*2.2. Corrosion Model*

In an isothermal closed liquid metal loop, corrosion may come to a halt because the corrosion products achieve saturation. However, in a non-isothermal loop where the saturation concentration of the corrosion products depends on temperature, continuous dissolution will occur in the relatively high-temperature part, whereas deposition of corrosion products will occur in the relatively low-temperature part. A kinetic equilibrium will be reached in which the amount of corrosion is balanced by the amount of deposition; thus, the precipitation sustains the corrosion [29]. This is particularly pertinent where metal corrosion occurs in an HLM environment, as this is a physical or physical-chemical process, which involves species dissolution, species transport, and chemical reaction between corrosion products and impurities, rather than an electrochemical process, as is usually the case in an aqueous environment [21,22].

In LBE environment, the corrosion process of a steel can be divided into two types according to the oxygen concentration in LBE [23,29]:

(1) If the oxygen concentration in LBE is sufficiently low, as in the present experimental study, there will be no effective oxide protective layer formed on the steel surface. In this situation, the steel contacts with the LBE directly, and the main constituents of the steel are thus dissolved into the LBE directly.

(2) If the oxygen concentration is within an appropriate range, oxidation of steel will occur, and an active oxide film (*Fe*3*O*4-based) will eventually be formed on the steel surface. Direct dissolution of steel into LBE will be prevented due to separation of the oxide film. In this case, the iron diffuses from the base metal, and the oxygen transfers from the bulk flow to the oxide/LBE interface to engage in the oxidation–reduction chemical reaction ( 3*Fe* + 2*O*<sup>2</sup> ⇔ *Fe*3*O*<sup>4</sup> ).

For long-term steady-state LBE loop operation without an oxide protective layer, the steel corrosion process in the control depends on the LBE flow speed. If the LBE flow speed is rapid enough, resulting in the mass transfer rates of constituents in liquid greater than their dissolution reaction rates at the solid/LBE interface, then the corrosion will be controlled by the dissolution rate; otherwise, it will be controlled by the mass transfer rate in fluid [21,23]. For the mass transfer-controlled corrosion, if the diffusion flux of the corrosion product in the solid is less than the mass transfer rate in the liquid, surface recession will consequently occur [22].

The selective corrosion of Cr and Ni in the steel superficial layer is common for a 316L SS contacting with LBE that contains a low oxygen concentration. Subsequently, a phase change from austenite to ferrite might occur in the selective corrosion layer, due to the depletions of Cr and Ni in this layer, and as a result, a ferritic layer might be formed on the steel surface, where the concentrations of Cr and Ni are considerably low [30,31]. Both austenite and ferrite crystal mainly consist of iron, so that the dissolution of iron atoms will liberate other element atoms of the crystal into LBE, and therefore, it is reasonable to assume that the corrosion rate of iron determines the surface recession rate [22,32]. Hereafter, the surface recession rate is referred to as corrosion rate (CR) and the surface recession depth is called corrosion depth. The concentration of Cr and Ni in the solid at the solid/LBE interface can be assumed to be zero [22]. By contrast, the concentration of Fe in the solid at the solid/LBE interface is assumed to be equal to its saturation solubility in LBE, which will be described and discussed in detail in Section 4.2.

The corrosion process is illustrated schematically in Figure 3, and described by the following expression [33]:

$$Fe\_{(s)} \iff Fe\_{(Sol)}\tag{5}$$

The dissolved iron is transferred into the diffusion boundary layer, where the transfer of iron is governed by the molecular diffusion process, and thus, the transfer rate is low. Above the diffusion boundary layer in the viscous sublayer, where the transfer rate of iron grows dramatically. The thickness of the viscous sublayer is determined by the turbulence level of LBE flow, while the thickness of the diffusion boundary layer for each constituent is determined by not only the turbulence level of LBE flow, but also the Schmidt number (*Sc*) of each constituent. Therefore, the thickness of diffusion boundary layers of each constituent are independently different from each other. Therefore, lack of Ni and Cr involvement in the present corrosion model has insignificant effects on the thickness of the viscous sublayer and the diffusion boundary layer of Fe.

For a fully developed flow in a simple geometry, the mass transfer coefficient of a constituent across the wall can be indicated by the Sherwood number, *Sh* = *aRebScc*. However, for complex geometries, there are no such empirical relationships, and CFD analysis must be used to aid its calculation.

The mass flux of iron, *JFe*, from the reaction location to the bulk flow can be given by

$$J\_{F\varepsilon} = \mathcal{K}\_{\mathfrak{C}} (\mathcal{C}\_{\mathfrak{w}} - \mathcal{C}\_{\mathfrak{b}}),\tag{6}$$

where *Kc* is the mass transfer coefficient, *Cw* is the concentration of iron at the wall, and *Cb* is the concentration of iron in the bulk fluid. In the simulation, if the first node (the layer of node closest to the wall in the mesh structure) is located in the diffusion boundary layer, then mass transfer between the wall and the first node is controlled solely by molecular diffusion. The effects of locations of the first node are insignificant, and can be positioned extremely close to the wall or at the end of the diffusion boundary layer. According to Fick's law, the mass flux between the wall and the first node can be expressed as

$$J\_{F\varepsilon} = \frac{D\_m}{y\_0} (\mathbb{C}\_w - \mathbb{C}\_0)\_\prime \tag{7}$$

where *y*<sup>0</sup> is the distance from the wall to the first node, *Dm* is the molecular diffusion coefficient of iron, and *C*<sup>0</sup> is the iron concentration at the first node.

As mentioned previously, the rate of mass transfer in the diffusion boundary layer is very low, so it controls the total rate of mass transfer from the wall to the bulk flow. Therefore, the mass flux of iron is the same in Equations (7) and (6). By substituting Equation (7) into Equation (6), the mass transfer coefficient can be expressed as

$$K\_{\mathbb{C}} = \frac{D\_m}{y\_0} \frac{(\mathbb{C}\_w - \mathbb{C}\_0)}{(\mathbb{C}\_w - \mathbb{C}\_b)}. \tag{8}$$

In diffusion-controlled mass transfer, the diffusion coefficient of iron to or from the reaction site controls the rate of corrosion. The corrosion rate (in mm/h) can thus be defined as follows [34,35]:

$$CR = \frac{K\_c(\mathbb{C}\_w - \mathbb{C}\_b)M\_{F\varepsilon}}{\rho\_{F\varepsilon}} = \frac{D\_m(\mathbb{C}\_w - \mathbb{C}\_0)M\_{F\varepsilon}}{y\_0\rho\_{F\varepsilon}} \times 60 \times 60 \times 1000,\tag{9}$$

where *MFe* is the molar mass of iron, and *ρFe* is the density of iron. In the above equation, *C*<sup>0</sup> can be obtained in the numerical simulation by solving the mass transport equations described in the last subsection.

**Figure 3.** Schematic drawing of steel corrosion in lead–bismuth eutectic (LBE) at low oxygen concentration.
