**2. Mass Transfer during the Melting Rate of Additions (Solid–Liquid and Gas–Solid–Liquid Systems)**

The effect of the stirring conditions on the mass transfer coefficient (*mtc*) during the melting rate of additions in ladles has been investigated in detail using a rotating cylinder electrode (RCE) immersed in a liquid and has resulted in many semi-empirical correlations involving dimensionless numbers, as shown in Table 1. Most of these correlations involve three dimensionless numbers; Sherwood (Sh), Reynolds (Re), and Schmidt (Sc). This is the result of a simple dimensional analysis, describing the mass transfer coefficient (k) as a function of the fluid's kinematic viscosity (ν), fluid's velocity (U), diffusivity (D) of transferred species, as well as a characteristic length of the reactor (*l*), under isothermal conditions.

$$\mathbf{k} = f(\mathbf{U}, \mathbf{D}, \mathbf{v}, l) \tag{5}$$

This system involves five variables and can be described with two dimensions (L,T), therefore, in accordance with the π-theorem, it can be defined with three π-dimensionless groups, as follows:

$$
\pi\_1 = \mathbf{k}(l)^{\mathbf{a}\_1} (\mathbf{D})^{\mathbf{b}\_1} \tag{6}
$$

$$
\pi\_2 = \mathsf{U}(l)^{\mathsf{a}\_2} (\mathsf{D})^{\mathsf{b}\_2} \tag{7}
$$

$$
\pi\_{\mathfrak{P}} = \mathbf{v}(l)^{\mathfrak{a}\_{\mathfrak{B}}} (\mathbf{D})^{\mathfrak{b}\_{\mathfrak{B}}} \tag{8}
$$

Applying the principle of dimensional homogeneity, the resulting π-groups are:

$$\frac{\text{k}l}{\text{D}} = f \left( \frac{\text{U}l}{\text{D}} \right)^{\text{a}} \left( \frac{\text{v}}{\text{D}} \right)^{\text{b}} \tag{9}$$

Alternatively:

$$\text{Sh} = f \text{(Re}^{\text{a}} \text{Sc}^{\text{b}}) \tag{10}$$

If the stirring conditions produce slag emulsification, the previous analysis should be extended to include surface tension [7]. In the mass transfer model developed by Oeters and Xie [8] this relationship holds for two cases under non-turbulent flow; a liquid in contact with a free surface and a liquid in contact with a solid wall. In the first case the velocity at the interface is the same as the velocity in the bulk and in the second case the velocity of the liquid at the interface is zero. Both are limiting cases for the liquid–liquid interface.

The first systematic correlation involving Sherwood (Sh), Reynolds (Re), and Schmidt (Sc) numbers, describing the *mtc* under turbulent flow without gas injection was reported by Eisenberg et al. in 1955. It has been confirmed to remain acceptable for the dissolution rate of iron into liquid steel by subsequent investigations [9–12]. An important parameter is the velocity of the fluid. If the experiments are carried out without gas injection, that velocity can be estimated from the peripheral velocity of the RCE. Under multi-phase flow conditions, the velocity components of the fluid are needed in order to compute the *mtc*. This information can be obtained with the development of mathematical models [13–16], by direct measurements, for example with particle image velocimetry (PIV) or laser doppler velocimetry (LDV) [17], by photographic analysis [18,19], and also with an energy balance [18].

Another group of correlations have been reported using the mass transfer Stanton number (St). It has been applied in the dissolution rate of solid lime into liquid slag [20–22].

The gas injection position is an important variable because it affects mixing phenomena. Wright [23] reported that the dissolution rate of a steel rod under natural convection was higher when placed in the center in comparison with an off-center position, on the contrary Koria [19] reported a higher dissolution rate when the rod was located off-center, under central bottom gas injection conditions. Alloy additions in the ladle should be made under conditions that enhance its melting rate.


**Table 1.** Mass transfer correlations for solid–liquid and gas–solid–liquid systems.

\* natural convection, where: Sh = kmL/D, St = km/U, Re = ρUL/μ, Sc = μ/ρD, Ti = U2 rms/U0. km is the convective *mtc*, D represents mass diffusivity, μ is the dynamic viscosity of the fluid, ρ is the density of the fluid, U the velocity of the fluid, U0 is the velocity at the center line of the rising two phase plume and Urms is the rms or fluctuating velocity.

#### **3. Mass Transfer due to Gas Absorption (Gas–Liquid System)**

Gas absorption is an important phenomenon in steelmaking that covers the absorption of undesirable gases from the atmosphere. Maruoka et al. [24] investigated the removal of oxygen by water modeling with different layouts of gas injection. The whole experimental data was described by a relationship between the *vmtc* and the product of the ladle eye times the bubble velocity, subsequently, the *vmtc* was defined in terms of the gas flow rate and the number of nozzles <sup>∝</sup> Q0.87N0.13 . Kato et al. [25] measured the absorption of oxygen in a water model and found that the *mtc* is higher for bottom gas injection in comparison with top gas injection. Another group of studies have been carried out on the absorption of CO2 in aqueous-NaOH solutions [26,27]. Inada et al. [26] reported that increasing the number of nozzles decreases the *vmtc* per one nozzle. They compared one, three, and five nozzles. These reports found an exponential relationship between the *mtc* and stirring energy, with an exponent in the range from 0.65 to 0.8. Rui et al. [28] also measured the rate of absorption of CO2 and found that the *mtc* is higher for one oval snorkel in comparison with a circular snorkel, if the nozzle radial position is located between the center and half radius.

Mass transfer in solid–liquid, gas–liquid, and liquid–liquid systems, even if gas stirring is not involved, has many similarities. In all of these systems mass transfer is controlled by diffusion coefficients, velocities (solid, liquid or gas phases), physical properties for the phase involved, etc. At the same time there are important differences. If mass transfer from a solid is involved, the first step is melting and then in a second step is the dissolution process. Mass transfer from a gas phase is different to the case that involves mass transfer due to chemical reaction at the slag/metal interface, not only because the phases involved are different but the chemical reaction itself. The main point is to understand that different variables operate in those processes. In the following sections the review will focus on the previous work that has been developed to identify the variables that affect the rate of mass transfer in gas–liquid–liquid systems.
