Phase transition and flow-rate behavior of merging granular flows

Merging of granular flows is ubiquitous in industrial, mining, and geological processes. However, its behavior remains poorly understood. This paper studies the phase transition and flow-rate behavior of two granular flows merging into one channel. When the main channel is wider than the side channel, the system shows a remarkable two-sudden-drops phenomenon in the outflow rate when gradually increasing the main inflow. When gradually decreasing the main inflow, the system shows obvious hysteresis phenomenon. We study the flow-rate-drop phenomenon by measuring the area fraction and the mean velocity at the merging point. The phase diagram of the system is also presented to understand the occurrence of the phenomenon. We find that the dilute-to-dense transition occurs when the area fraction of particles at the joint point exceeds a critical value ${\varphi }_c=0.65\pm 0.03$.

Figure 1

(a) Top view and (b) side view of the granular merging flow channels. (c) Red box and green line show the positions where the area fraction and mean velocity are determined.

Figure 2

Snapshots of typical granular flow patterns. (a) LL state $(D_1$ = 20 mm, $D_2$ = 20 mm); (b) LH state $(D_1$ = 30 mm, $D_2$ = 30 mm); (c) HL state $(D_1$ = 40 mm, $D_2$ = 18 mm); (d) HH state $(D_1$ = 40 mm, $D_2$ = 25 mm).

Figure 3

Inflow rate from the channels vs channel gate width $D_1$ or $D_2$. The flow rate is fitted with the Beverloo's fit and the Janda's fit as formulated by Eqs. (1) and (2). Data are obtained by averaging the results of 10 independent experiments.

Figure 4

Outflow rate vs main channel gate width $D_1$, with fixed $D_2=30$ mm and initially empty channels. The two vertical dotted lines are guides for the phase transition points. Data are obtained by averaging the results of 10 independent experiments.

Figure 5

Evolutions of the area fraction $\varphi$ (solid squares) and the mean velocity $V$ (solid triangles) with main channel gate width $D_1$. Here, the side channel gate width $D_2$ remains at 30 mm. The standard deviation of $V$ is about 0.1.

Figure 6

Outflow rate vs main channel gate width $D_1$, with fixed $D_2=30$ mm and initially full channels. The two vertical dotted lines are guides for the phase transition points.

Figure 7

Phase diagrams of the system: (a) initially empty channels; (b) initially full channels. For most data points, we have done several independent granular flow experiments. The first 60 s of experiment is regarded as the transient. For the data at the boundaries, we have done 10 experiments of 5 min to make sure the granular flow is not a transient state.

Figure 8

Outflow rate vs side channel gate width $D_2$, with fixed $D_1=40$ mm. The dotted line is guide for the phase transition point.

Figure 9

Illustration of the LH state and the HL state. The dark gray area is with high density. The light gray area is with low density. Note that the whole merging area is dense in the HL state.