Numerical Simulation of Destabilizing Heterogeneous Suspensions at Vanishing Reynolds Numbers

Numerical simulation of destabilizing heterogeneous suspensions at vanishing Reynolds numbers zur Erlangung des akademischen Grades eines DOKTORS DER INGENIEURWISSENSCHAFTEN (Dr.–Ing.) der Fakultät fürChemieingenieurwesen und Verfahrenstechnik des Karlsruher Instituts fürTechnologie (KIT) genehmigte DISSERTATION von Dipl.–Math. techn. Fabian Bülow aus Rheinfelden Referent: Prof. Dr.–Ing. H. Nirschl Korreferent: Prof. Dr. W. Dörfler Tag der mündlichen Prüfung: 10.12.2015 Acknowledgements First, I would like to express my gratitude to my two advisors, Professor Nirschl and Professor Dörfler, for giving me the opportunity to pursue my PhD studies in the field of process engineering. Both were always willing to help when I had questions. They complemented each other very well, each supporting me with knowledge in their field of expertise. Profes- sor Nirschl taught me an engineering approach to problems and that the process of writing successful proposals for research projects takes time. Professor Dörfler helped me improving my manuscripts by very detailed corrections, sharpening my eye for details, and answered any mathrelated question I approached him with. Most of all, I thank my advisors for giving me the immense freedom I had with respect to my research. I highly acknowledge this. Worth mentioning is definitely the financial support from the German Research Foundation (DFG) and the computational support from the Steinbuch Centre for Computing (SCC), which provided the necessary computational power. Of course I thank all my colleagues, foremost my office mates, for making the last years very enjoyable. I was never running out of chocolate or someone to discuss any, not necessarily researchrelated subject with. Everyone in the two workgroups I was working in was always very help ful and open for discussions. I would like to thank our technicians, my colleagues from the particle measurement group and Kerstin Bertsch for making conducted experiments possible. I also thank the students, who contributed to this work and who trusted me as their supervisor during their thesis or their job as research assis- tant. Namely, these are Philipp Hamberger, Nicolas Hesse, Daniel Knapp, Jonas Bretz, Amirah Bajawi, Sebastian Bäum, Marc Neumann, Adam Zi- etak and Jonas Hansert. Special thanks go to my family, who always supports me. Without my father, my mother and my sister I would not be where I am now. Last but not least, I thank my wife for her constant support and for her great patience during the time in which this work has been written. Abstract The present work deals with the investigation of destabilizing suspensions. The stability of such is determined by two basic processes: Clustering and sedimentation. Elementary effects, which influence suspension stability, are explained in an introductory part. Because numerical simulations of destabilizing suspensions rely on the mathematical description of these effects, modeling approaches are addressed as well. After a discussion of currently used hindered settling functions, an overview of the most widely used methods for suspension simulation is given. The methods are de scribed in such a way that similarities between different methods become clear. A rough characterization of existing methods can be made by means of the modeling approaches for different phases. This results in a classi- fication of methods in Eulerian, Lagrangian and Eulerian–Lagrangian methods. Results presented in this work have been obtained using a Lagrangian method, the socalled Stokesian Dynamics (SD) method. This method allows for a simple inclusion of models for particleparticle interactions of nonhydrodynamic nature. Moreover, hydrodynamic interactions of particles are accounted for without a direct solution of the basic equations describing the fluid phase. This is an advantage but also a limitation of the SD method. Especially the inclusion of external geometries, for instance the geometries of settling tanks, centrifuges, or CSTRs (continu ously stirred tank reactors) pose extreme difficulties for the method. On the other hand, hydrodynamic interparticle interactions in polydisperse systems can be computed very efficiently. The results section of this work is confined to investigations of problems on the particle level, such that the described drawback plays no role. Certain effects in suspensions can only be observed by means of numerical experiments when the number of simulated particles is high enough. Especially when it comes to polydisperse particle systems, reported direct numerical simulations, i.e. simulations in which hydrodynamic interac tions are described by as few modeling as possible, are often limited to systems of less than a thousand particles. This number does not suffice to show e.g. for isolated settling hydrodynamic particle clusters, also known as particle clouds, an experimentally and numerically verified behaviour. Even though the SD Method is one of the most efficient methods for the simulation of such problems, a sequential implementation requires simu lation times of several days up to weeks and is typically limited to not more than one or two thousand particles. This limitation gave rise to the parallelization of the opensource simulation framework RYUON as part of this work. The aim of this parallelization is the numerical simulation of destabilizing suspensions on CPUbased supercomputers in a reasonable time span. Results concerning this aim are presented in the first part of the results section of this work. It is shown that a parallelization pushes the limits of particle numbers which can be simulated within days from about a thou sand to a million. The exact number of particles in systems which can be simulated within a given time span depends on the number of used CPUs. Computational time is also for a parallelized SD method quadratically de pendent on the number of simulated particles. To reduce this complexity, the SD method has been coupled with the socalled Barnes–Hut method in the present work. The Barnes–Hut method has been developed for the solution of stellar dynamics problems and is nowadays applied to simulations in all fields where longranged interparticle interactions play a role. In the context of this method, particles can be stars, particles met in the field of process engineering, plasma ions, etc. The hybrid method resulting from the coupling of the SD and the Barnes–Hut method is described in the second part of the results section of this work. Apart from a detailed description of the method, results are presented, which show that the resulting method can indeed exhibit a better scaling than the parallel method, when it comes to increasing particle numbers. The resulting SSD method spatially groups particles during computation of hydrodynamic interactions according to a fixed criterion, which yields the reduction in computational time. Because of the motion of particles it is difficult to make a clear statement about the resulting complexity of the method. Yet, it is demonstrated in this work that complexity lies some where between linear complexity and the complexity of the parallelized SD method. Due to the dynamical grouping of particles, the timestep width is reduced when only hydrodynamic interparticle interactions are considered (and if an adaptive timestepping scheme is applied). When additionally interactions due to electrostatics and London–van–der–Waals attraction are considered, no reduction is observable. In the third part of the results section of this work, results concerning the numerical investigation on the settling behaviour of polydisperse particle clouds in viscous fluids are presented. The gained possibility to simulate large particle systems allows for the demonstration that polydisperse particle clouds undergo the same characteristic motion as monodisperse clouds of equal size. Because of the mentioned limitation of particle numbers in earlier works, consensus up to this work was that polydisperse systems would not be stable enough to show the typical behaviour. Apart from this, numerical experiments conducted in the context of this work indicate that the radius ratio of particles in an initially unmixed bidisperse cloud has a strong influence on the effect of small particles being dragged along by larger particles forming a hydrodynamic cluster. If both fractions are initially mixed, a larger number of small particles is dragged along by the fraction of large particles which determines the overall motion of the cluster. Another result of the described investigations is an analytical expression, which can be used to determine the maximum settling veloc ity of a polydisperse cloud. This expression yields a substantially better prediction than thus far applied formulas, which do not consider polydis- persity. The description of these results concerning hydrodynamic clustering are followed by an analysis of clustering due to heterocoagulation induced by hydrodynamic and nonhydrodynamic interactions which can be described by means of DLVO theory. On the example of orthokinetic heterocoagulation of two aluminiumoxide particles in water, it is set out how the standard procedure for the derivation of the socalled capture efficiency can be accelerated. The accurate determination of the capture efficiency is fundamental to a successful simulation of processes on a technical scale by means of population balance equations, e.g. in a crystallizer or in a settling tank. In this work, an expression for the capture efficiency of orthokinetic heterocoagulation

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