## Introduction

The availability of advanced commercial computational fluid dynamics (CFD) software and of faster computer processors have revolutionized scientific research in the field of multiphase flow. CFD has become an indispensable tool, for researchers and engineers alike, in solving many complex problems of academic and industrial interest in areas such as fluidization, combustion, oil flow assurance as well as aerospace science. In the field of fluidization, in particular, the use of CFD has pushed the frontier of fundamental understanding of fluid–solid interactions and has enabled the correct theoretical prediction of various macroscopic phenomena encountered in fluidized beds. Indeed many CFD simulations of monocomponent fluidized systems have been carried out by researchers covering the whole range of Geldart classified powders with great success.1–5 More recently, the simulation of industrial monosize powders has also been successfully tackled by Owoyemi et al.6 using a Eulerian–Eulerian approach. The various two-fluid models employed by the above-named authors have common origins in the spatially averaged Eulerian–Eulerian equations of motion first put forward by Anderson and Jackson7 and successively rigorously derived by several other researchers.8–14 Such equations, as it is well known, are not mathematically closed and necessitate empirical closures for contributions mainly related to the internal stress associated to each phase and to the fluid–particle interaction force. The need for suitable closures, indispensable for predicting the dynamics of multiphase systems with a satisfactory degree of accuracy, has stimulated a vast amount of original research that has led to the pervasiveness of many empirical closures in literature today.

A monosize system of particles seldom occurs in a fluidized bed of practical importance. Industrially operated fluidized beds typically consist of particles, which have a wide size distribution as well as different densities. The phenomenon of mixing and segregation becomes of major importance in these nonideal systems. A vast amount of experimental work has already been carried out by many authors looking at the behavior of binary mixtures in gas-fluidized beds and in particular at the segregation rate, which they display.15–18 The rate of segregation is often measured using the bed freeze test, where particles are taken from the fluidized bed layer by layer and sieved separately to measure the concentrations of different species.19 The computational modeling of binary systems, conversely, has not met with the same resounding success. The continuum modeling of binary mixtures is typically carried out using two approaches. The first is characterized by the use of separate momentum equations to cater for each particle species, whilst the second makes use of averaged mixture properties for the formulation of a mixture momentum equation coupled with the use of averaged constitutive relations.

Each of the two modeling approaches described above has been applied separately to simulate particulate flow patterns within gas-fluidized beds. The first approach based on separate momentum equations has been employed by a few authors.20–23 Their investigations, carried out using Geldart Group D particles, report good model predictions with a limited set of experimental measurements. The second approach based on the mixture momentum balance has been employed by van Wachem et al.24 to predict the flow of a binary-fluidized suspension of Geldart Group B particles. Results from the work showed good predictions with regard to bed expansion and minimum fluidization velocity.

The relatively little research conducted in the field of CFD involving binary mixtures and more generally mixtures with a particle size distribution (PSD) does not yet allow accurate theoretical predictions of the flow behavior of such systems. The work presented in this article is primarily concerned with the modeling of noncohesive binary-fluidized suspensions of solid particles; the approach adopted is that based on separate equations of motion as opposed to the mixture approach previously described. The authors propose to address some conceptual questions that naturally arise when modeling binary mixtures and which relate, for instance, to the nature of the particle–particle drag and of the internal stress associated to each phase.

The use of separate momentum equations for each particulate phase in the modeling of binary mixtures requires an extra term to account for the collisions between particles that belong to different size classes. This extra contribution should be termed *particle–particle interaction force*, but is often referred to as *particle–particle drag force*. Indeed, strictly speaking, the two forces are not equivalent, since the former might encompass several contributions, of which the drag is just one. The earliest attempt to quantify the nature of this force was made by Soo25 where a derivation was given for the force acting on a single particle of species F_{1} in a cloud of colliding particles of species F_{2}. This was followed by a similar development by Nakamura and Capes.26 An experimentally related theoretical development was carried out by Arastoopour et al.,27 where a semiempirical expression was derived for dilute gas–solid systems. Several expressions have since been put forward by many authors20–22, 28, 29 with most correlations being variations of earlier pioneering developments.

In the first section of this article, a derivation of the Eulerian–Eulerian averaged equations of motion for binary mixtures of particles in Newtonian fluids is presented. These equations are an extension of those originally put forward by Anderson and Jackson7 and Jackson8, 9 for systems of monosize particles in Newtonian fluids, where the averaged equations of conservation for mass, linear and angular momentum were derived and phase interactions modeled phenomenologically. In our derivation, however, only the mass and linear momentum equations of conservation shall be considered. These suffice to provide a rich description of the fluid–solid interaction forces at play in binary mixtures and a logical justification for the emergence of terms such as the particle–particle interaction force or the stress tensors associated with each phase.

A brief description of a new fluid dynamic model developed by the authors for binary-fluidized mixtures and implemented in the commercial CFD code CFX 4.4 follows thereafter. In this model, the fluid dynamic interaction between fluid and solid is based on the “elastic force” concept originally proposed by Wallis.30 The solid stress tensors are neglected, and a new numerical scheme is proposed to control the solid compaction in each particle phase.

A substantiation of the effect of the particle–particle drag force on the mixing and bubble dynamics of a binary gas–solid fluidized bed concludes the work. Here, three different closures available in literature and catering for this contribution are considered. The simulations are compared with a reference simulation where interphase particle–particle interactions are entirely neglected, and with dedicated experimental results. The particles used for experiments and computational studies belong to the Geldart Group B classification and are 200 and 350 μm in diameter respectively and have a density of 2500 kg/m^{3}.