**3. Results**

Figure 3 shows the temporal change in the three-dimensional distributions of the liquid iron (blue color) and molten slag (green color) as a result of the model calculation. The packed bed structure formed by irregularly shaped cokes is geometrically complex, and the liquid iron flow through the continuous void drops in the form of strings or droplets. From the viewpoint of localized flow behavior, several regions do not receive any flow, and it is assumed that liquid iron flows only through the limited region. However, the flow behavior of liquid iron is influenced by the presence of slag with different coke wettability, as shown in Figure 3b,c. As shown in Figure 3b, for a coke surface with poor wettability containing molten slag, the molten slag drops along with the liquid iron. However, the coke surface with good wettability shown in Figure 3c retains molten slag at the upper part of the packed bed owing to a high attractive force acting against the molten slag.

**Figure 3.** Liquid iron and molten slag distributions on vertical cross-sections of a coke bed. (**a**) Without slag, (**b**) θ*s* = 160◦, and (**c**) θ*s* = 30◦.

Figure 4 shows the volume distribution of liquid iron and molten slag in the height direction at *t* = 0.20 and 1.20 s. In Figure 4a, liquid iron almost has the same distribution at *t* = 0.20 s, irrespective of the presence of slag and the change in θ*s*. Conversely, at *t* = 1.20 s, the liquid iron with slag exists at a lower height compared to "without slag" iron. In the region shown in (a-1), liquid iron is concentrated regardless of θ*s*, and in the region shown in (a-2), liquid iron is concentrated when θ*s* = 30◦. In Figure 4b, molten slag exists at a lower height when θ*s* = 160◦ than that at 30◦. The attractive force due to the wettability of the coke surface prevents slag from dripping in the direction of gravity. This change in wettability affects the flow form of the liquid iron shown in (a-1) and (a-2). It is necessary to determine the time change of each melt velocity to consider the dripping mechanism leading to this result. The average velocity in the gravitational direction per unit time is given, considering the temporal change in the center of gravity of each phase, as follows [44]:

$$\mathbf{v}\_z = \frac{1}{V\Delta t} \int \left(\mathbf{r}\_{i,t+\Delta t} - \mathbf{r}\_{i,t}\right)dV\tag{15}$$

**Figure 4.** Dripping profiles of liquid iron and molten slag. Calculation domain of coke bed was divided into 34 control volumes in the height direction (Δz = 0.005 m), and the profiles were derived by counting the number of liquid particles existing in each control volume as time passes.

Figure 5 displays the average velocity of each melt. At *t* > 0.5 s, the liquid iron almost reaches a steady state and almost corresponds to the flow at the lower part of the blast furnace presented by Sugiyama et al. [45] and also appears to follow the Darcy-type equation. From a detailed perspective, however, the liquid iron tends to increase the dripping rate because of the presence of slag. Moreover, the velocity of liquid iron is affected by the wettability between the slag and coke. The angle θ*s* = 30◦ yields the maximum liquid-iron velocity within range (A); in contrast, θ*s* = 160◦ represents the maximum velocity in region (B). This reversal occurs at *t* = 0.31 s. As observed from Figure 4, the liquid iron dripping rate is higher than that of molten slag. Due to the difference in density between the two melts, the gravitational force of liquid iron was 2.7 times higher than that of molten slag. As molten slag has a high viscosity and low density, it appears that it prevents liquid iron from descending. However, when θ*s* = 30◦, the coke bed surface is wet by slag immediately and facilitates the sliding of the liquid iron. With time, this "lubrication" by molten slag is lost, because the liquid iron moves below the molten slag. When θ*s* = 160◦, the effect of wetting becomes weak, and the molten slag drips along with the liquid iron. Thus, the "lubrication" between the molten slag and liquid iron remains on the coke surface. Since the molten slag drops slowly, owing to the 0.37 times density and 53 times viscosity coefficient of the liquid iron, these observations appear as slight differences.

**Figure 5.** Time variations in the mean dripping velocity. These profiles were obtained using a space integral by considering the center of gravity of all the droplets each time.

The variation in the wettability between the coke and molten slag modifies the interfacial area between the four-phase iron-slag-coke-gas by the following mechanism. When some droplets are dispersed in the system under isothermal conditions, the Helmholtz free energy, *F* = σ*ijAi*, of the system increases. Since the potential energy corresponding to the flow of the liquid iron droplets through the packed bed is equal for each case, the kinetic energy corresponding to the secondary droplet formation may be the same for all the conditions. However, the total energy in the system is not conserved due to the viscous damping from this calculation, but it is useful to clarify the effect of wettability between the coke and slag on the interfacial area of each phase. As our interest is focused on the effect of wettability between the slag and coke on the interfacial energy of each phase, the interfacial area for each phase should be estimated. The main advantage of the SPH method is its rapid prediction of the interface area *A* from its initial condition *A*<sup>0</sup> and the interface-judged particle number *n*.

$$A(t) \cong \frac{n(t)}{n\_0} A\_0 \tag{16}$$

*A*<sup>0</sup> is geometrically determined from the initial conditions. In this study, the free surface of the liquid iron and slag are 0.04992 and 0.01248 m2, respectively, and the initial iron-slag interface area is 0.02110 m2. See Section 2 and a previous report [46] for the counting procedure of *n*. Figure 6 depicts the time change of the estimated interfacial area of each phase. At *t* = 0.1 s, the melt penetrates the coke bed, and both interfacial areas significantly vary simultaneously. In Figure 6a, for "without slag" liquid iron, the initial iron-gas surface area is significantly different from that of "with slag", but the area becomes almost equal to other conditions at *t* = 0.1 s. The case of θ*s* = 160◦ sometimes indicates a larger iron-gas surface area than that of "without slag", because the iron slides down first and forms a contact interface at a lower height. Beyond *t* = 0.1 s, "without slag" iron has the lowest iron-gas surface area. This behavior can be explained as follows. As shown in Figure 6b, since the liquid is not affected by molten slag in "without slag" liquid iron, the iron-coke interfacial area significantly increases compared to that of "with slag" at *t* > 0.1 s; in other words, the iron-coke interface decreases with the presence of the molten slag. From Figure 6c, although the iron-slag interface area has a larger value at θ*s* = 160◦

than at θ*s* = 30◦, the slag-coke interface area exhibits the opposite trend, as observed in Figure 6d. Since the coke surface with good wettability is covered with slag, it will likely increase the contact area between the free surface of the slag and iron-slag in a static state. In the case of θ*s* = 160◦, the slag drops with the liquid iron due to gravity. This dripping occurs because the attractive force acting between the coke and slag is less than the gravitational force. However, for θ*s* = 30◦, the attractive force acting between the coke-slag interface dominates, and the iron-slag interface is preferentially detached. Here, the effect of momentum in the direction of gravity appears. Since a "quasi-stable" interface is formed by the balance between the melt momentum and the force acting on each interface, the state predicted by the equilibrium theory is not always achieved during dripping [47]. This result indicates that static hold-up of the molten slag may promote the smooth dripping of liquid iron, as variations in the wettability between the coke and slag transiently change the flow field of the liquid iron. It emphasizes the distinctive interfacial features that can depend on interactions with the wettability of nonuniform packed cokes and the transient behavior of two melts. In the future, the calculation region of this model can be expanded and the model may be applied to continuous dripping, and the application of the model can also be extended to the entire bottom of the blast furnace.

**Figure 6.** Time change of estimated interfacial area between each phase.
