*2.5. Physical Properties of Liquid Iron and Molten Slag*

The condition of the lower part of the blast furnace was examined to determine the physical properties of liquid iron (ρ*m*, μ*m*, and σ*m*) and molten slag (ρ*s*, μ*s*, and σ*s*). Assuming that the molten iron at the lower part of the blast furnace has a sufficiently low oxygen concentration and is in a carbon-saturated state, the molten iron can be regarded as chemically stable at 1773 K. Hence, it can be assumed as follows: ρ*<sup>m</sup>* = 6800 kg/m3, μ*<sup>m</sup>* = 0.01 Pa s, and σ*<sup>m</sup>* = 1.25 N/m [38]. The contact angle between carbon-saturated iron and coke is reported to exceed 120◦. However, the physical properties of molten slag vary over a wide range. The typical slag composition in the dripping zone depends on the CaO-SiO2-Al2O3-MgO system at 1773 K; the basicity (CaO/SiO2 mass %) ranges from 0.7–2.0 and decreases as the slag descends in the furnace. When the slag contains MgO, the reduction of MgO may proceed preferentially over that of SiO2, thereby influencing the wetting behavior of the molten slag on the coke substrate [8]. In this study, the liquid phase composition was considered as the simplest 40 mol% CaO-40 mol% SiO2-20 mol% Al2O3-0 mol% MgO slag, in which the physical property data were assumed in a single phase at a constant temperature of 1773 K. Since the physical properties of molten slag exhibit large deviations among the reported values, typical datasets from the same research group are adopted [39,40]: ρ*<sup>s</sup>* = 2574 kg/m3, μ*<sup>s</sup>* = 0.53 Pa s, and σ*<sup>s</sup>* = 0.49 N/m. According to a previous study, there is no agreement on the contact angle between the CaO-SiO2-Al2O3-based molten slag and the carbonaceous material. On one hand, the contact angle decreases when CaO/SiO2 - 1 [11,41]; on the other hand, the contact angle remains constant at approximately 160◦ in the case of graphite, regardless of the basicity [10]. These differences are thought to be due to the different compositions of the carbonaceous material. The change in these contact angles might be due to the reduction reaction of SiO2 between the carbonaceous material and molten slag, as well as the formation of SiC as the interfacial product because an initial contact angle of 45◦ between SiC and the slag was reported [11]. Therefore, in this study, the contact angle between the molten slag and coke was defined by taking θ*<sup>m</sup>* = 160◦ for low-reactivity interface and θ*<sup>m</sup>* = 30◦ for high-reactivity interface [8,10].

#### *2.6. Calculation Condition*

Detailed packed-bed-structure digital data consisting of multiple coke samples was constructed to simulate the lower part of the blast furnace. Assuming that the coke distribution was just above the raceway, representative coke samples with an average equivalent spherical volume diameter *D* = 0.0247 m was selected [42]. By using a three-dimensional scanning technique [26], 100 pieces of representative coke three-dimensional surface shape dataset were numerically obtained. About 300,000 surface points on each coke sample were obtained with a minimum resolution of 0.43 mm. The obtained coordinates were converted to standard triangulated language, and the surface shape was polygonal with a triangular mesh. The natural packed structure comprising these particles can be obtained by a DEM-based scheme. The basic format of the DEM is to track spherical particles. Multi-sphere (MS) DEM is a method utilizing a DEM contact force model that is expanded to handle the motion of freely shaped solids. It arrays spherical particles and expresses complex shapes to enable intuitive mounting. The position and rotation angles of each coke sample were determined by using a pseudorandom number, and the packed bed structure was determined by the MS-DEM simulation of a box-type container with 0.12-m sides, similar as a previous report [43]. Next, two immiscible liquids

with a 0.12-m width and 0.02-m height were placed immediately above the packed bed to simulate the liquid iron-molten-slag trickle flow in the coke-packed bed. Position of liquid iron and molten slag was determined by using a pseudorandom number. The volume ratio was set as 8:2 to achieve the conditions similar to that of the actual operation. In SPH simulations for multi-phase flow that include a gas, the pressure differential becomes large between liquid and gas, and, consequently, the pressure gradient becomes excessive, thus making convergent calculation difficult. Therefore, in this research, a gas phase is assumed to exist in spaces where particles do not exist. The calculation domain is depicted in Figure 2. The well-known Courant–Friedrichs–Lewy condition was applied to determine *dt*. In this study, a calculation particle diameter *dp* of 1.00 mm was adopted as a constant value in all calculation processes. Thus, the following values were determined: *dt* = 1.0 <sup>×</sup> <sup>10</sup>−<sup>5</sup> s, the analysis time is 10 s, and the number of total particles is 1,402,280. All the programs were coded by the author. Each computer code was written in Fortran 90/95 and compiled and executed by an Intel Fortran compiler on a Windows system. The CPU used in this work was Intel Core i7-7820 × (3.6 GHz, 8 cores). A basic parallelized algorithm was applied using a multi-core processor and OpenMP. Simulation time was 1 week or more.

**Figure 2.** Schematic diagram of initial conditions: (**a**) three-dimensional view and (**b**) horizontal view of packed structure and liquid iron and molten slag.
