Botond Szilagyi

Graphical processing unit (GPU) acceleration for numerical solution of population balance models using high resolution finite volume algorithm

Abstract Population balance modeling is a widely used approach to describe crystallization processes. It can be extended to multivariate cases where more internal coordinates i.e., particle properties such as multiple characteristic sizes, composition, purity, etc. can be used. The current study presents highly efficient fully discretized parallel implementation of the high resolution finite volume technique implemented on graphical processing units (GPUs) for the solution of single- and multi-dimensional population balance models (PBMs). The proposed GPU-PBM is implemented using CUDA C++ code for GPU calculations and provides a generic Matlab interface for easy application for scientific computing. The case studies demonstrate that the code running on the GPU is between 2–40 times faster than the compiled C++ code and 50–250 times faster than the standard MatLab implementation. This significant improvement in computational time enables the application of model-based control approaches in real time even in case of multidimensional population balance models.

Coupled Population Balance-CFD Modelling of a Continuous Precipitation Reactor

Abstract Precipitation of amorphous calcium phosphate in a Y-mixer-tubular reactor device is investigated using a population balance equation including nucleation, growth and agglomeration of particles and a three-dimensional CFD flow model. The population balance equation is reduced into a moment equations system the quadrature form of which is coupled with the detailed CFD model. Simulation results obtained by the detailed CFD flow model provided results comparable to those produced by assuming ideal mixing conditions. This comparison reveals that under the given process conditions of precipitation of amorphous calcium phosphate application of a simple ideal mixing model, for instance in controlling such processes can be justified.

Model-based analysis of stirred cooling crystallizer of high aspect ratio crystals with linear and nonlinear breakage

Abstract 2D population balance model is presented for stirred cooling crystallizer of high aspect ratio crystals including primary and secondary nucleation, size-dependent growth of the two characteristic crystal facets and linear and nonlinear breakage along the crystal length. Three breakage mechanisms are modelled: crystal-impeller and crystal-wall collisions linear and crystal–crystal collisions nonlinear breakage processes. The 2D population balance equation is reduced into a system of moment equations for the mixed moments of length and width variables which is closed applying the quadrature method of moments and solved by using a three point QMOM-ODE method. It was shown that strong interactions exist between the secondary nucleation, crystal growth and breakage processes connected by the CSD. The stirring rate has strong impact on crystallization of high aspect ratio crystals forming the crystalline product directly by breakage. The crystal-impeller breakage proved to be the dominant process but the crystal–crystal breakage also play significant role.

Graphical Processing Unit (GPU) Accelerated Solution of Multi-Dimensional Population Balances Using High Resolution Finite Volume Algorithm

Abstract Population balance modeling is a widely used approach to describe, crystallization processes, taking into accountnot only the primary phenomena like nucleation and growth, as well asparticle agglomeration and breakage which can be extended to multivariate cases where more internal coordinates i.e. particle properties can be used. The solution of the generated partial differential equation is not trivial due to its hyperbolic nature which causes numerical dispersion and diffusion. The high resolution finite volume algorithms are able to solve multidimensional population balance equations avoiding these unwanted phenomena. Unfortunately, the computational time is significantly larger, compared to e.g. moments based solutions. More recently there is an increased interest to apply parallel computations using GPU-s due to its massively parallel hardware architecture. The current study presents two accelerating possibilities for finite volume solution of population balance equations: compiled C++ code executed on CPU and CUDA C++ code running on the GPU. The case studies demonstrate that the code running on the GPU was between 6–35 times faster than the compiled C++ code and 4–6 times faster than the standard MatLab function. This significant improvement in computational time enablesthe application of model-based control approaches in real time even in case of multidimensional population balance models.

Model Based Estimation of 2D Crystallization Kinetics From Concentration and CLD Measurements

Due to the fact that crystal size and shape influence relevant macroscopic properties of solid particles, the understanding and control of these quantities have increasing importance in particulate science. Crystallization, the primary crystal formation and purification process, is usually tracked real-time, in situ, by spectroscopic techniques and Focused Beam Reflectance Measurement (FBRM). This sensor can measure the chord length distribution (CLD) of a population of particles suspended in a solution. The CLD is related to both the size and shape of the particles and it is measured using a rotating infrared laser beam that emanates through the probe window inserted in the suspension. During its rotation, the beam hits the particles within the sample and is reflected back to the probe. The calculated length of laser-crystal intersection is the so-called chord length. FBRM can provide a large amount of useful information during crystallization processes, however, since the CLD is significantly different compared to the actual crystal size and shape distribution (CSD), it is normally not used for the quantification of the kinetics of crystal growth and nucleation. Usually off-line techniques (laser diffraction, microscopy, ultrasound) are exploited. In this study we develop and present a new, projection based forward 2D CSD➔CLD transformation technique. In addition, a 2D population balance model is employed to simulate the evolution of 2D CSD and solute concentration. The model equations are solved by a high resolution finite volume method, involving GPU acceleration for improved simulation time. Such model allows the use of FBRM data for the estimation of the kinetics of crystallization, without relying on off-line measurements of CSD. As model system succinic acid is used. This forms prism-like crystals in the presence of growth rate modifiers. Crystal breakage is minimized through reduced mixing rate and the kinetics of primary and secondary nucleation as well as the growth and dissolution of individual crystal facets were estimated by developing and solving a process optimization problem. The result of the parameter regression was a calibrated model, which simulates fairly the concentration and CLD variations too.