Particles and bubbles suspended in homogeneous isotropic turbulence are tracked and their collisions frequency is determined as a function of particle Stokes number. The carrier phase velocity fluctuations are determined by Direct Numerical Simulations (DNS). The effects of the dispersed phases on the carrier phase are neglected. Particles and bubbles of sizes on the order of Kolmogorov length scale are treated as point masses. In addition to Stokes drag, the pressure gradient in the carrier phase and added-mass forces are also included. Equations of motion of dispersed phases are integrated simultaneously with the equations of the carrier phase using the same time stepping scheme. The collision model used here allows overlap of particles and bubbles. Simulations for three turbulence Reynolds numbers Re_Λ = 57, 77, and 96 have been performed. Collisions kernel, radial relative velocity, and radial distribution function found by DNS are compared to theoretical models over a range of particle Stokes number. Comparisons are made with Zaichik et al. [22] model, which is applicable to heavy particles, and Zaichik et al. [23] model which is valid for an arbitrary Stokes number. Zaichik et al. [23] is essentially a model for the radial relative velocity, and for the purpose of computing the collision kernel, it assumes the radial distribution function to be one. In general, good agreement between DNS and Zaichik et al. models is obtained for radial relative velocity for both particle-particle and particle-bubble collisions. The DNS results show that around Stokes number of unity particles of the same group undergo expected preferential concentration while particles and bubbles are segregated. The segregation behavior of particles and bubbles leads to a radial distribution function that is less than one. Existing theoretical models do not account for effects of this segregation behavior of particles and bubbles on the radial distribution function.

A new boundary condition treatment has been devised for two-phase flow numerical simulations in a self-aerated minerals flotation machine and applied to a Wemco 0.8 m³ pilot cell. Airflow rate is not specified a priori but is predicted by the simulations as well as power consumption. Time-dependent simulations of two-phase flow in flotation machines are essential to understanding flow behavior and physics in self-aerated machines such as the Wemco machines. In this paper, simulations have been conducted for three different uniform bubble sizes (d_b = 0.5, 0.7 and 1.0 mm) to study the effects of bubble size on air holdup and hydrodynamics in Wemco pilot cells. Moreover, a computational fluid dynamics (CFD)-based flotation model has been developed to predict the pulp recovery rate of minerals from a flotation cell for different bubble sizes, different particle sizes and particle size distribution. The model uses a first-order rate equation, where models for probabilities of collision, adhesion and stabilization and collisions frequency estimated by Zaitchik-2010 model are used for the calculation of rate constant. Spatial distributions of dissipation rate and air volume fraction (also called void fraction) determined by the two-phase simulations are the input for the flotation kinetics model. The average pulp recovery rate has been calculated locally for different uniform bubble and particle diameters. The CFD-based flotation kinetics model is also used to predict pulp recovery rate in the presence of particle size distribution. Particle number density pdf and the data generated for single particle size are used to compute the recovery rate for a specific mean particle diameter. Our computational model gives a figure of merit for the recovery rate of a flotation machine, and as such can be used to assess incremental design improvements as well as design of new machines.

Two-phase (water and air) flow in the forced-air mechanically-stirred Dorr-Oliver machine has been investigated using computational fluid dynamics (CFD). A 6 m³ model is considered. The flow is modeled by the Euler-Euler approach, and transport equations are solved using software ANSYS-CFX5. Unsteady simulations are conducted in a 180-degree sector with periodic boundary conditions. Air is injected into the rotor at the rate of 2.63 m³/min, and a uniform bubble diameter is specified. The effects of bubble diameter on velocity field and air volume fraction are determined by conducting simulations for three diameters of 0.5, 1.0, and 2.0 mm. Air volume fraction contours, velocity profiles, and turbulent kinetic energy profiles in different parts of the machine are presented and discussed. Results have been compared to experimental data, and good agreement is obtained for the mean velocity and turbulent kinetic energy profiles in the rotor-stator gap and in the jet region outside stator blades.

This chapter develops finite-element models for a class of boundary-value problems governed by a second-order ordinary differential equation. It considers a standard form of the differential equation which is known as Model Problem. Formulation of the finite element method using global basis functions establishes a strong connection between the weighted residual method or the weak form and the finite element method. However, construction of global basis functions and evaluations of integrals are not practical for complex geometries particularly for multidimensional problems. Boundary conditions are specified on the boundary of the whole domain, and no boundary conditions are available for any of the elements except for an element whose boundary falls on the global boundary. The weak form before application of boundary conditions Equation is valid for each element provided that the integrals are evaluated over the element domain only and the secondary variables are evaluated at the element boundary points.