scholarly journals Particle-size and -density segregation in granular free-surface flows

2015 ◽  
Vol 779 ◽  
pp. 622-668 ◽  
Author(s):  
J. M. N. T. Gray ◽  
C. Ancey
Keyword(s):  
Particle Size ◽  
Steady State ◽  
Parameter Space ◽  
Peclet Number ◽  
Particle Density ◽  
Critical Concentration ◽  
Free Surface Flows ◽  
Size Segregation ◽  
Péclet Number ◽  
Concentration Changes

When a mixture of particles, which differ in both their size and their density, avalanches downslope, the grains can either segregate into layers or remain mixed, dependent on the balance between particle-size and particle-density segregation. In this paper, binary mixture theory is used to generalize models for particle-size segregation to include density differences between the grains. This adds considerable complexity to the theory, since the bulk velocity is compressible and does not uncouple from the evolving concentration fields. For prescribed lateral velocities, a parabolic equation for the segregation is derived which automatically accounts for bulk compressibility. It is similar to theories for particle-size segregation, but has modified segregation and diffusion rates. For zero diffusion, the theory reduces to a quasilinear first-order hyperbolic equation that admits solutions with discontinuous shocks, expansion fans and one-sided semi-shocks. The distance for complete segregation is investigated for different inflow concentrations, particle-size segregation rates and particle-density ratios. There is a significant region of parameter space where the grains do not separate completely, but remain partially mixed at the critical concentration at which size and density segregation are in exact balance. Within this region, a particle may rise or fall dependent on the overall composition. Outside this region of parameter space, either size segregation or density segregation dominates and particles rise or fall dependent on which physical mechanism has the upper hand. Two-dimensional steady-state solutions that include particle diffusion are computed numerically using a standard Galerkin solver. These simulations show that it is possible to define a Péclet number for segregation that accounts for both size and density differences between the grains. When this Péclet number exceeds 10 the simple hyperbolic solutions provide a very useful approximation for the segregation distance and the height of rapid concentration changes in the full diffusive solution. Exact one-dimensional solutions with diffusion are derived for the steady-state far-field concentration.

scholarly journals MODELING OF PARTICLE SIZE SEGREGATION: CALIBRATION USING THE DISCRETE PARTICLE METHOD

2012 ◽  
Vol 23 (08) ◽  
pp. 1240014 ◽  
Author(s):  
ANTHONY THORNTON ◽  
THOMAS WEINHART ◽  
STEFAN LUDING ◽  
ONNO BOKHOVE
Keyword(s):  
Particle Size ◽  
Peclet Number ◽  
Particle Method ◽  
Size Ratio ◽  
Size Segregation ◽  
Discrete Particle ◽  
Péclet Number ◽  
Particle Properties ◽  
Chute Flow ◽  
Discrete Particle Method

Over the last 25 years a lot of work has been undertaken on constructing continuum models for segregation of particles of different sizes. We focus on one model that is designed to predict segregation and remixing of two differently sized particle species. This model contains two dimensionless parameters: Sr, a measure of the segregation rate, and Dr, a measure of the strength of diffusion. These, in general, depend on both flow and particle properties and one of the weaknesses of the model is that these dependencies are not predicted. They have to be determined by either experiments or simulations. We present steady-state, periodic, chute-flow simulations using the discrete particle method (DPM) for several bi-disperse systems with different size ratios. The aim is to determine one parameter in the continuum model, i.e. the segregation Péclet number (ratio of the segregation rate to diffusion, Sr/Dr) as a function of the particle size ratio. Reasonable agreement is found; but, also measurable discrepancies are reported; mainly, in the simulations a thick pure phase of large particles is formed at the top of the flow. Additionally, it was found that the Péclet number increases linearly with the size ratio for low values, but saturates to a value of approximately 7.7.

Approximate analysis of the containment of contaminated sites prior to remediation

2000 ◽  
Vol 42 (1-2) ◽  
pp. 319-324 ◽  
Author(s):  
H. Rubin ◽  
A. Rabideau
Keyword(s):  
Steady State ◽  
Peclet Number ◽  
Dimensionless Parameter ◽  
Contaminated Sites ◽  
Unsteady State ◽  
Péclet Number ◽  
Vertical Barrier ◽  
Time Period ◽  
Barrier Performance ◽  
Vertical Barriers

This study presents an approximate analytical model, which can be useful for the prediction and requirement of vertical barrier efficiencies. A previous study by the authors has indicated that a single dimensionless parameter determines the performance of a vertical barrier. This parameter is termed the barrier Peclet number. The evaluation of barrier performance concerns operation under steady state conditions, as well as estimates of unsteady state conditions and calculation of the time period requires arriving at steady state conditions. This study refers to high values of the barrier Peclet number. The modeling approach refers to the development of several types of boundary layers. Comparisons were made between simulation results of the present study and some analytical and numerical results. These comparisons indicate that the models developed in this study could be useful in the design and prediction of the performance of vertical barriers operating under conditions of high values of the barrier Peclet number.

scholarly journals Asymmetric flux models for particle-size segregation in granular avalanches

2014 ◽  
Vol 757 ◽  
pp. 297-329 ◽  
Author(s):  
P. Gajjar ◽  
J. M. N. T. Gray
Keyword(s):  
Particle Size ◽  
Small Particle ◽  
Volume Fraction ◽  
Free Surface Flows ◽  
Two Dimensions ◽  
Combined Processes ◽  
Maximum Point ◽  
Size Segregation ◽  
Flux Function ◽  
Large Particles

AbstractParticle-size segregation commonly occurs in both wet and dry granular free-surface flows through the combined processes of kinetic sieving and squeeze expulsion. As the granular material is sheared downslope, the particle matrix dilates slightly and small grains tend to percolate down through the gaps, because they are more likely than the large grains to fit into the available space. Larger particles are then levered upwards in order to maintain an almost uniform solids volume fraction through the depth. Recent experimental observations suggest that a single small particle can percolate downwards through a matrix of large particles faster than a large particle can be levered upwards through a matrix of fines. In this paper, this effect is modelled by using a flux function that is asymmetric about its maximum point, differing from the symmetric quadratic form used in recent models of particle-size segregation. For illustration, a cubic flux function is examined in this paper, which can be either a convex or a non-convex function of the small-particle concentration. The method of characteristics is used to derive exact steady-state solutions for non-diffuse segregation in two dimensions, with an inflow concentration that is (i) homogeneous and (ii) normally graded, with small particles above the large. As well as generating shocks and expansion fans, the new asymmetric flux function generates semi-shocks, which have characteristics intersecting with the shock just from one side. In the absence of diffusive remixing, these can significantly enhance the distance over which complete segregation occurs.

scholarly journals Stable solutions of a scalar conservation law for particle-size segregation in dense granular avalanches

2008 ◽  
Vol 19 (1) ◽  
pp. 61-86 ◽  
Author(s):  
M. SHEARER ◽  
J. M. N. T. GRAY ◽  
A. R. THORNTON
Keyword(s):  
Particle Size ◽  
Free Surface ◽  
Shear Flow ◽  
Free Surface Flows ◽  
Size Segregation ◽  
Small Particles ◽  
Surface Flows ◽  
Scalar Conservation Law ◽  
Large Particles ◽  
Scalar Conservation

Dense, dry granular avalanches are very efficient at sorting the larger particles towards the free surface of the flow, and finer grains towards the base, through the combined processes of kinetic sieving and squeeze expulsion. This generates an inversely graded particle-size distribution, which is fundamental to a variety of pattern formation mechanisms, as well as subtle size-mobility feedback effects, leading to the formation of coarse-grained lateral levees that create channels in geophysical flows, enhancing their run-out. In this paper we investigate some of the properties of a recent model [Gray, J. M. N. T. & Thornton, A. R. (2005) A theory for particle size segregation in shallow granular free-surface flows. Proc. R. Soc. 461, 1447–1473]; [Thornton, A. R., Gray, J. M. N. T. & Hogg, A. J. (2006) A three-phase mixture theory for particle size segregation in shallow granular free-surface flows. J. Fluid. Mech. 550, 1–25] for the segregation of particles of two sizes but the same density in a shear flow typical of shallow avalanches. The model is a scalar conservation law in space and time, for the volume fraction of smaller particles, with non-constant coefficients depending on depth within the avalanche. It is proved that for steady flow from an inlet, complete segregation occurs beyond a certain finite distance down the slope, no matter what the mixture at the inlet. In time-dependent flow, dynamic shock waves can develop; they are interfaces separating different mixes of particles. Shock waves are shown to be stable if and only if there is a greater concentration of large particles above the interface than below. Constructions with shocks and rarefaction waves are demonstrated on a pair of physically relevant initial boundary value problems, in which a region of all small particles is penetrated from the inlet by either a uniform mixture of particles or by a layer of small particles over a layer of large particles. In both cases, and under a linear shear flow, solutions are constructed for all time and shown to have similar structure for all choices of parameters.

scholarly journals Phoretic self-propulsion at finite Péclet numbers

2014 ◽  
Vol 747 ◽  
pp. 572-604 ◽  
Author(s):  
Sébastien Michelin ◽  
Eric Lauga
Keyword(s):  
Particle Size ◽  
Theoretical Analysis ◽  
Peclet Number ◽  
Chemical Interaction ◽  
Computational Results ◽  
Parallel Study ◽  
Péclet Number ◽  
Interaction Layer ◽  
Small Scales ◽  
The Impact

AbstractPhoretic self-propulsion is a unique example of force- and torque-free motion on small scales. The classical framework describing the flow field around a particle swimming by self-diffusiophoresis neglects the advection of the solute field by the flow and assumes that the chemical interaction layer is thin compared to the particle size. In this paper we quantify and characterize the effect of solute advection on the phoretic swimming of a sphere. We first rigorously derive the regime of validity of the thin-interaction-layer assumption at finite values of the Péclet number (${Pe}$). Under this assumption, we solve computationally the flow around Janus phoretic particles and examine the impact of solute advection on propulsion and the flow created by the particle. We demonstrate that although advection always leads to a decrease of the swimming speed and flow stresslet at high values of the Péclet number, an increase can be obtained at intermediate values of ${Pe}$. This possible enhancement of swimming depends critically on the nature of the chemical interactions between the solute and the surface. We then derive an asymptotic analysis of the problem at small ${Pe}$ which allows us to rationalize our computational results. Our computational and theoretical analysis is accompanied by a parallel study of the influence of reactive effects at the surface of the particle (Damköhler number) on swimming.

scholarly journals Time-dependent solutions for particle-size segregation in shallow granular avalanches

2006 ◽  
Vol 462 (2067) ◽  
pp. 947-972 ◽  
Author(s):  
J.M.N.T Gray ◽  
M Shearer ◽  
A.R Thornton
Keyword(s):  
Particle Size ◽  
Logarithmic Singularity ◽  
Free Surface Flows ◽  
Time Dependent ◽  
Homogeneous Material ◽  
Recent Theory ◽  
Size Segregation ◽  
Rock Falls ◽  
Wide Range ◽  
Graded Layers

Rapid shallow granular free-surface flows develop in a wide range of industrial and geophysical flows, ranging from rotating kilns and blenders to rock-falls, snow slab-avalanches and debris-flows. Within these flows, grains of different sizes often separate out into inversely graded layers, with the large particles on top of the fines, by a process called kinetic sieving. In this paper, a recent theory is used to construct exact time-dependent two-dimensional solutions for the development of the particle-size distribution in inclined chute flows. The first problem assumes the flow is initially homogeneously mixed and is fed at the inflow with homogeneous material of the same concentration. Concentration shocks develop during the flow and the particles eventually separate out into inversely graded layers sufficiently far downstream. Sections with a monotonically decreasing shock height, between these layers, steepen and break in finite time. The second problem assumes that the material is normally graded, with the small particles on top of the coarse ones. In this case, shock waves, concentration expansions, non-centred expanding shock regions and breaking shocks develop. As the parameters are varied, nonlinearity leads to fundamental topological changes in the solution, and, in simple-shear, a logarithmic singularity prevents a steady-state solution from being attained.

THE STABILITY OF SPATIALLY NONHOMOGENEOUS STEADY STATES IN A TUBULAR CHEMICAL REACTOR

1993 ◽  
Vol 03 (06) ◽  
pp. 1477-1486
Author(s):  
JAMES M. ROTENBERRY ◽  
ANTONMARIA A. MINZONI
Keyword(s):  
Steady State ◽  
Peclet Number ◽  
Chemical Reactor ◽  
Higher Order ◽  
Steady States ◽  
Complex Conjugate ◽  
Heat Of Reaction ◽  
Péclet Number ◽  
The Stability ◽  
Steady State Solutions

We study the axial heat and mass transfer in a highly diffusive tubular chemical reactor in which a simple reaction is occurring. The steady state solutions of the governing equations are studied using matched asymptotic expansions, the theory of dynamical systems, and by calculating the solutions numerically. In particular, the effect of varying the Peclet and Damköhler numbers (P and D) is investigated. A simple expression for the approximate location of the transition layer for large Peclet number is derived and its accuracy tested against the numerical solution. The stability of the steady states is examined by calculating the eigenvalues and eigenfunctions of the linearized equations. It is shown that a Hopf bifurcation of the CSTR model (i.e., the limit as the P approaches zero) can be continued up to order 1 in the Peclet number. Furthermore, it is shown numerically that for appropriate values of the Peclet number, the Damköhler number, and B (the heat of reaction) these Hopf bifurcations merge with the limit points of an "S–shaped" bifurcation curve in a higher order singularity controlled by the Bogdanov–Takens normal form. Consequently, there must exist a finite amplitude, nonuniform, stable periodic solution for parameter values near this singularity. The existence of higher order degeneracies is also explored. In particular, it is shown for D ≪ 1 that no value of P exists where two pairs of complex conjugate eigenvalues of the steady state solutions can cross the imaginary axis simultaneously.

Effect of Drop Deformation on Heat Transfer to a Drop Suspended in an Electrical Field

1998 ◽  
Vol 120 (3) ◽  
pp. 682-689 ◽  
Author(s):  
M. A. Hader ◽  
M. A. Jog
Keyword(s):  
Nusselt Number ◽  
Steady State ◽  
Heat Transport ◽  
Peclet Number ◽  
Drop Deformation ◽  
Dielectric Fluid ◽  
Electrical Field ◽  
Drop Shape ◽  
Péclet Number ◽  
Internal Problem

Heat transfer to a drop of a dielectric fluid suspended in another dielectric fluid in the presence of an electric field is investigated. We have analyzed the effect of drop deformation on the heat transport to the drop. The deformed drop shape is assumed to be a spheroid and is prescribed in terms of the ratio of drop major and minor diameter. Results are obtained for both prolate and oblate shapes with a range of diameter ratio b/a from 2.0 to 0.5. The internal problem where the bulk of the resistance to the heat transport is in the drop, as well as the external problem where the bulk of the resistance is in the continuous phase, are considered. The electrical field and the induced stresses are obtained analytically. The resulting flow field and the temperature distribution are determined numerically. Results indicate that the drop shape significantly affects the flow field and the heat transport to the drop. For the external problem, the steady-state Nusselt number increases with Peclet number for all drop deformations. For a fixed Peclet number, the Nusselt number increases with decreasing b/a. A simple correlation is proposed to evaluate the effect of drop deformation on the steady-state Nusselt number. For the internal problem, for all drop deformations, the maximum steady-state Nusselt number becomes independent of the Peclet number at high Peclet number. The maximum steady-state Nusselt numbers for an oblate drop are significantly higher than that for a prolate drop.

Sign in / Sign up

Export Citation Format

Share Document