**1. Introduction**

Many studies have been conducted to describe transport phenomena in ladles and the results are summarized in several reviews [1–4]. The previous reviews have included mass transfer to a limited extent. The reviews from Mazumdar et al. [1,2] were focused on solid–liquid systems. Sichen [3] discussed limitations of the two-film theory and Liu et al. [4] summarized mixing phenomena by physical and mathematical modeling due to gas stirring in ladles, indicating that a limited number of studies involved mass transfer. Ghotli et al. [5] reviewed liquid–liquid mass transfer in mechanically stirred vessels. Mechanical and gas stirring involve different operational parameters. In steelmaking, in particular in ladle metallurgy, its rate of production and final steel quality rely on mixing efficiency both liquid steel and liquid slag. The current review will primarily focus on the physical and mathematical modeling work that involves liquid–liquid mass transfer due to bottom gas injection.

The molar flux of species j (Nj) is proportional to the concentration gradient according with Fick's law for diffusion without convection and shown below. This form of the equation is valid for dilute systems, typical of steelmaking. The proportionality constant is the *mass transfer coe*ffi*cient* (*mtc*). Depending on the experimental conditions the units for the *mtc* can change. To avoid confusion the units are briefly revised. If the concentration units for Cj are in mol/cm<sup>3</sup> then the units for kj are cm/s [6].

$$\mathbf{N}\_{\rangle} = -\mathbf{D}\_{\rangle} \begin{pmatrix} \frac{\partial \mathbf{C}\_{\rangle}}{\partial \mathbf{y}} \end{pmatrix} = \begin{pmatrix} \mathbf{D}\_{\rangle} \\ \frac{\partial \mathbf{C}\_{\rangle}}{\partial \mathbf{m}} \Delta \mathbf{C}\_{\rangle} \end{pmatrix} = \mathbf{k}\_{\rangle} \begin{pmatrix} \mathbf{C}\_{\rangle}^{b} - \mathbf{C}\_{\rangle}^{cq} \end{pmatrix} \tag{1}$$

where Nj is the molar flux in mol/cm2·s, Cj is the concentration in mol/cm3, Dj is the diffusion coefficient in cm2/s, y is distance in cm, δ<sup>m</sup> is the diffusion boundary layer thickness in cm, kj is the mass transfer coefficient in cm/s. Superscripts *b* and *eq* represent bulk and equilibrium values, respectively. Subscript j represents a chemical species.

Since the molar flux (Nj) is the amount of material transferred per unit area and unit time, the experimental measurement of the *mtc* requires knowledge of the interfacial area (A). In most cases this value is unknown. In this condition, the mass transfer coefficient is reported as the product, (kj·A) called *volumetric mass transfer coe*ffi*cient* (*vmtc*), with units cm3/min. If the volume of the liquid remains constant, we denote this *vmtc* as kj·*a* , with units min−<sup>1</sup>

$$\mathbf{N}\_{\mathbf{j}} = \frac{1}{\mathbf{A}} \frac{\partial \mathbf{n}\_{\mathbf{j}}}{\partial \mathbf{t}} = \mathbf{k}\_{\mathbf{j}} \Big(\mathbf{C}\_{\mathbf{j}}^{b} - \mathbf{C}\_{\mathbf{j}}^{c\eta}\Big) \tag{2}$$

$$\frac{\partial \mathbf{r}\_{\rangle}}{\partial \mathbf{t}} = \mathbf{V} \frac{\partial \mathbf{C}\_{\mathbf{j}}}{\partial \mathbf{t}} \ = \begin{pmatrix} \mathbf{k}\_{\rangle} \mathbf{A} \end{pmatrix} \begin{pmatrix} \mathbf{C}\_{\mathbf{j}}^{b} - \mathbf{C}\_{\mathbf{j}}^{cq} \end{pmatrix} \tag{3}$$

$$\frac{\partial \mathbf{C}\_{\mathbf{j}}}{\partial \mathbf{t}} = \left( \mathbf{k}\_{\mathbf{j}} \cdot \frac{\mathbf{A}}{\mathbf{V}} \right) \mathbf{C}\_{\mathbf{j}}^{b} - \mathbf{C}\_{\mathbf{j}}^{a\eta} \right) \\ = \left( \mathbf{k}\_{\mathbf{j}} \cdot \mathbf{a} \right) \left( \mathbf{C}\_{\mathbf{j}}^{b} - \mathbf{C}\_{\mathbf{j}}^{a\eta} \right) \tag{4}$$

The effect of the stirring conditions on the mass transfer coefficient has been experimentally measured for different systems: (i) gas–solid–liquid system: a typical example is the melting of additions by mechanical stirring (solid-liquid system) or by gas stirring (gas-solid-liquid system), (ii) gas–liquid system: some examples are gas absorption from the atmosphere into liquid steel, absorption of elements when different types of gases are injected (nitrogen, carbon dioxide, etc.) or oxygen injection for decarburization, and (iii) gas–liquid–liquid system: the main examples are slag/metal interfacial reactions. To clarify terms used in this work, in the gas–liquid system the gas is a reacting species that dissolves in the liquid in contrast to the gas–liquid–liquid system where the gas is an inert species and where an impurity dissolved in the lower liquid phase diffuses to the upper liquid phase. This review will focus primarily on the study of the *mtc* in the gas–liquid–liquid system, however, as an introduction the other systems are briefly reviewed in the beginning in Sections 2 and 3, followed by a detailed review in Section 4 on liquid–liquid mass transfer involving both physical and mathematical modeling studies. Section 5 provides a final assessment of our current understanding on this subject and suggest guidelines for further research.
