Metallurgical and Materials Transactions B

, Volume 51, Issue 1, pp 412–416

# Discussion on “Review of Physical and Numerical Approaches for the Study of Gas Stirring in Ladle Metallurgy”

As research work in this area continues, it is perhaps premature to side with a particular class of flow modeling procedures, given that exhaustive comparisons between industrial/pilot scale operations and numerical predictions are still largely pending. Indeed, depending on the phenomena of interest, even a very simple flow modeling approach can work satisfactorily. For example, if the primary goal of numerical computations is to predict bulk liquid mixing rates in an inert gas-stirred ladle, the quasi-single-phase procedure is known to be equally effective as the more elaborate, two-phase flow modeling approaches.

Referring to Figure 3, it is seen that two different, complimentary approaches have been advocated by Liu et al., for predicting de-S and mass transfer phenomenon from the first principles. Since slag–metal mass transfer was not included in the scope of their review, some discussion and analysis of this are therefore naturally warranted. Mass transfer coefficients, as well as slag–metal interfacial areas, as one would readily acknowledge, are the corner stone in the computation of melt-phase, transport-controlled, mass transfer processes. Recent work of Hoang et al. indicate that mass transfer coefficient can be estimated from the small eddy theory in terms of specific turbulence kinetic energy dissipation rate, ε, and the kinematic viscosity of the bulk liquid, ν, according to:

$$k = 0.4\sqrt D \left( {\frac{\varepsilon }{\nu }} \right)^{0.25}$$
(1)

An appropriate turbulence model is therefore a pre-requisite to the prediction of mass transfer coefficient in gas-stirred ladle systems. Mass transfer obeys first-order kinetics and the corresponding rate expression is conveniently represented as:

$$\mathop m\limits^{\text{o}} = - \frac{{{\text{d}}C_{i} }}{{{\text{d}}t}} = k\left( {\frac{{A_{\text{int}} }}{V}} \right)\left( {C_{i} - C_{i}^{e} } \right)$$
(2)

In Eq. , C is the molar concentration (kg mol/m3), i represents the transferring species, $$\mathop m\limits^{\text{o}}$$ is the mass transfer rate (kg mol/m3 s), k is the mass transfer coefficient (m/s), Aint is the interfacial area (m2), and V is the volume of the melt (m3). Evidently, an estimate of Aint is needed in addition to Eq. , if the mass transfer rate is to be predicted via Eq. . Toward this, the need for a multiphase flow modeling procedure, such as VOF + DPM or E–E, is readily apparent, as has been appropriately pointed out by Liu et al. It is, however, important to mention that in a vast majority of two-fluid mass transfer studies of ladle metallurgy steelmaking, the estimation of slag–metal interfacial area has not been rigorous. Rather, it is generally deduced by considering an artificially enhanced, planar slag–metal interfacial area. In such a context therefore, the adequacy of two- or three-phase modeling procedures, such as the VOF + DPM or the E–E, and various auxiliary models, is not known with any certainty. Clearly, more experimental work and plant scale measurements, as well as numerical simulations, are needed to substantiate such model study programs further. In addition to the above, the recommendation of a VOF + laminar flow model, as suggested in Figure 3 (see beneath the base of the ladle schematic) for investigating bubble behavior in the plume zone, does not appear to be sufficiently convincing and needs some clarification, since bubbles are known to exacerbate turbulence within an upwelling gas–liquid plume.

Two- (melt–gas) and three-phase (melt–slag and gas) computations of gas-stirred ladle systems have been reported by many researchers in recent years and most of these have been referred to by Liu et al. (see, for example, Tables IX and X in Reference 4). There, the authors emphasized four different flow modeling approaches, i.e., (i) quasi-single-phase approximations, (ii) VOF (volume of fluid), (iii) Euler–Euler (E–E), and (iv) Eulerian–Lagrangian (E–L), giving practically no attention to the newly suggested, VOF + DPM flow calculation procedure. A concise description and analysis of the combined VOF + DPM approach is therefore presented here: the VOF and the DPM calculation procedures, as one will note here, rely on fundamentally different concepts. As such, their straight-forward coupling, as advocated by many investigators, may not be scientifically sound. To substantiate this observation, the governing equations of mixture and discrete-phase motions, in the coupled VOF + DPM formalism, represented in compact vector notation, are considered below;

Equation of mixture motion:
$$\frac{\partial }{\partial t}\left( {\rho_{\text{mix}} \bar{v}_{\text{mix}} } \right) + \nabla .\left( {\rho_{\text{mix}} \bar{v}_{\text{mix}} \bar{v}_{\text{mix}} } \right) = - \nabla p + \nabla .\left( {\mu_{\text{e}} \nabla \bar{v}_{\text{mix}} } \right) + S_{{v_{\text{mix}} }} + \rho_{\text{mix}} \overline{g} + \overline{F}_{\text{B}} + f_{\sigma }$$
(3)
Equation of discrete-phase motion:
$$\frac{{{\text{d}}\bar{V}_{b} }}{{{\text{d}}t}} = \bar{F}_{\text{drag}} + \left( {1 - \frac{{\rho_{\text{l}} }}{{\rho_{\text{g}} }}} \right)\bar{g}$$
(4)
The two preceding equations are then explicitly coupled via the drag force term $$\bar{F}_{\text{drag}}$$, defined as:
$$\bar{F}_{\text{drag}} = \frac{3\mu }{{4d_{\text{b}}^{2} \rho_{\text{g}} }}C_{\text{D}} \text{Re} \left( {\bar{V}_{\text{liq}} - \bar{V}_{\text{b}} } \right)$$
(5)
Note that the drag force per unit volume of the liquid (represented as $$\bar{F}_{\text{B}}$$), and the drag force per unit mass ($$\bar{F}_{\text{drag}}$$), appearing in Eqs.  and  above, are interrelated. Furthermore, the Reynolds number, Re, appearing in Eq.  is expressed as:
$$\text{Re} = \frac{{d_{\text{b}} \rho_{\text{liq}} }}{\mu }\left| {\bar{V}_{\text{liq}} - \bar{V}_{\text{b}} } \right|$$
(6)

Estimation of Re and $$\bar{F}$$drag (and hence, $$\bar{F}_{\text{B}}$$), as seen from Eqs.  and , requires, from a rigorous stand point, the liquid velocity, $$\bar{V}_{\text{Liq}}$$, but this latter entity is not known explicitly from the flow equation (i.e., Eq. ), which is only specifically applicable to the gas–liquid mixture phase. As such, the VOF model is formulated on the basis of a single velocity scale, i.e., the mixture velocity. Therefore, in the combined VOF + DPM calculation scheme, such as those employed in the metallurgical engineering literature, the Reynolds number and the drag force in the bubbly region ($$\bar{F}_{\text{B}}$$) can, at best, be estimated on the basis of a “mixture velocity.” Incorporation of a “mixture velocity,” in lieu of a liquid velocity, to estimate Re and $$\bar{F}_{\text{B}}$$ is an ad hoc strategy and appears to be a serious conceptual limitation of the combined VOF + DPM formulations advocated recently in metallurgical engineering literature. Experimentally measured gas–liquid flows in water models are needed, as they could provide further insight and may justify, albeit not from a phenomenological stand point, the practicality of such approximations.

The three major reviews, published during the last three decades and a half, confirm significant progress on physical and mathematical modeling of ladle metallurgy steelmaking to show that sufficiently advanced mathematical models have been formulated and their implications worked out successfully. Despite such advances, modeling of non-isothermal, reacting, multiphase flow phenomena, which are of paramount importance in actual industrial practice, have been few and far between. It is therefore still not known with any certainty, to what extent different classes of mathematical models perform in real-life process simulations. While efforts must continue to include more complex physics into existing models and thereby enhance their predictive capabilities further, the need for parallel water modeling, as well as high temperature field trials, should not be neglected, or undermined. Widespread application and validation of mathematical models on the shop floor are the need of the hour, so one can make a pragmatic assessment of the various advances in the published literature.

So, as a final point, the archival importance of the review by Liu and coworkers, according to the present authors, could have been increased substantially, had a brief summary of practical ladle metallurgy steelmaking been included as a preamble. Similarly, a concise and critical account of notable previous contributions on such issues as gas–liquid coupling, bubble distribution and voidage, long- and short-term wandering of the plume, turbulence production by bubbles, etc., would have helped in maintaining continuity with previous reviews. It is also equally desirable that an authoritative review, beyond compiling the state of the art, also points out various gaps in the literature and the preferred directions along which future research ought to move. These suggestions are advanced, not as criticisms, but to improve the articulation of future review articles, so as to render them more comprehensive and effective.

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