Powders and Grains 2001, Kishino (ed.), p. 399-402 Dense granular flows in a vertical chute François Chevoir, Michaël Prochnow, Pascal Moucheront, Fréderic da Cruz, François Bertrand, Jean-Pierre Guilbaud, Philippe Coussot, Jean-Noël Roux LMSGC, UMR LCPC-CNRS, 2 allée Kepler, 77 420 Champs sur Marne, France ABSTRACT : Dense flows of dry grains in a vertical chute controlled by the orifice consist in a steady and uniform plug flow with a thin shear layer near the rough walls. Using magnetic resonance imaging, we present detailed measurements of velocity profiles, and discuss the influence of the shear rate and of the wall roughness in the shear zone. 1 INTRODUCTION Apart from being a usual handling device for bulk materials in the industry, the vertical chute is a simple geometry on which to study dense rapid flows of dry non cohesive grains (with lasting or collisional frictional contacts, and solid fraction near close packing). Far above the orifice which controls the flow rate, a steady and uniform plug flow is observed with a thin shear layer near the rough walls (Nedderman & Laohakul 1980, Natarajan et al. 1995, Pouliquen & Gutfraind 1996). The understanding of this shear layer is not satisfying in the frame of elasto-plastic models inspired from soil mechanics and relevant for slow deformations (Nedderman 1992). Cosserat theories (Tejchman & Gudehus 1993, Mohan et al. 1999), and models emphasizing the fluctuations of stress (Pouliquen & Gutfraind 1996) or strain (Savage 1998) have been proposed. Validation of these rheological models requires experimental measurements (Tüzün et al. 1982, Menon & Durian 1997) as well as numerical simulations (Pouliquen & Gutfraind 1996, Denniston & Li 1999). The interest of magnetic resonance imaging (MRI) for granular materials has been shown in the last ten years (rotating drum (Nakagawa et al. 1993), Couette shear cell (Mueth et al. 2000), mixing (Porion et al. 2000) or convectiondiffusion under vibration (Ehrichs et al. 1995)). Using MRI velocimetry, we present detailed measurements of velocity profiles inside such flows. tween two glass walls 100 mm apart and two rough walls distant of L = L*d. This quasi two-dimensional geometry allows measurement of solid fraction by gamma-densitometry and velocity by imagery at the glass wall, through correlation of reflects on the grain of a stroboscopic flash synchronized with a high-speed camera (Prochnow 2001). MRI measurements have been performed in a rough cylindrical chute of internal diameter 54 mm. The grains are respectively glass beads of mean diameter 1.5 mm (mass density 2700 kg.m-3) and mustard oilseeds (rich in protons responsible for the magnetic resonance signal) nearly spherical and of mean diameter 1.3 mm (mass density 1200 kg.m-3). The roughness is made by gluing grains of diameter dR on the walls with a double sided adhesive tape, and is described by the ratio R = dR /d. In the cylindrical chute, the wall grains are mustard seeds (R = 1), small glass beads (0.8 mm, R = 0.6) and large glass beads (3 mm, R = 2.3). 2 EXPERIMENTAL SET UP Figure 1 shows our two vertical chutes 1600 mm long, fed with grains of diameter d by a hopper tank (0.25 m3). A first series of experiment was performed in a rectangular chute, where grains flow be- Figure 1 : Vertical chute in the MRI, with the velocity profile. Figure 2 : MRI horizontal cut of a piling of mustard seeds in a cylinder of diameter 40 mm. The flow rate is controlled by the width D = D*d of the orifice (circular or rectangular) according to Beverloo’s law (Brown & Richards 1970). The shear rate near the wall, adimensioned by the characteristic frequency (g/d)1/2 in the gravity field g, is deduced from the flow rate by considering the velocity profile as flat with a shear layer of around 6 grains (see below) and an average solid fraction equal to 0.6 (as measured by gamma-densitometry): γ * ≈ 7.1 -5 10 (D* – 2)5/2 for the circular channel, and γ * ≈ 3.6 10-3 (Dh* – 1,4)5/2 L*-1 where Dh* = 133 D*/(D*+67) is the hydraulic diameter of the orifice of the rectangular chute. In the circular chute, we note that when D* < 8 the flow stops, and when D* > 20 the flow is too rapid for the MRI technique used here. 3 MRI VELOCIMETRY The distinctive features of our MRI device are a vertical magnet, and a large internal channel (cylinder of diameter 20 cm) and field of observation (sphere of diameter 18 cm). The static magnetic field B0 is 0.5 T in this experiment. Figure 3: Phase pictures at rest and in flow (D* = 9.6). the encoding direction, it becomes possible to measure the full three-dimensional velocity field. The measurements are realized in longitudinal cut plane containing the vertical axis 15 cm long and 8 mm thick, at the center of the chute (far from the hopper tank and from the orifice). The acquisition time for such a two-dimensional cut is around ten minutes. Figure 3 compares pictures at rest and in flow. The grey level is proportional to the phase shift, which is clearly displayed by adding a constant phase gradient before Fourier transform (zebra imagery technique). By correlating vertical lines of these pictures at rest and in flow, the profiles of phase shift are measured in pixels. A calibration of the velocity scale is performed by translating vertically a cylinder full of mustard seeds at a wellknown velocity. The flow rate deduced by integration of this velocity profile is in good agreement with the flow rate measured at the exit. A careful average of the measurements performed in several cut planes (6 to 12) is performed, so that it becomes possible to slightly enhance the spatial resolution of the initial pictures (256 x 256 pixels). 4 VELOCITY PROFILES The tomographic principle of MRI makes it possible the measurement not only of the solid fraction of protons (Fig. 2) but also of their motion. We have used the spin echo sequence with an echo time of 30 ms, to be compared with the relaxation time of the mustard seeds T2 = 345 ms. Among the various techniques of MRI velocimetry (tagging, time of flight…), the phase method (Weeden et al. 1985, Fukushima 1999) relies on the phase shift of the transverse magnetization (the quantity measured in MRI) due to motion in a magnetic field gradient : this shift is simply proportional to the velocity component in the cut plane along the direction of the magnetic field gradient. Varying the cut plane and We first emphasize that for high shear rates ( γ * ≥ 0.1), the flow becomes unsteady (Prochnow 2001). Detailed measurements in the rectangular chute have shown that the grains undergo quasiperiodically (period around 50 ms) a succession of sudden accelerating and decelerating motion around the mean velocity, with a typical acceleration comparable to gravity. This phenomenon, reminiscent of "silo music" (Tejchman & Gudehus 1993), indicates strong fluctuations associated to the reorganisation of the contact network inside the flow (Pouliquen & Gutfraind 1996, Veje et al. 1999). Even if we could not measure it in the cylindrical chute, the same phe- 35 1.2 30 1.0 * D = 13.4 25 0.8 20 * V 0.6 * V 15 [mm/s] D = 11.5 0.4 10 * D = 9.6 5 0 -25 -20 -15 -10 -5 0 5 x [mm] 0.2 10 15 20 25 Figure 4 : Velocity profiles with error bars for R = 1 and three shear rates ( γ * ≈ 0.01, 0.02 and 0.03). nomenon was revealed by vibrations of the tube and of the air, and was responsible for weakening the MRI signal. Note that the rotation of the grains in the shear layer also disturb the measurement of the translation velocity. The velocity profiles showed in the following figures correspond to a plug flow at the centre of the chute with thin shear layers near the rough walls. The extreme value of the x-coordinate (25 mm in Fig. 4 or 0 in Fig. 5-7) corresponds to the extreme position of the center of the grains near the wall. Figure 4 shows complete velocity profiles, with their error bars for three different flow rates. The precision is of the order of ± 4 % for the maximum velocity. Figure 5 is a zoom in the shear zone, for various roughness and shear rates. The width of the shear layer is of the order of six grains, with a strong influence of the roughness in the very first layers. For R = 1, we observe that the first layer is trapped in the interstices of the roughness, whereas there is a 0.0 0 2 4 6 x/d 8 10 Figure 6 : Adimensioned velocity profiles (R = 1) for various shear rates in the cylindrical chute (thin solid line) and in the rectangular chute (thick solid line). significant sliding velocity for R = 0.6. It is also possible to distinguish the relative sliding of the two or three first layers close to the wall. Dividing the velocity by the maximum velocity (in the plug) helps to compare the shape of the profiles. Figure 6 shows that this shape does not depend on the shear rate, which would indicate a rate independent constitutive law in the range considered ( γ *between 0.05 and 0.03). We also compare these profiles with a velocity profile measured at the wall in the rectangular chute with a roughness made of the same glass beads in the same range of shear rate ( γ * ≈ 0.03). The agreement is fair, especially far from the wall. In the very first layers, we note a small sliding velocity in the rectangular chute. The influence of the roughness on the shape of the velocity profile is shown in Figure 7, for the same shear rate ( γ * ≈ 0.01). 1.2 15 1.0 0.8 10 V [mm/s] * D = 9.6 R = 0.6 R=1 R = 2.3 * V 0.6 5 0 0.4 0.2 0 1 2 x/d 3 4 5 Figure 5 : Velocity profiles in the shear layer. Influence of the shear rate (R = 1, D* = 7.7, 9.6, 11.5, 13.4 - solid line), and of the roughness (D* = 9.6, R = 0.6 - short dotted line, and D* = 9.6, R = 2.3 - dash dotted line). 0.0 0 2 4 6 8 10 x/d Figure 7: Adimensioned velocity profiles. Influence of roughness. We should now study these velocity profiles quantitatively, and compare them with the predictions of various models (Pouliquen & Gutfraind 1996, Mohan et al. 1999, Savage 1998), as well as with experimental models obtained in other shear geometries (Mueth et al. 2000, Veje et al. 1999). 5 CONCLUSION These preliminary results on a simple geometry are promising for more thorough study of granular flows in vertical chute. Other MRI techniques must be used to measure transients, solid fraction and velocity fluctuations. Truly three dimensional flows are now under study, especially the converging geometry of a hopper and the area of the orifice. 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