Two-dimensional Silo Research Articles

Elongated particles discharged with a conveyor belt in a two-dimensional silo

Elongated particles discharged with a conveyor belt in a two-dimensional silo

A technique for the estimation of percolation thresholds in lattice systems: Application to a problem of granular flow through an orifice

A simple method to estimate percolation thresholds pc in lattice models is presented. The technique is based on the calculation of the probabilities RLX(p) for finding a percolating cluster of type X [X could be horizontal (H), vertical (V), average (A), intersection (I), or union (U)] on a finite lattice of side length L at concentration p of occupied sites. The functions RLX(p) are obtained by numerical simulations. Then, (1) effective percolation thresholds pcX,r(L) are estimated from the condition RLX(p)=r (where r is a parameter ranging between 0 and 1); and (2) the percolation threshold in the thermodynamic limit is determined by extrapolating the effective percolation thresholds to infinite L. The methodology presented herein contains and extends the classical scheme developed by Yonezawa, Sakamoto and Hori, offering a more complete and versatile theoretical/simulation framework to calculate percolation thresholds. The approach is validated by successfully comparing with well-known results obtained for the problem of random site percolation on square lattices. Taking advantage of the behavior of the curves of pcX,r(L) for different values of the parameter r, more general scaling relations are proposed for the effective percolation thresholds. Finally, the technique is applied to model experimental data of clogging transitions in a two-dimensional silo with a vibrated base.

Collisional regime during the discharge of a two-dimensional silo.

The present work reports an investigation into the collisional dynamics of particles in the vicinity of the outlet of a two-dimensional silo using molecular dynamics simulations. Most studies on this granular system focus in the bulk of the medium. In this region, contacts are permanent or long-lived, so continuous approximations are able to yield results for velocity distributions or mass flow. Close to the exit, however, the density of the medium decreases and contacts are instantaneous. Thus, the collisional nature of the dynamics becomes significant, warranting a dedicated investigation as carried out in this work. More interesting, the vicinity of the outlet is the region where the arches that block the flow for small apertures are formed. It is found that the transition from the clogging regime (at small apertures) to the continuous flow regime is smooth in collisional variables. Furthermore, the dynamics of particles as reflected by the distributions of the velocities is as well unaffected. This result implies that there is no critical outlet size that separates both regimes, as had been proposed in the literature. Instead, the results achieved support the alternative picture in which a clog is possible for any outlet size.

Granular mixtures discharging through a silo with lateral orifice

Here, we studied the flow dynamics of a mixture of dumbbells and disks flowing through an orifice situated on the lateral wall of a two-dimensional silo using the discrete element method. When two constituent parts of a dumbbell are simultaneously in contact with either a disk or one part of another dumbbell, it hinders the relative motion of both the particles. An increase in the fraction of dumbbells increases the number of contacts exhibiting the aforementioned mechanism, thus increasing the dynamic friction. This leads to a decrease in the flow rate with an increase in the fraction of dumbbells. To relate the flow rate with the fraction of dumbbells and the orifice width, we proposed modified Beverloo's law scaling. Moreover, we presented coarse-grained flow fields, which reveal the presence of stagnant zones that hinder the free flow of particles adjacent to them.

Experimental study on the discharge flow rate of binary mixture in a two-dimensional silo

In this work, the discharging process of the binary mixture composed of sphere and sphere-paired particles in a two-dimensional silo was studied. High-speed camera and self-developed particle tracking velocimetry (PTV) program were used to capture the flow behaviors of all particles. The key parameters of mixed flow, including coordination number, horizontal displacement and mechanical energy loss in the discharge process, were highlighted. It was found that the increase of sphere-paired particles can decrease the average coordination number of particles during the discharging process. The analysis about the loss of mechanical energy and the horizontal displacement of particles indicated that sphere-paired particles preferentially squeezed out sphere particles from fast flow field above the outlet. Moreover, an empirical formula was proposed to assess the influence of the proportion of sphere-paired particles on the discharge flow rate. Sphere-paired particles tended to hinder the discharging process, which was caused by the rotation around their centroids and the angular deflection close to angle of the hopper.

An experimental investigation on jamming and critical orifice size in the discharge of a two-dimensional silo with curved hopper

During the silo discharging progress, if the orifice size is not large enough, the orifice of the silo would be jammed by granules due to the formation of an arch. The silo with a curved hopper can effectively prevent jamming because the contraction rate is constant. To clarify the critical orifice size of the curved hopper which does not affect the outpouring of cohesionless granules,the jamming at different orifice sizes was analyzed. It was found that the change in jamming probability is similar at different contraction rates. Moreover, the relationship between the structure of the arch and the critical orifice size was clarified. Later on, a prediction model for the critical orifice size of the curved hopper was proposed. Finally, the effectiveness of the model was verified by experiments. Overall, the results are useful for understanding the jamming during discharging andfurther providing guidelines for the appropriate design of curved hoppers.

Nonergodicity in silo unclogging: Broken and unbroken arches.

We report an experiment on the unclogging dynamics in a two-dimensional silo submitted to a sustained gentle vibration. We find that arches present a jerking motion where rearrangements in the positions of their beads are interspersed with quiescent periods. This behavior occurs for both arches that break down and those that withstand the external perturbation: Arches evolve until they either collapse or get trapped in a stable configuration. This evolution is described in terms of a scalar variable characterizing the arch shape that can be modeled as a continuous-time random walk. By studying the diffusivity of this variable, we show that the unclogging is a weakly nonergodic process. Remarkably, arches that do not collapse explore different configurations before settling in one of them and break ergodicity much in the same way than arches that break down.

Friction-mediated flow and jamming in a two-dimensional silo with two exit orifices.

We show that the interparticle friction coefficient significantly influences the flow and jamming behavior of granular materials exiting through the orifice of a two-dimensional silo in the presence of another orifice located in its vicinity. The fluctuations emanating from a continuous flow through a larger orifice results in an intermittent flow through the smaller orifice consisting of sequential jamming and flowing events. The mean time duration of jammed and flow events, respectively, increase and decrease monotonically with increasing interparticle friction coefficient. The frequency of unjamming instances (n_{u}), however, shows a nonmonotonic behavior comprising an increase followed by a decrease with increasing friction coefficient. The decrease on either side of the maximum, then, represents a system moving progressively towards a permanently jammed or a permanently flowing state. The overall behavior shows a systematic dependence on the interorifice distance, which determines the strength of the fluctuations reaching the smaller orifice leading to unjamming instances. The probability distributions of jamming and flowing times are nearly similar for different combinations of friction coefficients and interorifice distances studied and, respectively, exhibit exponential and power-law tails.

Effect of hopper angle on granular clogging.

We present experimental results of the effect of the hopper angle on the clogging of grains discharged from a two-dimensional silo under gravity action. We observe that the probability of clogging can be reduced by three orders of magnitude by increasing the hopper angle. In addition, we find that for very large hopper angles, the avalanche size (〈s〉) grows with the outlet size (D) stepwise, in contrast to the case of a flat-bottom silo for which 〈s〉 grows smoothly with D. This surprising effect is originated from the static equilibrium requirement imposed by the hopper geometry to the arch that arrests the flow. The hopper angle sets the bounds of the possible angles of the vectors connecting consecutive beads in the arch. As a consequence, only a small and specific portion of the arches that jam a flat-bottom silo can survive in hoppers.

Clogging in two-dimensions: effect of particle shape

We study the discharge of a two-dimensional silo through small openings via discrete element simulations. We consider the effect of the shape of the grains on the clogging of the flow, focusing on regular polygons and disks. We show that the clogging probability presents a non-linear response as a function of the number of vertexes of the polygons. Also, the clogging probability is strongly dependent on the protocol used to trigger the flow after each clog. We found that the size and shape of the blocking arches display a much larger variability for some polygons, particularly for squares, than for disks.

Slow relaxation dynamics of clogs in a vibrated granular silo.

We experimentally explore the vibration-induced unclogging of arches halting the flow in a two-dimensional silo. The endurance of arches is determined by carrying out a survival analysis of their breaking times. By analyzing the dynamics of two morphological variables, we demonstrate that arches evolve toward less regular structures and it seems that there may exist a certain degree of irregularity that the arch reaches before collapsing. Moreover, we put forward that σ (the standard deviation of all angles between consecutive beads) describes faithfully the morphological evolution of the arch. Focusing on long-lasting arches, we study σ calculating its two-time autocorrelation function and its mean-squared displacement. In particular, the apparent logarithmic increase of the correlation and the decrease of the mean-squared displacement of σ when the waiting time is increased reveal a slowing down of the dynamics. This behavior is a clear hallmark of aging phenomena and confirms the lack of ergodicity in the unclogging dynamics. Our findings provide new insights on how an arch tends to destabilize and how the probability that it breaks with a long sustained vibration decreases with time.

Stress distribution in two-dimensional silos.

Simulations of a polydispersed two-dimensional silo were performed using molecular dynamics, with different numbers of grains reaching up to 64 000, verifying numerically the model derived by Janssen and also the main assumption that the walls carry part of the weight due to the static friction between grains with themselves and those with the silo's walls. We vary the friction coefficient, the radii dispersity, the silo width, and the size of grains. We find that the Janssen's model becomes less relevant as the the silo width increases since the behavior of the stresses becomes more hydrostatic. Likewise, we get the normal and tangential stress distribution on the walls evidencing the existence of points of maximum stress. We also obtained the stress matrix with which we observe zones of concentration of load, located always at a height around two thirds of the granular columns. Finally, we observe that the size of the grains affects the distribution of stresses, increasing the weight on the bottom and reducing the normal stress on the walls, as the grains are made smaller (for the same total mass of the granulate), giving again a more hydrostatic and therefore less Janssen-type behavior for the weight of the column.

Flow and clog in a silo with oscillating exit.

When grains flow out of a silo, flow rate W increases with exit size D. If D is too small, an arch may form and the flow may be blocked at the exit. To recover from clogging, the arch has to be destroyed. Here we construct a two-dimensional silo with movable exit and study the effects of exit oscillation (with amplitude A and frequency f) on flow rate, clogging, and unclogging of grains through the exit. We find that, if exit oscillates, W remains finite even when D (measured in unit of grain diameter) is only slightly larger than one. Surprisingly, while W increases with oscillation strength Γ≡4π^{2}Af^{2} as expected at small D, W decreases with Γ when D≥5 due to induced random motion of the grains at the exit. When D is small and oscillation speed v≡2πAf is slow, temporary clogging events cause the grains to flow intermittently. In this regime, W depends only on v-a feature consistent to a simple arch breaking mechanism, and the phase boundary of intermittent flow in the D-v plane is consistent to either a power law: D∝v^{-7} or an exponential form: D∝e^{-D/0.55}. Furthermore, the flow time statistic is Poissonian whereas the recovery time statistic follows a power-law distribution.

Role of particle size in the kinematic properties of silo flow.

We experimentally analyze the effect that particle size has on the mass flow rate of a quasi two-dimensional silo discharged by gravity. In a previous work, Janda etal. [Phys. Rev. Lett. 108, 248001 (2012)PRLTAO0031-900710.1103/PhysRevLett.108.248001] introduced a new expression for the mass flow rate based on a detailed experimental analysis of the flow for 1-mm diameter beads. Here, we aim to extend these results by using particles of larger sizes and a variable that was not explicitly included in the proposed expression. We show that the velocity and density profiles at the outlet are self-similar and scale with the outlet size with the same functionalities as in the case of 1-mm particles. Nevertheless, some discrepancies are evidenced in the values of the fitting parameters. In particular, we observe that larger particles lead to higher velocities and lower packing fractions at the orifice. Intriguingly, both magnitudes seem to compensate giving rise to very similar flow rates. In order to shed light on the origin of this behavior we have computed fields of a solid fraction, velocity, and a kinetic-stress like variable in the region above the orifice.

Influence of particle size in silo discharge

Recently Janda et al. [Phys. Rev. Lett. 108 , 248001 (2012)] reported an experimental study where it was measured the velocity and volume fraction fields of 1 mm diameter stainless steel beads in the exit of a two-dimensional silo. In that work, they proposed a new expression to predict the flow of granular media in silos which does not explicitly include the particle size as a parameter. Here, we study if effectively, there is not such influence of the particle size in the flux equations as well as investigate any possible effect in the velocity and volume fraction fields. To this end, we have performed high speed motion measurements of these magnitudes in a two-dimensional silo filled with 4 mm diameter beads of stainless steel, the same material than the previous works. A developed tracking program has been implemented to obtain at the same time both, the velocity and volume fraction. The final objective of this work has been to extend and generalize the theoretical framework of Janda et al. for all sizes of particles. We have found that the obtained functionalities are the same than in the 1 mm case, but the exponents and other fitting parameters are different.

Dynamics of breaking arches under a constant vibration

Granular flow through an orifice can be suddenly halted by the formation of arches in the vicinity of the outlet, which are stable under the action of gravity. They may be broken when an external driving (for instance, vibration) is applied. With the aim of shedding light on the dynamics of arch destruction, we built an experiment consisting of a vertical two-dimensional silo filled with monodisperse beads, to which a constant vibration is applied. It was previously found that an important parameter to predict the robustness of the arch is the angle between consecutive beads. We focus on long-enduring arches and study the angles among the beads along time. We have found that in many cases the dynamics of the largest angle determines the breaking of the arch; it does not only determine where the “weakest link” is, but also the process that leads to the final destabilization. This is interesting because it can provide information about whether the flow will resume in a well-defined time or not, which is especially useful for industrial processes that have to constantly deal with the possible emergence of clogs.

Optimal designs for model discrimination and fitting for the flow of particles

During the discharge of a two-dimensional silo, the flow of grains through an opening is arrested if the size of the outlet is not large enough. In the outpouring of grains, jamming occurs due to the formation of an arch at the outlet. After breaking the arch, the grains fall until a new arch is formed. Several models have been proposed to explain this process. The goal of this work is twofold. First, we developed a comparative study of the models proposed by Janda et al. (Europhys Lett 84(4):44002-1–44002-6, 2008) and To (Phys Rev E 71(6):060301-1–060301-4, 2005). We have computed D-optimal and c-optimal designs for the most important parameter of the models. Secondly, using the criterion of KL-optimality developed by Lopez-Fidalgo et al. (J R Stat Soc Ser B 69(2):231–242, 2007), optimal designs for discriminating between these models have been computed.

Stress profile in a two-dimensional silo: Effects induced by friction mobilization.

The effects of friction mobilization on the stress profile within a two-dimensional silo are investigated via simulations of discrete elements. Friction mobilization is driven by cyclic vertical displacement of the sidewalls. Two regimes have been observed for small filling height, with stress profiles identified as saturated (Janssen's profile) and exponentially growing. The transition between these regimes is denoted by an almost linear stress profile, similar to that of a hydrostatic system, with a significantly greater characteristic height compared to the height of the column of grains. For tall columns, the process of friction inversion is more complex. A partial inversion of friction mobilization is observed when the motion is reversed from upward to downward, which results in two coexisting zones of opposite mobilization. These zones are separated by a wide compaction front with a gradual upward progression sustained by the displacement of the walls. Conversely, if the motion is reversed, the two opposing friction mobilization zones retract, the transition zone becomes smooth, and the system rapidly transforms from two coexisting mobilization states to a Janssen-like regime. In both regimes, the general characteristics from the resulting stress profiles are depicted by generalizing Janssen's equation to include partial mobilization through the varying effective friction coefficient along the silo walls.

Optimal Experimental Designs in the Flow Rate of Particles

This article focuses on analyzing the process of jam formation during the discharge by gravity of granular material stored in a two-dimensional silo. The aim of the article is two-fold. First, optimal experimental designs are computed, in which four approaches are considered: D-optimality, a combination of D-optimality and a cost/gain function, Bayesian D-optimality, and sequential designing. These results reveal that the efficiency of the design used by the experimenters can be improved dramatically. A sensitivity analysis with respect to the most important parameter is also performed. Second, estimation of the unknown parameters is done using least squares, that is, assuming normality, and also via maximum likelihood assuming the exponential distribution. Simulations for the designs considered in this article show that the variance, the mean squared error, and the bias of the estimators using maximum likelihood are in most cases lower than those using least squares. Supplementary materials for this article are available online.

Flow of magnetic repelling grains in a two-dimensional silo

During a typical silo discharge, the material flow rate is determined by the contact forces between the grains. Here, we report an original study concerning the discharge of a two-dimensional silo filled with repelling magnetic grains. This non-contact interaction leads to a different dynamics from the one observed with conventional granular materials. We found that, although the flow rate dependence on the aperture size follows roughly the power-law with an exponent 3/2 found in non-repulsive systems, the density and velocity profiles during the discharge are totally different. New phenomena must be taken into account. Despite the absence of contacts, clogging and intermittence were also observed for apertures smaller than a critical size determined by the effective radius of the repulsive grains. Received: 10 July 2015, Accepted: 7 August 2015; Edited by: L. A. Pugnaloni; Reviewed by: J. R. Darias, Universidad Simón o Bolívar, Caracas, Venezuela; DOI: http://dx.doi.org/10.4279/PIP.070013Cite as: G Lumay, J Schockmel, D Henández-Enríquez, S Dorbolo, N Vandewalle, F Pacheco-Vázquez, Papers in Physics 7, 070013 (2015)This paper, by G Lumay, J Schockmel, D Henández-Enríquez, S Dorbolo, N Vandewalle, F Pacheco-Vázquez, is licensed under the Creative Commons Attribution License 3.0.