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Paulo L.c. Lage - One of the best experts on this subject based on the ideXlab platform.
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Solution of the Population Balance Equation using parallel adaptive cubature on GPUs
Computers & Chemical Engineering, 2013Co-Authors: Fabio P. Santos, Inanc Senocak, Jovani L. Favero, Paulo L.c. LageAbstract:Abstract The Dual Quadrature Method of Generalized Moments (DuQMoGeM) is an accurate moment method for solving the Population Balance Equation (PBE). The drawback of DuQMoGeM is the high computational cost associated with numerical integrations of the PBE integral terms in which each integrand can be integrated independently and, therefore, amenable to parallelization on GPUs. In this work, two parallel adaptive cubature algorithms were implemented on a hybrid architecture (CPU–GPU) to accelerate the DuQMoGeM. The speedup and scalability of these parallel algorithms were studied with different types of Genz's test functions. Then, we applied these parallel numerical integration algorithms in the DuQMoGeM solution of the PBE for three bivariate cases, obtaining speedups between 11 and 15.
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Solution of the Population Balance Equation by the direct dual quadrature method of generalized moments
Chemical Engineering Science, 2013Co-Authors: Fabio P. Santos, Jovani L. Favero, Paulo L.c. LageAbstract:The Direct Dual Quadrature Method of Generalized Moments (D2uQMoGeM) was formulated for the solution of the Population Balance Equation. It mixes the properties of the Direct Quadrature Method of Moments (DQMoM) and the Dual Quadrature Method of Generalized Moments (DuQMoGeM). The weights and weighted abscissas are tracking directly as in DQMoM and the quadrature errors are controlled by an adaptive quadrature as in DuQMoGeM. The D2uQMoGeM was implemented and tested for several different problems with analytical solutions. It was shown to be more accurate than DQMoM with a reasonable increase in computational time.
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Implementation and analysis of numerical solution of the Population Balance Equation in CFD packages
Computers & Chemical Engineering, 2008Co-Authors: Luiz Fernando L.r. Silva, R. B. Damian, Paulo L.c. LageAbstract:Abstract Simulation of polydisperse flows must include the effects of particle–particle interaction, as breakage and aggregation, coupling the Population Balance Equation (PBE) with the multiphase modelling. In fact, the implementation of efficient and accurate new numerical techniques to solve the PBE is necessary. The direct quadrature method of moments, known as DQMOM, is a moment-based method that uses an optimal adaptive quadrature closure and came into view as a promising choice for this implementation. In the present work, DQMOM was implemented in two CFD packages: the commercial ANSYS CFX, through FORTRAN subroutines, and the open-source OpenFOAM, by directly coding the PBE solution. Transient zero-dimensional and steady one-dimensional simulations were performed in order to explore the PBE solution accuracy using several interpolation schemes. Simulation cases with dominant breakage, dominant aggregation and invariant solution (equivalent breakage and aggregation) were simulated and validated against an analytical solution. The solution of the Population Balance Equation was then coupled to the two-fluid model, considering that all particles classes share the same velocity field. Momentum exchange terms were evaluated using the local instantaneous Sauter mean diameter of the size distribution function. The two-dimensional tests were performed in a backward facing step geometry where the vortex zones traps the particles and provides high rates of breakage and aggregation.
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a numerical method for solving the transient multidimensional Population Balance Equation using an euler lagrange formulation
Chemical Engineering Science, 2003Co-Authors: F.b. Campos, Paulo L.c. LageAbstract:Abstract A numerical method was developed to solve the Population Balance Equation for transient multidimensional problems including particle–particle interactions. The Population Balance Equation was written in a mixed Euler–Lagrange formulation which was solved using the discretization method that represents the number density function by impulse functions, an operator splitting method and a remeshing procedure for the internal variable that conserves the mass and the number of particles. This method was successfully tested against analytical and semi-analytical solutions for pure breakage, pure coalescence, breakage and coalescence, pure advection, advection with absorption, advection with binary uniform breakage and with constant or linear absorption. The method was also applied to a free-boundary transient one-dimensional gas-phase model in a bubble column reactor with simplified hydrodynamics. Accurate solutions were obtained for several simulation conditions for the bubble column, including gas absorption, bubble breakage, bubble coalescence and variable gas density effects. The results showed that the numerical method is adequate and robust for solving transient Population Balance problems with spatial dependence and particle–particle interactions.
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A numerical method for solving the transient multidimensional Population Balance Equation using an Euler–Lagrange formulation
Chemical Engineering Science, 2003Co-Authors: F.b. Campos, Paulo L.c. LageAbstract:Abstract A numerical method was developed to solve the Population Balance Equation for transient multidimensional problems including particle–particle interactions. The Population Balance Equation was written in a mixed Euler–Lagrange formulation which was solved using the discretization method that represents the number density function by impulse functions, an operator splitting method and a remeshing procedure for the internal variable that conserves the mass and the number of particles. This method was successfully tested against analytical and semi-analytical solutions for pure breakage, pure coalescence, breakage and coalescence, pure advection, advection with absorption, advection with binary uniform breakage and with constant or linear absorption. The method was also applied to a free-boundary transient one-dimensional gas-phase model in a bubble column reactor with simplified hydrodynamics. Accurate solutions were obtained for several simulation conditions for the bubble column, including gas absorption, bubble breakage, bubble coalescence and variable gas density effects. The results showed that the numerical method is adequate and robust for solving transient Population Balance problems with spatial dependence and particle–particle interactions.
Menwer Attarakih - One of the best experts on this subject based on the ideXlab platform.
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on the solution of the Population Balance Equation from global to local constrained maximum entropy method
Chemical Engineering Science, 2019Co-Authors: Menwer Attarakih, Hans-jörg Bart, Mazen M AbukhaderAbstract:Abstract We propose continuous approximations to the Population Balance Equation based on maximization of the Shannon entropy subject to the expected properties of the particle size distribution (PSD). This solution is used to close the source term of the PBE with careful sampling of the PSD at prescribed points as roots of the Nth-degree Legendre polynomial. Being a maximum entropy functional, the solution is unique and converges to the exact solution as the number of sampling points increases with accurate calculation of PSD integral properties. The accuracy and efficiency of the method are demonstrated by trying different analytical case studies (particle aggregation, aggregation and growth, and particle breakage) where we show it is not restricted to prespecified particle kinetics and functional forms. As practical case study, we modelled the coupled hydrodynamics and mass transfer in different liquid extraction columns and compared the calculated results with published steady state experimental data.
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a meshfree maximum entropy method for the solution of the Population Balance Equation
Computer-aided chemical engineering, 2015Co-Authors: Menwer Attarakih, Abdelmalek Hasseine, Hans-jörg BartAbstract:Abstract In this work, the number density function in the Population Balance Equation (PBE) is approximated in terms of field nodes through a complete set of orthogonal basis functions in a semi-logarithmic space. We proposed the functional values at these field nodes to satisfy the maximum entropy solution. This hybridization of function approximation and information theories based on Shannon Maximum Entropy principle, allowed us to construct a sequence of positive continuous approximations of the PBE. The Lagrange multipliers, which result from the maximization of the Shannon entropy subject to the available average information, was estimated by solving a well-conditioned linear system of algebraic Equations. As an application, this meshfree solution of the PBE is validated using an analytical solution of the microbial cell dynamics in a constant abiotic environment with simultaneous cell growth and division for which the analytical solution was derived by using the Adomian method.
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solution of the Population Balance Equation using the differential maximum entropy method dmaxentm an application to liquid extraction columns
Chemical Engineering Science, 2014Co-Authors: Menwer Attarakih, Hans-jörg BartAbstract:Abstract The Population Balance Equation (PBE) is an integro-partial differential Equation with nonlinear source term. The PBE is known to admit analytical solutions only for a few cases with restricted forms of interaction kernels. We propose for the first time a novel converging sequence of continuous approximations to the number concentration function as a solution to the Population Balance Equation (PBE). These approximations are internally consistent with respect to any finite number of desired moments. The uniqueness and convergence of such a sequence are assured by being an optimal solution to the constrained NLP, which maximizes the constrained Shannon entropy function. The solution is an optimal functional containing the maximum missed information about the distribution. This entropy maximization problem is a convex program and is solved by converting the constrained NLP into a set of transport Equations in terms of the optimal Lagrange multipliers. Since differential form of the Lagrange multipliers is used, the method is given the name the Differential Maximum Entropy (DMaxEnt) method. The DMaxEnt method is tested using many standard and even complex liquid–liquid extraction processes, where the Population Balance modeling is needed.
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A Novel MaxEnt Method for the Solution of Two-Dimensional Population Balance Equation with Particle Growth
Computer Aided Chemical Engineering, 2014Co-Authors: Menwer Attarakih, Hans-jörg BartAbstract:Abstract The Population Balance Equation for particle growth finds many applications in chemical process industries and physical sciences. It is a hyperbolic partial differential with few known analytical solutions. We propose in this paper a novel converging sequence of continuous approximations to this Equationfor the case of one- and two- dimensional particle growth. The uniqueness and convergence of such a sequence are assured by maximizing the Shannon entropy function, which is associated witha set of Lagrange multipliers. In contrast to the classical Maximum Entropy Method (MaxEntM), the Lagrange multipliers are estimated using a meshless method by point wise sampling of the continuous distribution. The proposed method provides local information about this distribution and is consistent with its low-order moments.
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integral formulation of the Population Balance Equation application to particulate systems with particle growth
Computers & Chemical Engineering, 2013Co-Authors: Menwer AttarakihAbstract:Abstract Numerical solution of the Population Balance Equation (PBE) is widely used in many scientific and engineering applications. Available numerical methods, which are based on tracking Population moments instead of the distribution, depend on quadrature methods that destroy the distribution itself. The reconstruction of the distribution from these moments is a well-known ill-posed problem and still unresolved question. The present integral formulation of the PBE comes to resolve this problem. As a closure rule, a Cumulative QMOM (CQMOM) is derived in terms of the monotone increasing cumulative moments of the number density function, which allows a complete distribution reconstruction. Numerical analysis of the method show two unique properties: first, the method can be considered as a mesh-free method. Second, the accuracy of the targeted low-order cumulative moments depends only on order of the CQMOM, but not on the discrete grid points used to sample the cumulative moments.
Hugo A. Jakobsen - One of the best experts on this subject based on the ideXlab platform.
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an hp adaptive quadrature method for irregular integrands application to the Population Balance Equation birth term
Chemical Engineering Science, 2019Co-Authors: Mathias Engh, Jannike Solsvik, Hugo A. JakobsenAbstract:Abstract The solution of the Population Balance Equation requires the integration of several source terms. In the numerical weighted residuals methods, Gaussian quadrature is a natural candidate for numerical integration. Previous works using the weighted residuals methods for solving the Population Balance Equation did use a fixed grid of quadrature points. This work shows that the use of adaptive quadrature points for the numerical integration can lead to more efficient and accurate solutions of the Equation. For cases where the integrand shows a high degree of irregularity, the hp-optimization method distributes the quadrature points such that the method becomes more efficient than with a fixed grid. An additional improvement is that the amount of quadrature points changes to fit the need for each integral present, rather than having one set of quadrature points for all cases. A simple Population Balance model demonstrates the use of the adaptive quadrature approach.
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numerical solution of the drop Population Balance Equation using weighted residual and finite volume methods
Journal of Dispersion Science and Technology, 2016Co-Authors: Jannike Solsvik, Per Julian Becker, Nida Sheibatothman, Hugo A. JakobsenAbstract:This article presents a comparison of numerical results obtained by two different approximations of Population Balances—the spectral orthogonal collocation and finite volume methods. In particular, the Population Balance Equation for a homogeneous dispersed liquid–liquid system in a batch reactor was considered in the present numerical study. The focus was placed on the accuracy of the numerical approximation of the particle property density distribution. An advantage of the finite volume method is the easy of distributing the points in a nonuniform discretization. It is supposed that the spectral-element orthogonal collocation method may benefit by dividing the computational domain into elements of various polynomial orders. For the present problems studied, the orthogonal collocation in the spectral framework does not perform as well as the finite volume method.
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The Foundation of the Population Balance Equation: A Review
Journal of Dispersion Science and Technology, 2014Co-Authors: Jannike Solsvik, Hugo A. JakobsenAbstract:In dispersed multi-phase flow modeling using Population Balances (PBs), the dispersed phase system is considered as a Population of entities of the dispersed phase distributed not only in physical space but also in an abstract property space. Different frameworks exist for the formulation of the Population Balance Equation (PBE): (i) continuum mechanical principles, (ii) statistical Boltzmann-like Equation, or (iii) probability principles. The source terms, that is, birth and death of the entities in the Population, are defined from mechanistic principles. This article presents a review of the foundation of the PBE.
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evaluation of breakage kernels for liquid liquid systems solution of the Population Balance Equation by the least squares method
Canadian Journal of Chemical Engineering, 2014Co-Authors: Jannike Solsvik, Per Julian Becker, Nida Sheibatothman, Zsolt Borka, Hugo A. JakobsenAbstract:The breakage frequency and daughter size distribution functions by Coulaloglou and Tavlarides[1] are frequently adopted closures in Population Balance (PB) modelling. A survey of the extensions and modifications of the Coulaloglou and Tavlarides[1] breakage frequency function is provided. Furthermore, the daughter size distribution functions within the statistical category, herein the model proposed by Coulaloglou and Tavlarides[1], are outlined. Most of the breakage models available in literature commonly assume binary breakage only. Thus, the daughter size distribution function suggested by Diemer and Olson[2] is of interest as higher order breakage can be modelled. The breakage closures are evaluated solving the Population Balance Equation (PBE) for a liquid–liquid emulsification system in a stirred tank. The results obtained from a least-squares solver are compared with the experimental data when available.
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Evaluation of the Least-Squares Method for the Solution of the Population Balance Equation
Journal of Dispersion Science and Technology, 2014Co-Authors: Jannike Solsvik, Hugo A. JakobsenAbstract:Spectral methods are evaluated for the solution of Population Balance problems. Both a simplified Population Balance Equation (PBE) with an analytical solution available and a rigorous PBE are considered in this numerical analysis. In comparison with the orthogonal collocation, tau, and Galerkin methods, the least-squares method does not produce the same favorable results. On the other hand, the least-squares method with a direct minimization solver is capable of producing equally accurate results of the model Equations as the orthogonal collocation, tau, and Galerkin methods. Compared to the conventional least-squares method, the direct minimization formulation is better conditioned but does not produce a symmetric positive-definite system matrix.
Pascale Domingo - One of the best experts on this subject based on the ideXlab platform.
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a hybrid stochastic fixed sectional method for solving the Population Balance Equation
Chemical Engineering Science, 2019Co-Authors: Alexandre Bouaniche, Luc Vervisch, Pascale DomingoAbstract:Abstract The dynamics of flowing non-inertial particles undergoing nucleation, surface growth/loss, agglomeration and sometimes breakage, is usually characterised by the particle size distribution function. This distribution evolves according to a Population Balance Equation. A novel approach combining Monte Carlo and fixed-sectional methods is proposed to minimise the discretisation errors when solving the surface growth/loss term of the Population Balance Equation. The approach relies on a fixed number of stochastic particles and sections, with a numerical algorithm organised to minimise errors even for a moderate number of stochastic particles and sections. Canonical test cases featuring nucleation, agglomeration, and surface growth/loss are simulated. Results against the analytical solutions confirm the improvement in accuracy of the novel approach compared with fixed-sectional methods for the same computational effort. The hybrid method is thus of particular interest for simulating problems where surface growth/loss dominates the particles physics.
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A hybrid stochastic/fixed-sectional method for solving the Population Balance Equation
Chemical Engineering Science, 2019Co-Authors: Alexandre Bouaniche, Luc Vervisch, Pascale DomingoAbstract:Abstract The dynamics of flowing non-inertial particles undergoing nucleation, surface growth/loss, agglomeration and sometimes breakage, is usually characterised by the particle size distribution function. This distribution evolves according to a Population Balance Equation. A novel approach combining Monte Carlo and fixed-sectional methods is proposed to minimise the discretisation errors when solving the surface growth/loss term of the Population Balance Equation. The approach relies on a fixed number of stochastic particles and sections, with a numerical algorithm organised to minimise errors even for a moderate number of stochastic particles and sections. Canonical test cases featuring nucleation, agglomeration, and surface growth/loss are simulated. Results against the analytical solutions confirm the improvement in accuracy of the novel approach compared with fixed-sectional methods for the same computational effort. The hybrid method is thus of particular interest for simulating problems where surface growth/loss dominates the particles physics.
Michael A. Henson - One of the best experts on this subject based on the ideXlab platform.
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Achieving Target Emulsion Drop Size Distributions Using Population Balance Equation Models of High-Pressure Homogenization
2015Co-Authors: Shashank N. Maindarkar, Hans Hoogland, Michael A. HensonAbstract:Population Balance Equation (PBE) models have been used extensively to predict drop size distributions (DSDs) of dispersed phase systems. In our previous publications, we have used the PBE framework to develop increasingly sophisticated process models for oil-in-water emulsification in high-pressure homogenizers. The goal of this study was to utilize these PBE models for integrated emulsion product and process design through the formulation and solution of homogenizer optimization problems. For a specified number of homogenization passes, the initial amount of surfactant and the pressure of each pass were determined by solving a nonlinear least-squares optimization problem such that the target DSD were achieved. Three alternative objective functions that differed with respect to the distribution specification and the penalty on surfactant usage were formulated and solved for different target DSDs. The model predictions were successfully validated by performing homogenization experiments using the optimized formulation and homogenization variables
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achieving target emulsion drop size distributions using Population Balance Equation models of high pressure homogenization
IFAC Proceedings Volumes, 2013Co-Authors: Shashank N. Maindarkar, Michael A. HensonAbstract:Abstract A Population Balance Equation (PBE) model that accounts for drop breakage and coalescence in high pressure homogenization was used for emulsion product design. Six adjustable parameters were estimated by nonlinear optimization from measured drop volume distributions at a specified operating condition. The values of two parameters were estimated at four different homogenization pressures and interpolated to allow improved prediction over a range of pressures. Using two alternative objective functions, the parameterized model was used to determine the pressure of each homogenization pass needed to achieve the target drop size distribution at the final pass. Homogenization experiments performed to validate the model predictions produced measured distributions in very good agreement with two target distributions.
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experimental investigation and Population Balance Equation modeling of solid lipid nanoparticle aggregation dynamics
Journal of Colloid and Interface Science, 2012Co-Authors: Yihui Yang, Alessandro Corona, Michael A. HensonAbstract:Solid lipid nanoparticles (SLNs) have applications in drug delivery and the encapsulation of bioactive, lipophilic compounds. However, SLNs tend to aggregate when stored due to the lipid crystals undergoing a polymorphic transformation from the unstable α form to the stable β form. We developed a Population Balance Equation (PBE) model for prediction of average polymorph content and aggregate size distribution to better understand this undesirable behavior. Experiments with SLNs stored at room temperature showed that polymorphic transformation was the rate determining step for our system, SLNs with smaller initial size distributions aggregated more rapidly, and aggregates contained particles with both α and β crystals. Using parameter values estimated from our data, the PBE model was able to capture the bimodal nature of aggregate size distributions, the α-to-β polymorph ratio, and the faster aggregation dynamics of SLNs with smaller initial size distributions. However, the model was unable to adequately capture the fast disappearance rate of primary particles, the broad size distributions of formed aggregates, and the significant α content of aggregating particles. These discrepancies suggest that a PBE model which accounts for polymorph content as an internal variable along with aggregate size may be required to better reproduce experimental observations.
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Distribution control of particulate systems based on Population Balance Equation models
Proceedings of the 2003 American Control Conference 2003., 1Co-Authors: Michael A. HensonAbstract:Manufacturing processes in which the desired product takes the form of individual particles are ubiquitous in the chemical, pharmaceutical and agricultural industries. Particulate processes often are modeled using a form of the Population Balance Equation (PBE) which describes the evolution of the particle distribution. In many applications, control of the particle distribution is necessary to achieve the desired product properties. In this paper, a model predictive control strategy based on a discretized representation of a general one-dimensional PBE is proposed for particle distribution control. The controller is formulated to minimize the least squares difference between the predicted and target distribution measurements. The proposed method is applied to the problem of cell mass distribution control in a continuous yeast fermentor.