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• A sliding-window based signal processing method for characterizing particle clusters. • Particle clusters in high-density circulating fluidized bed are characterized. • Time-dependent characteristics of clusters are obtained. • Higher solids holdup and lower velocity of the clusters are found in HDCFB. • More and looser clusters for better gas-solids contacting exist in HDCFB. Particle clusters in CFB risers were identified from the instantaneous solids holdup signals by a new sliding-window based signal processing method. By shifting the time window and calculating the mean and the standard deviation within it, a non-linear threshold curve for identifying the clusters was derived instead of the conventional constant threshold. The optimal sliding window size was determined as W b = 1024 data points by the bisection method on the entire piece of signals. Using the proposed method, a more realistic characterization of the particle clusters in both HDCFB and LDCFB was obtained by considering the bulk fluctuation of the gas-solids flow. The clusters in HDCFB have higher solids holdup and lower velocity than that in the LDCFB. The HDCFB is also found to have a greater number of loose clusters for better gas-solids contacting and exchanges in the center of the riser. [ABSTRACT FROM AUTHOR]

• Consider permeability in drag correlations for monodisperse arrays of spheres and porous spheres. • Obtain drag data for porous spheres by settling experiments and sphere arrays by direct numerical simulations. • Use symbolic regression method to derive the formula forms of drag correlations. • Symbolic regression based drag correlations are generic and physically sounded. An accurate drag correlation accounting for multiscale heterogeneous porous structures is a prerequisite for reliable CFD simulation of fluidized beds. Though particle clusters in fluidized beds are usually modeled as porous particles, particle-resolved direct numerical simulation (PR-DNS), in which monodisperse arrays of spheres are taken as model systems, has been widely used as a first-principle approach to derive drag correlations. This work is to bridge the gap of drag correlations for monodisperse arrays of spheres and porous spheres by considering permeability effects and derive new drag correlations using symbolic regression (SR) methods. Firstly, experimental porous spheres settling data were utilized to identify the most important features affecting drag force using Support Vector Machine (SVM), in which the permeability β was packed up in addition to the solid fraction ϕ and Reynolds numbers Re. A new drag correlation for porous spheres based on ϕ , R e , and β , which has physical terms, high prediction accuracy and correct limiting cases, is automatically generated using SR method. Then, PR-DNS data from open sources were used to distill drag correlations for monodisperse arrays of spheres by incorporating the extra permeability parameter by SR method, demonstrating solid physical basis with high accuracy. It is further shown the SR based on drag correlations for porous spheres and monodisperse arrays of spheres can be reduced to limiting cases of a single solid sphere and Stokes flow. The proposed new drag correlations not only provide a way to use permeability as a simple yet physically sounded parameter to quantify heterogeneous structures in fluidized beds, but also open a venue for directly validating drag correlations obtained purely from PR-DNS simulations with experimental data of flow around porous spheres. [ABSTRACT FROM AUTHOR]