Elmer M. Tory

Computer simulation of isotropic, homogeneous, dense random packing of equal spheres

Abstract A computer program generated 3000 interpenetrating spheres in a cubic container (with periodic boundary conditions which ensured complete continuity at all boundaries) and used a relaxation technique to reduce overlaps. When the average overlap fell below (exceeded) an empirical value, sphere radii were increased (decreased). This simulated vibration caused the nominal solids fraction to increase slowly to 0.6381. Further iterations with a very gradual decrease in sphere radii produced an overlap-free packing with a density of 0.6366. This packing, which is isotropic and homogeneous, is characterized by very few contacts and many non-touching neighbours. The radial distribution function is similar to that obtained experimentally. In particular, the first peak is split into two maxima.

Simulation of random packing of spheres

PACKS simulates the very slow settling of rigid spheres (as in sedimentation) from a dilute suspen sion into a randomly packed bed. Spheres are intro duced one at a time in a potential field and fall or roll until they occupy one of the available sites. The position of an incoming sphere is calculated at each of the significant events in its history. Thus, probabilities for each site are assigned on a realis tic basis. Periodic boundary conditions eliminate wall effects (smooth walls induce partial ordering) and a rough floor reduces the disturbance at the bottom. These features make a greater proportion of simulated spheres truly random in final position and so available for analysis. A new search routine greatly reduces the time required to assemble a pack ing. Preservation of the addresses of supporting spheres simplifies the subsequent analysis for near est neighbors. Packings of 100 000 or more spheres are feasible. The simulation has potential applica tions in crystallography, soil engineering, biology, nuclear engineering, and petroleum engineering.

On upper rarefaction waves in batch settling

We present a complete solution of the batch sedimentation process of an initially homogeneous ideal suspension where the Kynch batch flux density function is allowed to have two inflection points. These inflection points can be located in such a way that during the sedimentation process, the bulk suspension is separated from the supernate by a rarefaction wave or concentration gradient. This observation gives rise to two new modes of sedimentation as qualitative solutions of the batch sedimentation problem that had not been considered in previous studies. A reanalysis of published experimental data indicates that several observed upper concentration gradients can actually be interpreted as a rarefaction wave, and therefore be included in the framework of Kynchs theory. A numerical example shows an upper rarefaction wave in the settling of a flocculated suspension, to which Kynchs theory applies if the solid particles are in hindered settling.

On nonlocal conservation laws modelling sedimentation

The well-known kinematic sedimentation model by Kynch states that the settling velocity of small equal-sized particles in a viscous fluid is a function of the local solids volume fraction. This assumption converts the one-dimensional solids continuity equation into a scalar, nonlinear conservation law with a nonconvex and local flux. This work deals with a modification of this model, and is based on the assumption that either the solids phase velocity or the solid–fluid relative velocity at a given position and time depends on the concentration in a neighbourhood via convolution with a symmetric kernel function with finite support. This assumption is justified by theoretical arguments arising from stochastic sedimentation models, and leads to a conservation law with a nonlocal flux. The alternatives of velocities for which the nonlocality assumption can be stated lead to different algebraic expressions for the factor that multiplies the nonlocal flux term. In all cases, solutions are in general discontinuous and need to be defined as entropy solutions. An entropy solution concept is introduced, jump conditions are derived and uniqueness of entropy solutions is shown. Existence of entropy solutions is established by proving convergence of a difference-quadrature scheme. It turns out that only for the assumption of nonlocality for the relative velocity it is ensured that solutions of the nonlocal equation assume physically relevant solution values between zero and one. Numerical examples illustrate the behaviour of entropy solutions of the nonlocal equation.

Sedimentation is container-size dependent

Abstract In very dilute, bounded dispersions of identical spheres, the initial distributions of particle and fluid velocities in small containers differ enormously from those in very large ones. In the latter, local interactions are dwarfed by the massive collective influence of distant spheres. Large regions move downward or upward virtually en masse. Since particles and fluid in these clusters move with almost the same velocity, the mean of the initial distribution is scarcely affected by the much larger container size, but individual velocities can exceed the mean by one or more orders of magnitude. Experimental evidence of cluster settling, including the effect of container size, is ancient and abundant.

Extensions and refinements of a Markov model for sedimentation

Abstract The Markov model developed in previous papers is applied to the study of the slurry-supernate interface in batch sedimentation. The variability of particle velocities explains the induction period and the fuzziness of interfaces. Using the velocity-concentration curve indicated by experimental measurements of individual particles in the interior of dispersions, the Markov model predicts that slurries with initial concentrations in a critical range will suffer a depletion of particles near the top of the slurry. This results in an attenuated interface. The remnant concentration is the same for all attenuated interfaces. All attenuated interfaces have the same ultimate velocity. Outside this critical range, the ultimate velocity of the interface is equal to the mean of the steady-state pdf of particle velocities. Earlier papers introduced the concept of a parametric concentration as a weighted average of local solids concentrations. The introduction of suitable metrics for this parameter and particle personality (size and shape) extends theorems for monodisperse systems to slurries containing a countable number of species and shows that a family of nearly identical particles can be uniformly approximated by a single species. These results hold across realizations from continuous distributions of personality and initial position. Finally, sufficient conditions are established for classification in dilute slurries of multifarious particles.

Stochastic sedimentation and hydrodynamic diffusion

Molecular collisions with very small particles induce Brownian motion. Consequently, such particles exhibit classical diffusion during their sedimentation. However, identical particles too large to be affected by Brownian motion also change their relative positions. This phenomenon is called hydrodynamic diffusion. Long before this term was coined, the variability of individual particle trajectories had been recognized and a stochastic model had been formulated. In general, stochastic and diffusion approaches are formally equivalent. The convective and diffusive terms in a diffusion equation correspond formally to the drift and diffusion terms of a Fokker–Planck equation (FPE). This FPE can be cast in the form of a stochastic differential equation (SDE) that is much easier to solve numerically. The solution of the associated SDE, via a large number of stochastic paths, yields the solution of the original equation. The three-parameter Markov model, formulated a decade before hydrodynamic diffusion became fashionable, describes one-dimensional sedimentation as a simple SDE for the velocity process {V(t)}. It predicts correctly that the steady-state distribution of particle velocities is Gaussian and that the autocorrelation of velocities decays exponentially. The corresponding position process {X(t)} is not Markov, but the bivariate process {X(t),V(t)} is both Gaussian and Markov. The SDE pair yields continuous velocities and sample paths. The other approach does not use the diffusion process corresponding to the FPE for the three-parameter model; rather, it uses an analogy to Fickian diffusion of molecules. By focusing on velocity rather than position, the stochastic model has several advantages. It subsumes Kynch’s theory as a first approximation, but corresponds to the reality that particle velocities are, in fact, continuous. It also profits from powerful theorems about stochastic processes in general and Markov processes in particular. It allows transient phenomena to be modeled by using parameters determined from the steady-state. It is very simple and efficient to simulate, but the three parameters must be determined experimentally or computationally. Relevant data are still sparse, but recent experimental and computational work is beginning to determine values of the three parameters and even the additional two parameters needed to simulate three-dimensional motion. If the dependence of the parameters on solids concentration is known, this model can simulate the sedimentation of the entire slurry, including the packed bed and the slurry–supernate interface. Simulations using half a million particles are already feasible with a desktop computer.

The distribution of kth nearest neighbours and its application to cluster settling in dispersions of equal spheres

Abstract The spatial distribution of spheres in a very dilute suspension is described approximately by a Poison distribution, but much better approximations result when the volume of the spheres is taken into account. For solids fractions up to 0.32, the distribution of k th nearest neighbours calculated from the radial distribution function for a hard-sphere gas agreed closely with that determined from a computer simulation. This distribution provides insight into the concentration threshold for cluster settling and the increase in sedimentation rate with time in dilute dispersions. Two models provide rough explanations for the maintenance of cluster velocities greater than the Stokes velocity.

A random packing structure of equal spheres — statistical geometrical analysis of tetrahedral configurations

Abstract A random packing structure constructed by a computer simulation of the slow settling of rigid spheres into a randomly packed bed is investigated, where the bulk-mean volume fraction of the particles is 0.582. Each sphere settled on three others forms a tetrahedron, whose detailed geometrical configuration is successfully examined by a statistical geometrical analysis.

A three-parameter markov model for sedimentation II. Simulation of transit times and comparison with experimental results

Abstract An adequate description of the global behavior of sedimenting suspensions requires a knowledge of not only the mean velocity, but also the variation and autocorrelation of the velocities of individual particles. Much of the published information on these two parameters is encoded in transit times. For example, Koglin measured the length of time for individual spheres to traverse a fixed distance x and found that sedimentation velocities based on these transit times satisfied a log-normal distribution. Computer simulations, which yield the distribution of transit times as a function of the three parameters of the parent distribution, are an effective method of decoding these data. Our three-parameter Markov model yields first-passage times which are approximately log-normal, and the convergence to log-normality as x increases is remarkably fast. Thus, Koglins data provide strong support for the Markov model, experimentally verifying theoretical predictions based (via simulations) on it.