**1. Introduction**

Titanium is the fourth most abundant structural metal on Earth, with a content of about 0.6 % in the crust [1]. The most common forms found in nature are rutile (TiO2) and ilmenite (FeTiO3). The density of titanium and its alloys is about 60% of that of nickel or iron alloys, but their tensile strength is comparable; they also have grea<sup>t</sup> corrosion resistance in different environments [2].

The main problem limiting the wider use of titanium and its alloys is the high cost of its production. In the metallurgical industry, electron beam cold hearth melting (EBHM) is often used to produce titanium and titanium alloy ingots; this has an excellent ability to remove high and low density inclusions [3]. This device uses high-energy electron beams to scan the feedstock. The feedstock melts and flows into a cold hearth for refining and then into a crystallizer, where it solidifies and is removed under a high vacuum (Figure 1). Refining is a very important process. Some of the inclusions floating on the surface of the molten pool are melted by the high-energy electron beams and evaporated; some flow into the crystallizer, while others settle on the solidification shell during the cold hearth.

The inclusions with their size, locations, and sources often affect the quality of titanium and its alloy products. With the development of computer technology, many scholars have gradually started using numerical simulations to study the behavior of the inclusions in the mold [4]. Li et al. [5] simulated the fractal profile of two-dimensional and three-dimensional inclusions by using the diffusion-limited aggregation (DLA) model of inclusion growth. Yu et al. [6] calculated the heat transfer, flow, solidification, and movement behavior of the inclusions in molds with different corner structures using mathematical models; they found that inclusions larger than 100 μm were removed easily. Lei et al. [7] used a

solidification model and fluid flow coupled with the discrete phase model (DPM) to predict inclusions trapped in the solidification front of high-speed continuous casting slab and their distribution in the inner surface layer. Liu et al. [8,9] developed a DPM and used a Euler–Euler model to predict the quasi-four phases of argon-steel slag-air in a slab mold.

**Figure 1.** Schematic of electron beam cold hearth melting (EBCHM).

The Ti-0.3Mo-0.8Ni alloy is a near α alloy and has grea<sup>t</sup> properties, such as good welding property, crevice corrosion resistance, and excellent processing plasticity [10]. It is widely used in the chemical industry; its applications include being a crystallizer for salt production, plate brine coolers, heaters in the chlor-alkali industry, tubular reactors for the oxidizing acid tank, a reactor for the treatment of wastewater with high-chlorine content by using wet oxidation, and heat exchanger in treating strong corrosive and thermally concentrated chlorides [11]. Li et al. [12] studied the flow field motion and segregation of the Ti-0.3Mo-0.8Ni alloy in a crystallizer at different casting processes during EBCHM. Truong et al. [13] studied the effect of the electron beam scanning strategy in aluminum volatilization of Ti-6Al-4V during the cold hearth. Bellot et al. [14] used mathematical models to simulate the melting process of titanium in EBCHM and simulated the behavior of hard α inclusions in the melting process, including the dissolution trajectory and kinetics. Kroll-Rabotin et al. [15] investigated inclusion interactions in the plane shear flow, computing the fully resolved hydrodynamics at finite Reynolds numbers, using a lattice Boltzmann method with an immersed boundary method. In order to determine the collision efficiency, different initial conditions, different shear values, and different sizes of inclusions were studied.

The main research focus is the movement of inclusions during the cold hearth; the removal mechanism of different specifications of inclusions by cold hearth can be theoretically modeled and understood. A literature review found that the current study on the movement of inclusions and melt flow in cold hearth is still not involved for the Ti-0.3Mo-0.8Ni alloy during EBCHM; this study will systematically research the movement of inclusions in the cold hearth. The three-dimensional numerical model considers the melt flow field, solidification process, and inclusions movement of the Ti-0.3Mo-0.8Ni alloy; the DPM in ANSYS was used to quantitatively analyze the movement of inclusions in the cold hearth.

The purpose of this study was to analyze the melt flow field and inclusions movement in the cold hearth. Two main aspects were studied in detail: (1) the movement of different inclusions in the cold hearth and (2) the melt flow field of the cold hearth.

#### **2. Mathematical Model**
