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375355 Effect of Particle-to-Fluid Density Ratio on the Drag Force in Fluidized Beds

In order to effectively study fluid-particle systems on a large scale, it is vital to accurately model the fluid-particle interaction force on a microscopic level. In a fluidized bed, the drag force, which is the most dominant contribution to the interactions between the fluid and particles, acts to oppose the downward force of gravity on a particle, and thus provides the main mechanism for fluidization. The fluid-particle drag therefore plays a very significant role in determining the dynamics of a fluidized bed. Consequently, the drag force constitutive model that is applied to simulations of a large-scale system, where the flow is not resolved to the scale of the particle surface, must faithfully represent the interactions between the fluid and particles that are observed at the level of the particle surface.

Drag models that are employed in large-scale simulations of fluidized beds are typically based on one of two types of systems. In one type, the drag models are developed from the results of fine-scale lattice Boltzmann simulations of beds in which the particles are fixed in place, such as in the case of Hill, Koch, Ladd^{1} and Beetstra et al^{2}. Such drag models are particularly applicable to systems with high particle-to-fluid density ratios, such as in ambient pressure gas-solid fluidized beds, where the particles are slow to respond to the fluid due to inertial effects. In the other type, the drag models are based on data arising from experiments of the sedimentation of solid particles in liquid, such as in the case of Wen and Yu^{3}. Such drag models effectively represent the dynamics of fluidized beds with a low particle-to-fluid density ratio, where the particles translate and rotate according to the motion of the surrounding fluid. Fluidized beds with moderate particle-to-fluid density ratios (~ 10 - 100), such as in the case of gas-solid systems at elevated pressures, are therefore not directly represented by these commonly-used drag models. High-pressure gas-solid fluidized beds have a wide range of industrial applications, including a number of polymerization processes, and so there is a clear need to address this gap in the drag models.

In this work, the effect of the particle-to-fluid density ratio on the drag force is studied using fully-resolved lattice Boltzmann simulations of a system composed of fluid and spherical particles. The simulation scheme is based on the work of Derksen and Sundaresan^{4}. In contrast to prior drag force simulation studies, the particles are allowed to translate and rotate in these simulations in order to assess the effect of the density ratio. Through these simulations, the drag force coefficient, which is defined to be the ratio of the observed drag force to the Stokes drag force on an isolated particle, has been found to increase slower with particle volume fraction for systems with moderate particle-to-fluid density ratios compared to systems with high ratios. Such a result has the potential to further explain the fact that smoother fluidization has been experimentally observed in systems at elevated pressures^{5}. Overall, this work seeks to develop a drag force model that accounts for the particle-to-fluid density ratio.

- R.J. Hill, D.L. Koch, A.J.C. Ladd. (2001). The first effects of fluid inertia on flows in ordered and random arrays of spheres.
*J. Fluid Mech*.,**448**, 213-241. - R. Beetstra, M.A. van der Hoef, J.A.M. Kuipers. (2007). Drag Force of Intermediate Reynolds Number Flow Past Mono- and Bidisperse Arrays of Spheres.
*AIChE J.*,**53**(2), 489–501. - C.Y. Wen, Y.H. Yu, Mechanics of fluidization,
*Chem. Eng. Prog.**Symp.*Ser. 62 (1966) 100. - J.J. Derksen, S. Sundaresan. (2007). Direct numerical simulations of dense suspensions: wave instabilities in liquid-fluidized beds.
*J. Fluid Mech.*,**587**, 303–336. - J.G. Yates. (1996). Effects of temperature and pressure on gas-solid fluidization.
*Chem. Eng. Sci.*,**51**(2), 167-205.

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