Granular Drainage from a Narrow Rectangular Conduit

Granular materials are aggregate of discrete particulate matter where the size of the individual particles is larger than 100 µm¹. Sand, grain, corn, sawdust, rock, snow are common examples of granular material. These are frequently encountered in industrial and natural processes and are the second largest used material in industry after water¹.

Granular flow constitutes one of the subsections of multiphase flow. Flow occurs when the particles are set into motion. Dry granular materials during flow may behave like a solid, a liquid or a gas and more intriguingly, all three behaviors may be present simultaneously. A granular aggregate is assumed to be in a solid phase, when the relative velocity of particle is negligible and is often represented by plasticity models. This phase is also known as quasi-static region². A gas like behaviour described by kinetic theory of gases is observed when the medium is very sparse, the distance between particles are more and the particles interact mainly by binary collision. This is known as the grain inertia flow regime^2,3. In between solid and gaseous phase, a dense regime exists where the inter-particle gap is less and the medium flows like liquid. In this intermediate transitional regime, particles interact both by collision and friction^4,5. Dense granular flow is abundant in nature and is a very common mode of granular motion in diverse industrial applications. Such flows, exhibited during gravity drainage from closed conduits can present a challenging problem of hydrodynamics. Though the drainage of liquid from various systems have been studied extensively^6–11, apart from silo discharge, the drainage of granular material is yet grossly unexplored.

During the rapid motion of granular material through confined environment, some unique phenomena are observed. Boycott effect is one of them. The granular material flows faster in an inclined tube than in a vertical tube¹². Another interesting phenomenon is the formation of voids or bubbles in vertical and inclined channels during rapid downward motion of grains. This has been reported for circular tubes filled with granular material when the medium is vibrated^13,14 or simply during gravity driven flow when a semi filled closed tube is suddenly inverted^15,16. Nevertheless, not much is known about dense granular flow in non-circular conduits.

The present study reports experiments to investigate drainage from the open end of a closed top rectangular conduit of small cross-section. Such flows (through narrow channel) can present a pseudo-2-D case easy to analyze. Further due to confinement, sidewall friction may play a vital role and the flow behavior can be complex. However, there is not much experimental evidence on granular drainage from an inclined closed top rectangular channel. The 120 cm long conduit, 2.5 cm high has a 0.4 cm gap between the two sidewalls. The flow is observed from the transparent (Perspex) side wall for different inclination angles (θ) with respect to horizontal.

With an aim to study the effect of size and shape of the grains on the dynamics of discharge, both natural sand (irregular shape) and glass beads (spherical shape) with different grain size (d) have been taken as the granular medium. The diameter range of glass beads is 180-300 µm, 300-425µm and 400-600 µm and the diameter range of sand particles is 150-250 µm, 250-300µm and 300-500 µm. Before each experiment, we fill the conduit completely with grains. To ensure that the compaction is uniform and the initial conditions are same for each experiment, we tap the sidewalls and slowly turn the front end down several times during the filling process. The sliding gate is then opened instantaneously to initiate draining. The granular motion within the conduit is recorded using a high speed camera (MotionBLITZ cube 7) and the flow phenomena is analyzed using image processing (Image pro-plus^® and Matlab-2013^®).

We observe that the material flows down creating an uphill front or avalanche¹⁷. Depending on θ, different behaviors of the avalanche are observed. When θ < θ_r (angle of repose), granular drainage resembles dam breakage^18,19. However, the developed avalanche is short-lived and transforms into a stationary inclined surface with no further discharge of grains. The final erosion free surface profiles are self-similar in nature. For θ slightly above θ_r, the avalanche starts moving from the lower open end to the closed upper end but often the motion is intermittent. Probably the friction at the side and top wall of the conduit prevents continuous sliding of the grains in the avalanche even at θ>θ_r.

In general, when θ exceeds θ_r by 5°-6° for sand, the avalanche continuously moves towards the upper end. With increase in θ, the grain velocity increases and the higher kinetic energy results in a smooth flow by overcoming intergranular friction as well as friction at the different walls. Spatial and temporal variation of the drainage pattern is observed during the emptying process. We observe a zone with constant flow rate, termed as the quasi-steady zone. In this zone, a layer of material slides over stagnant particles and the moving layer have a wedge like shape. The free surface exhibits self-similarity. The time of drainage exhibits power law dependence with θ and the avalanche velocity and static layer erosion rate exhibits power law dependence with inclination angle (θ), grain size (d) and channel height (W). This can be attributed to the ubiquitous tendency of natural phenomenon to follow hidden laws of self-similarity.

At higher θ, rapid movement of grains also results in merging of voids. There is fast growth of voids and they try to move towards the channel roof, often coalescing with the neighboring voids. Ultimately these coalesced voids or bubbles reach the channel roof and start translating uphill, i.e. in a direction opposite to the direction of grain motion. These coalesced voids resemble the elongated bubbles rising through an inclined tube in gas-liquid systems. The propagation of bubbles inside granular medium is mainly governed by the transition between dynamic and static friction of the medium above the bubble¹⁶. The velocity of bubble inside granular medium increases with increase in conduit inclination till a critical θ close to 75°-80° and then decreases with further increase in θ close to 90°.

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