Cluster Reynolds number

If the reduced scale models faithfully reproduce the dynamics of the exact case, the cluster dimensions should scale directly with the linear dimensions of the bed. Thus, a one-quarter linear scale model that has a velocity one-half that of the exact case will have a cluster Reynolds number (Re c) one-eighth that of the exact bed. From the relationship of Co with Re the change of Cd with model scale at a given Reynolds number of the exact bed can be determined. Figure 16... 

T 0 determine the hydrodynamic radius Rfj of the clusters, one has to solve the Stokes equations that govern the low Reynolds number flow of Newtonian fluids. By taking two opposite limits (namely, the solid concentration goes to zero and the size of the aggregates goes to infinity), it could be shown that the hydrodynamic drag F on an aggregate scale is... 

The Kirkwood-Riseman theory allows for an approximate calculation of the hydrodynamic drag of arbitrary clusters of small objects (molecules, particles) with undefined shape. It is based on the fact that, for low Reynolds-numbers, the velocity perturbations originating from the primary particles superpose linearly. The hydrodynamic force on a primary particle j results from the total perturbation field around j (yj) and from the undisturbed flow u°°. To first approximation, it can be assumed that the presence of j does not afl ect y, i.e. the spatial extension of particle j is neglected. Moreover, the perturbation of the velocity field at the place of j can be related to the perturbation forces acting on the other particles ... 

A.V. Filippov, Drag and torque on clusters of N arbitrary spheres at low Reynolds number. 

Calculation of the terminal velocity of a porous sphere is useful and important in applications in water treatment where settling velocities of a floe or an aggregate are estimated. It is also important in estimation of terminal velocities of clusters in fluidized bed applications. The terminal velocity of a porous sphere can be quite different from that of an impermeable sphere. Theoretical studies of settling velocity of porous spheres were conducted by Sutherland and Tan (1970), Ooms et al. (1970), Neale et al. (1973), Epstein and Neale (1974), and Matsumoto and Suganuma (1977). The terminal velocity of porous spheres was also experimentally measured by Masliyah and Polikar (1980). In the limiting case of a very low Reynolds number, Neale et al. (1973) arrived at the following equation for the ratio of the resistance experienced by a porous (or permeable) sphere to an equivalent impermeable sphere. An equivalent impermeable sphere is defined to be a sphere having the same diameter and bulk density of the permeable sphere. 

Fig. 13 Number of equal-sized primary particles per agglomerate as a function of the averaged radius of gyration including its extreme deviation (horizontal bars) as well as a the particle Stokes number St and b the particle Reynolds number Rep. Results obtained by LBM-based direct numerical simulations (DNS, open symbols) and numerical predictions for agglomerates formed by particle-cluster (P-C, solid line) and cluster-cluster (C-C, dashed line) collisions based on Langevin dynamics [4]...

Having done a detailed comparison between methods for calculating hydrodynamics, some words should be mentioned about the need for expensive calculations. While FEM was not considered in this study due to applicability, the FDA was compared with the results from SD (see Fig. 5). Hydrodynamic differences are clearly seen through the critical shear rate defined by the onset of restructuring. In low Reynolds number flows, the disturbance decays proportional to resulting in drag force reduction of particles within isolated clusters. Also, the... 

Collins LR, Keswani A Reynolds number scaling of particle clustering in turbulent aerosols. New J Phys 6 119, 2004. http //dx.doi.Org/10.1088/1367-2630/6/l/119. 

Perhaps a more useful means of quantifying structural data is to use a similarity measurement. These are reviewed by Ludwig and Reynolds (1988) and form the basis of multivariate clustering and ordination. Similarity measures can compare the presence of species in two sites or compare a site to a predetermined set of species derived from historical data or as an artificial set comprised of measurement endpoints from the problem formulation of an ecological risk assessment. The simplest similarity measures are binary in nature, but others can accommodate the number of individuals in each set. Related to similarity measurements are distance metrics. Distance measurements, such as Euclidean distance, have the drawbacks of being sensitive to outliers, scale, transformations, and magnitudes. Distance measures form the basis of many classification and clustering techniques. 

By using rather than C/g, the effective Reynolds mmiber is reduced so that the calculated Nusselt numbers tend toward the conservative conduction limit of 2. Obviously this approach greatly simplifies the actual phenomena, and disregards the complications of particle acceleration, clustering, and downflow at walls. Estimates of the particle/gas heat transfer coefficient (/jp) from Eqs. (11) and (52) should be considered only as approximate. 

---