Mass transfer of sulfur from liquid iron into lime-saturated CaO-Al2O3-MgO-SiO2 slags

Laboratory experiments on the kinetics of sulfur transfer from aluminium-deoxidized liquid iron into lime-saturated CaO-Al₂O₃-MgO-SiO₂ slags were carried out at 1600°C in MgO crucibles with 1500 g iron and 180 to 250 g slag. The mass-transfer coefficients of sulfur were determined under defined flow conditions of liquid metal induced by gas stirring. A square-root dependence of measured rate constants on flow velocity of metal was found. This is interpreted as normal liquid-liquid mass transfer caused by the forced convection of gas stirring. It could be described by boundary-layer theory applied to the metal-slag interface. With increasing sulfur contents of metal mass-transfer coefficients became larger. This is interpreted as mass transfer by interfacial convection superimposed to normal mass transfer. Interfacial convection was confirmed by microscopic observation of quenched metal-slag interface samples. If the sulfur content in liquid iron exceeds a critical value, calcium sulfide crystals precipitate in slag at the slag-metal interface. The precipitation inhibits generation of interfacial convection.     

Kinetics of iron oxide reduction from CaO-MgO-FeOn-SiO2 slags by silicon dissolved in liquid iron

At steelmaking temperatures, the kinetics of slag-metal reactions is usually determined by mass transfer. This occurs in two ways: normal mass transfer which is induced by stirring, and mass transfer by interfacial convection induced by interfacially active elements like oxygen and sulphur. In the present work, mass transfer during the reduction of iron oxide from a basic slag by silicon dissolved in liquid iron was studied under defined conditions of gas stirring by argon in MgO crucibles with 1500 g iron and 250 g slag. The variations of the FeO content in the slag and the silicon content in the iron during the reaction were measured by sampling. Trials were carried out with stirring gas flow rates between 1 and 20.4 l/h(STP). The experimental data were evaluated with the multi-component transport model in order to determine the mass transfer coefficients of the reaction components. Simultaneously, the coefficients of normal mass transfer were calculated with the boundary layer theory of liquid-liquid mass transfer for non-turbulent flow conditions. The measured mass transfer coefficients were by a factor 2.5 larger than the theoretically calculated. The difference indicates the presence of mass transfer by interfacial convection. Mass transfer by interfacial convection is superimposed to normal mass transfer.     

Water model study on inclusion removal from liquid steel by bubble flotation under turbulent conditions

Abstract

Inclusion removal from liquid steel by bubble flotation under turbulent conditions is analysed using a water model. Turbulence is realised by impeller stirring in a water containing vessel. First, the effects of variables such as filter pore size, gas flowrate, NaCl concentration, and stirring intensity on bubble size are investigated. Second, particle removal by bubble flotation is studied using the water containing vessel system. The results indicate that particle removal rate by bubble flotation is controlled by non-first order kinetics. The factors affecting the particle removal rate constant k ₁ are discussed and a final empirical equation is derived as follows: -dc/dt = k ₁ c^1·3665 and k ₁ = A(d _p/d _B)^2·65?0·104 Q _g^1·630, where c is particle number density, t is time, A is a constant parameter, d _p and d _B are the particle and bubble diameter respectively, ? is the turbulent energy dissipation rate, and Q _g is the gas flowrate.     

Mixing and solid-liquid mass-transfer rates in a creusot-loire uddeholm vessel: A water model case study

The process of mixing and solid-liquid mass transfer in a one-fifth scale water model of a 100-ton Creusot-Loire Uddeholm (CLU) converter was investigated. The modified Froude number was used to relate gas flow rates between the model and its protoype. The influences of gas flow rate between 0.010 and 0.018 m³/s and bath height from 0.50 to 0.70 m on mixing time were examined. The results indicated that mixing time decreased with increasing gas flow rate and increased with increasing bath height. The mixing time results were evaluated in terms of specific energy input and the following correlation was proposed for estimating mixing times in the model CLU converter: T _mix=1.08Q ^−1.05 W ^0.35, where Q (m³/s) is the gas flow rate and W (tons) is the model bath weight. Solid-liquid mass-transfer rates from benzoic acid specimens immersed in the gas-agitated liquid phase were assessed by a weight loss measurement technique. The calculated mass-transfer coefficients were highest at the bath surface reaching a value of 6.40 × 10⁻⁵ m/s in the sprout region. Mass-transfer coefficients and turbulence parameters decreased with depth, reaching minimum values at the bottom of the vessel.     

Dissolution of solid antimony in molten bismuth under static conditions

The dissolution of solid antimony in molten bismuth was studied under static and isothermal conditions using the reaction couple and immersion methods between 623 and 773 K. The dissolution of antimony under the antimony upper position, using reaction couple method, was governed by diffusion, and the diffusion coefficient of antimony in molten bismuth was obtained as follows:D _L = 2.08 X 10-⁸exp(-ll kJ mol^-1/RT), (m²/s). The dissolution of antimony under the immersed condition was governed by the natural convection resulting from the density difference in the melt, and the dissolving antimony was distributed uniformly by natural convection in molten bismuth. The apparent activation energy for the dissolution of solid antimony in molten bismuth was the same as that for the diffusion of antimony in the melt.     

Mass transfer during dissolution of a solid into liquid in the iron-carbon system

The rates of dissolution of solid pure iron and a solid iron-carbon alloy into molten iron-carbon alloys were studied “isothermally” as a function of temperature, carbon content and fluid dynamic (stirring) conditions. Experiments were carried out in a resistance-heated tube furnace under an inert atmosphere of argon gas at temperatures ranging from 2165 to 2563°F. A cylindrical specimen of the solid was preheated to the liquid temperature, immersed in the liquid bath contained in a graphite or alumina crucible, and rotated at speeds ranging up to 1800 rpm. The diameter of the specimen after partial solution was determined by direct measurement or calculated from the weight loss of the specimen. The experimental data were fitted with nondimensional correlations of mass transfer for a vertical cylinder, both stationary and rotating. When the specimen was stationary the mass transfer was dominated by natural convection in the liquid bath. When the specimen was rotated, forced convection prevailed and controlled the mass transfer rate. This paper is based in part on a thesis submitted by Y-U. Kim in partial fulfillment of the requirements for the degree Doctor of Philosophy at the University of Michigan.     

Model study of mixing and mass transfer rates of slag-metal in top and bottom blown converters

Water model experiments have been conducted to clarify mixing rates of molten steel and mass transfer rates between slag and metal in LD and Q-BOP furnaces using six different circular tuyere arrangements. Splashing and ‘spitting’ were also examined with a view to finding a quiet bath with minimum mixing time and maximum mass transfer rate. Froude’s similarity criterion was fulfilled to determine gas flow rate and bath depth. Complete mixing time of water determined by tracer technique had been 0.9 second to 1.8 seconds for Q-BOP as compared to 6 seconds to 13 seconds for LD. This shows that the stirring intensity in Q-BOP is remarkably larger than that of LD. A simple relationship τ = 5.9(Q/N) ^−0.49 was obtained with gas flow rateQ and number of tuyereN. This indicates that flow rate of gas per tuyere should be intensified to realize better mixing. Mass transfer coefficient K_Ba for bottom blowing was found to be almost double that for top blowing. Of all the tuyere configurations studied for Q-BOP’s, a half circular tuyere arrangement was found to be the best considering all aspects of mixing, mass transfer, and bath agitation.     

Hydrodynamics of gas stirred melts: Part I. Gas/liquid coupling

A hydrodynamic model of submerged gas injection systems and their effects on liquid metal stirring is presented. It is argued that hydrodynamic conditions at the nozzle, tuyere, or plug are not critical to flow recirculation produced in large cylindrical vessels(i.e., furnaces or ladles). An analysis of a buoyancy driven plume generated through gas injection shows that gas voidages are usually quite low (less than 10 pct). By equating the energy supplied by rising bubbles to turbulent energy losses within the bath, it is shown that mean plume velocities can be predicted using the relationship,U _p α (Q ^1/3 L ^1/3)/R^1/3 whereU _p equals mean plume velocity,Q is gas flow rate (at mean height and temperature),L is depth of liquid, andR is radius of the vessel. Associated rates of liquid turnover as a function of vessel dimensions and gas flow rate can also be predicted and these are similarly presented.     

Mixing time and correlation for ladles stirred with dual porous plugs

Bulk mixing times up to a degree of 95 pct were measured in three different, cylindrical-shaped water model ladles (D=0.60 m, 0.45 m, and 0.30 m, respectively) in which, water was agitated by air introduced through two tuyeres/nozzles placed diametrically opposite at the base of the vessels at ±1/2 R positions. To this end, the electrical conductivity measurement technique was applied. A range of gas flow rates and liquid depths were investigated (viz. 0.7≤L/D≤1.2 and 0.002≤ɛ _m (watt/kg)≤0.01) and these were so chosen to conform to the practical ladle refining conditions. In the beginning, extensive experimental trials were carried out to assess the reliability of the measurement technique. In addition, some experiments were carried out to determine the location of the probe in the vessel such that measured mixing times could be interpreted as the bulk mixing times. It was observed that for smaller gas flow rates (or specific energy input rates), 95 pct bulk mixing times tend to decrease appreciably with increasing gas flow rates (e.g., τ _mix∞Q ^−0.58. However, for relatively higher flow rates, the dependence was found to be less pronounced, mixing times decreasing nearly in proportion to a third power of gas flow rates. Similarly, it was found that there exists a critical gas flow rate for any given vessel beyond which mixing times in dual plug stirred configuration are somewhat shorter than those in equivalent axi-symmetrical systems. A dimensional analysis followed by multiple regression of the experimental data (for ɛ _m ≥0.07 W/kg) indicated that mixing times in ladles fitted with dual plugs located diametrically opposite at ±R/2 locations could be reasonably described via τ _{mix, 95 pct}=15Q ^−0.38 L ^−0.56 R ^2.0 in which L is the depth of liquid (m), R is the vessel radius (m), and Q is the ambient flow rate (referenced to mean height and temperature of the liquid). Finally, the adequacy and appropriateness of the correlation was demonstrated with reference to the experimental data derived from a 0.20 scale, tapered cylindrical-shaped water model of a 140 T industrial ladle as well as scaling equations and modeling criteria reported in the literature.     

Experimental investigation of dissolution rates of carbonaceous materials in liquid iron-carbon melts

Interactions of carbonaceous materials in liquid Fe-C melts have been investigated experimentally by determining the rates of dissolution at temperatures ranging from 1623 to 1935 K. The rates of dissolution of spectroscopic graphite and an industrial coke obeyed the correlation for natural convection under turbulent conditions. The experimental data for the graphite suggested that the rate of dissolution was controlled by mass transfer in liquid boundary layer adjacent to the solid sample. The value of the empirical parameter correlating the dissolution coefficient and the operating variables was found to be 0.19, which was close to that reported in the literature. The comparison of the results obtained for coke and low-volatile coal char samples with those for the graphite revealed that impurities and porosity of the samples can effect the dissolution rates. The values ofk ₁, for coke decreased with increasing the dissolution time. The examination of some of the partially dissolved coke samples by electron micro-scopy revealed that a thin, viscous ash layer was forming on the sample surface, which must be the main reason for the behavior. The dissolution rates were controlled by both mass transfer and phase boundary reactions when sulfur was present in the bath. The extent of devolatilization and dissolution of coal particles when they were injected into an Fe-C melt depended on the particle size and location. Formerly Visiting Professor, Massachusetts Institute of Technology. Formerly with the Department of Materials Science and Engineering, Massachusetts Institute of Technology Formerly with the Department of Materials Science and Engineering, Massachusetts Institute of Technology, is deceased.     

Rates of dissolution of solid iron,cobalt, nickel,and silicon in liquid copper and diffusion rate of iron from liquid Cu-Fe alloy into liquid copper

Consideration is given to the previously reported data on the observed dissolution rate of the vertical peripheries of cylindrical iron, cobalt, and nickel in liquid copper at temperatures in the range of 1468 to 1653 K under natural convection. The observed steady-state rate is close to the rate calculated from an equation expressed in terms of the activity of solute for dissolution rate controlled by mass transfer through a boundary layer in the liquid. The dissolution rate of the horizontal bottom face of cylindrical cobalt in liquid copper is determined at 1573 to 1574 (±6) K in the absence of fluid flow. The decrease in height of the cylinder, z(m), obeys z = α √t where t is time (s) and α is 4.18 × 10⁻⁶ m · s^−1/2. This value is in close agreement with the calculated one based on an equation expressed in terms of the activity of solute for dissolution rate controlled by non-steady-state diffusion in the liquid. Experiments have been carried out to explore the dissolution rates of one vertical edge of a square silicon plate and the horizontal bottom face of a silicon disk in liquid copper at ∼1473 K. The rate at the lowest part of the vertical edge ranges from 39 to 69 mol · m⁻² · s⁻¹, and the calculated rate based on the former equation is close to the average of the maximum and minimum observed values. The ratios of these observed dissolution rates to those of iron, cobalt, or nickel under natural convection are in the range of 74 to 910. In the dissolution of the horizontal bottom face, the value of α is estimated from another equation in which the densities of solid and liquid different from each other and the activity of solute are taken into account. This estimated value is in the range of the observed ones. The total amounts of iron diffused from liquid Cu-Fe alloy into liquid copper within capillaries at 1574 K are determined. Their average is expressed as a function of the activity of iron.     

Kinetics of mass transfer of manganese and silicon between liquid iron and slags

The kinetics of mass transfer of Mn and Si between liquid iron and slags were investigated in laboratory experiments at 1600°C in MgO crucibles with 1500 g iron and 250 g slag. Three different slags consisting of CaO-MgO-MnO-SiO₂, MgO-MnO-SiO₂ and MgO-MnO-Al₂O₃-SiO₂ were used. The concentration-vs.-time curves, experimentally measured under defined flow conditions generated by gas stirring, were evaluated by application of a multi-component transport model in order to obtain the mass transfer coefficients. The numerical values of the thus determined measured mass transfer coefficients were compared with values calculated by a theory of mass transfer at liquid-liquid interfaces. The measured and theoretical values were in good agreement with each other in the case of reduction of MnO from the slag by Si in the metal, provided that the measurements had been carried out below a critical stirring intensity, above which metal droplets were emulsified in the slag. Experiments, where sulphur was dissolved in the metal melt and where the sulphur contents were systematically varied, showed no changes of the mass transfer coefficient in comparison to sulphur-free melts. The experimental mass transfer coefficients for the reduction of silica from the slag by manganese in the metal were smaller than those calculated by the mentioned mass transfer theory. This could be explained by inhibition of surface renewal under the influence of solid reaction products precipitated at the interface.     

Reaction mechanism of gold dissolution with bromine

The dissolution of gold with elemental bromine was studied by using a rotating disc technique. The main parameters studied were bromine and bromide concentrations, stirring speed, pH, and temperature. The effect of various salts, manganese, and hydrogen peroxide was also examined. The dissolution kinetics of gold with Br₂ and NaBr mixture is complex. The reaction mechanism is a function of solution composition, which determines the kind of adsorbing species. For an excess concentration of bromide ions, the rate expression is Rate = (2k _cl7 k _al6)^1/2 K ₁₅ [Br ₃ ⁻ ] and for an excess concentration of bromine, the rate expression is Rate = (2k _c27 k _a29)^1/2 [Br⁻]^1/2 {K₂₅ [Br₂]³/(1 +K ₂₅ [Br₂]³)}^1/2 Gold in bromine solutions dissolves according to electrochemical/chemical (EC) mechanisms. The electrochemical component of the mechanism is responsible for the formation of AuBr₂. In the chemical component of the mechanism, this monovalent gold bromide disproportionates into gold and stable AuBr ₄ ⁻ , which reports into solution. With respect to pH, there are two characteristic dissolution regions. In the pH range of 1 to 7, gold dissolution rates were insensitive to pH. Above pH 7, gold dissolution rates decreased with increase of pH.     

Mass transfer between a horizontal,submerged gas jet and a liquid

The rate of reaction between a horizontal, submerged gas jet and a liquid has been measured in a model system under conditions where mass transfer in the gas phase is rate limiting. The gas was 1 pct SO₂ in air, and the liquid was a 0.3 pct solution of hydrogen peroxide in water. SO₂ absorption rates were measured as a function of jet Reynolds number (10,000 < N_Re < 40,000) and jet orifice diameter (0.238 < d₀ < 0.476 cm). The product of the gas phase mass transfer coefficient and the interfacial area per unit length of jet trajectory, k_SO2 α was found to increase linearly with increasing Reynolds number and to be a strong function of the orifice diameter. The ratio of k_so2 α to volumetric gas flow rate was shown to be independent of Reynolds number for a given orifice diameter. Extrapolated values of k_so2 α are lower than the coefficients measured for vertical CO jets blown upward through liquid copper. Extrapolation of the measured mass transfer data to the jet conditions in copper matte converting and in the gaseous deoxidation of copper has indicated that the gas utilization efficiencies in these processes should approach 100 pct if gas phase mass transport is rate controlling.     

Kinetics of reduction of ferric iron in Fe2O3-CaO-SiO2-Al2O3 slags under argon, CO-CO2, or H2-H2O

The rates of reduction of ferric iron in Fe₂O₃-CaO-SiO₂-Al₂O₃ slags containing 3 to 21 wt pct Fe₂O₃ under impinging argon, CO-CO₂, or H₂-H₂O have been studied at 1370 °C under conditions of enhanced mass transfer in the slag using a rotating alumina disc just in contact with the slag surface. For a 6 wt pct Fe slag at a stirring speed of 900 rpm the observed reduction rates by 50 pct H₂-H₂O were a factor of 2 to 3 times higher than those by 50 pct CO-CO₂ and more than one order of magnitude higher than those under pure argon. The observed rates were analyzed to determine the rate-controlling mechanisms for the present conditions. Analysis of the rate data suggests that the rates under 50 pct H₂-H₂O are predominantly controlled by the slag mass transfer. The derived values of the mass-transfer coefficient followed a square-root dependence on the stirring speed for a given slag and, at a given stirring speed, a linear function of the total iron content of the slags. The rates of oxygen evolution during reduction under pure argon were shown to be consistent with a rate-controlling mechanism involving a fast chemical reaction at the interface and relatively slow mass transfer in the gaseous and the slag phases. The rates of reduction by CO-CO₂ (pCO=0.02 to 0.82 atm) were found to be likely of a mixed control by the slag mass transfer and the interfacial reaction. A significant contribution of oxygen evolution to the overall rates was observed for more-oxidized slags and for experiments with relatively low values of pCO. Assuming a parallel reaction mechanism, the estimated net reduction rates due to CO were found to be of the first order in pCO, with the first-order rate constants being approximately a linear function of the ferric concentration. This article is based on a presentation made in the “Geoffrey Belton Memorial Symposium” held in January 2000, in Sydney, Australia, under the joint sponsorship of ISS and TMS. The original symposium appeared in the October 2000 Vol. 31B issue.     

Kinetic model for reduction of iron oxide in molten slags by iron-carbon melt

Rate of reduction of iron oxide in iron and steelmaking slags by mass contents of dissolved carbon (>3%) in molten iron depends upon activity of FeO, temperature, mixing of bulk slag and other experimental conditions. A general kinetic model is developed by considering mass transfer of FeO in slag, chemical reaction at gas-metal interface and chemical reaction at gas-slag interface, respectively, as the three rate controlling steps. A critical analysis of the experimental data reported in literature has been done. It is shown that in the case of slags containing mass contents of less than 5% FeO, the reduction of FeO is controlled by mass transfer of FeO in slag plus chemical reaction at gas-metal interface; when slags contain more than 40% FeO, the reduction of FeO is controlled by chemical reaction at gas-metal interface plus chemical reaction at gas-slag interface; at intermediate FeO mass contents (between ～ 5 and 40% FeO), the reduction of FeO is controlled by all three steps, namely, mass transfer of FeO in slag, chemical reaction at gas-metal interface and chemical reaction at gas-slag interface. The temperature dependence of rate constant for the gas-slag reaction is obtained as: In k₂ = –32345.4(&6128)/ T + 19.0(&3.42); σ_lnk2,1/T = &0.3. where k₂ is expressed in mol m^-2 s^-1 bar^-1. The mass transfer coefficient of iron oxide in bulk slag is found to vary in the range 1.5 × 10^-5 to 5.0 × 10^-5 m/s, depending upon the slag composition as well as experimental conditions.     

Reaction rates for sulfur fixation with iron at 1100 to 1275 K

The rates of sulfidizing iron in a simulated coal gasification atmosphere were studied. Mixtures of H₂S and CO were passed through fixed beds of coal char and prereduced iron ore, and effluent gas compositions were measured as a function of time. These mixtures ranged from 2.5 pct to 10 pct H₂S at various flow rates, with temperatures from 1100 K to 1275 K and iron ore sizes from 10 mesh down to 100 mesh. Experimental conditions were established to form a steady state reaction profile in the fixed bed. Analysis of the exit gas provided a measurement of the profile. The slope of the profile was used directly as a measure of the reactivity of the solids in the bed. The development of this experimental technique and its experimental design requirements are discussed. The observed sulfidization rate of thein situ reduced iron ore is characterized by a single rate constantm (minutes^-1), which varies primarily with temperature and particle size and is substantially independent of gas flow rate, bed configuration, and H₂S content of the incoming gas. Accordingly, the rate constant m can be applied in the design of a combined sulfur fixation, coal gasification reactor to estimate the solids retention time, and the minimum mass of iron required per cross sectional area of reactor. CRAIG B. SHUMAKER, formerly a Graduate Student in the School of Materials Engineering, Purdue University     

Effects of process conditions on mixing between molten iron and slag in smelting reduction vessel via water model study

Abstract

The present study has been conducted to investigate the effects of operating conditions, which include gas flowrate, tuyere size, tuyere number, and height of iron phase, on the extent of mixing between molten iron and molten slag in the direct iron ore smelting reduction process. A transparent acrylic water model, 30% of the size of the actual smelter, was constructed to study the mass transfer phenomena. In the water model, water and spindle oil were used to simulate molten iron and molten slag, respectively, while air was used to replace the bottom blown nitrogen gas. In addition, thymol (C₁₀H₁₄O₆) was used as a tracer material in the water model, added to the water at the beginning of the experiment. As mixing between water and spindle oil proceeded owing to stirring by the bottom blown gas, the concentration of thymol in the water decreased and that in the spindle oil increased. Water samples were taken from the bottom and 12 cm above the bottom of the water model at various operating times. Concentrations of thymol were then measured using a diode array ultraviolet visible spectrophotometer. By analysing the concentration data, the mass transfer rate k_wA, which is a direct index for evaluating the mixing efficiency, could be derived. The process conditions under investigation included 40-500 L min^-1 gas flowrate, 0·3^-1 cm tuyere size, four or five tuyeres, and 20-30 cm height of the water phase. The test results indicate that when the gas flowrate increases, the value of k_wA increases, which indicates better mixing between oil and water phases. However, as the gas flowrate approaches 40 L min^-1, the improvement becomes less obvious. The smaller tuyere gives better mixing, and the design of five tuyeres results in better mixing compared with four tuyeres when they are blown with the same total gas flowrate. However, mixing efficiency decreases with increased height of the water phase. Also, as the gas flowrate of bottom blowing approaches 40 L min^-1, gas blowing from the top has little effect on the mixing behaviour in the liquid bath. For a four tuyere system, the process conditions of height of oil phase 5 cm, height of water phase 25 cm, diameter of tuyere 0·75 cm, and gas flowrate for each tuyere 40 L min^-1, appear to be the optimal design.     

Modelling criteria for flow simulation in gas stirred ladles: experimental study

Abstract

Criteria for modelling isothermal flows encountered in typical gas stirred ladles have been investigated both theoretically and experimentally. To this end, the phenomena of fluid mixing in ladles have been investigated in order to deduce the relationship between model and full scale gas flowrates, needed for maintaining dynamic similarity between the two. Starting with the governing equation for material transport, mixing times in geometrically and dynamically similar gas stirred systems were first correlated theoretically. On the basis of this, it is shown that, in the Froude dominated flow regime (typical of industrial ladle refining operations), the ratio of mixing times in geometrically and dynamically similar gas stirred systems can be represented in terms of the geometrical scale factor λ(=L _mod /L _fs ) according to τ_m,mod /τ_{m,fs = λ}^1/2. To assess the adequacy and appropriateness of various scaling equations reported in the literature (namely Q_{mod /Q fs} =λⁿ, proposed values of nbeing 1·5, 2·5, and 2·75, respectively), extensive experimental measurements of mixing times were carried out in three differently sized water model ladles. To measure mixing times, the conventional conductivity measurement technique was adopted. Comparisons of experimental ratios of mixing times with the corresponding theoretical ratio (=λ^1/2 ) confirm that, in the Froude dominated flow regime, the most appropriate criterion for dynamic similarity between model and full scale ladles is Q _mod /Q _fs = λ^2·5. Such findings were also corroborated through consideration of empirical mixing time correlations reported for Froude dominated ladle flows.