Self-leveling behavior of mixed solid particles in cylindrical bed using gas-injection method

Figures & data

Figure 1. Illustration of self-leveling behavior.

Figure 2. Schematic diagram of experimental setup for self-leveling experiments.

Figure 3. Definition of measured bed mound height.

Figure 4. Observed self-leveling behavior of binary-mixed particle bed (Al₂O₃ (25%) and SS (75%) particles, $d_p$ = 2 mm, $Q_g$ ~ 50 L/min, $V$ = 7.0 L, $D$ = 0.31 m).

Table 1. Physical properties of particles

Type	No.	Material	Volume mixing ratio	${\rho }_p$[kg/m^3]	$d_p$[mm]	$\varphi$	$\epsilon$	$u_t$[m/s]	$R{e}_p$
Sphere	H1	Al2O3		3720	6.10	0.99	0.403	0.676	4600
	H2	ZrO2		6240	1.00	0.96	0.386	0.331	371
	H3				2.00	0.92	0.386	0.453	1020
	H4				3.90	0.93	0.388	0.646	2820
	H5				6.00	0.95	0.399	0.847	5640
	H6	SS		8050	2.00	1.0	0.377	0.633	1410
	H7				4.00	0.95	0.390	0.806	3580
	H8				6.00	0.92	0.402	0.915	6130
Non-sphere	H9	Zn		7140	2.00	0.72	0.386	0.299	679
	H10	SS		7800	1.18	0.97	0.337	0.435	575
	H11				2.20	0.84	0.385	0.443	1100
	H12				2.20	0.81	0.373	0.409	1010
	H13	Cu		8710	1.90	0.91	0.299	0.523	1110
Binary mixture	M1	Al2O and SS with ${\mathit{d}}_{\mathit{p}}$ ~ 2 mm	1:3	6980	2.00	0.99	0.381	0.573	1290
	M2	Al2O3 and SS with ${\mathit{d}}_{\mathit{p}}$ ~ 4 mm	1:1	5910	4.00	0.94	0.392	0.648	2930
	M3		1:3	6980	4.00	0.97	0.395	0.792	3610
	M4	Al2O3 and SS with ${\mathit{d}}_{\mathit{p}}$ ~ 6 mm	3:1	4800	6.00	1.0	0.401	0.821	5530
	M5		1:1	5890	6.00	0.89	0.406	0.707	4780
	M6		1:3	6970	6.00	1.0	0.396	1.03	7010
	M7	Al2O3 and ZrO2 with ${\mathit{d}}_{\mathit{p}}$ ~ 2 mm	1:3	5610	2.00	1.0	0.378	0.512	1160
	M8	Al2O3 and ZrO2 with ${\mathit{d}}_{\mathit{p}}$ ~ 4 mm	3:1	4360	4.00	0.98	0.395	0.587	2590
	M9		1:1	4990	4.00	0.93	0.386	0.573	2560
	M10		1:3	5620	3.90	0.96	0.390	0.664	3010
	M11	Al2O3 and ZrO2 with ${\mathit{d}}_{\mathit{p}}$ ~ 6 mm	3:1	4350	6.00	1.0	0.400	0.768	5150
	M12		1:3	5610	6.00	1.0	0.396	0.907	6140
	M13	SS with ${\mathit{d}}_{\mathit{p}}$ ~ 2 and 4 mm	1:1	8050	2.40	0.89	0.361	0.532	1440
	M14	Spherical and non-spherical SS with ${\mathit{d}}_{\mathit{p}}$ ~ 2 mm	1:1	7930	2.10	0.82	0.399	0.408	953
	M15	ZrO2 with ${\mathit{d}}_{\mathit{p}}$ ~ 1 and 2 mm	1:1	6240	1.21	0.89	0.357	0.315	427
Ternary mixture	T1	SS with ${\mathit{d}}_{\mathit{p}}$ ~ 2, 4, and 6 mm	3:2:2	8050	2.54	1.0	0.338	0.718	2041

Figure 5. Comparison of bed mound variation between intermittent and continuous gas-injection methods ($Q_g$ ~ 46 L/min, $V$ = 7.0 L, $D$ = 0.21 m).

Figure 6. Effect of cylindrical bed diameter on mound height variation ($V$ = 7.0 L).

Figure 7. Effect of particle volume on mound height variation ($D$ = 0.21 m).

Figure 8. Effect of mixing ratio on mound height variation in various particle mixtures ($V$ = 7.0 L, $D$ = 0.31 m).

Table 2. Experimental cases used to determine model parameters

Particle Type	$Q_g$ [L/min] (${u_g}^{\prime }$ [m/s])
H1	~50 ($1.1\times {10}^{-2}$)	~100 ($2.3\times {10}^{-2}$)
H2	~10 ($2.8\times {10}^{-3}$)	~20 ($4.5\times {10}^{-3}$)
H3	~20 ($4.8\times {10}^{-3}$)	~50 ($1.1\times {10}^{-2}$)	~100 ($2.2\times {10}^{-2}$)
H4	~100 ($2.3\times {10}^{-2}$)
H5	~100 ($2.3\times {10}^{-2}$)	~200 ($4.7\times {10}^{-2}$)	~500 ($1.3\times {10}^{-1}$)
H6	~50 ($1.1\times {10}^{-2}$)	~100 ($2.3\times {10}^{-2}$)	~200 ($4.6\times {10}^{-2}$)
H7	~50 ($1.2\times {10}^{-2}$)	~200 ($4.7\times {10}^{-2}$)	~300 ($6.9\times {10}^{-2}$)
H8	~300 ($6.9\times {10}^{-2}$)	~500 ($1.1\times {10}^{-1}$)
H9	~50 ($1.2\times {10}^{-2}$)	~100 ($2.3\times {10}^{-2}$)
H10	~50 ($1.1\times {10}^{-2}$)	~200 ($4.5\times {10}^{-2}$)
H11	~50 ($1.2\times {10}^{-2}$)	~100 ($2.3\times {10}^{-2}$)	~200 ($4.6\times {10}^{-2}$)
H12	~50 ($1.1\times {10}^{-2}$)	~100 ($2.3\times {10}^{-2}$)
H13	~50 ($1.2\times {10}^{-2}$)	~100 ($2.5\times {10}^{-2}$)	~200 ($4.7\times {10}^{-2}$)
M1	~50 ($1.2\times {10}^{-2}$)	~100 (2.4$\times {10}^{-2}$)
M2	~50 ($1.2\times {10}^{-2}$)	~100 ($2.3\times {10}^{-2}$)
M3	~50 ($1.2\times {10}^{-2}$)	~100 ($2.3\times {10}^{-2}$)	~200 (4.5$\times {10}^{-2}$)
M4	~200 ($4.6\times {10}^{-2}$)	~300 ($6.8\times {10}^{-2}$)
M5	~200 ($4.6\times {10}^{-2}$)	~300 ($6.7\times {10}^{-2}$)
M6	~200 ($4.5\times {10}^{-2}$)	~300 ($6.7\times {10}^{-2}$)	~500 ($1.1\times {10}^{-1}$)
M7	~50 ($1.2\times {10}^{-2}$)
M8	~50 ($1.1\times {10}^{-2}$)
M9	~50 ($1.1\times {10}^{-2}$)
M10	~50 ($1.2\times {10}^{-2}$)	~100 ($2.3\times {10}^{-2}$)	~200 ($4.6\times {10}^{-2}$)
M11	~300 ($6.8\times {10}^{-2}$)
M12	~200 ($4.6\times {10}^{-2}$)	~300 ($6.8\times {10}^{-2}$)
M13	~100 ($2.2\times {10}^{-2}$)	~200 ($4.6\times {10}^{-2}$)
M14	~50 ($1.1\times {10}^{-2}$)	~100 ($2.3\times {10}^{-2}$)	~200 ($4.6\times {10}^{-2}$)
M15	~20 ($5.3\times {10}^{-3}$)	~50 ($1.2\times {10}^{-2}$)

Figure 9. Comparison of model parameters between experiments and predictions.

Figure 10. Comparison of mound height between experiments and empirical model.

Figure 11. Effect of mixing ratio on transient mound height of various binary particle mixtures ($V$ = 7.0 L, $D$ = 0.31 m).

Figure 12. Self-leveling behavior of ternary mixture (spherical SS particles with $d_p$ = 2.00, 4.00, and 6.00 mm; $V$ = 7.0 L; $D$ = 0.31 m).

Table

$A_p$	actual particle surface area (m^2)
$Ar$	gas-phase Archimedes number
$D$	bed diameter (m)
$d_p$	volume-equivalent spherical diameter (mm)
$g$	gravitational acceleration (m/s^2)
$H_0$, $H_{b0}$	reference mound height at $t=0$ (m)
$H_{eq}$, $H_{be}$	equilibrium mound height at $t=$ ∞ (m)
$H_m\left(t\right)$, $H_b\left(t\right)$	bed mound height at time $t$ (m)
$Q_g$	gas flow rate (L/min)
$Re$	gas Reynolds number
$R{e}_p$	particle Reynolds number at terminal velocity in stagnant liquid
$t$	time (s)
$u_0$, ${u_0}^{\prime }$	Self-leveling velocity (m/s)
$u_g$	superficial gas velocity (m/s)
${u_g}^{\prime }$	effective superficial velocity of gas (m/s)
$u_t$	terminal velocity of a single particle in stagnant liquid (m/s)
Greek letters
$\epsilon$	Packed-bed porosity
${\mu }_g$	gas viscosity (Pa·s)
${\mu }_l$	liquid viscosity (Pa·s)
${\rho }_g$	gas density (kg/m^3)
${\rho }_l$	liquid density (kg/m^3)
${\rho }_p$	particle density (kg/m^3)
$\tau$	characteristic time constant (s)
$\varphi$	particle sphericity