Simulation of Mono- and Bidisperse Gas-Particle Flow in a Riser with a Third-Order Quadrature-Based Moment Method

Gas-particle flows can be described by a kinetic equation for the particle phase coupled with the Navier−Stokes equations for the fluid phase through a momentum exchange term. The direct solution of the kinetic equation is prohibitive for most applications due to the high dimensionality of the space of independent variables. A viable alternative is represented by moment methods, where moments of the velocity distribution function are transported in space and time. In this work, a fully coupled third-order, quadrature-based moment method is applied to the simulation of monoand bidisperse gas-particle flows in the riser of a circulating fluidized bed. Gaussian quadrature formulas are used to model the unclosed terms in the moment transport equations. A Bhatnagar−Gross−Krook (BGK) collision model is used in the monodisperse case, while the full Boltzmann integral is adopted in the bidisperse case. The predicted values of mean local phase velocities, rms velocities, and particle volume fractions are compared with the Euler−Lagrange simulations and experimental data from the literature. The local values of the time-average Stokes, Mach, and Knudsen numbers predicted by the simulation are reported and analyzed to justify the adoption of high-order moment methods as opposed to models based on hydrodynamic closures. ■ INTRODUCTION Gas-particle flows in risers have been the topic of extensive research in order to develop reliable computational models capable of describing their features. In general, two kinds of approaches are possible to describe the particle phase: a Lagrangian approach, where each particle trajectory is resolved independently, applying the fundamental laws of Mechanics; and an Eulerian approach, in which the particle phase is described by transport equations for moments of the particle velocity distribution function. The computational convenience and the absence of statistical noise characteristic of Eulerian models make them very attractive both for research and applications, and significant effort to improve their formulation has been spent in the last two decades. Nevertheless, when properly formulated, the predictions from Lagrangian and Eulerian models for the particle phase should be identical. Hydrodynamic models for circulating-fluidized-bed (CFB) reactors have been developed to account for heat transfer and to introduce a normal stress modulus for the particle phase. By adopting an appropriate drag correlation, one can properly predict flow regimes typical of CFB risers. If all the interactions between the phases are accounted for and correlated to the averaged and fluctuating components of the phase velocity fields, a hydrodynamic model is able to properly predict the behavior of gas−solid flow in a riser. A stationary hydrodynamic model is able to predict the particle phase stresses through the kinetic theory of granular flow as a function of the particle fluctuating energy (granular energy). A modified model has been proposed and validated against experimental data, showing a high sensitivity to the value of the restitution coefficient, whose reduction may lead to a wrong prediction of the particle segregation patterns inside the duct. In general, such sensitivity to model parameters points to a breakdown of the hydrodynamic description for relatively dilute flows. For turbulent flow, a one-equation turbulence model has been used to describe the gas-phase turbulence by adopting standard wall-functions for the zone near the wall. A zeroequation closure for the gas-phase turbulence has also been proposed. Extending previous work to arbitrarily inclined ducts, a model accounting for the effects of particle sliding and rotation has been developed. A low Reynolds number two-equation k-ε model for the gas phase has been proposed to eliminate the need for wall functions. The influence of turbulence both on the transport equations and on the kinetic theory closure equations has been studied, leading to a reformulation of the dissipation term of granular energy, which resulted in a reduced sensitivity of the model to the value of the particle restitution coefficient. A CFB riser has been simulated using the kinetic theory of granular flow, neglecting the gas-phase turbulence. A model that couples a two-equation turbulence model with a set of two equations for the particle-phase turbulent kinetic energy and for the gas-particle velocity correlation has also been proposed. It is noteworthy that RANS turbulence models for gas-particle flows are rarely compared to Lagrangian simulations and, in fact, rely on adjustable parameters to close the interaction terms between the two phases. Following the ideas behind large-eddy simulations (LES), LES models have been introduced using a standard subgrid stress model to describe the turbulence of the gas phase. Recently, a Lagrangian description of the particle phase coupled to an LES model for the gas phase has been compared to a multi-fluid model of a circulating fluidized bed. For dense Special Issue: L. T. Fan Festschrift Received: February 16, 2012 Revised: August 30, 2012 Accepted: August 31, 2012 Published: August 31, 2012 Article