Quantifying silo flow using MRI velocimetry for testing granular flow models

In this work we present experimental results of the gravity-driven discharge of poppy seeds from three-dimensionally (3D) printed silos. The velocity fields of the flowing poppy seeds are measured using magnetic resonance imaging (MRI) velocimetry techniques. Crucially, this approach allows the velocity field to be determined throughout the flow domain, unlike visual techniques such as particle image velocimetry and related methods where only the flow at or near the wall is accessible. We perform the experiment three times; with 3D-printed silos of cone half angles ${30}^{\circ }$ and ${50}^{\circ }$, respectively, and then repeat the ${30}^{\circ }$ silo experiment but with a layer of poppy seeds glued to the silo wall to create a “rough wall” condition. In our experiments, we observe and quantify velocity fields for three well-known granular flow regimes; mass flow, funnel flow, and rat-holing. The results of the experiments are compared to equivalent output of numerical simulations. In this mathematical model, the well-known $\mu \left(I\right)$ friction law is used to define an effective granular viscosity, and the flow is solved using a standard Navier-Stokes type solver. While the results are generally encouraging, it is noted that some aspects of the model are lacking and should be improved; in particular, the rat-holing effect observed in one of the MRI experiments was not predicted by the model, nor was the exact volumetric flow rate from any of the silos. Suggestions for model improvement are discussed.

Figure 1

Scanning electron microscope images of a sample of poppy seeds. It is apparent from the image that the seeds are nonspherical with a kidney shape. The surface of the seeds is also seen to be textured. A scale is included at the bottom of each image. (a) An image of multiple poppy seeds. (b) A close up of a single poppy seed.

Figure 2

A sketch of the piping and silo in the experimental set-up (not to scale). (a) The system is loaded from above. The seeds flow through the largest pipe 1 into the more narrow pipe 2 through the test silo section and out through pipe #3. (b) A close up of the test silo section.

Figure 3

The log of the magnitude of the vertical component of velocity ($v$) is plotted for each of the three silos. Mass flow is observed in the ${30}^{\circ }$ silo, funnel flow in the ${50}^{\circ }$, and rat-holing in the ${30}^{\circ }$ silo with rough walls (with particles glued to the silo wall). Yellow regions indicate rapid flow, while purple/blue areas indicate slow to stagnant zones.

Figure 4

The log of the magnitude of the horizontal component of velocity ($u$) for the ${30}^{\circ }$ and ${50}^{\circ }$ silos.

Figure 5

The volumetric flow rate, $Q\left(z\right)$, for each of the three silo experiments as a function of height above the silo opening.

Figure 6

The vertical velocity MRI measurements (solid circles) compared with those predicted by the numerical model (lines) for the ${30}^{\circ }$ silo.

Figure 7

The horizontal velocity MRI measurements (solid circles) compared with those predicted by the numerical model (lines) for the ${30}^{\circ }$ silo at the same locations as in the vertical velocity figure.

Figure 8

A comparison of the normalized vertical velocity measured along the axial centerline of the silo compared with that predicted by the model for the ${30}^{\circ }$ silo.

Figure 9

The vertical velocity MRI measurements (solid circles) compared with those predicted by the numerical model (lines) for the ${50}^{\circ }$ silo.

Figure 10

The horizontal velocity MRI measurements (solid circles) compared with those predicted by the numerical model (lines) for the ${50}^{\circ }$ silo at the same locations as in the vertical velocity figure.

Figure 11

A comparison of the normalized vertical velocity measured along the axial centerline of the silo compared with that predicted by the model for the ${50}^{\circ }$ silo.

Figure 12

The vertical velocity MRI measurements (solid circles) compared with those predicted by the numerical model (lines) for the ${30}^{\circ }$ silo with roughened walls.

Figure 13

A comparison of the normalized vertical velocity measured along the axial centerline of the silo compared with that predicted by the model for the ${30}^{\circ }$ silo with roughened walls.

Figure 14

Sensitivity analysis of the numerical model to parameters $I_0$ and ${\mu }_2-{\mu }_1$. The contour plots display the value of the slope found by performing a least-squares linear regression between the experimental and numerical normalized vertical velocity data. The left graph is the analysis for the ${30}^{\circ }$ silo and the right for the ${50}^{\circ }$ one. The red dot in the plots indicated the value of the parameters used in the current work, while the fine red line in the left plot is the contour of slope $=1$ (indicating a perfect fit of the numerical to experimental data).