Mathematical modelling of an electrostatic oiling machine for steel strips

Figures & data

Figure 1. Main components of the considered hot-dip galvanizing line.

Figure 2. Electrostatic oiling machine. (a) Cross section. (b) Front view of the electrostatic oiling blades.

Figure 3. Cross section of the oiling blades.

Figure 4. Block diagram of the model.

Figure 5. Flow chart of the numerical model implemented in ANSYS Fluent.

Figure 6. Photograph of the oil spray at the upper blade (lateral view, captured through inspection window of the EOM) with detail showing the measurement of the droplet velocity and diameter, scale: 1px $\widehat{=}$ 28 μm.

Figure 7. Atomization process at the blade edge (detail from Figure 2(b)).

Table 1. Operating points for the parameter identification, $v_{strip}=0\mathrm{m}{\mathrm{s}}^{-1}$, $l_u=305\mathrm{m}\mathrm{m}$, and $l_l=135\mathrm{m}\mathrm{m}$.

$n$	$U$ (kV)	$I$ ($\mu$ A)	$Q$ (gs ${ }^{-1}$)	$r_0$ (μm)
Upper Blade
1	30	15	4.7	160
2	60	25	6.8	120
3	90	33	5.6	80
Lower Blade
4	−30	−20	5.1	170
5	−60	−27	6.8	145
6	−90	−55	8.5	110

Figure 8. Simulation results for $U=75\mathrm{k}\mathrm{V},I=29\mu \mathrm{A},Q=6.8\mathrm{g}{\mathrm{s}}^{-1}$, $r_0=95\mu \mathrm{m}$, $v_{strip}=0\mathrm{m}{\mathrm{s}}^{-1}$, and $l_u=305\mathrm{m}\mathrm{m}$. (a) Droplet radii. (b) Droplet velocities.

Table 2. Comparison of (simulated) forces per unit mass acting on the droplets, positions defined in Figure 8(a).

Position	Electrostatic force	Drag force
	($Nk{g}^{-1}$)	($Nk{g}^{-1}$)
Blade edge	$5.4\cdot {10}^4$	$1.5\cdot {10}^{-1}$
A	$1.7\cdot {10}^3$	$2.1\cdot {10}^3$
B	$8.5\cdot {10}^2$	$8.9\cdot {10}^2$

Table 3. Comparison of measured and simulated absolute (mean) droplet velocities $u_A$, $u_B$ and (mean) droplet radii $r_A$, $r_B$ at the positions A and B, respectively, $v_{strip}=0\mathrm{m}{\mathrm{s}}^{-1}$, $l_u=305\mathrm{m}\mathrm{m}$, and $l_l=135\mathrm{m}\mathrm{m}$.

Quantity	Measured	Simulated	Relative error (%)	Conditions
Upper blade
$u_A$ (m/s)	$7.3$	$7.6$	4.1	$U=50\mathrm{k}\mathrm{V}$
$u_B$ (m/s)	$6.4$	$6.7$	4.7	$I=23\mu \mathrm{A}$
$r_A$ (μm)	$90$	$94$	4.4	$Q=6.5\mathrm{g}{\mathrm{s}}^{-1}$
$r_B$ (μm)	$75$	$72$	−4.0	$r_0=140\mu \mathrm{m}$
$u_A$ (m/s)	$9.3$	$9.0$	−3.2	$U=75\mathrm{k}\mathrm{V}$
$u_B$ (m/s)	$8.3$	$7.9$	−4.8	$I=29\mu \mathrm{A}$
$r_A$ (μm)	$70$	$72$	2.9	$Q=6.8\mathrm{g}{\mathrm{s}}^{-1}$
$r_B$ (μm)	$60$	$65$	8.3	$r_0=95\mu \mathrm{m}$
Lower blade
$u_A$ (m/s)	$7.7$	$8.1$	5.2	$U=-50\mathrm{k}\mathrm{V}$
$u_B$ (m/s)	$6.8$	$7.4$	8.8	$I=-20\mu \mathrm{A}$
$r_A$ (μm)	$65$	$62$	−4.6	$Q=4.1\mathrm{g}{\mathrm{s}}^{-1}$
$r_B$ (μm)	$55$	$49$	−10.9	$r_0=150\mu \mathrm{m}$
$u_A$ (m/s)	$8.6$	$9.2$	7.0	$U=-75\mathrm{k}\mathrm{V}$
$u_B$ (m/s)	$7.4$	$7.8$	5.4	$I=-52\mu \mathrm{A}$
$r_A$ (μm)	$45$	$49$	8.9	$Q=9.3\mathrm{g}{\mathrm{s}}^{-1}$
$r_B$ (μm)	$40$	$45$	12.5	$r_0=125\mu \mathrm{m}$

Figure 9. Simulation results without a break-up model for $U=75\mathrm{k}\mathrm{V},I=29\mu \mathrm{A},Q=6.8\mathrm{g}{\mathrm{s}}^{-1}$, $r_0=62\dots 95\mu \mathrm{m}$, $v_{strip}=0\mathrm{m}{\mathrm{s}}^{-1}$, and $l_u=305\mathrm{m}\mathrm{m}$. (a) Droplet radii. (b) Droplet velocities.

Table 4. Simulated absolute (mean) droplet velocities $u_A$, $u_B$ and (mean) droplet radii $r_A$, $r_B$ at the positions A and B, respectively, without a break-up model and $v_{strip}=0\mathrm{m}{\mathrm{s}}^{-1}$, $l_u=305\mathrm{m}\mathrm{m}$, and $l_l=135\mathrm{m}\mathrm{m}$.

Quantity	Simulated	Conditions
Upper blade
$u_A$ (m/s)	$8.5$	$U=75\mathrm{k}\mathrm{V}$
$u_B$ (m/s)	$7.0$	$I=29\mu \mathrm{A}$
$r_A$ (μm)	$79$	$Q=6.8\mathrm{g}{\mathrm{s}}^{-1}$
$r_B$ (μm)	$79$	$r_0=62\dots 95\mu \mathrm{m}$

Table 5. Comparison of calculated droplet radius limits $r_R$ according to (15) (for K = 0.7) with measured and simulated (mean) droplet radii at position B, $v_{strip}=0\mathrm{m}{\mathrm{s}}^{-1}$, $l_u=305\mathrm{m}\mathrm{m}$, and $l_l=135\mathrm{m}\mathrm{m}$.

$r_R$	$r_B$	$r_B$
($\mu$ m)	Measured	Simulated	Conditions
	(μm)	(μm)
Upper blade
			$U=50$ kV
77	75	72	$I=23\mu$ A
			$Q=6.5\mathrm{g}{\mathrm{s}}^{-1}$
			$r_0=140\mu$ m
			$U=75$ kV
67	60	65	$I=29\mu$ A
			$Q=6.8\mathrm{g}{\mathrm{s}}^{-1}$
			$r_0=95\mu$ m
Lower blade
			$U=-50$ kV
61	55	49	$I=-20\mu$ A
			$Q=4.1\mathrm{g}{\mathrm{s}}^{-1}$
			$r_0=150\mu$ m
			$U=-75$ kV
56	40	45	$I=-52\mu$ A
			$Q=9.3\mathrm{g}{\mathrm{s}}^{-1}$
			$r_0=125\mu$ m

Figure 10. Simulated electric field strength (log scale) for $U=75\mathrm{k}\mathrm{V},I=29\mu \mathrm{A},Q=6.8\mathrm{g}{\mathrm{s}}^{-1}$, $r_0=95\mu \mathrm{m}$, $v_{strip}=0\mathrm{m}{\mathrm{s}}^{-1}$, and $l_u=305\mathrm{m}\mathrm{m}$.

Figure 11. Simulated droplet radii for $U=100\mathrm{k}\mathrm{V},I=36\mu \mathrm{A},Q=6.8\mathrm{g}{\mathrm{s}}^{-1}$, $r_0=70\mu \mathrm{m}$, $v_{strip}=0\mathrm{m}{\mathrm{s}}^{-1}$, and $l_u=305\mathrm{m}\mathrm{m}$.

Figure 12. Simulation results for $U=75\mathrm{k}\mathrm{V},I=29\mu \mathrm{A},Q=6.8\mathrm{g}{\mathrm{s}}^{-1}$, $r_0=95\mu \mathrm{m}$, $v_{strip}=5\mathrm{m}{\mathrm{s}}^{-1}$, and $l_u=305\mathrm{m}\mathrm{m}$. (a) Air flow velocity. (b) Droplet velocities.

Table 6. Dependence of the output variables on the input variables.

Input	Output
	$r$	$u$	$q$	$t_{bs}$	$w$	$v$
$U\uparrow$	$\downarrow$	$\uparrow$	$\uparrow$	$\downarrow$	$\uparrow$	$\uparrow$
$Q\uparrow$	$\simeq$	$\simeq$	$\simeq$	$\uparrow$	$\uparrow$	$\uparrow$
$v_{strip}\uparrow$	$\simeq$	$\simeq$	$\simeq$	$\simeq$	$\simeq$	$\uparrow$

Figure 13. Laterally inclined upper blade.

Figure 14. Reduced unused air volume in the rotated upper blade.

Figure 15. Simulated droplet radii for $U=50\mathrm{k}\mathrm{V}$, $I=25\mu \mathrm{A}$, $Q=6.1\mathrm{g}{\mathrm{s}}^{-1}$, $r_0=135\mu \mathrm{m}$, $v_{strip}=5\mathrm{m}{\mathrm{s}}^{-1}$, $l_u=305\mathrm{m}\mathrm{m}$, and $\alpha ={20}^{\circ }$.

Figure 16. Measured oil coating weight on the upper strip surface for $U=50\mathrm{k}\mathrm{V}$, $I=29\mu \mathrm{A}$, $Q=6.5\mathrm{g}{\mathrm{s}}^{-1}$, $r_0=135\mu \mathrm{m}$, $v_{strip}=5\mathrm{m}{\mathrm{s}}^{-1}$, and $l_u=305\mathrm{m}\mathrm{m}$.