Gas-fluidized beds with particles of different sizes and/or densities are common in industry, and known to exhibit segregation under some operating conditions. Numerous studies have been carried out to investigate the segregation behavior in fluidized beds. The degree of segregation depends on the differences in density and size of the particles, as well as the gas velocity (Chiba et al., 1979). For systems composed of equal density particles, small particles tend to concentrate near the surface of the bed while large particles fall to the bottom (Rowe and Nienow, 1976; Wu and Baeyens, 1998; Goldschmidt et al., 2003). Segregation by size increases with increasing bed height, decreasing size of fines, increasing mean size, and as the gas velocity approaches the minimum fluidization velocity of the smaller particle (Geldart et al., 1981). For systems composed of different density particles, denser particles tend to fall to the bottom of the bed. For these systems, a large degree of segregation is present at low gas velocities and this degree of segregation decreases as velocity increases (Rowe and Nienow, 1976). A phenomenon which is not well understood is layer inversion, which refers to systems in which a given species may behave as either flotsam or jetsam, depending on the operating condition (Rasul et al., 1999).

            Mathematical models have been used to study segregation in gas-solid fluidized beds. Both Eulerian (van Wachem et al., 2001a; Huilin et al., 2003; Cooper and Coronella, 2005) and Lagrangian models (Hoomans et al., 2000; Limtrakul et al., 2003; Feng et al., 2003; and Bokkers et al., 2004; Dahl and Hrenya, 2005) have been able to predict segregation with a certain degree of success and have provided insight on the contributing mechanisms. Van Wachem et al. (2001a) used binary models for drag and solid phase stress in an Eulerian framework and was able to model layer inversion. The inversion phenomena was explained through the dominating mechanisms of the system, namely at low gas velocities the system is dominated by gravity and drag force and at high velocities by pressure drop and gradients in the granular temperature.

            A key component of both Eulerian and Lagrangian models is the drag force, which couples the fluid and solid phases. Numerous experimental studies have been carried out to measure drag force in different monodisperse systems and several correlations have been proposed. These correlations have been developed from packed-bed measurements (Ergun, 1952 and Macdonald et al., 1979), settling experiments (Richardson and Zaki, 1954, Syamlal and O'Brien, 1994), fluidized-bed experiments (Wen and Yu, 1966), and Lattice-Boltzmann simulations (Koch and Hill, 2001). Recently, Pirog (1998) developed a drag force correlation for polydiperse systems based on a voidage-velocity correlation obtained from settling and creaming experiments in liquid-solid systems. Van der Hoef et al. (2005) developed a drag force correlation for binary mixtures based on Lattice-Boltzmann simulations of low gas flow past arrays of random spheres. Van Wachem et al. (2001a) and Dahl and Hrenya (2005) used Pirog's correlation in conjunction with the Ergun (1952) and the Wen and Yu (1966) drag models to describe drag force in binary systems.

            The use of different drag force models significantly impacts the simulation results, modifying the bed expansion and solid concentration in the bed (van Wachem, et al., 2001b). Monodisperse models have been traditionally employed in polydisperse systems, adapting them by replacing the particle diameter by a species diameter, the slip velocity of the monodisperse system by that of a single species in a polydisperse system, and assuming that the individual species drag force is equal to the drag force of a monodisperse system at the same volume fraction (van der Hoef et al., 2005). These assumptions have no physical basis but have been used due to the lack of adequate drag models for polydisperse systems. The focus of the present work is to determine the impact of various drag laws on fluidized beds composed of binary mixtures, with special emphasis on the prediction of segregation patterns. As described below, the simulations performed as part of this work indicate the critical role of the drag force in predicting segregation. Furthermore, a companion experimental work (Joseph et al., 2005) demonstrates the relative abilities of the existing drag laws, as applied to binary systems.       In order to isolate the effect of the drag force and hence minimize the contribution of the collisional (particle) stress, the simulation study was performed at low velocities, in which the drag force is one of the dominating forces, as noted by van Wachem et al. 2001a). A new drag force model is proposed which combines the approach taken by Gidaspow (1994) and the recent work of van der Hoef et al. (2005).  The former is a “stitching” together of monodisperse drag laws for the packed and fluidized regions, whereas the latter provides a correction to monodisperse drag laws for binary systems in order to account for the presence of particles with different diameters. A comparison between drag laws with and without the binary correction was completed.

            To assess the relative merits of the various drag laws, simulations were performed using a novel alternative for modeling fluid-solid systems, namely the Multi-Phase Particle-in-Cell (MP-PIC) method (Andrews and O'Rourke, 1996). This implementation is a combination of the Eulerian and Lagrangian models for the solid phase and possesses advantages of both approaches; particles are grouped in parcels that are individually tracked, but the particle-phase stress is calculated from an Eulerian description, which thereby eliminates the need to resolve individual collisions between particles. Since particles are tracked in groups and individual collisions are not resolved, the modeling of many-particle systems is performed in a more computationally efficient manner than a strict Lagrangian treatment. Arena-flow^TM is being used as a framework.

            To assess the impact of the various drag force treatments on the segregation behavior, simulations were performed for a defluidizing bed in a 10 cm diameter column. The initial condition was a bed 40 cm high with 100 and 200 micron particles, both at a solid volume fraction of 0.15 and uniformly mixed. The density of both components was 2600 kg/m³. The initial gas velocity was set to 0.25 m/s, and was decreased in steps of 0.025 m/s in periods of 0.2 s. The gas velocity was then kept constant for 2 s, and the time average properties were calculated over the second half of this period. Both drag laws predicted the minimum fluidization velocity around 0.05 m/s. For the drag law without the binary correction, the simulation results show total segregation (i.e., complete separation of species) for all gas velocities. The particles segregated, with the big particles on top of the small particles, at the initial velocity and remained unmixed. When using the drag law with the binary correction, however, a relatively homogeneous mixture is obtained for the higher velocities and segregation is observed as the gas velocity is decreased from 0.1 m/s. The final state, at a zero gas velocity, presents partial mixing; i. e., a layer of small particles is present at the top of the bed, a layer of large particles is present at the bottom, and the middle portion of the bed has both components present. For gas velocities above 0.05 m/s, the pressure drop equals the total weight of the bed divided by its area. As the gas velocity decreases, a defluidized layer of coarse particles forms in the bottom of the bed. This defluidized layer is no longer supported by the fluid. For a gas velocity of 0.025 m/s a smaller pressure drop than that of the weight divided by its area was calculated by the drag law with the binary correction. More particles had fallen into the defluidized layer in the simulation without the binary correction causing an even smaller pressure drop. In summary, the MP-PIC simulation results indicate that the form of the drag law plays a crucial role in the qualitative and quantitative nature of segregation predictions. A test of the correctness of the drag laws employed, and a validation of the simulation results discussed here, are determined via a direct comparison between simulations and experiments, as is detailed in the companion contribution by Joseph et al. (submitted).

            To verify that the results obtained with the MP-PIC approach were not influenced by the semi-empirical, solid-phase stress model (Snider, 2001) in Arena-flow^TM, two-dimensional, discrete-particle simulations were carried out following the development by Dahl and Hrenya (2005).  In this treatment, individual particle collisions are resolved and hence no solid-phase stress model is required (unlike the MP-PIC approach). For this strictly Lagrangian formulation, a higher degree of segregation was observed when using the model without the binary correction; this behavior qualitatively mimics that obtained in the MP-PIC simulations.