Population Balance Systems Research Articles

Simulations of an ASA flow crystallizer with a coupled stochastic-deterministic approach

A coupled solver for population balance systems is presented, where the flow, temperature, and concentration equations are solved with finite element methods, and the particle size distribution is simulated with a stochastic simulation algorithm, a so-called kinetic Monte-Carlo method. This novel approach is applied for the simulation of an axisymmetric model of a tubular flow crystallizer. The numerical results are compared with experimental data.

Optimal control of univariate and multivariate population balance systems involving external fines removal

We propose an algorithm for optimal control of univariate and multivariate population balance systems governed by a class of first-order linear partial differential equation of hyperbolic type involving size dependent particle growth and filtering. In particular, we investigate the impact of filtering on the optimal control outcomes. To this end, we apply a recently developed polynomial method of moments and a standard steepest descent gradient-based optimization scheme to batch crystallization benchmark problems involving external fines removal. Numerical evaluations for various case-studies aiming at minimizing the mass of grown nuclei in the context of crystal shape manipulation demonstrate the effectiveness of fines dissolution. We also validate the accuracy of the proposed method of moments by utilizing a known numerical moving grid discretization method as a reference. Our results are of interest for the control of a wide range of population balance systems in various applications such as pharmaceutics, chemical engineering, particle shape engineering.

A comparative study of a direct discretization and an operator-splitting solver for population balance systems

A direct discretization approach and an operator-splitting scheme are applied for the numerical simulation of a population balance system which models the synthesis of urea with a uni-variate population. The problem is formulated in axisymmetric form and the setup is chosen such that a steady state is reached. Both solvers are assessed with respect to the accuracy of the results, where experimental data are used for comparison, and the efficiency of the simulations. Depending on the goal of simulations, to track the evolution of the process accurately or to reach the steady state fast, recommendations for the choice of the solver are given.

Generalizing ODE modeling structure for multivariate systems with distributed parameters

We propose an approximate model in form of a system of ordinary differential equations (ODEs) for a class of first-order multivariate linear partial differential equations (PDEs) of the hyperbolic type. The resulting scheme utilizes the method of moments and least-square approximations over orthogonal polynomial bases for the factors of PDE depending on the spatial coordinate of the PDE. The class of examined PDEs appears typically in population balance systems with fines removal. The proposed modeling approach is generally of interest for control and optimization of multivariate systems with distributed parameters.

Approximate ODE models for population balance systems

We propose an approximate polynomial method of moments for a class of first-order linear PDEs (partial differential equations) of hyperbolic type, involving a filtering term with applications to population balance systems with fines removal terms. The resulting closed system of ODEs (ordinary differential equations) represents an extension to a recently published method of moments which utilizes least-square approximations of factors of the PDE over orthogonal polynomial bases. An extensive numerical analysis has been carried out for proof-of-concept purposes. The proposed modeling scheme is generally of interest for control and optimization of processes with distributed parameters.

An efficient method for calculating the moments of multidimensional growth processes in population balance systems

AbstractMultidimensional growth processes play an important role in many fields of applications, such as crystallization processes and cell culture engin‐ eering. The numerical solution of the corresponding multivariate population balance equations is quite challenging as standard discretization‐based methods are not efficient for high dimensional problems. For this reason, the distribution dynamics is often characterized by its moments. However, the moment dynamics generally cannot be calculated in closed form. Existing approximation techniques cannot be implemented efficiently in the multivariate framework, in particular for high dimensional problems. This contribution presents an alternative methodology for the efficient approximate computation of moments for multidimensional growth processes using Monomial Cubatures and the Method of Characteristics (MOC). The procedure will be shown to reproduce the moments accurately for one‐ and two‐dimensional examples which feature nonlinear growth rates and coupling to a continuous phase, respectively. Furthermore numerical effort and accuracy will be analyzed for a five‐dimensional benchmark problem.

Method of moments over orthogonal polynomial bases

A method for the design of approximate models in the form of a system of ordinary differential equations (ODE) for a class of first-order linear partial differential equations of the hyperbolic type with applications to monovariate and multivariate population balance systems is proposed in this work. We develop a closed moment model by utilizing a least square approximation of spatial-dependent factors over an orthogonal polynomial basis. A bounded hollow shaped interval of convergence with respect to the order of the approximate ODE model arises as a consequence of the structural and finite precision computation numerical errors. The proposed modeling scheme is of interest in model-based control and optimization of processes with distributed parameters.

Direct discretizations of bi-variate population balance systems with finite difference schemes of different order

The accurate and efficient simulation of bi-variate population balance systems is nowadays a great challenge since the domain spanned by the external and internal coordinates is five-dimensional. This report considers direct discretizations of this equation in tensor-product domains. In this situation, finite difference methods can be applied. The studied model includes the transport of dissolved potassium dihydrogen phosphate (KDP) and of energy (temperature) in a laminar flow field as well as the nucleation and growth of KDP particles. Two discretizations of the coupled model will be considered which differ only in the discretization of the population balance equation: a first order monotone upwind scheme and a third order essentially non-oscillatory (ENO) scheme. The Dirac term on the right-hand side of this equation is discretized with a finite volume method. The numerical results show that much different results are obtained even in the class of direct discretizations.

An operator-splitting finite element method for the efficient parallel solution of multidimensional population balance systems

A finite element method for solving multidimensional population balance systems is proposed where the balance of fluid velocity, temperature and solute partial density is considered as a two-dimensional system and the balance of particle size distribution as a three-dimensional one. The method is based on a dimensional splitting into physical space and internal property variables. In addition, the operator splitting allows to decouple the equations for temperature, solute partial density and particle size distribution. Further, a nodal point based parallel finite element algorithm for multi-dimensional population balance systems is presented. The method is applied to study a crystallization process assuming, for simplicity, a size independent growth rate and neglecting agglomeration and breakage of particles. Simulations for different wall temperatures are performed to show the effect of cooling on the crystal growth. Although the method is described in detail only for the case of d=2 space and s=1 internal property variables it has the potential to be extendable to d+ s variables, d=2, 3 and s ≥ 1 .

Optimal control of multidimensional population balance systems for crystal shape manipulation

AbstractOptimal control of multidimensional particulate processes for crystal shape manipulation based on the minimum principle and the method of characteristics is discussed. The posed optimal control problems consist in achieving a desired morphology for a seed crystal population within a free or a limited time scope, while minimizing the net mass of the nucleated crystal particles. An ODE system is obtained by neglecting the natural feedback of the nucleation mass into the crystallization kinetics. Optimal solutions resulting thereof reduce to boundary value problems in only one or two unknown parameters. Moreover, for the original optimal control problem, involving the full process dynamics, a simple feasible sub-optimal solution and an upper bound for the cost function are proposed. The proposed computational approach is compared with dynamic optimization techniques in terms of the efficiency and accuracy by means of a numerical example.

On the impact of the scheme for solving the higher dimensional equation in coupled population balance systems

AbstractPopulation balance systems are models for processes in nature and industry that lead to a coupled system of equations (Navier–Stokes equations, transport equations, etc.) where the equations are defined in domains with different dimensions. This paper will study the impact of using different schemes for solving the three‐dimensional (3D) equation of a precipitation process in a two‐dimensional flow domain. The numerical schemes for the 3D equation are assessed with respect to the median of the volume fraction of the particle size distribution and the computational costs. It turns out that in the case of a structured flow field with small variations in time all schemes give qualitatively the same results. For a highly time‐ dependent flow field, the evolution of the median of the volume fraction differs considerably between first order and higher order schemes. Copyright © 2010 John Wiley & Sons, Ltd.

Simulations of population balance systems with one internal coordinate using finite element methods

The paper presents an approach for simulating a precipitation process which is described by a population balance system consisting of the incompressible Navier–Stokes equations, nonlinear convection–diffusion–reaction equations and a transport equation for the particle size distribution (PSD). The Navier–Stokes equations and the convection–diffusion–reaction equations are discretized implicitly in time and with finite element methods in space. Two stabilization techniques for the convection–diffusion–reaction equations are investigated. An explicit temporal discretization and an upwind finite difference method are used for discretizing the equation of the PSD. Simulations of the calcium carbonate precipitation in a cavity are presented which study the influence of the flow field on the PSD at the outflow. It is shown that variations of the positions of the inlets change the volume fraction of the PSD at the center of the outlet. The corresponding medians of the volume fraction differ up to a factor of about three. In addition, it is demonstrated that the use of the two different stabilized finite element methods for the convection–diffusion–reaction equations leads to completely different numerical results.

Integral approximation—an approach to reduced models for particulate processes

In this contribution, a model reduction technique for population balance systems describing particulate processes is presented. This technique is based on integral approximation and allows the derivation of highly accurate moment models. In contrast to other model reduction methods which can be found in literature, this integral approximation technique can be applied for arbitrarily complex phenomena specifications. The applicability of the presented method will be demonstrated for different example processes by comparing the dynamic behavior of the original population balance models with those of the derived reduced models of moments.