Silo discharge of binary granular mixtures

We present numerical and experimental results on the mass flow rate during the discharge of three-dimensional silos filled with a bidisperse mixture of grains of different sizes. We analyzed the influence of the ratio between coarse and fine particles on the profile of volume fraction and velocity across the orifice. By using numerical simulations, we have shown that the velocity profile has the same shape as that in the monodisperse case and is insensitive to the composition of the mixture. On the contrary, the volume fraction profile is strongly affected by the composition of the mixture. Assuming that an effective particle size can be introduced to characterize the mixture, we have shown that previous expression for the mass flow rate of monodisperse particles can be used for binary mixtures. A comparison with Beverloo's correlation is also presented.

Figure 1

(a) Photograph of the experimental setup. The interchangeable disks are used to modify the aperture size. (b) Outline of the experimental exit apertures scaled with the particle sizes, $r_c=0.15\text{cm}$ (in red) and $r_f=0.10\text{cm}$ (in blue). The arrows denote the smallest ($R=0.65\text{cm}$) and largest ($R=1.95\text{cm}$) studied outlet radius. (c) Snapshot of the simulated case B, in which the particle size ratio was increased to $r_c/r_f=2.5$.

Figure 2

(a) Experimental mass flow rate $W$ vs the outlet radius $R$ for different fine mass ratios ${\chi }_f$. The lines are a guide to the eye. Inset: Values obtained from the numerical simulations (case A). The systematic difference between experimental and numerical results is due to the dependence of the bulk density with the filling protocol. (b) Downward free surface velocity as a function of $R$. In both panels, the standard deviations of the data are smaller than the symbol sizes.

Figure 3

(a) Experimental mass flow rate against the vertical downward speed of the top free surface. The slopes of different linear fits provide an estimation of the average bulk volume fraction of each mixture [displayed as gray circles in Fig. 3]. (b) Box plot of the experimental bulk volume fraction obtained from all the studied outlet orifices for different values of ${\chi }_f$. The crosses are the mean values obtained for the different outlet orifices. The up-down boxes correspond to the percentiles $25\text{th}$ and $75\text{th}$, respectively and the whiskers shows the $10\%$ and $90\%$ limits of the volume fraction distribution. The stars indicate the bulk volume fraction of the simulated case B.

Figure 4

Dimensionless mass flow rate against the dimensionless outlet size. All experimental data collapse on a single straight line assuming that there exists an effective particle size $d_e$ describing each mixture [see Eq. (4)]. The line corresponds to the fitting the dimensionless Beverloo's expression [Eq. (3)] with the values shown in the legend.

Figure 5

(a) Profile of the vertical velocity corresponding to the coarse (left panel) and the fine (right panel) particles for case A and ${\chi }_f=0.50$. Both profiles are indistinguishable. The solid line corresponds to $\mathsf{v}\left(r\right)=\sqrt{2gR}{[1-{(r/R)}^2]}^{1/2}$. (b) Velocity profiles corresponding to all ${\chi }_f$ studied were scaled with $\sqrt{2gR}$. The radial coordinate was normalized by the corresponding outlet radius (left panel: case A, $R=1.5$ cm; right panel: case B, $R=7.5$ cm). Apart from some minor differences, the exit velocities are marginally affected by the mass or size ratio.

Figure 6

(a) Normalized number of fine particles passing through a flat annulus of radius $r$ and thickness $\mathrm{\Delta }r=r_f$ against $r$ in the simulation (case A). The probability of observing fine particles is almost independent of the radial position and equal to the bulk value (see horizontal lines). (b) Volume fraction of both simulated situations (cases A and B) can be fitted with the same function introduced for the monodisperse limit [5]: $\varphi \left(r\right)={\varphi }_c{[1-{(r/R)}^2]}^{1/4}$ (shown for the two monodisperse limits by dashed lines). The error bars, corresponding to the standard deviation of the data are similar in all of the cases but they are displayed only for one series of data.

Figure 7

(a) Dimensionless mass flow rate against the outlet radius $R$. Each set of data has been fitted using the exponential saturation indicated on the label. (b) All data can be collapsed in a single curve when plotted against the normalized radius, $R^{\prime }=\frac{R}{r_e}$. The inset shows the values used to fit the collapsed results.

Figure 8

Collapsed data corresponding to the simulated and experimental mass flow rates. For the sake of convenience, values of $W$ and $R$ have been normalized by the radius $r_e$ to compare all the results. The continuous line corresponds to Eq. (8) with the parameter $C^{\prime }$ calculated using ${\varphi }_{\infty }=0.638$ and $\alpha =8.524$ and the corresponding effective radius of $r_e$. The dashed line represents Berverloo's correlation using the parameters introduced in Fig. 4. Inset: log-log plot of the same data where the small difference between both fitting protocols becomes noticeable.