On the dual effect of obstacles in preventing silo clogging in 2D

The placement of obstacles has been suggested as a strategy to prevent clog formation when discrete systems flow through constrictions. Originally proposed for enhancing pedestrian evacuation through small exits, obstacle placement has been widely reproduced in models, yet robust experimental evidence in pedestrian dynamics remains contradictory. Conversely, experiments with animals and granular matter show clearer benefits. Granular silo discharge offers a simple platform for detailed analysis due to inert grain interactions and high repeatability. Prior silo experiments showed that a circular obstacle optimally placed above the orifice can reduce clogging probability by over two orders of magnitude without altering flow rate; too close an obstacle increases clogging by enabling arches between the obstacle and bottom. A key mechanism previously identified is the creation of an empty region below the obstacle, which destabilizes arches; increased ratios of upwards particle motions and enhanced horizontal granular temperature near the outlet have been linked to clogging reduction. Existing explanations emphasize dynamic effects; thus, one might expect diminished obstacle impact as grain motion is minimized. Here, using a conveyor belt extractor to achieve quasi-static discharge, we show that even in this limit an obstacle reduces clogging by introducing anisotropy in the contact fabric tensor above the outlet, in addition to dynamic effects at higher extraction rates.

Prior studies reported obstacle-facilitated flow improvements in models of pedestrian dynamics (e.g., Helbing et al.; Kirchner et al.; others), though experimental results for humans are mixed. Analogous benefits were experimentally observed in animals (sheep, mice) and in granular silo discharge (Zuriguel et al.; Lozano et al.). For silos, an obstacle placed at an optimal height reduces clogging probability drastically without affecting flow rate; if too close, arches form between obstacle and base, worsening clogging. Endo et al. showed that obstacles shape the flow field and can increase horizontal granular temperature. Theoretical and experimental work on silo clogging has addressed jamming transitions, avalanche size distributions, and the fraction of clogging configurations sampled during flow. Conveyor-belt driven silo discharge allows decoupling geometric and kinematic contributions (Gella et al.), revealing regimes from belt-controlled to free-discharge-like saturation. Studies on pressure, particle deformability, and shape anisotropy indicate that soft or non-spherical particles can change clogging and arch stability mechanisms.

Experimental setup: A two-dimensional silo is formed by two parallel glass sheets (70 cm wide, 160 cm high), separated by 0.4 cm aluminum gauges (with thin cardboards) as lateral sides so that only a single layer of particles fits. The granular material consists of monodisperse AISI 420 stainless steel spheres, diameter d = 0.4000 ± 0.0002 cm, density ρm = 7.75 g/cm³. The bottom has two wedge-shaped stainless steel pieces whose separation defines the orifice size D (experiments used D = 1.33 cm and 1.53 cm). A conveyor belt is placed below the orifice, with a fixed separation of 0.67 ± 0.03 cm from the silo bottom; belt velocities vb range 0.1–16 cm/s. A circular methacrylate obstacle of 4 cm diameter is glued above the exit, centered laterally, at a vertical distance 1.56 ± 0.03 cm from its lowest edge to the orifice center. A hopper at the top enables manual filling. Measurement protocol for clogging probability: An automated protocol synchronizes a balance and bucket at the belt end, a web camera monitoring the outlet region, and a vibration system to break clogs. After filling, the belt runs at set velocity and the camera tracks pixel intensity below the arch-formation zone. A clog is deemed stabilized if the region remains empty (above a threshold) for 4 s. Upon clogging, the belt delivers discharged grains to the bucket, stops, the avalanche size s is recorded by mass, the shaker breaks the arch, and the belt restarts. Approximately 1000 avalanches are collected per configuration. The survival function ccdf(s) is exponential; ln[ccdf(s)] is fit linearly to obtain the negative slope p_c, the probability a bead clogs while crossing the orifice. Kinematic data: High-speed videos (Photron FASTCAM-1024PCI, 500 fps, ~7 s each; three videos per configuration) record particle motion near the orifice and obstacle with backlighting. Spatial resolution yields ~40 pixels per particle diameter. Custom tracking detects bead centers with subpixel accuracy; velocities are computed as frame-to-frame displacements over 1/500 s. Mean velocities are averaged over characteristic regions: (i) a blue box at the outlet to obtain mean exit velocity vo, and (ii) a 1.2 × 1.2 cm² red box above the orifice and below the obstacle to compute horizontal and vertical velocity distributions, average |vx| and |vz|, the proportion of upward motions Vz−/Vz, and the 2D solid fraction φ. Statics and fabric tensor: To characterize quasi-static microstructure, particle–particle contacts are analyzed in temporally intermittent flow. The kinetic energy Ek of all grains in view is monitored; snapshots are sampled each time grains rearrange and stop without forming a stable clog. Contacts are defined for pairs with center separations < 1.05 d. For each particle i, the second-order contact fabric tensor is computed as fαβ = (1/Ni) Σj n_ij^α n_ij^β, with n_ij the normalized branch vector from i to contacting neighbor j. Coarse-grained continuous fields fαβ(r,t) are built using a Gaussian kernel φ(r − ri(t)) = exp(−|r − ri|²/2w²) with w = d/2, truncated beyond 3w. Fields are averaged over configurations to obtain maps of fxx, fzz, and their rescaled difference (fxx − fzz)/(fxx + fzz). Dynamic regime characterization: The mean particle velocity at the outlet vo (measured in the blue box) parameterizes system dynamics and is related nonlinearly to belt speed vb, exhibiting saturation at higher vb (free-discharge-like regime), with lower saturation for smaller D.

- Obstacle reduces clogging probability across all extraction rates, including the quasi-static limit. For both D = 1.33 cm and D = 1.53 cm, p_c decreases with vo and is consistently lower with an obstacle than without it. - The empirical relation p_c = (a + b vo)^(−D/4) fits the data. Fit parameters: with obstacle a = 1.406 ± 0.004, b = 0.0153 ± 0.0004 s cm−1; without obstacle (from prior work) a = 1.33, b = 0.0128 s cm−1. Thus, the obstacle modifies both the quasi-static (geometric) parameter a and the dynamic parameter b. - At fixed D, the obstacle reduces p_c to less than 50% of the no-obstacle values across vo, and notably remains lower as vo → 0 (quasi-static regime). In quasi-static experiments at vo ≈ 0.1 cm/s, p_c vs D/d follows Eq. (1) with v = 0 and a = 1.406 (obstacle), confirming a persistent static effect. - Kinematics in the outlet vicinity (red box) with obstacle: distributions of horizontal velocity vx broaden; average |vx| increases with vo relative to no obstacle, while average |vz| decreases, consistent with particle detouring and reduced vertical motion. Vertical velocity pdf tails extend more to negative values, indicating enhanced upward motions. - Proportion of upward displacements Vz−/Vz: without obstacle, rises markedly as vo decreases (approaching ~15% at lowest vo), but is negligible at high vo. With obstacle, the entire curve shifts upward (more upwards motions at high vo) but converges toward the no-obstacle values as vo → 0, indicating diminishing kinematic influence in the quasi-static limit. - Solid fraction φ in the arch-formation region (between obstacle and orifice) is reduced by about 0.1 due to the obstacle, relatively independent of vo. This creates an empty gap below the obstacle, consistent with reduced contact probability and/or pressure, aiding arch destabilization. - Static contact network anisotropy (quasi-static regime): Maps show increased horizontal fabric component fxx and decreased vertical component fzz below the obstacle compared to the no-obstacle case. The rescaled anisotropy (fxx − fzz)/(fxx + fzz) is near zero above the outlet without obstacle (isotropic) but clearly positive with obstacle, indicating horizontally biased contacts. This anisotropy implies arches are loaded more from the sides than vertically, likely reducing their stability. - Dynamics of discharge: vo increases roughly linearly with vb at low speeds (belt-controlled regime) and saturates at higher vb (free-discharge-like regime), with lower saturation for smaller D, consistent with known D^1/2 scaling.

The study asks how an obstacle above a silo exit affects clogging as grain dynamics are progressively reduced to the quasi-static limit. Results demonstrate a dual mechanism. At higher extraction rates, the obstacle alters kinematics near the outlet: it induces detouring (increased |vx|, decreased |vz|), lowers local packing fraction by creating an empty gap, and enhances upward particle motions. These features hinder formation and stabilization of clogging arches; frequent horizontal collisions and reduced vertical confinement lead to upward ejections that break nascent arches. As extraction becomes quasi-static, kinematic differences between obstacle and reference diminish, yet clogging reduction persists. Static analysis reveals that the obstacle imposes a pronounced anisotropy in the contact fabric above the orifice (fxx > fzz), effectively screening vertical load and promoting horizontally oriented contacts. Assuming force orientations align with contact networks, arches (typically semicircular on average) become predominantly side-loaded and thus less stable, lowering clogging probability. Both mechanisms are captured by a single empirical relation for p_c in terms of vo and D, but with different parameters a and b when an obstacle is present, indicating separate geometric (quasi-static) and dynamic contributions. These insights help reconcile differing reports on obstacle efficacy in other systems (pedestrians, animals, active matter) by emphasizing the role of dynamics (limit velocities vs gravity-driven) and local microstructure.

Placing a circular obstacle above a 2D silo exit reduces clogging through two mechanisms whose relative importance depends on extraction rate. In the quasi-static regime, the obstacle induces an anisotropic contact fabric (fxx > fzz) above the orifice, diminishing vertical contacts and destabilizing arches. At higher extraction rates, the obstacle creates a low-density gap and enhances horizontal motion, promoting upward ejections that prevent arch stabilization. The clogging probability is described by p_c = (a + b vo)^(−D/4), with obstacle-specific parameters (a, b) distinct from the no-obstacle case, suggesting dependence on obstacle properties and placement. Future work should quantify how obstacle geometry and position tune these parameters, and extend to particles with complex shapes and deformability to assess generality and pressure response.

- Geometry and dimensionality: Experiments are in a 2D monolayer silo with hard, monodisperse spherical steel beads; 3D systems or polydisperse, non-spherical, or soft particles may behave differently. - Obstacle configuration: A single obstacle shape (4 cm diameter disk) and a fixed position (1.56 cm above orifice center) were tested; other shapes/positions may alter outcomes. - Orifice sizes and dynamic range: Only two orifice sizes (D = 1.33, 1.53 cm) and the available belt speed range were explored. - Region of analysis: Kinematic and fabric metrics were averaged over specific windows; gradients near boundaries may affect quantitative values. - Assumption linking contact orientation to force orientation is inferred rather than directly measured; direct force measurements were not performed.