Structural Characteristics Of Randomly Packed Beds Of Spheres

Abstract

Packed beds find extensive application in a wide variety of industries to cany out a large number of diverse processes. The main objective of the present work is to develop models to predict the arrangement of particles and based on them, to determine and evaluate the structural characteristics of packed beds. These problems have received only a limited attention in the literature. As a first attempt, spheres of uniform size are considered.

Beds of aspect ratio up to 2 (referred to as low aspect ratio beds) are analyzed by application of principles of analytical geometry. Expressions are derived for the location of particles and for the structural characteristics of the beds, both of which show periodicity. This leads to the concept of a unit cell which is the repetitive section of the bed whose characteristics are the same as those of the complete bed. The beds fall into three distinct groups — those with aspect ratio between 1 and l√3⁄2, between 1√3⁄2 and 2, and with aspect ratio 2. Equations are distinct for each group. The aspect ratio shows marked influence on the structural characteristics of the beds. Agreement of the predictions on the overall void fraction with the available experimental data is excellent. Radial void fraction profiles are estimated by defining a concentric cylindrical channel (CCC) of an arbitrary thickness and with the cylindrical surface through the radial position of interest located at the middle of the CCC, and by accounting for the solid volumes of all the segments (in this CCC) of spheres with centers lying within a distance of a particle radius on either side of the cylindrical surface. The curved boundaries of the sphere segments are rigorously accounted for. The results show that the entire bed is filled with variations in the void fraction, starting from a value of unity at the wall and zero (or close to zero) towards the axis of the bed.

Monte Carlo model for the simulation of high aspect ratio beds has not proved successful even with any of a wide variety of distribution functions for the coordinates of the sphere dropping point. With uniform distribution, the only distribution used in all the reports so far, and with normal distribution, there is not even a qualitative agreement with the reported data on void fraction variations. Distributions with asymmetric density functions such as exponential, Weibull, gamma and beta, show considerable improvement; beta distribution being the best. However even the best results with beta distribution show satisfactory agreement with the experimental data only up to about 2dp from the wall.

Simulations with the cluster growth model, modified to account for the confining nature of the wall, lead to more satisfactory results. The proposed algorithm consists of building up the cluster, sphere by sphere, by calculating all possible interior and wall sites for placing an incoming sphere in a stable and non-overlapping position on the current cluster. A preference parameter is defined to place the new sphere at locations along the cross section of the column at which the experimental void fraction profiles show prominent minima, that is, locations around which the bed has relatively high solid volume. Void fraction profiles in beds of various aspect ratios simulated by this model show good agreement with the corresponding experimental data. The structural characteristics of the high aspect ratio beds thus simulated are evaluated. The number of spheres per unit length, Ni is correlated with the aspect ratio. It becomes proportional to the square of the aspect ratio, with the proportionality constant being close to 0.9, for aspect ratios greater than about 10. This follows since in these beds the overall void fraction becomes constant at 0.4.

Majority of the spheres have contacts (with neighboring spheres) between 4 and 7, with the lower and upper limits for the coordination number being 2 and 9. The radial profile of the average coordination number (averaged over the height of the bed at the given radial position) shows small oscillations about a mean value of about 6 over almost the entire bed cross section starting from a distance of about ldp from the wall. At a distance of 0.5dp from the wall the predominant number of contacts is four while the mean value is about 4.3. The overall coordination number (averaged over the entire bed) shows inverse dependence on the aspect ratio. For random packings, that is, as the aspect ratio becomes infinity, the overall coordination number tends to six which corresponds to regular cubic arrangement.

Cumulative number fraction, CNf is a global measure of the arrangement of spheres in beds of high aspect ratio. Its radial variation shows four distinct regions whose locations are independent of the aspect ratio The CNf values in each region are correlated with aspect ratio The correlations combined with that of NL lead to a very useful and effective model for predicting void fraction profiles in a bed of any specified aspect ratio The validity of the predictive model is demonstrated