### Constitutive model development for polydisperse gas-solid suspensions

Graduate student researcher: Bill Holloway

Rigorous solution of the motion of gas-particle mixtures requires solution of the Navier-Stokes equations for the fluid phase and Newton's equation of motion for each particle. Due to the large number of particles in industrial devices, solutions of this type are both expensive and of limited value. It is more practical to construct locally averaged equations for fluid and particle phases, and solve them to study large scale flows. These locally averaged equations, however, contain several terms for which constitutive relations are needed. Such terms include fluid-particle drag force, particle-particle drag force, particle phase stress, etc. Among these, the fluid-particle drag is especially important; a lot of research has already been devoted to developing constitutive models for it for the case where the particles are all of uniform size. Distribution of particle size routinely appears in practical devices and in many case this polydispersity affects the quality of fluidization and flow of gas-particle mixtures. In this project, we investigate the effect of polydispersity on the fluid-particle drag and develop constitutive relations. Towards this end, we perform detailed computations of fluid-flow through frozen and sedimenting, polydisperse assemblies of particles. The simulation results are then used to construct constitutive models. We then investigate the consequences of these refined constitutive relations on the predicted flow characteristics of polydisperse gas-particle mixtures in devices such as fluidized beds and risers.

### Coursened equations of motion for two- and multiphase flows

The closure relations for the averaged equations of motion for dispersed two-phase flows, which are available in the literature, are usually valid for nearly homogeneous suspensions. Analysis of the averaged equations with these closures reveals that, in gas-particle flows and buoyant rise of bubbly suspensions, the dominant instability through which an uniform state breaks into meso-scale structures occurs at a length scale of the order or 10 - 50 particle (or bubble) diameters. This severely limits the value of the averaged equations as a tool for studying flows in large process vessels, as the required spatial resolution renders the calculations far beyond what is feasible today. With this in mind, we are developing coarsened equations of motion, where the averaging is first performed at the level of the individual particles, and then over the meso-scale structures. This second-stage filtering is necessary as our recent work has demonstrated that the fluctuations associated with the meso-scale structures exert a significant influence on the macro-scale flow characteristics and cannot be ignored. Through detailed computational experiments, we are gathering information on the statistical and scaling properties of the meso-scale structures, and using them to devise meso-scale structural models for coarse simulations of two-phase flows.

### Rheological behavior of dense assemblies of granular materials

Graduate student researcher: Sebastian Chialvo

Flow of dense assemblies of granular materials is encountered in many devices used to handle and mix particles, and in fluidized beds. Frictional stresses transmitted through sustained contact of particles with multiple neighbors play an important role in the flow behavior obtained in such devices. For example, a stable operation of a circulating fluidized bed is possible only when the frictional contact between the particle assembly and the walls of the standpipe can provide a stabilizing effect.

Assemblies of granular materials behave differently when they are flowing rapidly from when they are slowly deforming. The behavior of rapidly flowing granular materials, where the particle-particle interactions occur largely through binary collisions, is commonly related to the properties of the constituent particles through the kinetic theory of granular materials. The same cannot be said for slowly moving or static assemblies of granular materials, where enduring contacts between particles are prevalent. For instance, a continuum description of the yield characteristics of dense assemblies of particles in the quasistatic flow regime cannot be written explicitly on the basis of particle properties, even for cohesionless particles. Continuum models for this regime have been proposed and applied, but these models typically assume that the assembly is at incipient yield and they are expressed in terms of the yield function, which we do not yet know how to express in terms of particle-level properties. The description of the continuum rheology in the intermediate regime is even less understood. Yet, many practically important flows in nature and in a wide range of technological applications occur in the dense flow regime and at the transition between dilute and dense regimes; the lack of validated continuum rheological models for particle assemblies in these regimes limits predictive modeling of such flows. This research project is aimed at developing such rheological models.

The overall objective of this project is to construct: (i) validated continuum models for frictional flow of granular materials in the quasi-static and intermediate regimes, including transitions from stagnant-to-shearing, quasistatic-to-intermediate and intermediate-to-inertial regimes; and (ii) closures for the parameters appearing in the models in terms of particle scale properties. We perform discrete element simulations of simple shear, triaxial and biaxial tests involving dense assemblies of spherical and ellipsoidal particles to track the evolution of the stress and microstructure; these results are being used to construct continuum constitutive models that explicitly include particle properties. The consequences of these constitutive models are then studied using test problems.

### Accelerating Discrete Particle Simulations

Graduate student researcher: Matthew Girardi

Discrete particle simulations aim on tracking individual particles in a flow domain, e.g., catalyst particles in a reactor. These simulations have several advantages over locally averaged simulation approaches, e.g., they are better suited to study rare events in an agglomeration process. However, an inherent challenge for this simulation approach is the extremely large number of particles that must be tracked when studying industry-relevant flow problems. We face this challenge by using (i) graphic processing units (GPUs), as well as (ii) a sophisticated parcel-based simulation approach that tracks packages of particles (often called “parcels”) instead of single particles.

GPUs allow for relatively low-cost parallel processing of particle data, i.e., solving Newton's equation of motion for millions of particles (see Figure). In this first part of the project we use GPU-technology by programming an in-house code that runs on GPUs, and which is coupled with the fluid flow solver “OpenFOAM”. OpenFOAM is an open-source software tool for solving partial differential equations using the finite volume method (FVM).

Furthermore, we develop a novel parcel-based approach that allows us to track parcels of particles instead of individual particles. While parcel-based approaches have been in use for more than ten years, recent studies have shown that current parcel interaction models fail to correctly predict certain flow features (e.g., the fluctuation energy of the parcels). This second part of the project aims on developing a parcel interaction model that is consistent with the kinetic theory of granular flow in regions where the particle volume fraction is small, but is still able to describe the more complex particle interactions in closely packed regions.