# Numerical simulation of population balance systems

Stochastic-Deterministic Solver
 Clemens Bartsch, Viktoria Wiedmeyer, Zahra Lakdawala, Robert I.A. Patterson, Andreas Voigt, Kai Sundmacher, Volker John Stochastic-Deterministic Population Balance Modeling and Simulation of a Fluidized Bed Crystallizer Experiment, Chemical Engineering Science 208, Article 115102, 2019 The crystallization of potassium aluminum sulfate dodecahydrate (potash alum) in a fluidized bed crystallizer is studied both with experiments and simulations. A population balance system with three spatial coordinates and one internal coordinate (mass) is utilized as our model. The simulations are performed with a stochastic-deterministic method with novel extensions, where the fluid dynamics of the crystallizer (flow field, temperature, concentration) are solved deterministically and the particles are simulated with a stochastic method. In experiments of \SI{30}{min} duration, the average crystal diameter increases by growth and agglomeration from about \SI{130}{\mu m} to \SI{210}{\mu m}. This observation agrees qualitatively well with our simulation results. A quantitative difference between simulation and experiment leaves room for future improvements in modeling. Clemens Bartsch, Volker John, Robert Patterson Simulations of an ASA Flow Crystallizer with a Coupled Stochastic-Deterministic Approach, Comp. & Chem. Engrg. 124, 350-363, 2019 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.
Crystallization in a helically-coiled flow tube
 Viktoria, Wiedmeyer, Felix Anker, Clemens Bartsch, Andreas Voigt, Volker John, Kai Sundmacher Continuous Crystallization in a Helically-Coiled Flow Tube: Analysis of Flow Field, Residence Time Behavior and Crystal Growth, Ind. Eng. Chem. Res. 56, 3699 - 3712, 2017 A continuously operated helically-coiled flow tube (HCT) crystallizer is investigated for crystal growth. Inline video-imaging is used for crystal shape analysis and residence time estimation of potash alum. The main finding is that there is a size-dependent particle residence time. Large particles are moving faster through the HCT than small particles. Consequently, small crystals have more time to grow in the HCT. Physical reasons for this behavior are proposed e.g. small-scale flow characteristics. In a direct numerical simulation of the instationary Navier-Stokes equations, velocity fluctuations and a secondary flow are identified. The presented flow field may have a different impact on the particles and cause the size-dependent particle residence time. A particle-size dependent residence time may potentially narrow the crystal size and shape distribution in such a process, frequently a desired feature in solids' production.
Comparison of numerical methods
 Felix Anker, Sashikumaar Ganesan, Volker John, Ellen Schmeyer A Comparative Study of a Direct Discretization and an Operator-Splitting Solver for Population Balance Systems , Comp. & Chem. Engrg. 75, 95 - 104, 2015 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.
Simulation of a bivariate particle size distribution
 Volker John, Carina Suciu Direct Discretizations of Bi-Variate Population Balance Systems with Finite Difference Schemes of Different Order , Chemical Engineering Science 106, 39 - 52, 2014 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.
Simulation of droplet size distributions
 Ellen Schmeyer, Robert Bordas, Dominique Thevenin, Volker John Numerical Simulations and Measurements of a Droplet Size Distribution in a Turbulent Vortex Street, Meteorologische Zeitschrift 23(4), 387 - 396, 2014 A turbulent vortex street in an air flow interacting with a disperse droplet population is investigated in a wind tunnel. Non-intrusive measurement techniques are used to obtain data for the air velocity and the droplet velocity. The process is modeled with a population balance system consisting of the incompressible Navier--Stokes equations and a population balance equation for the droplet size distribution. Numerical simulations are performed that rely on a variational multiscale method for turbulent flows, a direct discretization of the differential operator of the population balance equation, and a modern technique for the evaluation of the coalescence integrals. After having calibrated two unknown model parameters, a very good agreement of the experimental and numerical results can be observed. Robert Bordas, Volker John, Ellen Schmeyer, Dominique Thevenin Numerical Methods for the Simulation of a Coalescence-Driven Droplet Size Distribution , Theor. Comp. Fluid Dyn. 27, 253 - 271, 2013 The droplet size distribution in a turbulent flow field is considered and modeled by means of a population balance system. This paper studies different numerical methods for the 4D population balance equation and their impact on an output of interest, the time-space-averaged droplet size distribution at the outlet, which is known from experiments. These methods include different interpolations of the experimental data at the inlet, various discretizations in time and space, and different schemes for computing the coalescence integrals. It will be shown that noticeable changes in the output of interest might occur. In addition, the computational efficiency of the studied methods is discussed. Robert Bordas, Volker John, Ellen Schmeyer, Dominique Thevenin Measurement and Simulation of a Droplet Population in a Turbulent Flow Field , Computers and Fluids 66, 52 - 62, 2012 The interaction of a disperse droplet population (spray) in a turbulent flow field has been investigated by combining wind tunnel experiments with simulations based on a population balance system. The behavior of the droplets is modeled numerically by a population balance equation. Velocities of the air and of the droplets are determined by non-intrusive measurements. A direct discretization of the 4D equation for the droplet size distribution is used in the simulations. Important components of the numerical algorithm are a variational multiscale method for turbulence modeling, stabilized finite difference schemes for the 4D equation and a pre-processing approach to evaluate the collision integrals. The simulations of this system accurately predict the modifications of the droplet size distribution from the inlet to the outlet of the measurement section. Since the employed configuration is simple and considering that all measurement data are freely available thanks to an internet-based repository, the considered experiment is proposed as a benchmark problem for the simulation of disperse two-phase turbulent flows.
Simulation of the synthesis of urea
 Wolfgang Hackbusch, Volker John, Aram Khachatryan, Carina Suciu A Numerical Method for the Simulation of an Aggregation-Driven Population Balance System , Int. J. Numer. Meth. Fluids 69, 1646 - 1660, 2012 A population balance system which models the synthesis of urea is studied in this paper. The equations for the flow field, the mass and the energy balances are given in a three-dimensional domain and the equation for the particle size distribution (PSD) in a four-dimensional domain. This problem is convection-dominated and aggregation-driven. Both features require the application of appropriate numerical methods. This paper presents a numerical approach for simulating the population balance system which is based on finite element schemes, a finite difference method and a modern method to evaluate convolution integrals that appear in the aggregation term. Two experiments are considered and the numerical results are compared with experimental data. Unknown parameters in the aggregation kernel have to be calibrated. For appropriately chosen parameters, good agreements are achieved of the experimental data and the numerical results computed with the proposed method. A detailed study of the computational results reveals the influence of different parts of the aggregation kernel.
Simulation of the calcium carbonate precipitation in 2D/3D and 3D/4D
 Volker John, Michael Roland On the impact of the scheme for solving the higher-dimensional equation in coupled population balance systems , Internat. J. Numer. Methods Engrg., 82, 1450 - 1474, 2010 Population balance systems are models for processes in nature and industry which lead to a coupled system of equations (Navier--Stokes equations, transport equations, $\ldots$) 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 equation of a precipitation process in a two--dimensional flow domain. The numerical schemes for the three--dimensional 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. Volker John, Michael Roland Simulations of 3D/4D Precipitation Processes in a Turbulent Flow Field , in Numerical Mathematics and Advanced Applications 2009, G. Kreiss et al. (eds.), Springer, 479 - 487, 2010 Precipitation processes are modeled by population balance systems. An expensive part of their simulation is the solution of the equation for the particle size distribution (PSD) since this equation is defined in a higher-dimensional domain than the other equations in the system. This paper studies different approaches for the solution of this equation: two finite difference upwind schemes and a linear finite element flux-corrected transport method. It is shown that the different schemes lead to qualitatively different solutions for an output of interest. Volker John, Michael Roland, Teodora Mitkova, Kai Sundmacher, Lutz Tobiska, Andreas Voigt, Simulations of population balance systems with one internal coordinate using finite element methods , Chemical Engineering Science 64, 733 - 741, 2009 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. 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 tested. An explicit temporal discretization and an upwind finite difference method are used for discretizing the equation of the particle size distribution. Simulations of the calcium carbonate precipitation in a cavity are presented which study the influence of the flow field on the particle size distribution at the outflow. It is shown that variations of the positions of the inlets change the volume fraction of the particle size distribution 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 stabilized finite element methods for the convection-diffusion-reaction equations leads to completely different numerical results.
 L.G.M. de Souza, G. Janiga, V. John and D. Thevenin Reconstruction of a distribution from a finite number of moments with an adaptive spline-based algorithm , Chemical Engineering Science 65, 2741 - 2750, 2010 An adaptive algorithm suitable for reconstructing a distribution when knowing only a small number of its moments is presented. This method elaborates on a previous technique presented in [JAOT07], but leads to many advantages compared to the original algorithm. The so-called ,,finite moment problem'' arises in many fields of science, but is particularly important for particulate flows in chemical engineering. Up to now, there is no well-established algorithm available to solve this problem. The examples considered in this work come from crystallization processes. For such applications, it is of crucial interest to reconstruct the Particle Size Distributions (PSD) knowing only a small number of its moments, obtained mostly from numerical simulations or from advanced experiments, but without any a priori knowledge concerning the shape of this PSD. This was already possible in many cases with the original algorithm of [JAOT07], but complex shapes could not be identified appropriately. The key of the advanced algorithm is the adaptive criterion for positioning dynamically the nodes in an appropriate manner. It turns out that the adaptive algorithm shows considerable improvements in the reconstruction of distributions with a quickly changing curvature or for non-smooth distributions. Since such configurations are quite often found in practice, the new algorithm is more widely applicable compared to the original method. V. John, I. Angelov, A.A. Öncül and D. Thevenin Techiques for the Reconstruction of a Distribution from a Finite Number of its Moments , Chemical Engineering Science 62, 2890 - 2904, 2007 The reconstruction of a distribution knowing only a finite number of its moments is an extremely important but in practice still unsolved question for many fields of science (chemical and process engineering, electronic engineering, nuclear physics, image analysis, biotechnology$\ldots$). Several methods have been proposed and corresponding mathematical formulations have been introduced in the literature during the last decades. Nevertheless, all these are generally limited to particular, often simple cases and require specific assumptions. It is indeed extremely difficult from a theoretical point of view (it is necessary, however not sufficient, that all moments are available for a correct reconstruction) as well as from a practical point of view (ill-posed inverse problem) to find an accurate and relatively fast method which can be applied to all scientific areas. In the present paper, different possible methods (prescribed functions, discrete method, spline-based reconstruction) allowing such a reconstruction are explained, compared in terms of efficiency and accuracy, and validated for chemical engineering applications using examples with different degrees of difficulty.