Développement d'un modèle Euler-Lagrange robuste pour la simulation des écoulements solide-liquide dans les opérations de mélange

Les opérations de mélange solide-liquide en cuves agitées jouent un rôle clef dans de nombreux procédés, que ce soit dans la fabrication de produits alimentaires, pharmaceutiques, cosmétiques, ou pour promouvoir l’homogénéité de suspensions, ce qui est particulièrement vital au bon fonctionnement des réacteurs chimiques employant un catalyseur solide. Malgré leur importance indéniable pour l’industrie chimique et les efforts considérables qui ont été déployés afin de mieux les comprendre, la conception et l’optimisation de ces opérations demeurent un grand défi. En effet, la quasi-totalité de la littérature se concentre sur le régime d’opération pleinement turbulent, malgré le fait que de nombreux procédés industriels soient opérés en régimes laminaire ou transitoire. Malheureusement, la littérature sur ces derniers régimes d’opération est quasi-inexistante. De plus, bien que le régime d’opération turbulent ait fait l’objet d’un grand nombre d’études, celles-ci se sont principalement concentrées sur la prédiction, à l’aide de corrélations empiriques ou semi-empiriques, de la vitesse nécessaire pour la suspension complète des particules (Njs), c’est-à dire la vitesse d’agitation nécessaire pour suspendre toutes les particules hors du fond de la cuve. Cependant, de nombreux procédés pourraient être opérés dans des conditions différentes telles que la suspension partielle ou complètement homogène. Le premier cas permettrait d’économiser de l’énergie et d’éviter des contraintes trop fortes sur l’agitateur, tandis que le second assurerait une cinétique de réaction et une qualité de produit nettement mieux contrôlée. Dans ces deux situations, la connaissance de Njs n’est que d’une aide limitée. Pour mieux opérer et concevoir ces unités, il est nécessaire de pouvoir prédire la distribution et la dispersion des particules solides ainsi que les patrons d’écoulement au sein de la cuve. L’étude des systèmes de mélange solide-liquide représente une difficulté importante, car l’opacité des suspensions solide-liquide limite fortement la mesure de variables locales telles que les profils de concentration. Ainsi, la majeure partie des travaux expérimentaux n’ont mesuré que des paramètres globaux, tels que le couple sur l’agitateur, la fraction de particules suspendues ou Njs. La mécanique des fluides numérique (CFD), complémentaire à l’expérience, permet quant à elle d’investiguer à la fois les paramètres globaux, mais aussi ce qui se passe localement en tout point de la cuve. Cependant, la modélisation d’écoulements multiphasiques renferme de nombreux défis compte tenu de l’interaction multiéchelle (de temps et d’espace) entre les phases. Les nombreux modèles capables de modéliser ces types d’écoulements sont décrits dans cette thèse, dans l’objectif de faire ressortir leurs forces ainsi que leurs limites. Parmi ceux-ci, il est montré que seuls les modèles à deux fluides, où la phase solide est modélisée comme un second fluide, ont été utilisés exhaustivement pour aborder le mélange solide-liquide. Cependant, ces modèles souffrent d’une incapacité à bien décrire les régimes rapides et denses d’écoulement granulaire (comportement de Burnett et de super Burnett) ainsi que de plusieurs difficultés à saturer la concentration de solide lorsque les particules sont à leur fraction maximale d’empilement. La CFD-DEM, une famille de modèles relativement récents qui combinent la CFD pour la phase fluide et la méthode des éléments discrets (DEM) pour les particules solides permet quant à elle de décrire la dynamique de la phase granulaire avec un grand degré de précision et a largement fait ses preuves dans l’étude de milieux solide-gaz. Cependant, cette méthode n’a jamais été employée rigoureusement pour l’étude d’écoulement solide-liquide dans des géométries complexes telles que des mélangeurs. Pour que ceci soit possible, de nombreux développements mathématiques sont nécessaires afin de s’assurer que le schéma soit stable, qu’il converge (en temps et en espace) et qu’il soit capable de simuler des géométries complexes en mouvement tels que les agitateurs. La conception d’un tel modèle et son application à l’étude de la dynamique du mélange solide-liquide et de la mise en suspension de particules solides est l’objectif principal de cette thèse. En premier lieu, un schéma volume fini de type Pressure Implicit with Splitting of Operator (PISO) pour résoudre les équations de Navier-Stokes moyennées volumiquement (VANS), nommé PISO-VANS, est établi. La résolution de ces équations est une partie essentielle d’un modèle CFD-DEM applicable à des écoulements concentrés. Afin de vérifier la cohérence du schéma PISO-VANS pour la résolution des équations VANS, une méthodologie basée sur la méthode des solutions manufacturées est développée afin d’établir des cas tests analytiques permettant d’effectuer des tests de convergence numérique. Ces tests, les premiers de ce genre, démontrent que le schéma proposé converge à la précision désirée, c’est-à-dire qu’il est bien de second ordre en temps et en espace. Ensuite, cette méthodologie est employée pour vérifier un nouveau schéma permettant de résoudre les équations VANS avec la méthode de Boltzmann sur réseau (LBM). Ce schéma est basé sur un nouvel opérateur de collision. Il est démontré, à l’aide d’une analyse de Chapmann-Enskogg, que la formulation proposée permet de retrouver les équations VANS. Cet opérateur est le premier permettant de résoudre les équations VANS avec la LBM lorsque la fraction volumique n’est pas constante dans l’espace, une capacité essentielle pour l’étude d’écoulements polyphasiques ou dans des milieux poreux. Dans la troisième partie de ce travail, une nouvelle méthode de condition immergée semiimplicite permettant de modéliser des corps rigides en rotation est développée. Cette méthode est conçue pour bien s’harmoniser avec le schéma PISO et pour être fonctionnelle sur un maillage non structuré polyédrique tout en demeurant parallèle. Cette méthode est tout d’abord vérifiée sur des cas tests académiques tels que l’allée de von Karman derrière un cylindre, ainsi qu’un écoulement de Taylor-Couette entre deux cylindres. Il est montré que le schéma peut bien reproduire les vortex de von Karman, mais qu’il dégrade l’ordre du schéma volume fini de 2 à 1.33 dans le cas de l’écoulement de Taylor-Couette. Le schéma est finalement validé expérimentalement et comparé à d’autres méthodes numériques permettant de simuler des géométries en rotation. Un accord quasi parfait est obtenu. La quatrième partie de ce travail résulte directement de la combinaison du schéma volume fini PISO-VANS avec la méthode de conditions immergées afin de simuler le mélange solideliquide, du démarrage au régime permanent, à l’aide de la CFD-DEM. Différentes stratégies de couplage entre les phases sont testées et il est montré que, contrairement au cas gaz-solide, un couplage explicite est préférable, car il atténue les erreurs de moyenne volumique. Il est aussi démontré qu’un modèle de rhéologie est nécessaire afin de considérer la dissipation à l’échelle inférieure à une taille de maille. Le modèle complet est ensuite validé qualitativement, en comparant des profiles particuliers d’écoulements obtenus au début de la suspension des particules, et quantitativement, à travers la comparaison avec l’expérience de la fraction de particules suspendues mesurée par la technique de pression de jauge (PGT). À nouveau, un excellent accord est obtenu entre les données expérimentales et les résultats du modèle. Dans la cinquième partie de ce travail, le modèle CFD-DEM est modifié afin de permettre l’étude des écoulements turbulents à l’aide de la simulation aux grandes échelles turbulentes (LES). Suite à une validation avec l’expérience, deux nouvelles techniques de mesure de la fraction de particules suspendues, l’analyse lagrangienne de fraction suspendue (LSFA) et l’analyse de fraction de décorrélation (DFA), sont introduites. Les résultats issus de ces méthodes sont ensuite comparés à ceux obtenus numériquement et expérimentalement par la méthode de pression de jauge. Il est montré que ces deux méthodes sont pratiquement aussi précises que la PGT, mais qu’elles sont aussi plus versatiles, car elles peuvent être appliquées à toutes les géométries et ne nécessitent pas de simulations sur des larges plages de vitesse d’agitation. Dans la sixième partie de ce travail, le modèle CFD-DEM est utilisé pour étudier en détail le mélange solide-liquide en régime laminaire et transitoire. Notamment, l’impact du dégagement de l’agitateur et de la présence de chicanes sur la fraction de particules suspendues et la dynamique du mélange solide-liquide est établi. Il est montré que de réduire le dégagement au fond de l’agitateur permet de prévenir l’apparition de zone mortes à haute vitesse. De surcroît, une étude de sensibilité sur les paramètres de la DEM est effectuée et montre que seule la friction entre les particules joue un rôle significatif sur la dynamique de la phase solide. Finalement, une brève discussion permet de résumer les résultats obtenus et de donner de nombreuses pistes de travaux pouvant faire suite à ce qui fut développé dans le cadre de cette thèse. ---------- Despite the fact that solid-liquid mixing plays a key role in the production of a wide variety of consumer goods such as pastes, paints, cosmetics, propellants, pharmaceuticals, and food products as well as in the operation of chemical reactors with solid catalysts, it still faces considerable challenges. Most research on solid-liquid mixing has focused on the fully turbulent regime of operation even though many industrial operations take place in the laminar or transitional regime. In particular, it is unclear how the rheology of a suspension, particle interactions, and a complex rotating geometry impact flow patterns and particles distribution and dispersion in these regimes. Although more is known about the turbulent regime of operation, most research on this type of regime has been devoted to the prediction of the just-suspended speed (Njs ), which is the impeller speed at which all particles are suspended in the liquid phase. However, numerous mixing operations require a different state of operation. For these processes, operating at Njs can lead to energy overconsumption, product fouling, or inhomogeneous reactions due to the presence of dead zones. Consequently, more information on the velocity patterns and distribution of particles in agitated vessels is required. To shed light on issues related to solid-liquid mixing, numerical and experimental investigations are essential. However, due to the opacity of most viscous suspensions, local measurements of the flow field using optical techniques are highly problematic. Consequently, almost all experimental measurements have been limited to determining the global characteristics of the mixing flow such as the fraction of suspended particles or the torque acting on the impeller. However, CFD simulations of these systems do not suffer from these drawbacks. A variety of models have been developed to simulate solid-liquid flows. These include the classic Eulerian-Eulerian (or two-fluid) model and the combination of the Discrete Element Method (DEM) for the particles and CFD method for the liquid phase (CFD-DEM). Although it possesses enormous potential due to its formulation, notably as regards to its natural capacity to reproduce the maximal packing fraction of solid particles, the ability of the CFD-DEM approach to accurately model solidliquid flows in complex geometries has not yet been proved. In addition, the method has not been validated experimentally for solid-liquid flows. However, this type of model could theoretically allow for a quantitative assessment of flow patterns, particle distributions, and the fraction of suspended particles. In this thesis, a CFD-DEM model is developed to model the suspension of particles in a stirred tank, from start-up to steady state and in all regimes of operation. The model is used to improve our understanding of solid-liquid mixing, notably the issue of predicting the fraction of suspended particles. It is shown to be a quantitative tool that can predict the state and dynamics of a suspension. A methodology is designed to verify a Pressure Implicit with Splitting of Operator (PISO) scheme for the volume-averaged Navier-Stokes (VANS) equations (Article 1). We recall that these equations are essential for the unresolved CFD-DEM method. The methodology, which is based on the method of manufactured solutions, is used to design analytical solutions for the VANS equations for which order of convergence analyses are carried out. The validity of the semi-implicit scheme is established by demonstrating the second-order convergence of the scheme for various complex 2D cases. A novel collision operator for the Lattice Boltzmann Method (LBM) is designed to solve the VANS equations (Article 2). It is demonstrated analytically that this operator solves the VANS equations with second-order accuracy. Numerical test cases designed using the process established in Article 1 were used to confirm these results. The model is able to solve cases where there are large void fraction gradients in the domain. To our knowledge, is the first time that this has been achieved. A semi-implicit immersed boundary method (PISO-IB) is developed to study rotating rigid bodies such as impellers (Article 3). The method is verified using academic test cases, namely the Taylor-Couette flow and the Von Karman vortex street behind a static and a moving cylinder. The scheme accurately reproduces vortex shedding, but degrades the order of convergence of the overall finite volume scheme from 2 to 1.33 in the Taylor-Couette case. The PISO-IB method is validated for single phase mixing, and good agreement is obtained between this method and experimental torque measurements. The methods developed in the first and third sections are then combined to formulate a CFD-DEM scheme for solid-liquid mixing (Article 4). The formulation of the model is analyzed, and two coupling approaches (implicit and explicit) are investigated. Explicit coupling leads to a more stable scheme for viscous fluids. However, due to unresolved hydrodynamic dissipation at the particle scale, a rheology model be introduced into the model. The complete model is validated qualitatively using photographs of the peculiar particle dynamics observed experimentally as well as quantitatively by comparing the fraction of suspended particles measured numerically with experimental data obtained using the pressure gauge technique. The validated CFD-DEM model is extended to the turbulent regime using a large-eddy simulation (LES) approach (Article 5). The model accurately reproduces the fraction of suspended particles measured experimentally. Two new techniques to calculate the fraction of suspended particles, the Lagrangian suspended fraction analysis (LSFA) and the decorrelated fraction analysis (DFA), are developed. These two techniques can be used to calculate the fraction of suspended particles for any vessel bottom, without requiring the simulation of numerous impeller velocities, something that cannot be accomplished using the pressure gauge technique. The entire set of tools described in the previous article is used to study the suspension of solid particles in the laminar and transitional regimes in detail (Articles 6). A parametric study of the model parameters is performed. It shows that only the coefficient of friction plays a role in the solid dynamics. Alternative geometries are also studied by varying the impeller clearance and by adding or removing baffles. These results show that reducing the clearance results in a better distribution of particles and prevents the creation of a dead zone below the impeller. This thesis finishes with a short discussion of the overall capabilities of the model and future research that could arise from it

Towards High-Order CFD-DEM: Development and Validation

CFD-DEM is used to simulate solid-fluid systems. DEM models the motion of discrete particles while CFD models the fluid phase. Coupling both necessitates the calculation of the void fraction and the solid-fluid forces resulting in a computationally expensive method. Additionally, evaluating volume-averaged quantities locally restricts particle to cell size ratios limiting the accuracy of the CFD. To mitigate these limitations, we develop a monolithic finite element CFD-DEM solver which supports dynamically load-balanced parallelization. This allows for more stable, accurate and time efficient simulations as load balancing ensures the even distribution of workloads among processors; thus, exploiting available resources efficiently. Our solver also supports high order schemes; thus, allowing the use of larger elements enhancing the validity and stability of the void fraction schemes while achieving better accuracy. We verify and validate our CFD-DEM solver with a large array of test cases: the Rayleigh Taylor instability, particle sedimentation, a fluidized bed, and a spouted bed

CFD-DEM investigation of viscous solid–liquid mixing: Impact of particle properties and mixer characteristics

In chemical engineering, numerous processes require the suspension of particles in a laminar or transitional regime. For such operations, predicting the fraction of suspended particles as well as their distribution and homogeneity is a major concern. In this work, the unresolved CFD-DEM model introduced by our group for solid–liquid mixing is used to investigate the mixing dynamics of viscous suspensions. The techniques chosen to characterize the degree of suspension, the homogeneity and the distribution of the particles are presented. They are used to assess the efficiency of a pitched blade turbine with a clearance of C = T/4. The impact of solid properties on mixing dynamics is investigated by varying the Young's modulus, the coefficient of restitution and the sliding friction coefficient in the DEM model. Lastly, five alternative configurations of the mixing rig are investigated by varying the clearance of the impeller and introducing baffles

On the use of the method of manufactured solutions for the verification of CFD codes for the volume-averaged Navier–Stokes equations

The volume-averaged Navier–Stokes (VANS) equations are a key constituent of numerous models used to study complex problems such as flows in porous medias or containing multiple phases (e.g. solid–liquid flows). These equations solve the mesoscopic scale of the flow without taking into account explicitly each individual solid particles, therefore greatly reducing computational cost. However, due to a lack of analytical solutions, the models using the VANS equations are generally validated directly against experimental data or empirical correlations. In this work, a framework to design analytical solutions and verify codes that solve the VANS equations via the method of manufactured solutions is presented for the first time. Three test cases of increasing complexity are designed with this method and used to assess the second-order convergence of a finite volume solver developed in OpenFOAM. The proposed approach is suitable for the verification of any code that solves the VANS equations with any CFD technique such as the finite element method or the lattice Boltzmann method

Development and validation of a stabilized immersed boundary CFD model for freezing and melting with natural convection

Numerous processes in the automotive, additive manufacturing or energy storage industries require an accurate prediction of the solidification (freezing) and melting (thawing) dynamics of substances. The numerical modeling of these phase changes is highly complex because it includes sharp moving interfaces and strong discontinuities in the material properties. This complexity is often exacerbated by the occurrence of natural convection, which induces a strong coupling between the motion of the liquid and the position of the solid–liquid interface. This leads to strongly coupled non-linear thermo-fluid problems which have to be solved in complex geometries. In this work, we introduce two novel stabilized finite element models to predict the phase change with natural convection. The first model uses a more classical viscosity approach to impose stasis in the solid region whereas the second one is based on an immersed boundary formulation to accurately describe the solid–fluid interface. The efficiency of the stabilization is first demonstrated by studying the Stefan problem. The two approaches to impose stasis are then compared using 2D test cases before they are both used to study melting in a rectangular (2D) and prismatic (3D) cavity. Significant differences are observed in the flow profiles and the solid–liquid interface position between the 2D and the 3D simulations

CFD-DEM simulations of early turbulent solid–liquid mixing: Prediction of suspension curve and just-suspended speed

Solid–liquid mixing as a unit operation still faces considerable challenges, notably regarding the prediction of the impeller speed required to suspend the particles (Njs), the fraction of suspended solids and the homogeneity of the suspension at a given speed. In this work, we extend to the turbulent regime, by means of large eddy simulation (LES), a CFD-DEM model developed recently in our group for solid–liquid mixing. The resulting model is used to study the mixing of glass particles in a baffled stirred tank equipped with a down-pumping pitched blade turbine. Various characteristics of the liquid dynamics as well as the distribution and motion of the solids are investigated. The fraction of suspended solid particles predicted by the model is validated against experimental data obtained via the pressure gauge technique (PGT). Two new methods to calculate the fraction of suspended particles in a Euler–Lagrange simulation, the so-called Lagrangian suspended fraction analysis (LSFA) and the decorrelated fraction analysis (DFA) techniques are introduced. The results obtained with these two methods, as well as with many others taken from the literature, are compared to the Zwietering correlation and to the results obtained by the PGT. It is found that some techniques proposed in the literature, namely the local concentration, the power consumption and the transient solids concentration analysis techniques, cannot be applied adequately in this case. On the other hand, the LSFA, DFA and PGT techniques are observed to predict accurately the fraction of suspended solids when compared to experimental PGT data

A conservative lattice Boltzmann model for the volume-averaged Navier–Stokes equations based on a novel collision operator

The volume-averaged Navier–Stokes (VANS) equations are at the basis of numerous models used to investigate flows in porous media or systems containing multiple phases, one of which is made of solid particles. Although they are traditionally solved using the finite volume, finite difference or finite element method, the lattice Boltzmann method is an interesting alternative solver for these equations since it is explicit and highly parallelizable. In this work, we first show that the most common implementation of the VANS equations in the LBM, based on a redefined collision operator, is not valid in the case of spatially varying void fractions. This is illustrated through five test cases designed using the so-called method of manufactured solutions. We then present an LBM scheme for these equations based on a novel collision operator. Using the Chapman–Enskog expansion and the same five test cases, we show that this scheme is second-order accurate, explicit and stable for large void fraction gradients

A semi-implicit immersed boundary method and its application to viscous mixing

Computational fluid dynamics (CFD) simulations in the context of single-phase mixing remain challenging notably due the presence of a complex rotating geometry within the domain. In this work, we develop a parallel semi-implicit immersed boundary method based on Open∇FOAM, which is applicable to unstructured meshes. This method is first verified on academic test cases before it is applied to single phase mixing. It is then applied to baffled and unbaffled stirred tanks equipped with a pitched blade impeller. The results obtained are compared to experimental data and those predicted with the single rotating frame and sliding mesh techniques. The proposed method is found to be of comparable accuracy in predicting the flow patterns and the torque values while being straightforwardly applicable to complex systems with multiples impellers for which the swept volumes overlap

Simulation of granular flow in a rotating frame of reference using the discrete element method

Over the years, the Discrete Element Method (DEM) has attracted significant attention for its capacity to simulate granular flows because it captures physical phenomena that cannot be observed using continuum methods. However, the simulation of granular systems with DEM is computationally demanding, especially in the case of systems in rotation. One solution is to perform simulations in a non-inertial rotating frame of reference, which requires the addition of fictitious velocity-dependent forces such as the Coriolis force. We assess the numerical feasibility and accuracy of such DEM simulations. We show that the velocity Verlet scheme in its classical form no longer defines a symplectic map and is no longer of second order when there are velocity dependent forces. Nevertheless, our study of a dense particle flow within a rotating hourglass shows that the relevant properties of such flow are accurately reproduced in a non-inertial frame and that computational performance is improved

Development of an unresolved CFD–DEM model for the flow of viscous suspensions and its application to solid–liquid mixing

Although viscous solid–liquid mixing plays a key role in the industry, the vast majority of the literature on the mixing of suspensions is centered around the turbulent regime of operation. However, the laminar and transitional regimes face considerable challenges. In particular, it is important to know the minimum impeller speed () that guarantees the suspension of all particles. In addition, local information on the flow patterns is necessary to evaluate the quality of mixing and identify the presence of dead zones. Multiphase computational fluid dynamics (CFD) is a powerful tool that can be used to gain insight into local and macroscopic properties of mixing processes. Among the variety of numerical models available in the literature, which are reviewed in this work, unresolved CFD–DEM, which combines CFD for the fluid phase with the discrete element method (DEM) for the solid particles, is an interesting approach due to its accurate prediction of the granular dynamics and its capability to simulate large amounts of particles. In this work, the unresolved CFD–DEM method is extended to viscous solid–liquid flows. Different solid–liquid momentum coupling strategies, along with their stability criteria, are investigated and their accuracies are compared. Furthermore, it is shown that an additional sub-grid viscosity model is necessary to ensure the correct rheology of the suspensions. The proposed model is used to study solid–liquid mixing in a stirred tank equipped with a pitched blade turbine. It is validated qualitatively by comparing the particle distribution against experimental observations, and quantitatively by compairing the fraction of suspended solids with results obtained via the pressure gauge technique