Reverse segregation in dense granular flow through narrow vertical channel

Abstract

Controlling flow-induced segregation in a granular mixture is highly relevant to many industrial settings. To enhance mixing or promote segregation, the continuous gravity flow of a bidisperse granular mixture through a series of narrow vertical channels with exit slots is investigated. The bidisperse mixture is composed of two different sizes of particles, but of the same density. In dense flow, segregation occurs, leading to formation of bands. The bands of large particles appear at a distance away from the walls. This finding is in contrast to that in shear-driven segregation in a dense flow where large particles segregate towards the walls. Using a phenomenological model, it has been shown that rolling and bouncing induced segregation is the dominant mechanism. When cylindrical inserts are placed to modify flow patterns, that significantly influences segregation patterns. The symmetrical placement of a cylindrical insert close to the exit slot vanishes the bands and enhances mixing. However, with two inserts placed symmetrically and close to the exit slot, the degree of segregation in the reverse direction is greatly enhanced compared to that without insert. In the former, small particles accumulate in thin regions adjacent to the walls, and large particles comprise the bulk of the domain and the flowing stream. The heap formation above the insert in a narrow channel, when the insert is close to the exit, enhances mixing in one configuration, whereas it amplifies reverse segregation in the other.

Figures (5)

• Figure 1: The first panel is the schematic of gravity flow through a series of vertical channels. The other panels represent schematics of one channel without and with inserts (of diameter $D$) at different locations.
• Figure 2: With no insert: (a) The spatial distribution of large particles at different time points. The background color represents the number fraction $n_f^l$ of large particles. (b) Streamlines of the flow at steady state. The background color represents the value ${\rm{log}_{10}}S$, where $S = (u_x^2 + u_y^2)^{1/2}$ is the magnitude of the flow. (c) Schematic of particles rolling and bouncing over the free surface. (d) Schematic of the mechanism of single particle rolling over stairs. (e) The increment $(\Delta u_{x,p})^a$ and decrease $(\Delta u_{x,p})^d$ in the horizontal velocity as functions of $u_{x,p}^0$ due to the acceleration and deceleration processes, respectively. The black curve represents $(\Delta u_{x,p})^a$, and the blue curves represent $(\Delta u_{x,p})^d$. The blue solid curve is for $\theta = 12^{\text{o}}$ and the blue dotted curve for $\theta = 30^{\text{o}}$. (f) Variation of the granular temperature with the vertical direction $y-$ at the mid-plane of the channel. The black, red, and blue curves represent $T$, $T_x$, and $T_y$, respectively. (g) The cumulative density function (CDF) of the horizontal velocity of individual particles (small and large) leaving from the left edge of the feed stream.
• Figure 3: With inserts. The spatial distributions of large particles at different times for one and two inserts at different depths are shown in (a), (c), (e), (g), (i) and (k); the background color represents the number fraction $n_f^l$ of large particles. Streamlines of the flows at steady state are shown in (b), (d), (f), (h), (j) and (l); the background color represents the value ${\rm{log}_{10}}S$, where $S = (u_x^2 + u_y^2)^{1/2}$ is the magnitude of the flow. The profiles in (a)--(f) are for one insert, and (g)--(l) for two inserts. In (a), (b), (g) and (h), $L/d_p = 45$; in (c), (d), (i) and (j), $L/d_p = 30$; in (e), (f), (k) and (l), $L/d_p = 15$.
• Figure 4: Variation of flow velocity $u_y$ with $y/d_p$ at the mid-plane $x/d_p = 25$ for one insert (a) and two inserts (b). (c) and (d) With two inserts: variation of $u_y$ and $u_x$ with $y/d_p$ at position $x/d_p = 12.5$. In (a), the filled $\diamond$ symbol represents the data for the case--no insert. With one insert and two inserts at different heights in (a)--(d), black $\circ$, $L/d_p = 45$; blue $\triangle$, $L/d_p = 30$; red $\square$, $L/d_p = 15$. The data are shown for the planes marked as vertical lines in the insets. (e) Schematic of a multi-tray vertical tower. Variation of $(n^l_f - n^s_f)$ with time $t$ in the green shaded exit region shown in (e) for the cases of one insert (f) and two inserts (g). In (f) and (g), solid $\circ$, no insert; $\triangle$, $L/d_p = 45$; $\diamond$, $L/d_p = 30$; $\square$, $L/d_p = 15$. The error bars represent 95% confidence limits of time-averaged values over a period $\Delta t = 125$. The red curves are the fits for the exponential function $y = b (1 - e^{-at} )$.
• Figure 5: The color maps of the shear rate $\dot{\gamma}$ ((a)--(c)), granular temperature $T$ ((d)--(f)), and difference in temperature $(T_y - T_x)$ ((g)--(i)) between $x-$ and $y-$ directions for different cases. The background color represents the $\rm{log_{10}(\star)}$ values of the above variables. No insert; (a), (d), (g). With one insert for different values of $L/d_p$; (b), (e), (h). With two inserts for different values of $L/d_p$; (c), (f), (i).