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Computational Fluid Dynamics Coupled to Discrete Element Method (CFD-DEM) is widely used in simulating a large variety of particulate flow system. This Eulerian-Lagrangian technique tracks all the particles included in the system by the application of point mass models in their equation of motion. CFD-DEM is a more accurate (and more expensive) technique compared to an Eulerian-Eulerian representation. Compared to Particle Resolved Simulations (PRS), CFD-DEM is less expensive since it does not require resolving the flow around each particles and thus can be applied to larger scale systems. Nevertheless, simulating industrial and natural scale systems is a challenge for this numerical technique. This is because the cost of CFD-DEM is proportional to the number of particles in the system under consideration. Thus, massively parallel codes are used to tackle these problems with the help of supercomputers. In this thesis, the CFD-DEM capability in the in-house code Generalized Incompressible Direct and Large Eddy Simulation of Turbulence (GenIDLEST) is used to investigate large scale dense particulate flow systems. Central to the contributions made by this work are developments to reduce the computational cost of CFD-DEM. This includes the development and validation of reduced order history force model for use in large scale systems and validation of the representative particle model, which lumps multiple particles into one, thus reducing the number of particles that need to be tracked in the system. Numerical difficulties in the form of long integration times and instabilities encountered in fully coupling the fluid and particle phases in highly energetic systems are alleviated by proposing a partial coupling scheme which maintains the accuracy of full-coupling to a large extent but at a reduced computational cost. The proposed partial-coupling is found to have a better convergence behavior compared to the full coupling in large systems and can be used in cases where full coupling is not feasible or impractical to use. Alternative modeling approaches for the tangential treatment of the soft-sphere impact model to avoid storing individual impact deformation are proposed and tested. A time advancement technique is developed and proposed for use in dense particulate systems with a hard-sphere impact model. The new advancement technique allows for the use of larger time steps which can speed-up the time to solution by as much as an order of magnitude. / PHD

<p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small;"><span style="font-family: Times New Roman;">This thesis consists of two different parts, related to two different fields in mathematical physics: wave propagation and kinetic theory of gases. Various mathematical and computational problems for equations from these areas are treated.</span></span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small; font-family: Times New Roman;"> </span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small;"><span style="font-family: Times New Roman;">The first part is devoted to high order finite difference methods for the Helmholtz equation and the wave equation. Compact schemes with high order accuracy are obtained from an investigation of the function derivatives in the truncation error. With the help of the equation itself, it is possible to transfer high order derivatives to lower order or to transfer time derivatives to space derivatives. For the Helmholtz equation, a compact scheme based on this principle is compared to standard schemes and to deferred correction schemes, and the characteristics of the errors for the different methods are demonstrated and discussed. For the wave equation, a finite difference scheme with fourth order accuracy in both space and time is constructed and applied to a problem in discontinuous media.</span></span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small; font-family: Times New Roman;"> </span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small;"><span style="font-family: Times New Roman;">The second part addresses some problems related to kinetic equations. A direct simulation Monte-Carlo method is constructed for the Landau-Fokker-Planck equation, and numerical tests are performed to verify the accuracy of the algorithm. A formal derivation of the method from the Boltzmann equation with grazing collisions is performed. The linear and linearized Boltzmann collision operators for the hard sphere molecular model are studied using exact reduction of integral equations to ordinary differential equations. It is demonstrated how the eigenvalues of the operators are found from these equations, and numerical values are computed. A proof of existence of non-zero discrete eigenvalues is given. The ordinary diffential equations are also used for investigation of the Chapman-Enskog distribution function with respect to its asymptotic behavior.</span></span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small; font-family: Times New Roman;"> </span></span></p>

Bei der Verwendung von Hartkugelpackungen als Modelle für verschiedene Systeme in Physik, Chemie und den Ingenieurwissenschaften kommen einige Fragen auf, z.B. nach dem Zusammenhang zwischen der Packungsdichte und der Radienverteilung der Kugeln bzw. der Packungsstruktur. Der erste Teil dieser Arbeit beschäftigt sich mit dem Problem der optimalen Packungsdichte von zufällig dichten Packungen. Es wird ein Optimierungsalgorithmus vorgestellt, der aus einer vorgegebenen Klasse von Radienverteilungen diejenige bestimmt, für die die Packungsdichte maximal wird. Die Packungsstruktur kann man durch verschiedene statistische Größen charakterisieren, die im zweiten Teil dieser Arbeit beschrieben werden. Dabei wird die Abhängigkeit dieser Größen von der Packungsdichte und der Radienverteilung untersucht und gezeigt, dass in monodispersen Packungen mit zunehmender Dichte erhebliche strukturelle Veränderungen auftreten: Im Dichteintervall zwischen 0,64 und 0,66 erfolgt offenbar ein Übergang von ungeordneten zu kristallinen Packungen, bei weiterer Verdichtung entwickelt sich schließlich eine FCC-Struktur.

The phase behavior of CO2 + polymer systems is of interest in polymer synthesis, flue and natural gas processing, polymer foam and nanoparticle processing, and drug delivery. Theoretical and experimental evidence suggests that CO2 is able to interact with electron donating functional groups in polymers to form weak Lewis acid – base or EDA (Electron Donor Acceptor) complexes. These complexes can have a significant effect on the phase behavior of associated CO2 + polymer systems. In spite of this, however, the phase equilibria of only a few associated CO2 + polymer systems have been measured. Some success in modeling the phase behavior of polymer solutions has been achieved by various versions of the Statistical Association Fluid Theory (SAFT), as well as by several Lattice Models. However, many of these models incorporate two to four adjustable parameters that often depend on temperature (T), pressure (P), and/or molecular weight (MW). As a result, a large amount of experimental data is required to apply these models. The goal of the present work was therefore to develop a new thermodynamic model for associating systems that would include no more than two temperature-independent adjustable parameters. The new model presented in this work is based on the Guggenheim-Huggins-Miller lattice and includes complex formation in the development of the partition function. The EOS obtained from the resulting partition function includes two mixture parameters – the enthalpy of association or complex formation and a reference value of the equilibrium constant for complex formation . Most importantly, can be obtained from in situ Attenuated Total Reflection Fourier Transform Infrared (ATR – FTIR) measurements. This work therefore demonstrates the use of ATR – FTIR spectra to obtain molecular level information regarding the interaction of CO2 and electron donating functional groups in polymers. Unlike other studies, this work uses the bending vibration of CO2 to estimate the enthalpies of association ( ) of CO2 + polymer systems. Values of were directly incorporated in the new model and were found to lie between -7 and -12 kJ/mol for the systems investigated in this work. They increased (i.e. became more negative) in the order: CO2 + PS-co-PMMA < CO2 + PMMA <CO2 + PBMA < CO2 + PSF < CO2 + PVAc < CO2 + EVA40 < CO2 + PEG. Values of the second parameter in the new EOS ( ) were obtained by fitting solubility data at one temperature. Both and were found to be temperature independent. The application of the new EOS was demonstrated by calculating the solubility (sorption) of CO2 in polymers, the extent of swelling of polymers due to CO2, and the solubility of polymers in CO2 (cloud points). Both sorption and cloud point behavior in CO2 + polymer systems could be calculated using a single value of for each binary system. Ionic Liquids (ILs) can also incorporate electron donating functional groups in their structure. Evidence for the interaction of such ILs with CO2 can be found in the large values of the enthalpies of absorption of CO2 in these ILs. The ALF EOS was therefore extended to CO2 + IL systems using the enthalpy of absorption as a measure of association ( ) in these systems. was again treated as an adjustable parameter in the calculation of the CO2 solubility in ILs. A single value of was sufficient to predict swelling in these systems within experimental error.

This thesis is devoted to several aspects of geometry and morphology in wetting problems and hard sphere packings. First, we propose a new method to simulate wetting and slip on nanostructured substrates: a phase field model associated with a dynamical density theory approach. We showed omniphobicity, meaning repellency, no matter the chemical properties of the liquid on monovalued surfaces, i.e. surfaces without overhangs, which is in contradiction with the macroscopic Cassie-Baxter-Wenzel theory, can produce so-called We checked systematically the impact of the surface parameters on omniphobic repellency, and we show that the key ingredient are line tensions, which emerge from needle shaped surface structures. Geometrical effects have also an important influence on glassy or jammed systems, for example amorphous hard sphere systems in infinite pressure limit. Such hard sphere packings got stuck in a so-called jammed phase, and we shall demonstrate that the local structure in such systems is universal, i.e. independent of the protocol of the generation. For this, robust order parameters - so-called Minkowski tensors - are developed, which overcome robustness deficiencies of widely used order parameters. This leads to a unifying picture of local order parameters, based on geometrical principles. Furthermore, we find with the Minkowski tensor analysis crystallization in jammed sphere packs at the random closed packing point

This thesis consists of two different parts, related to two different fields in mathematical physics: wave propagation and kinetic theory of gases. Various mathematical and computational problems for equations from these areas are treated.   The first part is devoted to high order finite difference methods for the Helmholtz equation and the wave equation. Compact schemes with high order accuracy are obtained from an investigation of the function derivatives in the truncation error. With the help of the equation itself, it is possible to transfer high order derivatives to lower order or to transfer time derivatives to space derivatives. For the Helmholtz equation, a compact scheme based on this principle is compared to standard schemes and to deferred correction schemes, and the characteristics of the errors for the different methods are demonstrated and discussed. For the wave equation, a finite difference scheme with fourth order accuracy in both space and time is constructed and applied to a problem in discontinuous media.   The second part addresses some problems related to kinetic equations. A direct simulation Monte-Carlo method is constructed for the Landau-Fokker-Planck equation, and numerical tests are performed to verify the accuracy of the algorithm. A formal derivation of the method from the Boltzmann equation with grazing collisions is performed. The linear and linearized Boltzmann collision operators for the hard sphere molecular model are studied using exact reduction of integral equations to ordinary differential equations. It is demonstrated how the eigenvalues of the operators are found from these equations, and numerical values are computed. A proof of existence of non-zero discrete eigenvalues is given. The ordinary diffential equations are also used for investigation of the Chapman-Enskog distribution function with respect to its asymptotic behavior.

Bei der Verwendung von Hartkugelpackungen als Modelle für verschiedene Systeme in Physik, Chemie und den Ingenieurwissenschaften kommen einige Fragen auf, z.B. nach dem Zusammenhang zwischen der Packungsdichte und der Radienverteilung der Kugeln bzw. der Packungsstruktur. Der erste Teil dieser Arbeit beschäftigt sich mit dem Problem der optimalen Packungsdichte von zufällig dichten Packungen. Es wird ein Optimierungsalgorithmus vorgestellt, der aus einer vorgegebenen Klasse von Radienverteilungen diejenige bestimmt, für die die Packungsdichte maximal wird. Die Packungsstruktur kann man durch verschiedene statistische Größen charakterisieren, die im zweiten Teil dieser Arbeit beschrieben werden. Dabei wird die Abhängigkeit dieser Größen von der Packungsdichte und der Radienverteilung untersucht und gezeigt, dass in monodispersen Packungen mit zunehmender Dichte erhebliche strukturelle Veränderungen auftreten: Im Dichteintervall zwischen 0,64 und 0,66 erfolgt offenbar ein Übergang von ungeordneten zu kristallinen Packungen, bei weiterer Verdichtung entwickelt sich schließlich eine FCC-Struktur.

Particulate systems in practical applications such as biomass combustion, blood cellular systems and granular particles in fluidized beds, have often been computationally represented using spherical surfaces, even though the majority of particles in archetypal fluid-solid systems are non-spherical. While spherical particles are more cost-effective to simulate, notable deficiencies of these implementations are their substantial inaccuracies in predicting the dynamics of particle mixtures. Alternatively, modeling dense fluid-particulate systems using non-spherical particles involves increased complexity, with computational cost manifesting as the biggest bottleneck. However, with recent advancements in computer hardware, simulations of three-dimensional particulate systems using irregular shaped particles have garnered significant interest. In this research, a novel Discrete Element Method (DEM) model that incorporates geometry definition, collision detection, and post-collision kinematics has been developed to accurately simulate non-spherical particulate systems. Superellipsoids, which account for 80% of particles commonly found in nature, are used to represent non-spherical shapes. Collisions between these particles are processed using a distance function computation carried out with respect to their surfaces. An event - driven model and a time-driven model have been employed in the current framework to resolve collisions. The collision model's influence on non–spherical particle dynamics is verified by observing the conservation of momentum and total kinetic energy. Furthermore, the non-spherical DEM model is coupled with an in-house fluid flow solver (GenIDLEST). The combined CFD-DEM model's results are validated by comparing to experimental measurements in a fluidized bed. The parallel scalability of the non-spherical DEM model is evaluated in terms of its efficiency and speedup. Major factors affecting wall clock time of simulations are analyzed and an estimate of the model's dependency on these factors is documented. The developed framework allows for a wide range of particle geometries to be simulated in GenIDLEST. / Master of Science / CFD – DEM (Discrete Element Method) is a technique of coupling fluid flow solvers with granular solid particles. CFD – DEM simulations are beneficial in recreating pragmatic applications such as blood cellular flows, fluidized beds and pharmaceutics. Up until recently, particles in these flows have been modeled as spheres as the generation of particle geometry and collision detection algorithms are straightforward. However, in real – life occurrences, most particles are irregular in shape, and approximating them as spheres in computational works leads to a substantial loss of accuracy. On the other hand, non – spherical particles are more complex to generate. When these particles are in motion, they collide and exhibit complex trajectories. Majority of the wall clock time is spent in resolving collisions between these non – spherical particles. Hence, generic algorithms to detect and resolve collisions have to be incorporated. This primary focus of this research work is to develop collision detection and resolution algorithms for non – spherical particles. Collisions are detected using inherent geometrical properties of the class of particles used. Two popular models (event-driven and time-driven) are implemented and utilized to update the trajectories of particles. These models are coupled with an in – house fluid solver (GenIDLEST) and the functioning of the DEM model is validated with experimental results from previous research works. Also, since the computational effort required is higher in the case of non – spherical particulate simulations, an estimate of the scalability of the problem and factors influencing time to simulations are presented.

Cette thèse porte sur différents aspects géométriques et morphologiques concernant des problèmes de mouillage et d'empilement de sphères. Nous proposons tout d'abord une nouvelle méthode de simulation pour étudier le mouillage et le glissement d'un liquide sur une surface nanostructurée: un modèle de champ de phase en lien avec la théorie de la fonctionnelle de la densité dynamique. Nous étudions grâce à cette méthode la possibilité de transformer une surface quelconque en surface omniphobe (c'est à dire qui repousse tous les liquides). Nous montrons que contrairement à la théorie classique de Cassie-Baxter-Wenzel, il est possible d'inverser la mouillabilité d'une surface en la texturant, et nous montrons qu'une surface monovaluée, i.e. sans constrictions, peut produire un comportement omniphobe c'est à dire repousser tous les liquides grâce à un effet de pointe. La géométrie a également un effet considérable dans les milieux vitreux ou bloqués. Les empilements aléatoires de sphères conduisent par exemple à des état bloqués ("jamming") et nous montrons que la structure locale de ces systèmes est universelle, c'est à dire indépendante de la méthode de préparation. Pour cela, nous introduisons des paramètres d'ordre - les tenseurs de Minkowski - qui suppriment les problèmes de robustesse qu'ont les paramètres d'ordre utilisés classiquement. Ces nouveaux paramètres d'ordre conduisent à une vision unifiée, basée sur des principes géométriques. Enfin, nous montrons grâce aux tenseurs de Minkowski que les empilements de sphères se mettent à cristalliser au delà du point d'empilement aléatoire le plus dense ("random close packing") / This thesis is devoted to several aspects of geometry and morphology in wetting problems and hard sphere packings. First, we propose a new method to simulate wetting and slip on nanostructured substrates: a phase field model associated with a dynamical density theory approach. We showed omniphobicity, meaning repellency, no matter the chemical properties of the liquid on monovalued surfaces, i.e. surfaces without overhangs, which is in contradiction with the macroscopic Cassie-Baxter-Wenzel theory, can produce so-called We checked systematically the impact of the surface parameters on omniphobic repellency, and we show that the key ingredient are line tensions, which emerge from needle shaped surface structures. Geometrical effects have also an important influence on glassy or jammed systems, for example amorphous hard sphere systems in infinite pressure limit. Such hard sphere packings got stuck in a so-called jammed phase, and we shall demonstrate that the local structure in such systems is universal, i.e. independent of the protocol of the generation. For this, robust order parameters - so-called Minkowski tensors - are developed, which overcome robustness deficiencies of widely used order parameters. This leads to a unifying picture of local order parameters, based on geometrical principles. Furthermore, we find with the Minkowski tensor analysis crystallization in jammed sphere packs at the random closed packing point