance equation (PBE). PBE is an integro-differential equation for the population density that is a function of time t, space coordinates and internal parameters. Usually, the particle is characterized by a unique parameter, e.g. the matter volume v. PBE consists of several terms: for instance, the growth rate and the aggregation rate. So, the growth rate is a function of v and t. In classical modelling, the growth and the aggregation are independently considered, i.e. they are not coupled. However, current applications occur where the growth and the aggregation are coupled, i.e. the change of the particle volume with time is depending on its initial value v0, that in turn is related to an aggregation event. As a consequence, the dynamics of the suspension does not obey the classical Von Smoluchowski equation. This paper revisits this problem by proposing a new modelling by using a bivariate PBE (with two internal variables: v and v0) and by solving the PBE by means of a numerical method and Monte Carlo simulations. This is applied to a physicochemical system with a simple growth law and a constant aggregation kernel.

Accurate and efficient solution of bivariate population balance equations using unstructured grids

In this work we demonstrate an application of the cell average technique (CAT) for solving bivariate population balance equations on unstructured triangular grids. Various strategies for improving unstructured grids are also discussed in this work. It is observed that the main unwanted feature of the triangular grid are elongated triangles. Strategies for removing such elements are suggested. It is shown that such an improvement reduces the number of pivots by 50% and leads to a drastic reduction of computational time without affecting the accuracy of the solution. Comparison of the numerical solution with the exact solution reveals that the cell average technique on unstructured triangular grid produces a better solution in comparison to the cell average technique on a structured triangular grid. It also produces better results in comparison to fixed pivot technique (FPT) on both structured triangular and unstructured triangular grids. In conclusion, the cell average technique with improved irregular triangular mesh can be seen as the smartest solution technique for the solution of bivariate population balance equations.

Modeling and dynamic analysis of a rotating disc contactor (RDC) extraction column using one primary and one secondary particle method (OPOSPM)

Modeling and dynamic analysis of liquid extraction columns are essential for the design, control strategies and understanding of column behavior during start up and shutdown. Because of the discrete character of the dispersed phase, the population balance modeling framework is needed. Due to the mathematical complexity of the full population balance model, it is still not feasible for dynamic and online control purposes. In this work, a reduced mathematical model is developed by applying the concept of the primary and secondary particle method (Attarakih et al., 2009b, Solution of the population balance equation using the one primary and one secondary particle method (OPOSPM), Computer Aided Chemical Engineering, vol. 26, pp. 1333–1338). The method is extended to solve the nonhomogenous bivariate population balance equation, which describes the coupled hydrodynamics and mass transfer in an RDC extraction column. The model uses only one primary and one secondary particles, which can be considered as Lagrangian fluid particles carrying information about the distribution as it evolves in space and time. This information includes averaged quantities such as total number, volume and solute concentrations, which are tracked directly through a system of coupled hyperbolic conservation laws with nonlinear source terms. The model describes droplet breakage, coalescence and interphase solute transfer. Rigorous hyperbolic analysis of OPOSPM uncovered the existence of four waves traveling along the column height. Three of these are contact waves, which carry volume and solute concentration information. The dynamic analysis in this paper reveals that the dominant time constant is due to solute concentration in the continuous phase. On the other hand, the response of the dispersed phase mean properties is relatively faster than the solute concentration in the continuous phase. Special shock capturing method based on the upwind scheme with flux vector splitting is used, with explicit wave speeds, as a time–space solver. The model shows a good match between analytical and numerical results for special steady state and dynamic cases as well as the published steady state experimental data.

On the solution of bivariate population balance equations for aggregation: Pivotwise expansion of solution domain

The solution of a bivariate population balance equation (PBE) for aggregation of particles necessitates a large 2-d domain to be covered. A correspondingly large number of discretized equations for particle populations on pivots (representative sizes for bins) are solved, although at the end only a relatively small number of pivots are found to participate in the evolution process. In the present work, we initiate solution of the governing PBE on a small set of pivots that can represent the initial size distribution. New pivots are added to expand the computational domain in directions in which the evolving size distribution advances. A self-sufficient set of rules is developed to automate the addition of pivots, taken from an underlying X-grid formed by intersection of the lines of constant composition and constant particle mass. In order to test the robustness of the rule-set, simulations carried out with pivotwise expansion of X-grid are compared with those obtained using sufficiently large fixed X-grids for a number of composition independent and composition dependent aggregation kernels and initial conditions. The two techniques lead to identical predictions, with the former requiring only a fraction of the computational effort. The rule-set automatically reduces aggregation of particles of same composition to a 1-d problem. A midway change in the direction of expansion of domain, effected by the addition of particles of different mean composition, is captured correctly by the rule-set. The evolving shape of a computational domain carries with it the signature of the aggregation process, which can be insightful in complex and time dependent aggregation conditions.

A method for estimating effective coalescence rates during emulsification from oil transfer experiments

The Oil Transfer Technique (OTT) was developed by Taisne et al. [1] to measure coalescence during emulsification and has been applied since in several studies. One of the main drawbacks of this technique is that it only gives a qualitative measure of coalescence. This paper proposes a new evaluation method of OTT experimental results for estimating qualitative coalescence rates, e.g. for investigating the scaling of coalescence with emulsification parameters (such as homogenizing pressure, and emulsifier concentration).The method is based on comparison with simulated OTT experiments using bivariate Population Balance Equation models. Simulations have been performed under a wide variety of conditions in order to investigate the influence of assumptions on coalescence and fragmentation kernels. These investigations show that the scaling of coalescence rates could be determined accurately when the scaling of efficient residence time of drops in the active region of homogenization is known. The proposed evaluation method is also exemplified by analyzing OTT data from two previously published studies.

Modeling and Optimization of Two Stage AC Electrostatic Desalter

Electrostatic desalters are commonly used to separate water from crude oil emulsions. In this study, a novel mathematical model based on a bivariate population balance equation has been proposed to model the steady state distribution of water droplet sizes and salt concentration along an Alternative Current (AC) electrostatic desalter. Pilot plant data for two heavy crude oils are used to validate the model. Effect of different operational parameters on the outlet dehydration efficiency and salt content of crude oil have been studied. Furthermore, the effect of two-stage desalting on the outlet Basic Sediment and Water (BS&W) and salt content of oil have been studied and the optimized parameters of such configuration have been found by differential evolution method to get the best performance of the electrostatic desalters.

On the solution of bivariate population balance equations for aggregation: X–discretization of space for expansion and contraction of computational domain

A new structured discretization of 2D space, named X-discretization, is proposed to solve bivariate population balance equations using the framework of minimal internal consistency of discretization of Chakraborty and Kumar [2007, A new framework for solution of multidimensional population balance equations. Chem. Eng. Sci. 62, 4112–4125] for breakup and aggregation of particles. The 2D space of particle constituents (internal attributes) is discretized into bins by using arbitrarily spaced constant composition radial lines and constant mass lines of slope −1. The quadrilaterals are triangulated by using straight lines pointing towards the mean composition line. The monotonicity of the new discretization makes is quite easy to implement, like a rectangular grid but with significantly reduced numerical dispersion. We use the new discretization of space to automate the expansion and contraction of the computational domain for the aggregation process, corresponding to the formation of larger particles and the disappearance of smaller particles by adding and removing the constant mass lines at the boundaries. The results show that the predictions of particle size distribution on fixed X-grid are in better agreement with the analytical solution than those obtained with the earlier techniques. The simulations carried out with expansion and/or contraction of the computational domain as population evolves show that the proposed strategy of evolving the computational domain with the aggregation process brings down the computational effort quite substantially; larger the extent of evolution, greater is the reduction in computational effort.

A competitive aggregation model for Flash NanoPrecipitation

Flash NanoPrecipitation (FNP) is a novel approach for producing functional nanoparticles stabilized by amphiphilic block copolymers. FNP involves the rapid mixing of a hydrophobic active (organic) and an amphiphilic di-block copolymer with a non-solvent (water) and subsequent co-precipitation of nanoparticles composed of both the organic and copolymer. During this process, the particle size distribution (PSD) is frozen and stabilized by the hydrophilic portion of the amphiphilic di-block copolymer residing on the particle surface. That is, the particle growth is kinetically arrested and thus a narrow PSD can be attained. To model the co-precipitation process, a bivariate population balance equation (PBE) has been formulated to account for the competitive aggregation of the organic and copolymer versus pure organic-organic or copolymer-copolymer aggregation. Aggregation rate kernels have been derived to account for the major aggregation events: free coupling, unimer insertion, and aggregate fusion. The resulting PBE is solved both by direct integration and by using the conditional quadrature method of moments (CQMOM). By solving the competitive aggregation model under well-mixed conditions, it is demonstrated that the PSD is controlled primarily by the copolymer-copolymer aggregation process and that the energy barrier to aggregate fusion plays a key role in determining the PSD. It is also shown that the characteristic aggregation times are smaller than the turbulent mixing time so that the FNP process is always mixing limited.

Solution of Population Balance Equations by the Finite Size Domain Complete Set of Trial Functions Method of Moments (FCMOM) for Inhomogeneous Systems

The FCMOM (finite size domain complete set of trial functions method of moments) is an efficient and accurate numerical technique to solve monovariate and bivariate population balance equations. It was previously formulated for homogeneous systems. In this paper, the FCMOM approach is extended to solve monovariate population balance equations for inhomogeneous (spatially not uniform) systems. In the FCMOM, the method of moments is formulated in a finite domain of the internal coordinates and the particle size distribution function is represented as a truncated series expansion by a complete system of orthonormal functions. The FCMOM is extended to inhomogeneous systems assuming that the particle-phase convective velocity is independent of the internal variables (particle size). The method is illustrated by applications to particle diffusion and to particle convection. In the case of particle convection, a gas−solid dilute flow in a pipe was simulated and the FCMOM equations were coupled with the governing...

On the solution and applicability of bivariate population balance equations for mixing in particle phase

New benchmarks are used to test two classes of discretization methods available in the literature to solve bivariate population balance equations (2-d PBEs), and the applicability of these mean-field equations to finite size systems. The new benchmarks, different from the extensions of their 1-d counterparts, relate to prediction of kinetics of mixing in particle phase under: (i) pure aggregation of particles, called aggregative mixing, and (ii) simultaneous breakup and coalescence of drops. The discretization methods for 2-d PBEs, derived from the widely used 1-d solution methods, are first classified into two classes. We choose one representative method from each class. The results show that the extensions based on minimum consistency of discretization perform quite well with respect to both the new and the old benchmarks, in comparison with the geometrical extensions of 1-d methods. We next revisit aggregative mixing using Monte-Carlo simulations. The simulations show that (i) the time variation of the extent of mixing in finite size systems has power law scaling with the system size, and (ii) the mean-field PBEs fail to capture the evolution of mixing for reduced population of particles at long times. The sum kernel limits the applicability of PBEs to substantially larger particle populations than that seen for the constant kernel. Interestingly, these populations are orders of magnitude larger than those at which the PBEs fail to capture the evolution of total particle population correctly.

Solution of Bivariate Population Balance Equations Using the Finite Size Domain Complete Set of Trial Functions Method of Moments (FCMOM)

The FCMOM (Finite size domain Complete set of trial functions Method Of Moments) is an efficient and accurate numerical technique to solve PBE (population balance equations) and was validated for monovariate PBE [Strumendo, M.; Arastoopour, H. Solution of PBE by MOM in Finite Size Domains. Chem. Eng. Sci. 2008, 63 (10), 2624]. In the present paper, the FCMOM is extended and used to solve bivariate PBE. In the FCMOM, the method of moments is formulated in a finite domain of the internal coordinates and the particle distribution function is represented as a truncated series expansion by a complete system of orthonormal functions. In the extension to bivariate PBE, the capabilities of the FCMOM are maintained, particularly as far as the algorithm efficiency and the accuracy in the bivariate particle distribution function reconstruction. The FCMOM was validated with the following bivariate applications: particle growth, particle dissolution, particle aggregation, and simultaneous aggregation and growth.

Solution of the population balance equation using the sectional quadrature method of moments (SQMOM)

A discrete framework is introduced for simulating the particulate physical systems governed by population balance equations (PBE) with particle splitting (breakage) and aggregation based on accurately conserving (from theoretical point of view) an unlimited number of moments associated with the particle size distribution. The basic idea is based on the concept of primary and secondary particles, where the former is responsible for distribution reconstruction while the latter is responsible for different particle interactions such as splitting and aggregation. The method is found to track accurately any set of low-order moments with the ability to reconstruct the shape of the distribution. The method is given the name: the sectional quadrature method of moments (SQMOM) and has the advantage of being not tied to the inversion of large sized moment problems as required by the classical quadrature method of moments (QMOM). These methods become ill conditioned when a large number of moments are needed to increase their accuracy. On the contrary, the accuracy of the SQMOM increases by increasing the number of primary particles while using fixed number of secondary particles. Since the positions and local distributions for two secondary particles are found to have an analytical solution, no large moment inversion problems are anymore encountered. The generality of the SQMOM is proved by showing that all the related sectional and quadrature methods appearing in the literature for solving the PBE are merely special cases. The method has already been extended to bivariate PBEs.