*3.3. Comprehensive Analysis of Mixing and Slag Entrapment*

The results of mixing time show that the optimum gas flow rate for the prototype ladle is 36–42 m3/h, the ladle with double porous plugs should be selected at the same time. Slag thickness has a significant influence on the entrapping depth of the slag and slag eye area. In actual production, slag entrapment is beneficial to improve the refining effect, therefore, more consideration should be given to the area of the slag eye and bath mixing.

The bubble motion in the water model experiment was shown in Figure 10. Bubble motion is accompanied by the process of coalescence, collapse and re-coalescence. In the process of bubble floating up, the size and shape of the bubble have changed. When the diameter of the bubble exceeds 1 cm, the shape of the bubble changes into a spherical corona. In low viscosity liquids, the rising velocity of the spherical coronal bubble is independent of the properties of liquids and can be calculated by Equation (7) [26].

$$
u\_b = 1.02 \left(\frac{\mathcal{g}d\_b}{2}\right)^{\frac{1}{2}}\tag{7}$$

where, *ub* is the bubble velocity, m/s; *g* is the acceleration of gravity, m/s2; *db* is the bubble diameter, m.

**Figure 10.** The bubble motion in the water model: (**a**) Q = 0.0475 m3/h; (**b**) Q = 0.095 m3/h; (**c**) Q = 0.143 m3/h; (**d**) Q = 0.191 m3/h; (**e**) Q = 0.238 m3/h.

In order to apply the research results to more ladles, a dimensionless treatment was carried out for the slag layer thickness, gas flow rate and mixing time. Assuming that the diameter of the bubble is 1 cm, the product of the bubble velocity and area of the gas ports is taken as the characteristic flow rate.

$$Q\_b = \mathbb{S} \cdot u\_b \tag{8}$$

$$S = \pi \left( r\_1^2 + r\_2^2 + \dots + r\_n^2 \right) \tag{9}$$

where, *Qb* is the characteristic flow rate, m3/s; *r*1, *r*2, ... , *rn* are the radius of gas ports, m; *S* is the area of gas ports of porous plugs, m2. The radius of gas port in the water model experiment is 12 mm.

The dimensionless gas flow rate was treated according to Equation (10), and the dimensionless slag layer thickness was treated according to Equation (11).

$$Q^\* = \frac{Q}{3600Q\_b} \tag{10}$$

$$h^\* = \frac{h}{h\_l} \tag{11}$$

where, *V\** is the dimensionless gas flow rate; *V* is the gas flow rate, m3/h; *h\** is the dimensionless thickness of the slag layer; *h* is the thickness of the oil layer, m; *hl* is the depth of the molten bath, m. In the water model, the water depth is 0.643 m.

Assuming that the shape of the bubble is a spherical corona and does not deform in the process of bubble flotation, the residence time *tb* of the bubble in the water model can be calculated. Taking residence time *tb* as the characteristic time, the dimensionless mixing time can be calculated by Equation (12).

$$t\_b = \frac{l\_l}{u\_b} \tag{12}$$

$$t^\* = \frac{t}{t\_b}$$

where, *tb* is the characteristic time, s; *hl* is the depth of the water, m; *t\** is the dimensionless time; *t* is the mixing time, s.

The relation between the dimensionless mixing time and dimensionless gas flow rate in the ladle with double porous plugs was shown in Figure 11, the 0, 0.0187, 0.0218, 0.0249, 0.028 and 0.0311 are dimensionless thickness in Figure 11 correspond to without oil and with oil layer of 12, 14, 16, 18, 20 mm. The trend of mixing time is consistent with Figure 4. In the water model, the optimum dimensionless flow rate is 0.157–0.183 for the ladle with double porous plugs.

**Figure 11.** The relation of dimensionless mixing time and dimensionless flow rate in the ladle with double porous plugs.

Using multiple linear regression, the relationship between dimensionless mixing time, dimensionless flow rate and dimensionless oil layer thickness was fitted by Equation (13). The coefficient of determination (R square) of fitting Equation (13) is equal to 0.942, which indicates that the equation has a high fitting degree and may be used to calculate the mixing time of the ladle with double porous plugs.

$$\mathbf{f}^\* = 125.3601 + 1476.75327 \mathbf{\hat{h}}^\* - 502.76857 \mathbf{\hat{Q}}^\* \tag{13}$$

The area of the slag eye in the water model with a single porous plug at different dimensionless flow rates and dimensionless oil thicknesses was shown in Figure 12. It can be seen from the figure that when the dimensionless flow rate increases to 0.366 (corresponding to the prototype ladle flow rate of 42 m3/h), further increasing flow rate has little effect on the slag eye area, the dimensionless flow rate corresponding to the prototype ladle is 37.595. Therefore, considering the mixing time and slag entrainment, the optimized injection mode for the prototype ladle is the double porous plug with a flow rate of 36–42 m3/h.

**Figure 12.** The areas of slag eye at different flow rates and oil thickness in water model with single porous plug.

Using multiple linear regression, the relationship between the area of the slag eye, dimensionless flow rate and dimensionless oil layer thickness was fitted by Equation (14). The coefficient of determination (R-square) of fitting Equation (14) is equal to 0.93188, which indicates that the equation has a high fitting degree and may be used to calculate the area of the slag eye of a single porous plug ladle.

$$S\_{\%} = 21.30098 - 660.60213 \cdot h^\* + 45.37523 \cdot Q^\* \tag{14}$$

where, *S%* is the percentage of the slag eye area to the molten bath surface area.

Equations (13) and (14) use dimensionless experimental data, and the determinant coefficient shows that equations have a high fitting degree, so for the general ladles, Equations (13) and (14) can be used to calculate the mixing time and bare steel area by flow rate and slag layer thickness. However, in the process of dimensionless data processing, the characteristic flow rate is a fixed value, geometric similarity ratio has an effect on dimensionless flow rate, so the equations can not be directly used in the ladle. Therefore, when using Equations (13) and (14) in generic ladles, the model flow rate calculated by the geometric similarity ratio of 1:5 can be used to calculate the mixing time and slag hole area.

#### **4. Conclusions**

Through the establishment of the water model, the influence of the bottom blowing flow rate on the mixing time and slag entrapment were studied. Through dimensionless treatment and multivariate linear regression, the equations which may be used to calculate the mixing time and slag eye area of the ladle are obtained and the conclusions are as follows:

(1) The bath mixing in the ladle is affected by the number of porous plugs, flow rate and slag layer. Under the same blowing flow rate, the mixing time of double porous plugs is shorter than that of a single porous plug. The mixing time of the two methods eccentric blowing is basically the same, and the mixing time decreases with the increase of the blowing flow rate, and increases with the increase of the slag layer thickness. There is an inflection point in the mixing time curve, the flow rate at the point is 0.333 m3/h (corresponding 42 m3/h in prototype), the mixing time before the inflection point changes significantly, but after the inflection point, the mixing time changes slowly. The mixing time of the ladle without the slag layer is significantly shorter than that with the slag layer.


**Author Contributions:** Conceptualization, Y.J., X.D.; Methodology, Y.J., F.Y.; Software, X.D., P.L.; Validation, P.L., F.Y. and X.D.; Formal analysis, Y.J., F.Y.; Investigation, P.L.; Resources, C.Z.; Data curation, L.S., J.P., and Q.C.; Writing—original draft preparation, F.Y.; Writing—review and editing, F.Y., Y.J.; Visualization, X.D.; Supervision, Y.L.; Project administration, Y.J.; Funding acquisition, C.Z., C.C.

**Funding:** This research was funded by the National Nature Science Foundation of China (NSFC), grant number 51674180 and grant number 51474163.

**Acknowledgments:** The technical assistance in the water model experiment provided by Yue Yu is greatly acknowledged.

**Conflicts of Interest:** The authors declare no conflict of interest.
