A Unified Approach to the Fluid Dynamics of Gas–Liquid Plumes in Ladle Metallurgy

Abstract

The paper focuses on the fundamental aspects of the fluid mechanics of unconstrained gas-liquid plumes relevant to Ladle Metallurgy Practice. A mathematical model previously proposed by the authors is justified by comparison with Eulerian-Lagrangian models. Further, a unified analytical framework is proposed to describe the fluid dynamic and similarity characteristics of two-phase plumes. Despite the apparent complexity of the system, the analysis demonstrates that the plume cross-sectional area, void fraction and gas and liquid velocities can be quantified in terms of two parameters: a dimensionless gas flow rate and the normalized axial height. The analysis clarifies the proper form of the Froude number similarity which is important for process design and scale-up.KEY WORDS: gas-liquid plume; ladle metallurgy; mathematical model; similarity analysis. above equations properly capture available experimental data. An additional outcome of the work is that this analysis is linked with dimensional analysis to provide a unified framework for gas-liquid plumes relevant to Ladle Metallurgy practice. Dimensionless Representation of Two-phase PlumesThe analysis starts by considering all the possible variables which can be broadly categorized as geometric factors (ladle diameter, height, and porous plug/lance location), physical properties (densities and viscosities) and external factors (gas flow rate and gravity). Some of these variables can be eliminated from the analysis, as discussed below.The diameter of the ladle has been shown in a survey of previous work 5,6) to be large enough to not affect plume dynamics. Furthermore, the plug is usually located far enough from the wall, so as not to interact with it, so the plume is free-rising. 7,8) It has been shown elsewhere that ladle flows are typically Froude dominated and viscous effects have only a secondary effect on the plume and bulk flows, 9,10) so the viscosity can be neglected. The gas and liquid densities drop out of the analysis, as explained in Appendix A. Interfacial tension is not considered because bubbles in water models as well as liquid metal systems are generally in the spherical-cap regime in which shape and velocity are determined by drag and buoyancy. Figure 2 shows schematically that the remaining independent variables: gas flow rate Q, metal height H, vertical position z and gravitational acceleration g influence the dependent plume variables: crosssectional area, A p , void fraction, ā, and liquid and gas velocities, Ū l and Ū g . The overbar indicates that these quantities are averaged over the plume cross sectional area.It is now convenient to introduce two dimensionless quantities that contain only the independent variables: Mathematical AnalysisThe present model (Eqs. (1) to (4)) is essentially a onedimensional model in which the variables change over the height of the bath, but are averaged over the cross-section of the plume. The conventional Eulerian-Lagrangian models are usually solved in 2 or 3 dimensions with computational fluid dynami...

Recent Advances in the Fluid Dynamics of Ladle Metallurgy

“…1. In previous work by the present authors on the dynamics of two-phase plumes, 1) it was proven that only two dimensionless quantities are required to characterize free-rising plumes as in ladle metallurgy applications. These are the non-dimensionalized gas flow rate (Q 2) where Q is the gas flow rate, H is the metal height, z is the The plume Froude number arose from the non-dimensionalization of the equation for momentum in the free-rising part of the plume.…”

“…These are the non-dimensionalized gas flow rate (Q 2) where Q is the gas flow rate, H is the metal height, z is the The plume Froude number arose from the non-dimensionalization of the equation for momentum in the free-rising part of the plume. In that study the plume Froude number was mathematically related to Q * : 1) . Further, by using the mathematical descriptions of twophase systems, viz., the well-known Eulerian-Lagrangian or bubble tracking models and momentum and continuity for the phases, a similarity framework was developed to correlate all the major plume parameters to Q * and z * on a fundamental basis.…”

“…Further, by using the mathematical descriptions of twophase systems, viz., the well-known Eulerian-Lagrangian or bubble tracking models and momentum and continuity for the phases, a similarity framework was developed to correlate all the major plume parameters to Q * and z * on a fundamental basis. 1) In this framework, the similarity parameters for the void fraction, plume radius, liquid and gas velocities, respectively are: (8) where the overbar indicates that the quantity is averaged over the plume cross-section and the superscript * indicates that that the variable is non-dimensional.…”

Scaling Physical Model Study on Dynamic Accumulation of Bottom Dross in a Molten Zinc Pot

“…7,9,10) The other maintained similar movement of inclusions on the basis of modified Froude number 11,12) or plume Froude number. 13,14) Few studies on physical model experiments for dynamic behaviors of bottom dross were conducted in literatures. The unique public report on interrelated research gotten by authors was performed by Jun Kurobe.…”

A Simple Mathematical Model for Estimating Plume Hydrodynamics of Metallurgical Ladles

“…Also, the feasibility and rationality of a proposed CFD model is probably more attributed to the joint adjustment of the term coefficients that typically differ from one model to another and therefore, the correlations based on air-water system are of minor importance. Figure 1 further illustrates the comparisons between measured data of average plume velocity in a 60-tonne ladle by Hsiao and co-workers 4) and the corresponding predicted ones using the correlations (for gas fraction, gas and liquid velocities at plume centerline) by Krishnapisharody and Irons, 5,6) who recently suggested a 'plume' Froude number for guaranteeing full similarity between a water model and its prototype. The average plume velocity in Fig.…”