Improvement of voidage prediction in liquid-solid fluidized beds by inclusion of the Froude number in effective drag relations

A novel effective drag relation for liquid-solid fluidisation is proposed, suitable for application in full-scale installations. This is achieved by presenting new insights related to the influence of the temporal-spatial heterogeneity on the effective hydrodynamic drag for large fluidised systems. While heterogeneous flow behaviour can be predicted increasingly accurately in CFD simulations that explicitly model the heterogeneous solids distribution, for the operation of many large-scale applications it is infeasible to perform such computationally intensive simulations. Therefore, there is a clear need for full-scale drag relations that effectively take into account the heterogeneous behaviour and irregular spatial particle distributions. Our new drag relation is based on a large set of experiments, which shows that the degree of overall expansion is not only dependent on the ratio of laminar-turbulent flow, but also on the amount of homogenous versus heterogeneous flow, which is not included in current full-scale drag relations. To include the effect of heterogeneity, the standard drag relation, based on the Reynolds number, is extended with a specific type of Froude number. Because fully turbulent flow regimes are rare in applications of liquid-solid fluidisation, our focus is not on the turbulent flow regime but instead on laminar and transitional flow regimes. In these regimes, three types of models are investigated. The first type is based on a theoretical similarity with terminal settling, the second is based on the semi-empirical Carman-Kozeny model, and the third is based on empirical equations using symbolic regression techniques. For all three types of models, coefficients are calibrated on experimental data with monodisperse and almost spherical glass beads. The models are validated with a series of calcium carbonate grains applied in drinking water treatment processes as well as data obtained from the literature. Using these models, we show that the voidage prediction average relative error decreases from approximately 5% (according to the best literature equations which use Reynolds number only) to 1-2% (using both Reynolds and Froude number). This implies that our new models are more suitable for operational control in full-scale fluidised bed applications, such as pellet softening in drinking water treatment processes.