Eulerian modelling and computational fluid dynamics simulation of mono and polydisperse fluidized suspension

i Abstract This research project is concerned with the Eulerian-Eulerian mathematical modelling of fluidized suspensions. We first derive new averaged equations of motion for particulate systems made up of a finite number of monodisperse particle classes; this clarifies the mathematical origin and physical meaning of the terms featuring in the equations and allows to attain a well-posed multiphase model. We then tackle the closure problem of the fluid-particle interaction force in monodisperse fluidized suspensions, laying emphasis on the buoyancy, drag and elastic forces. We analyze critically several constitutive relations used to express these forces, we identify their shortcomings and we advance new, and more accurate, closure equations. To validate them we study, analytically and computationally, the expansion and collapse of homogeneous fluidized beds and their transition to the bubbling regime, comparing the result with experimental data. We then address the mathematical modelling of polydisperse fluidized suspensions, which are characterized by a continuous distribution of the particle properties, such as size or velocity. Here we adopt a more powerful modelling approach based on the generalized population balance equation (GPBE). Whereas the classical transport equations of continuum mechanics are three-dimensional, the GPBE is usually higher-dimensional and incompatible with customary computational schemes. To solve it, we use the method of moments (MOM), which resorts to a limited number of GPBE moments to derive three-dimensional transport equations that can be handled by normal CFD codes. The limited set of equations, which replaces the single multidimensional GPBE, keeps the problem tractable when applied to complicated multiphase flows; the main obstacle to the method is that the moment transport equations are mathematically unclosed. To overcome the problem, we present two very efficient methods, the direct quadrature method of moments (DQMOM) and the quadrature method of moments (QMOM). Both approximate the volume density function (VDF) featuring in the GPBE by using a quadrature formula. The methods are very flexible: the number of nodes in the quadrature corresponds to the number of disperse phases simulated. The more the nodes, the better the quadrature approximation; more nodes, however, entail also more complexity and more computational effort. For monovariate systems, i.e., systems with only one internal coordinate in the generalized sense, the methods are entirely equivalent from a theoretical standpoint; computationally, however, they differ substantially. To conclude the work, we use DQMOM to simulate the dynamics of two polydisperse powders initially arranged as two superposed, perfectly-segregated packed systems. As fluidization occurs, the simulation tracks the evolution in time and physical space of the quadrature nodes and weights and predicts the mixing attained by the system. To validate the method, we compare computational predictions with experimental results.This research project is concerned with the Eulerian-Eulerian mathematical modelling of fluidized suspensions. We first derive new averaged equations of motion for particulate systems made up of a finite number of monodisperse particle classes; this clarifies the mathematical origin and physical meaning of the terms featuring in the equations and allows to attain a well-posed multiphase model. We then tackle the closure problem of the fluid-particle interaction force in monodisperse fluidized suspensions, laying emphasis on the buoyancy, drag and elastic forces. We analyze critically several constitutive relations used to express these forces, we identify their shortcomings and we advance new, and more accurate, closure equations. To validate them we study, analytically and computationally, the expansion and collapse of homogeneous fluidized beds and their transition to the bubbling regime, comparing the result with experimental data. We then address the mathematical modelling of polydisperse fluidized suspensions, which are characterized by a continuous distribution of the particle properties, such as size or velocity. Here we adopt a more powerful modelling approach based on the generalized population balance equation (GPBE). Whereas the classical transport equations of continuum mechanics are three-dimensional, the GPBE is usually higher-dimensional and incompatible with customary computational schemes. To solve it, we use the method of moments (MOM), which resorts to a limited number of GPBE moments to derive three-dimensional transport equations that can be handled by normal CFD codes. The limited set of equations, which replaces the single multidimensional GPBE, keeps the problem tractable when applied to complicated multiphase flows; the main obstacle to the method is that the moment transport equations are mathematically unclosed. To overcome the problem, we present two very efficient methods, the direct quadrature method of moments (DQMOM) and the quadrature method of moments (QMOM). Both approximate the volume density function (VDF) featuring in the GPBE by using a quadrature formula. The methods are very flexible: the number of nodes in the quadrature corresponds to the number of disperse phases simulated. The more the nodes, the better the quadrature approximation; more nodes, however, entail also more complexity and more computational effort. For monovariate systems, i.e., systems with only one internal coordinate in the generalized sense, the methods are entirely equivalent from a theoretical standpoint; computationally, however, they differ substantially. To conclude the work, we use DQMOM to simulate the dynamics of two polydisperse powders initially arranged as two superposed, perfectly-segregated packed systems. As fluidization occurs, the simulation tracks the evolution in time and physical space of the quadrature nodes and weights and predicts the mixing attained by the system. To validate the method, we compare computational predictions with experimental results.