Mixing particle softness in a two-dimensional hopper: Particle rigidity and friction enable variable arch geometry to cause clogging

Understanding the clogging of mixtures of soft and rigid particles flowing through hoppers becomes important as soft particle usage increases in consumer products. We investigate this clogging under varying particle size and rigid fraction by quantifying various properties of arches formed in the neck of a quasi-two-dimensional hopper. As more soft particles are added to the mixture, the arch tends to become both narrower and more curved. This effect arises from the fact that soft particles have less ability to sustain a clog than rigid particles. The clogging probability is seen to have a linear correlation with the span (width) of the arch. The angles between the arch particles are shown to have higher values as rigid fraction increases. The arch occasionally shows a partially convex shape at high rigid fractions when rigid particles are sitting next to each other, while soft particles can form angles of less than ${180}^{\circ }$ only. The relation between the span and aspect ratio (width to height) of the arch is theoretically formulated for three-particle arches and extended to arches of more than three particles, using an asymptotic parameter that represents the width of a flat arch. Finally, it is concluded that clogging probability closely correlates with both the arch span and the variation of other geometric arch properties.

Figure 1

A schematic of the working hopper. The inset shows arch properties definitions.

Figure 2

Clogging probability vs rigid fraction for different particle sizes while $d_s/d_r\approx 1$. Stochastic error bars are smaller than the symbol size due to the stabilized calculation strategy.

Figure 3

(a) Average number of particles in the arch and (b) average nondimensionalized span versus rigid fraction. Error bars in both subfigures show the standard error $\sigma /\sqrt{n}$ with $\sigma$ being the standard deviation and $n$ the sample size.

Figure 4

Clogging probability versus span of the arch for all particle sizes. Dashed line is the best linear fit $P_c=2.78\overline{s}$–3.37 with goodness-of-fit $R^2=0.82$.

Figure 5

(a) Normalized probability distribution of angles for different rigid fractions at $w/d=2.00$. Lines are added to guide the eye. Panels (b) and (c) show representative examples of a three-particle convex arch made entirely of soft particles and a four-particle arch made entirely of rigid particles exhibiting a region of concavity with one value of $\theta >{180}^{\circ }$. The arrows in (a) point to angles exhibited in the arches shown in (b) and (c).

Figure 6

Average aspect ratio versus rigid fraction for three different particle sizes. Error bars show standard error as defined in Fig. 3.

Figure 7

[(a) and (b)] Illustration of a triangle overlaid on a sample three- and four-particle arch, respectively, with the labels in (a) defining parameters used in Eqs. (4, 5, 6). (c) Center-to-center nondimensionalized span versus aspect ratio for the case of $w/d=2.00$ and ${\varphi }_r=0.8$. Symbols show experimental data with different colors representing different number of soft particles contributing to the arch. Solid lines represent the values predicted by Eq. (6) with $n_a=6$, 5, 4 and 3 from top to bottom, respectively.

Figure 8

Coefficient of variation for all arch properties investigated as a fraction of their values at ${\varphi }_r=1$ corresponding to the size $w/d=2.00$.