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vivo:departmentOrSchool "Applied Science, Faculty of"@en, "Materials Engineering, Department of"@en ;
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dcterms:creator "Sahajwalla, Veena"@en ;
dcterms:issued "2010-09-16T22:09:22Z"@en, "1988"@en ;
vivo:relatedDegree "Master of Applied Science - MASc"@en ;
ns0:degreeGrantor "University of British Columbia"@en ;
dcterms:description """The spout formed at the free surface of a gas-stirred liquid has received little attention even though it has both theoretical and practical significance. In steelmaking ladles, for example, the spout is the site of strong metal-slag-air mixing which affects: the kinetics of reactions at the slag-metal interface, the absorption of oxygen by the bath and the temperature drop of the bath. Notwithstanding its importance, the spout is usually neglected in flow models of gas-stirred baths because it has not been characterized quantitatively; assumption of a flat top surface, however, reduces the accuracy of the velocity and kinetic energy predictions, particularly close to the spout region.
Thus in this study, the spout of upwardly injected gas jets in water was characterized
experimentally in terms of gas fraction, bubble frequency and axial velocity distributions. The measurements were made with a two-element electroresistivity probe coupled to a microcomputer. Special hardware and software were developed to analyze
the signals generated by contact of the bubbles with the sensor, in real time, for the turbulent flow conditions prevailing in the jet plume and spout. Correlations of the gas fraction with axial and radial position for different gas flow rates have been established from the measurements. The dimensions of the spout were obtained from time-exposure photographs; when compared with the gas fraction measurements, the spout boundary always corresponded to values ranging from 0.82 to 0.86. The radial profiles of bubble frequency at different levels in the spout have a bell shape; the bubble frequency decreases with increasing height. The velocity of the bubbles in the spout drops linearly with increasing axial position. Measurements of bath velocity near the walls of the vessel were also conducted with a laser doppler velocimeter for comparison to model predictions.
The gas fraction data obtained for the spout then were incorporated into a mathematical
model of turbulent recirculatory flow with which predictions of velocity, kinetic energy and effective viscosity in the bath were made. Predictions of the model were compared with the experimental measurements as well as with predictions assuming a flat bath surface (no spout); and the importance of incorporating the spout thus was demonstrated. The variation of the total kinetic energy in the spout with gas flow rate was determined. The energy increased with flow rate, as expected, but at a critical value, the rate of increase abruptly rose. Based on photographs taken of the gas/liquid dispersion, the increased spout kinetic energy appears to be related to the location of bubble break-up and possibly to gas channeling. At lower flow rates below the critical
value, the bubble break-up occurs relatively close to the nozzle, whereas at higher flow rates bubble disintegration is nearer to the surface. At the lower flow rates the gas/liquid interaction was maximum which promoted the gas/liquid momentum transfer.
Moreover, at the higher flow rates the gas dispersion was observed intermittently to be a continuous chain of large envelopes which could permit a fraction of the gas to channel through the bath for a considerable distance. The channeling phenomenon could lead to an inefficient gas/liquid energy transfer resulting in a reduced efficiency of bath mixing and enhanced energy release at the surface. These results can explain the observations of previous investigators who found that beyond a critical gas injection rate, the rate of decrease of mixing time with flow rate decreased. The metallurgical consequences of the spout and its influence on the flow field, especially in the near-surface region, have been highlighted, thus unveiling the practical bearing of the spout on the gas injection process."""@en ;
edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/28515?expand=metadata"@en ;
skos:note "T H E S P O U T O F A I R J E T S U P W A R D L Y I N J E C T E D I N T O A W A T E R B A T H By Veena Sahajwalla B. Tech. (Metallurgical Eng.) Indian Institute of Technology, Kanpur, 1986 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES METALS AND MATERIALS ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 1988 © Veena Sahajwalla, 1988 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date 15TH AUG '8% Abstract The spout formed at the free surface of a gas-stirred liquid has received little attention eventhough it has both theoretical and practical significance. In steelmaking ladles, for example, the spout is the site of strong metal-slag-air mixing which affects: the kinetics of reactions at the slag-metal interface, the absorption of oxygen by the bath and the temperature drop of the bath. Notwithstanding its importance, the spout is usually neglected in flow models of gas-stirred baths because it has not been characterized quantitatively; assumption of a flat top surface, however, reduces the accuracy of the velocity and kinetic energy predictions, particularly close to the spout region. Thus in this study, the spout of upwardly injected gas jets in water was char-acterized experimentally in terms of gas fraction, bubble frequency and axial velocity distributions. The measurements were made with a two-element electroresistivity probe coupled to a microcomputer. Special hardware and software were developed to ana-lyze the signals generated by contact of the bubbles with the sensor, in real time, for the turbulent flow conditions prevailing in the jet plume and spout. Correlations of the gas fraction with axial and radial position for different gas flow rates have been established from the measurements. The dimensions of the spout were obtained from time-exposure photographs; when compared with the gas fraction measurements, the spout boundary always corresponded to values ranging from 0.82 to 0.86. The radial profiles of bubble frequency at different levels in the spout have a bell shape; the bubble frequency decreases with increasing height. The velocity of the bubbles in the spout drops linearly with increasing axial position. Measurements of bath velocity near the walls of the vessel were also conducted with a laser doppler velocimeter for comparison ii to model predictions. The gas fraction data obtained for the spout then were incorporated into a mathe-matical model of turbulent recirculatory flow with which predictions of velocity, kinetic energy and effective viscosity in the bath were made. Predictions of the model were compared with the experimental measurements as well as with predictions assuming a flat bath surface (no spout); and the importance of incorporating the spout thus was demonstrated. The variation of the total kinetic energy in the spout with gas flow rate was determined. The energy increased with flow rate, as expected, but at a critical value, the rate of increase abruptly rose. Based on photographs taken of the gas/liquid dispersion, the increased spout kinetic energy appears to be related to the location of bubble break-up and possibly to gas channeling. At lower flow rates below the criti-cal value, the bubble break-up occurs relatively close to the nozzle, whereas at higher flow rates bubble disintegration is nearer to the surface. At the lower flow rates the gas/liquid interaction was maximum which promoted the gas/liquid momentum trans-fer. Moreover, at the higher flow rates the gas dispersion was observed intermittently to be a continuous chain of large envelopes which could permit a fraction of the gas to channel through the bath for a considerable distance. The channeling phenomenon could lead to an inefficient gas/liquid energy transfer resulting in a reduced efficiency of bath mixing and enhanced energy release at the surface. These results can explain the observations of previous investigators who found that beyond a critical gas injection rate, the rate of decrease of mixing time with flow rate decreased. The metallurgical consequences of the spout and its influence on the flow field, especially in the near-surface region, have been highlighted, thus unveiling the practical bearing of the spout on the gas injection process. iii Table of Contents Abstract ii Table of Contents iv List of Tables vii List of Figures viii List of Symbols xiii Acknowledgement xv 1 Introduction 1 2 Literature Review 4 2.1 Physical and Mathematical Modelling Studies . 4 2.2 Mass-Transfer Studies 8 2.3 Mixing-Time Studies 9 3 Objectives of the Present Work 10 4 Experimental 12 4.1 Injection System 12 4.2 Electroresistivity Probe and Data Acquisition 13 4.3 Photographic Observations 15 4.4 Laser Doppler Velocimetry 16 iv 4.5 Experimental Conditions Investigated . . . 16 5 Observations and Experimental Correlation 18 5.1 High Speed Cinematographic Observation of the Spout 18 5.2 Boundary of the Spout 18 5.3 Gas-Fraction Profiles 19 5.3.1 General Equation for Gas Fraction in the Spout 20 5.4 Bubble-Frequency Profiles 22 5.5 Profiles of Axial Velocity 23 6 Mathematical Model 55 6.1 Governing Equations 55 6.2 Two-Phase Plume . 57 6.3 Two-Phase Spout 58 6.4 Boundary Conditions 58 6.5 Numerical Calculations 60 7 Model Predictions and Discussion 62 7.1 Comparison between Model Predictions and Measurements 62 7.2 Profiles of Turbulent Kinetic Energy and Viscosity Distribution 64 7.3 Turbulent Kinetic Energy of the Spout 65 7.4 Potential Energy of the Spout 66 7.5 Proposed Mechanism 67 7.6 Experimental Evidence 69 7.6.1 Axial Bubble Velocity Variation 69 7.6.2 Axial Bubble Frequency Variation 69 7.6.3 Mixing Time Measurements 70 v 8 Metallurgical Consequences 97 8.1 Turbulence at the Surface 97 8.2 Reactions at the Slag-Metal Interface 98 8.3 Mixing in the Bath 100 8.4 Variables Influencing the Critical Gas Flow Rate 100 8.5 Critical Value of Specific Input Power Density in the Bath 101 8.6 Critical Flow Rate in an Operating Ladle 102 9 Summary and Conclusions 106 Bibliography 108 vi List of Tables 4.1 Experimental conditions of the study 17 6.2 Values of constants in A; — e Turbulence Model 57 8.3 The variables and the critical specific input power density in two gas-stirred systems 101 8.4 Comparison between critical flow rates used in practice with values cal-culated from the critical specific input power density 103 vii List of Figures 4.1 Schematic diagram of experimental facility 17 5.2 Tracings of high-speed photographs of the spout, with its height remain-ing constant during the cycle 24 5.3 Tracings of high-speed photographs of the spout, with its height varying during the cycle 25 5.4 Time-exposure photographs of the spout (a) Q = 876 and (b) Q = 1257 Ncm 3 s - 1 26 5.5 Radial gas-fraction profiles at different axial distances from the nozzle in the spout, for Q = 371 Ncm 3s _ 1 27 5.6 Radial gas-fraction profiles at different axial distances from the nozzle in the spout, for Q = 876 Ncm 3s _ 1 28 5.7 Radial gas-fraction profiles at different axial distances from the nozzle in the spout, for Q = 1257 Ncm 3 s - 1 29 5.8 Contours of the gas void fraction in a vertical plane passing through the centreline of the spout, for Q = 371 Ncm 3 s - 1 30 5.9 Contours of the gas void fraction in a vertical plane passing through the centreline of the spout, for Q = 876 Ncm 3 s _ 1 31 5.10 Contours of the gas void fraction in a vertical plane passing through the centreline of the spout, for Q = 1257 Ncm 3 s _ 1 32 5.11 Normalized radial gas-fraction profiles in the spout at different axial distances from the nozzle, for Q = 371 Ncm 3 s - 1 33 viii 5.12 Normalized radial gas-fraction profiles in the spout at different axial distances from the nozzle, for Q = 876 Ncm 3 s _ 1 34 5.13 Normalized radial gas-fraction profiles in the spout at different axial distances from the nozzle, for Q = 1257 Ncm 3 s _ 1 35 5.14 Variation of the parameter \"A\" in Eq.(5.2) with reduced axial distance from the nozzle, for the spout 36 5.15 Variation of the parameter \"n\" in Eq.(5.2) with reduced axial distance from the nozzle, for the spout 37 5.16 Variation of axial gas fraction with distance from the nozzle 38 5.17 Variation of the core radius, r 0 , with reduced axial distance from the nozzle in the spout 39 5.18 Variation of the gas fraction at the spout centreline, am,-n, with reduced axial distance from the nozzle in the spout 40 5.19 Normalized radial gas-fraction profiles in the spout for different flow rates at 400 mm from the orifice 41 5.20 Normalized radial gas-fraction profiles in the spout for different flow rates at 410 mm from the orifice 42 5.21 Normalized radial gas-fraction profiles in the spout for different flow rates at 420 mm from the orifice 43 5.22 Normalized radial gas-fraction profiles in the spout for different flow rates at 430 mm from the orifice 44 5.23 Normalized radial gas-fraction profiles in the spout for different flow rates at 440 mm from the orifice 45 5.24 Normalized radial gas-fraction profiles in the spout for different flow rates at 450 mm from the orifice 46 ix 5.25 Radial bubble-frequency profiles in the spout at different axial distances from the nozzle, for Q = 371 N c m 3 s _ 1 . 47 5.26 Radial bubble-frequency profiles in the spout at different axial distances from the nozzle, for Q = 876 N c m 3 s - 1 48 5.27 Radial bubble-frequency profiles in the spout at different axial distances from the nozzle, for Q = 1257 N c m 3 s _ 1 49 5.28 Radial bubble-frequency profiles at the static level of the bath 50 5.29 Contours of the bubble frequency in a vertical plane passing through the centreline of the spout, for Q = 371 N c m 3 s _ 1 . 51 5.30 Contours of the bubble frequency in a vertical plane passing through the centreline of the spout, for Q = 876 N c m 3 s - 1 . 52 5.31 Contours of the bubble frequency in a vertical plane passing through the centreline of the spout, for Q = 1257 N c m 3 s _ 1 53 5.32 Profiles of axial velocity in the spout for the three gas flow rates 54 6.33 Schematic of the gas-stirred bath illustrating the boundary conditions used in the mathematical model 61 7.34 Predicted velocity profile in the bath at Q = 371 N c m 3 s _ 1 71 7.35 Predicted velocity profile in the bath at Q = 876 N c m 3 s _ 1 72 7.36 Predicted velocity profile in the bath at Q = 1257 N c m 3 s _ 1 . 73 7.37 Comparison between measurements and model predictions of near-wall bath velocity at Q = 371 N c r n V 1 74 7.38 Comparison between measurements and model predictions of near-wall bath velocity at Q = 876 N c m 3 s _ 1 75 7.39 Comparison between measurements and model predictions of near-wall bath velocity at Q = 1257 N c m 3 s _ 1 76 x 7.40 Comparison between measurements and model predictions of axial jet velocity at Q = 371 N c m 3 s _ 1 77 7.41 Comparison between measurements and model predictions of axial jet velocity at Q = 876 N c m 3 s _ 1 78 7.42 Comparison between measurements and model predictions of axial jet velocity at Q = 1257 N c m 3 s _ 1 79 7.43 Contours of predicted turbulent kinetic energy (Joules/kg) distribution in the plume, at Q = 371 N c m 3 s _ 1 80 7.44 Contours of predicted turbulent kinetic energy (Joules/kg) distribution in the plume, at Q = 876 N c m 3 s _ 1 ' 81 7.45 Contours of predicted turbulent kinetic energy (Joules/kg) distribution in the plume, at Q = 1257 N c m 3 s _ 1 82 7.46 Contours of predicted turbulent kinetic energy (Joules/kg) distribution in the spout, at Q = 371 N c m 3 s _ 1 83 7.47 Contours of predicted turbulent kinetic energy (Joules/kg) distribution in the spout, at Q = 876 N c m 3 s _ 1 84 7.48 Contours of predicted turbulent kinetic energy (Joules/kg) distribution in the spout, at Q = 1257 N c m 3 s _ 1 85 7.49 Contours of predicted effective viscosity non-dimensionalized with molec-ular viscosity in the plume, at Q = 371 N c m 3 s _ 1 86 7.50 Contours of predicted effective viscosity non-dimensionalized with molec-ular viscosity in the plume, at Q = 876 N c m 3 s - 1 . 87 7.51 Contours of predicted effective viscosity non-dimensionalized with molec-ular viscosity in the plume, at Q = 1257 N c m 3 s _ 1 88 7.52 Contours of predicted effective viscosity non-dimensionalized with molec-ular viscosity in the spout, at Q = 371 N c m 3 s _ 1 89 x i 7.53 Contours of predicted effective viscosity non-dimensionalized with molec-ular viscosity in the spout, at Q = 876 Ncm 3 s - 1 90 7.54 Contours of predicted effective viscosity non-dimensionalized with molec-ular viscosity in the spout, at Q = 1257 Ncm s s - 1 91 7.55 The variation of predicted total turbulent kinetic energy in the spout with flow rate 92 7.56 The variation of measured 2ghmax with flow rate 93 7.57 Photographs of the gas/liquid dispersions at a flow rate less than the critical value 94 7.58 Photographs of the gas/liquid dispersions at a flow rate higher than the critical value 95 7.59 The variation of local bubble frequency at the axis with height above the nozzle at varying modified Froude number as measured by Castillejos and Brimacombe [2,3] 96 8.60 Variation of the calculated critical flow rate with bath height for an 80 ton ladle , 104 8.61 Variation of the calculated critical flow rate with bath height for a 250 ton ladle 105 xii List of Symbols A d0 h Fr 9 G k Ks,Ki, Ki Kz, K4, K5 n Pi Patm Q , Q o Parameter in Eq.(5.2). Constants in the k — e turbulence model. Orifice diameter, mm. Local bubble frequency, s - 1 . T0, Tp ^amax/2 Modified Froude number — -TTT V Acceleration due to gravity, (9.81 m/s2). Generation term in the turbulence model. Bath depth; maximum height of the spout, mm. Turbulent kinetic energy, Joules/kg. Desulphurization rate constant; Constants in Eq.(5.3). Constants in Eq.(5.8). Parameter in Eq.(5.2); subindex in Eq.(4.1) indicating class interval with the largest central value in bubble velocity histogram. Pressure; atmospheric pressure, kg/m s2. Gas flow rate at STP; gas flow rate at given conditions (atmospheric pressure plus static head of water and 20°C), em's'1. Distance in the radial direction, mm. Core radius; statistic defined in Eq.(4.1). Half-value radius, mm. x i i i TL Temperature of the liquid. r Mixing time. Uax Vertical velocity of the fluid at the static level of the bath, along the axis. Uf, Local bubble velocity, m/s. Ur Radial component of mean velocity, m/s. Uz Axial component of mean velocity, m/s. W Weight of the bath. x Non-dimensionalized axial distance = z(-^j)1^ z Vertical distance from the nozzle, mm. a Local time-averaged gas fraction. &max Local gas fraction at plume centreline. ocmin Local gas fraction at spout centreline. e Dissipation rate of turbulent kinetic energy, J/kg s. e Mixing power density. es Specific input power density, Watts/kg. PL,PG Density of liquid; density of gas at orifice, kg/m 3. crjfc Effective Prandtl number for k. o~e Effective Prandtl number for e. Ii Molecular viscosity, kg/m s. Heff Effective viscosity, kg/m s. lit Turbulent viscosity, kg/m s. v Turbulent kinematic viscosity, m2/s. xiv Acknowledgement I would like to express my sincere gratitude to my supervisors, Professor J.K. Brima-combe and Professor M.E. Salcudean for their continuous assistance and guidance dur-ing the course of this research. My sincere thanks to Dr. A.H. Castillejos for his help in modifying the software and getting me aquainted with the experimental facility. My appreciation to Dr. P.E. Anagbo, Mr. Z. Abdullah, Mr. B. Hernandez, Mr. P. Wenman and to those fellow graduate students who helped me in my work. The generous support of the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. Finally I want to express my gratitude to my husband, R.B. Mahapatra, for his help and understanding during the course of my work; and to my mother Dr. Indira Sahajwalla for her encouragement without which it would have been difficult for me to initiate my graduate studies. xv Chapter 1 Introduction The importance of ladle metallurgy in steelmaking has now become established, as the demand for a better quality, clean steel increases. Ladle metallurgy involves various operations including homogenization with respect to temperature and chemical com-position, desulphurization, deoxidation, injection of alloying elements and removal of inclusions. In order to conduct these operations, submerged gas injection is the most widespread technique that has been utilized. Submerged gas injection can be achieved by various means through a tuyere, porous plug or lance; the gas may be injected from the bottom, top, laterally or in a combined mode. Although submerged gas in-jection techniques have been widely adopted owing to the efficient mixing of the bulk liquid, the large surface area produced for reaction, easy control of process variables such as gas flow rate, relative simplicity and low capital costs, a number of problems remain. These include severe bath slopping, refractory wear, poor utilization of certain additions, reoxidation and excessive cooling of the bath. To improve our understanding of these injection phenomena, physical and math-ematical modelling studies, which can give a clearer insight into the complex nature of the fluid flow in these systems, are necessary. A review of the literature, presented in the next chapter, shows that a gas-stirred melt, with a centrally placed nozzle at the bottom of the vessel has been extensively studied. Physical modelling has given a qualitative picture of the fluid flow in the bath, and experimental measurements have helped identify regions of high turbulence. Mathematical modelling has made it 1 Chapter 1. Introduction 2 possible to make quantitative predictions regarding mixing and homogenization times as well as regions of high turbulent kinetic energy which lead to accelerated refrac-tory wear. Over the years the mathematical models being developed have improved greatly. The turbulence in the bath has been studied utilizing the k — e model; the two-phase plume has been experimentally investigated and correlations, which can be used in models, have been developed to represent this region [1,2,3]. These are some of the difficult aspects in mathematically modelling turbulent recirculatory flow in a gas-stirred liquid. There are several areas in these systems which have not been addressed, one of which is the free surface of the liquid; this is the focus of attention in the present study. This free surface, which has a significant bearing on the process, generally has been assumed to be flat in mathematical models. This is a convenient but incorrect representation which can lead to inaccurate model predictions especially near the top of the bath. The free surface is important from a practical viewpoint because excessive turbulence in this region can lead to bath slopping and opening up of the slag layer which exposes the metal to reoxidation and heat losses. Moreover at the slag-metal interface, important reactions like desulphurization and deoxidation occur, and the refractories near the free surface region undergo accelerated wear. These phenomena can be better understood if the free surface is correctly represented in the mathematical model of the turbulent recirculatory flow in a gas-stirred liquid. This may permit better control of some of the process variables such as flow rate which can directly affect the phenomena occurring at the free surface discussed above as well as the efficiency of the overall gas-injection process. The aim of this investigation was to improve our understanding of the free surface, in particular the spout formed in a gas-stirred water bath. In the study, the gas-fraction distribution in the spout was characterized experimentally, and the data obtained then Chapter 1. Introduction 3 were incorporated into a mathematical model to make fluid flow predictions for the bath. Hence the importance of incorporating the free surface in a recirculatory flow model for a gas-stirred liquid bath has been examined. Chapter 2 Literature Review 2 .1 Physical and Mathematical Modelling Studies The gas-stirred liquid bath has been modelled by several investigators, in an attempt to comprehend the hydrodynamics of the system. Szekely and Asai [4,5] modelled the turbulent recirculatory flow in a gas-stirred ladle. Turbulence was simulated by the Kolmogorov-Prandtl one-equation model and qualitative agreement was found be-tween model predictions and experimental observations. Szekely et al. [6] undertook two-dimensional aqueous modelling of an argon-stirred ladle, in which the gas jet was modelled as a solid wall moving upwards at constant velocity; the turbulence was sim-ulated using the k — W model; qualitative agreement was found between measurements and model-predicted velocities [7]. Deb Roy et al. [8] and Grevet et al. [9] reported two-dimensional computations of argon bubbling in vessels, based on buoyancy-driven flow models, in which the plume was assumed to be cylindrical; average plume void fractions were estimated for both slip and non-slip conditions. For the slip condition, the gas was assumed to move with a \"terminal velocity\" relative to the liquid. Salcudean et al. [10] performed calculations for steady-state, buoyancy-induced, three-dimensional flow to simulate gas bubbling in ladles. The k — e model was used to represent turbulence in the bath. Salcudean et al. [12] developed a computational scheme for predicting flow fields and heat-transfer phenomena associated with asymmetric gas-driven flows 4 Chapter 2. Literature Review 5 in systems of cylindrical geometry. Mass, momentum and energy conservation equa-tions were solved numerically based on the Marker-and-Cell technique [13]. Specific solutions were obtained for direct comparison with other two-dimensional model pre-dictions. Flow patterns for centered and off-centered, conical plumes were presented. The convective effect on the temperature fields in an initially stagnant and fully strat-ified fluid due to gas injection at the bottom surface, was illustrated. Salcudean et al. [14] presented a three-dimensional model to simulate the transient temperature field and homogenization time for a gas-stirred liquid in a cylindrical vessel. Velocity fields, temperature distributions and mixing times were obtained using the algebraic models of Pun and Spalding [11], and of Sahai and Guthrie [15] and the k — e turbu-lence model. The velocity fields and mixing times were compared with experimental data. Sahai and Guthrie [16,17] developed a generalized expression for axisymmetric gas stirred systems; plume velocities and the average speed of liquid recirculation were related to vessel diameter, gas flow rate and liquid depth with the aid of an energy balance technique. In all the models mentioned above, the spout formation was not considered and the free surface was assumed to be flat. Tse-Chiang et al. [18] studied fluid flow in a 60 t melt stirred by argon. Direct measurements in liquid steel were conducted from which the surface velocity was found to vary with the gas flow rate to the power of 1/3. Sano and Mori [19] proposed a simple model to characterize fluid flow in a liquid bath agitated by gas bubbles. The model postulated that the bath consists of two zones, a central bubble plume moving upward and a liquid annular zone moving downward. The analysis was based on a steady-state energy balance for the plume, which states that the rate of energy dissipation due to liquid recirculation is equal to the rate of energy input. They assumed that the rate of energy dissipation is equal to the difference between the rate of kinetic energy associated with the liquid moving upward and that Chapter 2. Literature Review 6 of the liquid moving downward plus an energy dissipation due to the bubble slip. The equations then derived permitted the calculation of the liquid velocity in the plume zone, the liquid recirculation velocity and the mixing time as a function of gas flow rate, bath depth and cross-sectional area of the plume. The model predicted that the liquid velocity in the plume was directly proportional to the gas flow rate to the power of 1/3. Tacke et al. [20] applied an integral-profile technique to study vertical gas injection in water and mercury. This technique involved the utilization of the conservation equations of mass for the gas and liquid and the conservation equation for the vertical momentum in integrated form. The profiles of gas concentration and liquid velocity were taken to be Gaussian and the velocity difference between the rising bubbles and the liquid was introduced. The set of differential equations that resulted from the analysis was solved for a region separated from the nozzle where the maximum gas fraction had decreased to fifty percent. Reasonable predictions of axial bubble velocity profiles were obtained. Haida and Brimacombe [21] conducted water-model experiments involving gas in-jection through a single axial tuyere located at the bottom of a cylindrical vessel. They measured the shear stress on the bottom surface of the vessel by an electrochemical technique, and confirmed the presence of a maximum in the variation of the spatial av-erage shear stress with increasing gas flow rate. They attributed it to the phenomenon of \"chanelling\" in the plume region which lowers the efficiency of gas/liquid energy transfer. A comprehensive review of the literature has revealed that the spout of a submerged gas jet, has received little attention by previous researchers. Gray et al. [22] have qualitatively described the formation of two important regions in the upper portion of a gas-stirred bath, namely a disengagement zone and a splash zone. Splashing Chapter 2. Literature Review 7 and spitting were studied by Paul and Ghosh [23] for various tuyere arrangements in physical models of the LD and Q-BOP processes and comparisons between them were drawn. The LD process was found to exhibit greater splashing and spitting. Also, the influence of different injection positions and design of injectors on splashing was investigated by Robertson and Sabharwal [24]. Modelling of transient, free-surface liquid flow utilizing computer techniques such as MAC and SOLA has been applied to the filling of moulds with molten metal by Stoehr [25]. Fluid flow modelling in a water bath, with gas injection through horizontal tuyeres, has been conducted by Shook [26]. The free-surface profiles were obtained by digitizing the traces of the experimental surfaces; the velocity measurements in the bath were made with a laser doppler velocimeter. It was observed that the mean cell kinetic energy increased with flow rate up to 1500 em's\"1, beyond which it decreased abruptly. This phenomenon was found to occur even when the surface cells were not included in the calculation of the kinetic energy, indicating that at higher flow rates, the bulk of the liquid in the bath had received less energy from the incoming gas. Photographs of the gas/liquid dispersions were taken on either side of the transition flow rate mentioned above; the photographs at the flow rate smaller than the critical flow rate showed the presence of distinct bubbles, which did not interfere with each other. At a flow rate higher than the critical flow rate, the photographs indicated the presence of bubbles which coalesced continuously, and the gas channelled its way to the surface. This phenomenon was ascribed to be the cause of reduced efficiency of energy transfer. Recently, Mazumdar et al. [27] employed a physical model of oil and water to simulate slag and steel respectively. Measurements of the total specific kinetic energy of motion in the bath indicated that there was an energy decrease in the recirculat-ing liquid which was enhanced further with increasing flow rate; they proposed three Chapter 2. Literature Review 8 modes of energy dissipation to account for the loss of energy in the bath: 'slag' droplet creation, 'slag' droplet suspension and 'slag/metal' interface distortion. First-principle calculations were made which indicated that the first two modes involved a small mag-nitude of energy dissipation. Only the third mode was large enough to account for the energy decrease in the bath but even then it was still only half the required value; hence other modes accounting for the energy decrease in the bath have to be found. The mechanism responsible for the energy decrease in the bath has to be identified, since it leads to a reduced efficiency of mixing of alloying elements and their dissolution as well as an inefficient homogenization of bath temperature. 2.2 Mass-Transfer Studies The mass-transfer coefficient at slag-metal interfaces stirred by injected gas has been characterized by several investigators [28,29] in different physical models. The relation-ship between the mass-transfer parameter (mass-transfer coefficient x interfacial area), K, and flow rate, Q, has been determined to be K oc Qn. The value of the exponent, n, was found to increase abruptly at a critical flow rate. Hirasawa et al. [30] found that only at very high flow rates does the interfacial area play an important role; at other times, the dominant factor is the metal-side mass-transfer coefficient which, in turn, is governed by the velocity distribution in the spout region. Kim and Fruehan [31] studied liquid/liquid mass transfer with physical models involving oil and water. They found that the recirculation and the entraining depth of oil droplets in the liquid bath are the major parameters which increase mass transfer. This emphasizes again the need to understand the spout region and also to characterize the velocity distribution at the interface accurately. Chapter 2. Literature Review 9 2.3 Mixing-Time Studies The mixing time of refining vessels stirred by gas injection, at various gas flow rates, was studied by Asai et al. [32]. They observed that the mixing time decreased with flow rate according to the relationship: r oc e - 0 6 8 , up to a critical mixing power density beyond which the relationship changed to: r oc e - 0 3 2 , which indicated that the mixing in the bath had become less efficient. They also found that as the bath height decreased, the critical mixing power density at which the mixing became less efficient, decreased; thus at the same input power density the mixing process was more efficient in vessels with greater bath heights. Mazumdar and Guthrie [33] measured the mixing time in a water model of a gas-stirred ladle. They observed that the mixing time decreased with gas flow rate as: r oc e~048, up to a critical gas flow rate; beyond which the dependence of the mixing time on input power density changed to: T ' ' I—'— — 1 — I — ' — ' — I ' 1 1 8 10 12 14 16 18 20 Reduced axial position z [ g / Q 2 ] 1 / s o Figure 5.18: Variation of the gas fraction at the spout centreline, a m , „ , with reduced axial distance from the nozzle in the spout. Figure 5.19: Normalized radial gas-fraction profiles in the spout for different flow rates at 400 mm from the orifice. Chapter 5. Observations and Experimental Correlation 42 1.1 — i — . ' i i ' — i ' i ' — ' ' i 1 2 3 4 Reduced radial position r [g/Q*]1/5 Figure 5.20: Normalized radial gas-fraction profiles in the spout for different flow rates at 410 mm from the orifice. Figure 5.21: Normalized radial gas-fraction profiles in the spout for different flow rates at 420 mm from the orifice. Chapter 5. Observations and Experimental Correlation 44 1.1-c E c c o \"o a V) o cn o 3 •a a H 0.94 0.8-0.7H o.eH _ 0.5H o o o 0.4 0.3-1 L « v e l = * 3 0 m m h b = 4 0 0 m m d Q = 6 .35mm o A + F low Rate 371Ncm5s_ l 878Ncm3s~1 1257Ncm3B\"' 0.2-1 0 .H 4 2lV3 Reduced radial position r [g/0o] Figure 5.22: Normalized radial gas-fraction profiles in the spout for different flow rates at 430 mm from the orifice. Chapter 5. Observations and Experimental Correlation 45 Figure 5.23: Normalized radial gas-fraction profiles in the spout for different flow rates at 440 mm from the orifice. Chapter 5. Observations and Experimental Correlation 46 1.1-0.9-0.8-0.7-0.6-Lev«l=450mm hb=400mm d0=6.35mm A + F l o w R a t e 8 7 6 N c m 3 s \" 1 1 2 5 7 N c m s s \" ' 0.5-0.4-0.3-0.2-0.1-R e d u c e d r a d i a l p o s i t i o n r [ g / QQ] 4 2-,1/S Figure 5.24: Normalized radial gas-fraction profiles in the spout for different flow rates at 450 mm from the orifice. Chapter 5. Observations and Experimental Correlation 47 16 14-12-Q=371Ncm 33 - 1 hh=400mm d0=6.35mm >> 10-c o> z) O\" a> i_ J> XI •5 o o 14- + + • ++++ 12-4-+ + ++ + + + + + ^ A, + + + A AA + + 10- A AAA O O O oo oo o oo o 6- s 0 o O o o o 0 ° o o cv 'o o 4-Level=400mm hb= 400mm dn=6.35mm Flow Rate O 371Ncm3s_1 A 878Ncm3s_1 -|- 1257Ncm3s_1 I 1 1 1 I 50 100 Radial position r,mm 150 Figure 5.28: Radial bubble-frequency profiles at the static level of the bath. Chapter 5. Observations and Experimental Correlation 51 Figure 5.29: Contours of the bubble frequency in a vertical plane passing through the centreline of the spout, for Q = 371 Ncm 3s _ 1 . Chapter 5. Observations and Experimental Correlation 52 o 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 Radial position c m Figure 5.30: Contours of the bubble frequency in a vertical plane passing through the centreline of the spout, for Q = 876 Ncm 3 s - 1 . Chapter 5. Observations and Experimental Correlation 53 e Radial position, cm Figure 5.31: Contours of the bubble frequency in a vertical plane passing through the centreline of the spout, for Q = 1257 Ncm 3s - 1 . Chapter 5. Observations and Experimental Correlation 54 1.6 1.4-h b =400mm dQ = 6.35mm 1.2-1-o 0.8-1 A \\ o X < 0.6-0.4-0.2-F l o w R a t e O 371Ncm3s_I A 87eNcmss_1 + 1257Ncm38\"' i • • • i • • • i • : • i • • • i • • • 340 360 380 400 420 440 460 Distance from the nozzle z,mm Figure 5.32: Profiles of axial velocity in the spout for the three gas flow rates. Chapter 6 Mathemat ical Mode l 6.1 Governing Equations The turbulent recirculatory flow in a water bath, stirred with gas discharging from a bottom-centered nozzle, is assumed to be axisymmetric and buoyancy driven. The turbulence was simulated using the k — e differential model. Two transport equations are written for the turbulent kinetic energy \"k\", and the energy dissipation rate, \"e\" [38]. The two-phase plume was represented by an experimental correlation for gas-fraction distribution developed by Castillejos and Brimacombe [1,2,3]. The spout at the free surface is represented by the experimental correlation developed in the present study. The spatially variable density of the jet in the model was written as [39]: p = apG + (1 - a)pL (6.12) where a = local time-averaged gas-fraction pa = density of gas PL = density of liquid p = spatially variable density 55 Chapter 6. Mathematical Model 56 The following equations were written for the steady-state, axi-symmetrical flow in cylindrical coordinates: Equation of continuity: Id d -fr(prUr) + j;(pu*) = 0 (6-13) Equation of motion in axial direction: 1 d . T T T T . d . t t 2 . dP Id. dUz, -rTr^W + a P ® = S i + rTr^-\"-*\\ n d , dUz, 1 d . dUr, Equation of motion in radial direction: d , dUr. d , dU,. Uru.ff Equation for turbulent kinetic energy: where a* is the effective Prandtl number for k and G is the generation of turbulent kinetic energy expressed as follows: „ , ,,dUr.~ ,dU..» ,Ur.0. ,dUr dU,.~. , - \" ' l 2 \" - ^ ' + + ( f \" + < 1 7 + ~bT)] ( 6 ' 1 7 ) Chapter 6. Mathematical Model 57 Equation for dissipation rate of turbulent kinetic energy: where oe is the effective Prandtl number for e and Ci and C2 are constants. The effective viscosity, fieff is given as; Veff = M + M« (6.19) Ht = -^j— (6.20) After Launder and Spalding [40], the values given in Table 6.2 were assigned to the above-mentioned constants in the model; a variation in the values of these constants was explored and no significant effect on the results was observed. Table 6.2: Values of constants in k — e Turbulence Model. Cx c 2 cd Ok 1.43 1.92 0.09 1.00 1.30 6.2 Two-Phase Plume Since the flow is buoyancy dominated, it is important to have an accurate represen-tation of the two-phase plume. In the present model, an experimental correlation to represent the gas volume fraction in the two-phase plume developed by Castillejos and Brimacombe [1,2,3] was used. The correlation is as follows: Chapter 6. Mathematical Model 58 For N > 4 For N < 4 : = «p[-0.7(- )2 4] (6.21) \" m a x ' a max/2 r ^ / a ^ ) 1 ' 5 = 0 . 2 4 3 [ ( £ % - ^ i ) ° ^ ( f )«>•«] (6.22) < W = 2 9 3 . 7 7 [ ( ^ | — ^ i ) 0 2 6 ^ ^ ) 0 \" 3 ] - 1 (6.23) y o / j G d0 » r Figure 6.33: Schematic of the gas-stirred bath illustrating the boundary conditi used in the mathematical model. Chapter 7 Model Predictions and Discussion The predicted velocity profiles, for the three different flow rates investigated are pre-sented in Figs. 7.34, 7.35 and 7.36 respectively. These show that with increasing flow rate, the size of the spout and the magnitude of the velocity in the spout increases. 7.1 Comparison between Model Predictions and Measurements Figures 7.37 to 7.39 show for the three flow rates investigated, the near-wall velocities, as measured by laser doppler velocimetry, as predicted by the flat surface model (no spout) and the present model (including spout). Similar values for velocities near the walls of the vessel have been reported by Tse-Chiang et al. [18]. The agreement between the present model and measurements is good, whereas the flat-surface model overestimates the velocities. This is understandable because the flat-surface, free-slip model does not allow for the dissipation of kinetic energy in the free surface region. Thus the kinetic energy is over-estimated at the near-wall region. Figures 7.40 to 7.42 show a comparison between axial velocities measured by the electroresistivity probe and predicted by the two models. The flat-surface model can-not predict velocities beyond a certain height in the bath, since it does not consider the formation of the spout at the free surface. Discrepancies between the measurements and the predictions of both models can be observed. The differences arise because the electroresistivity probe measures the velocity of the gas bubbles moving vertically upwards, whereas the model assumes a homogenous, single-phase fluid with a spatially 62 Chapter 7. Model Predictions and Discussion 63 variable density. The model does not account for the relative velocity of the gas with respect to the liquid. Thus, the largest discrepancy is observed near the nozzle, caused probably by the assumption of zero gas jet momentum at the orifice. This assump-tion affects only a small region near the nozzle; since the kinetic energy of the gas is important only close to the nozzle. As the distance above the nozzle increases, the discrepancy is almost constant, since the axial velocity remains essentially unchanged in this region. In the near-surface region, the discrepancy between the present model and the measurements decreases since the gas bubbles undergo a deceleration in the spout. The flat-surface model underestimates the velocities in the near-surface region because the assumption of a non-slip wall requires zero normal velocity so that the fluid is forced to turn and flow radially outward toward the vessel wall. The present model predicts the velocity to be maximum at the axis and to decrease with radial distance, in the near-surface region. A similar radial distribution for the near-surface velocities has been measured by Tse-Chiang et al. [18] at different flow rates. The flat-surface model, on the other hand, predicts an almost uniform near-surface velocity along the radius. The flat-surface model thus underestimates the near-surface velocity at the axis and overestimates the velocity at the wall. The near-surface velocities predicted by the flat-surface model are close to horizontal, whereas those predicted by the present model have a comparatively large axial component and they are seen to deflect downwards, while moving radially outwards, at an angle of thirty to fifty degrees depending on the vertical location of the velocity vector. The incorrect velocity prediction by the flat surface model can have several conse-quences: 1. The velocity distribution obtained from the model is necessary to determine the Chapter 7. Model Predictions and Discussion 64 distribution of temperature and chemical composition in the bath, and to predict the times required for temperature and composition homogenization. Hence, an incorrect velocity distribution will lead to inaccurate predictions for both. 2. The overestimation of velocities in the region near the wall, especially close to the free surface, will result in predictions of locally high turbulence. Consequently the excessive erosion of refractories in that region may be attributed wrongly to high turbulence, rather than to the presence of a highly corrosive slag layer adjacent to the refractories. 3. The near-surface velocity distribution is of foremost importance in controlling the rate kinetics of various reactions like desulphurization and deoxidation taking place at the slag-metal interface, since the velocity of the liquid below the surface determines the rate at which the reactants will be transported toward the inter-face. Therefore a flat surface model can not predict correctly the rate of these reactions. This is because in these systems the reaction is usually controlled by the mass-transfer rate. 7.2 Profiles of Turbulent Kinetic Energy and Viscosity Distribution The maps of predicted turbulent kinetic energy in the plume (from the bottom of the vessel to the static level of the bath) are shown in Figs. 7.43 to 7.45. These contours show maximum turbulent kinetic energy at a small distance from the axis, where the velocity gradients are very large. The turbulent kinetic energy is seen to be comparatively high in the near-surface region close to the axis, and low in the near-wall region. For the spout, the maps of turbulent kinetic energy are shown in Figs. 7.46 to 7.48. These profiles indicate that the region of maximum turbulent kinetic energy is located close to the boundary of the spout. Close to the axis of the spout the kinetic Chapter 7. Model Predictions and Discussion 65 energy is an order of magnitude smaller than that close to the boundary; hence it has not been shown on the contours. The profiles of effective viscosity, non-dimensionalized with molecular viscosity, in the plume are shown in Figs. 7.49 to 7.51. These show that the effective viscosity increases with height above the nozzle, and attains a maximum value at the near-surface region (the static level of the bath) at a small distance from the axis. The effective viscosity decreases with radial position; and in the region close to the wall is an order of magnitude smaller than that at the axis. Profiles of effective viscosity, non-dimensionalized with molecular viscosity, in the spout are shown in Figs. 7.52 to 7.54. These profiles show that the region of maximum effective viscosity is located at the near-surface region (the static level of the bath), close to the axis. The effective viscosity decreases with height above the nozzle as well as with radial distance from the axis; hence the minimum viscosity was found close to the boundary of the spout. This result is expected since beyond the spout the turbulent kinetic energy must drop to zero, and hence the effective viscosity will approach the value of laminar viscosity at the boundary of the spout. The present results have been compared with predictions by Woo et al. [47]. Good agreement was observed in the central region of the bath. Near the wall Woo et al. [47] have predicted higher values of turbulent kinetic energy and effective viscosity than the present results. This is due to their assumption of a flat, free-slip top surface. 7.3 T u r b u l e n t K i n e t i c E n e r g y o f t h e S p o u t The turbulent kinetic energy of the spout is a key parameter and has important met-allurgical consequences, as will be shown in the next chapter. In order to quantify the effect of the spout on the metallurgical processes occurring in a gas-stirred vessel, the Chapter 7. Model Predictions and Discussion 66 variation of total turbulent kinetic energy in the spout with flow rate was analysed. The total turbulent kinetic energy in the spout was obtained by summation of turbulent kinetic energies at all nodal points within the flow domain. Its variation as a func-tion of gas flow rate is illustrated in Fig. 7.55. It can be seen that the slope increases abruptly at a \"critical\" value of the gas flow rate, which is 1350 c m 3 s _ 1 . Beyond the critical flow rate, a small increase in flow rate can lead to a large rise in the level of turbulence at the free surface. In a steel ladle this could accelerate reoxidation and moreover reduce the efficiency of temperature homogenization, dissolution and mixing of alloying additions as will be shown later in this chapter. Thus in order t o operate a gas-stirring vessel in an optimum manner, the flow rate range of reduced efficiency of bath recirculation, associated with excessive turbulence at the free surface, must be identified and consequently avoided. 7 .4 P o t e n t i a l E n e r g y o f t h e S p o u t The spout is formed at the free surface due to the kinetic energy of the fluid present at the static level of the bath, which is transformed into the potential energy of the spout. Hence the maximum height of the spout, which is observed to be at the axis, can be determined as follows (the energy losses through diffusion are assumed to be negligible and a constant mixture density is considered): U2 hmax = (7.26) where hmax = maximum height of the spout. Uax — vertical velocity of the fluid at the static level of the bath, along the axis. g = acceleration due to gravity. Chapter 7. Model Predictions and Discussion 67 The h-max is an important parameter from a practical point of view, in ladle furnaces where electrode reheating is conducted. The value of hmax should be smaller than the height at which the electrode is placed above the bath in order to avoid short circuiting between the metal and the electrode. The variation of 2ghmax, obtained from photographs of the spout, with flow rate is shown in Fig . 7.56. The relationship obtained is similar to that of total turbulent kinetic energy in the spout. A comparison between Fig. 7.56 and Fig . 7.55 shows that the abrupt increase of 2ghmax and of the turbulent kinetic energy occur at approximately the same flow rates. Hence 2ghmax also can be considered as a measure of the level of turbulence at the surface, since its variation reveals two distinct flow rate ranges. As indicated before, if in a ladle process, reoxidation is to be minimized, the higher flow rate range must be avoided, to prevent excessive turbulence at the free surface. Mazumdar et al. [27] measured the total specific kinetic energy of motion in the bath, and pointed out that there was an energy decrease in the recirculating liquid which was enhanced further with increasing flow rate. They attributed this energy decrease to 'slag/metal' interface distortion; but this mode of energy dissipation could account for only half the required value. Hence there exist other equally important reasons to explain the energy decrease in the bath. 7.5 P r o p o s e d M e c h a n i s m The critical gas flow rate in the gas injection process determines the degree of par-titioning of gas energy between the bulk liquid and the surface region. It has been observed in this work, that the rate of increase of the spout height increases with flow rate beyond a critical value. At the same time, in other studies, the efficiency of bath mixing has been found to decrease, at higher flow rates, as was mentioned in Chapter Chapter 7. Model Predictions and Discussion 68 2. These two phenomena are linked, and would certainly depend upon the manner in which the injected gas interacts with the liquid. Mechanisms have been proposed to explain these observed phenomena in a gas-stirred bath. To explain the phenomena occurring on either side of the \"transition\" (the critical flow rate), photographs of the gas/liquid dispersions were taken at an interval of one second as shown in Figs. 7.57 and 7.58 respectively. The series of photographs, in both cases, reveal distinct differences between the two flow rate ranges. At the smaller flow rate, Fig. 7.57, the large gas bubbles seen at the bottom of the bath, break up into distinct small bubbles, not far away from the orifice. They are able to interact with the liquid during a large fraction of their total travel time from the orifice to the surface, thus transmitting a major fraction of their total energy to the bath. This leads to efficient mixing, (as well as temperature homogenization, dissolution and dispersion of alloying elements in the bulk of the liquid). In contrast, at the higher flow rate, seen in Fig. 7.58, the bubble breakup does not occur until close to the bath surface. The gas pockets continue to move upwards from the orifice toward the surface, transmitting only a fraction of their energy to the liquid. A major fraction of the total energy content is released close to the surface, resulting in excessive turbulence and a prominent spout at the bath surface. It has been pointed out by Sahai and Guthrie [16] that a dynamic interaction between the gas and the liquid and a constant exchange of liquid between the plume and the single-phase liquid leads to the transfer of energy to the bulk. Such an exchange of liquid is more efficient in the case of the lower flow rates, since the bubbles break up relatively close to the nozzle and hence a greater volume of the bulk liquid is able to penetrate the gas column due to a comparatively earlier initiation of gas/liquid interaction. On the other hand at higher flow rates the intermittent formation of a chain of gas envelopes hinders the frequent penetration of the bulk liquid into the plume. Chapter 7. Model Predictions and Discussion 69 Channeling of gas could also be responsible for the different phenomena occurring on either side of the critical flow rate. Beyond the critical value, the large gas envelopes are closely spaced so that intermittently, gas channeling or bypass may be possible. This reduces the extent of gas/liquid interaction, and hence reduces the transfer of energy from the gas to the liquid, resulting in a reduced efficiency of bath mixing. Thus the energy of the gas is released close to the surface leading to the formation of a prominent spout. 7.6 Experimental Evidence 7.6.1 Axial Bubble Velocity Variation Axial bubble velocity measurements conducted by Castillejos and Brimacombe [2,3], shed light on the cause of the gas flow \"transition\". At lower flow rates, the velocity is seen to increase up to a certain height above the nozzle, beyond which it begins to decrease. This is due to the fact that distinct bubbles are formed at the orifice and they accelerate in the wake of the preceeding bubbles. At higher flow rates the bubble velocity decreases continuously with distance from the orifice because the gas forms a continuous series of envelopes from the orifice. These lose their momentum and decelerate on their upward path toward the surface. The modified Froude number(Fr) at which the transition in axial bubble velocity variation was observed by Castillejos and Brimacombe [2,3] is the same as the critical Fr observed in the present study. 7.6.2 Axial Bubble Frequency Variation The axial variation of local bubble frequency for different modified Froude numbers(irr), as measured by Castillejos and Brimacombe [2,3], is shown in Fig. 7.59. This plot re-veals that up to a particular value of Fr, the bubble frequency is always the least Chapter 7. Model Predictions and Discussion 70 at the smallest Fr at any height above the orifice. But at the higher Fr, this trend is maintained only up to a small distance from the orifice, beyond which, the trend reverses and the bubble frequency begins to decrease at a rapid rate and drops even below the value at the lower Fr. The Fr at which the transition in the variation of bubble frequency occurs, is the same as the critical Fr observed in the present study. These measurements directly support the assumption that at higher flow rates, the gas moves upward in the form of large gas envelopes which do not break into smaller size bubbles until close to the surface; whereas the higher frequency observed at lower flow rates imply that the gas is present in the form of a large number of small size bubbles. 7.6.3 Mixing Time Measurements Measurements of mixing time in gas-stirred vessels were conducted by Asai et al. [32] as described in Chapter 2. They found that the mixing time decreases with the mixing power density, e, according to the relationship: r oc e - 0- 6 8, up to a critical value of e. Beyond this value the rate of drop of mixing time, decreases to r oc e - 0 , 3 2 , indicating that above the critical value of mixing power density, the efficiency of the process has been reduced. This can be attributed again to the large gas envelopes as observed in the present study at high gas flow rates. These gas envelopes do not break up into bubbles until close to the surface thus transmitting only a small fraction of their total energy to the bath. They also found a similar observation for the fluid velocity, which varied with £ as v oc e0AS up to the critical value of e, which was the same as that observed in the mixing time measurements. At higher values of e the relationship changed to v oc eQU. Chapter 7. Model Predictions and Discussion 71 Q=371Ncm3i\"' hb=400mm d0=6.35mm r i o r 5 ' ' i i i * f i . . . i t v 1 \\ \\ 1 \\ \\ 1 v o CM / i i - ^ ^ < i o CM d o d N o a __ o CO o -tf .2? '3 as o d o t t ' l o o o Radial position, m Figure 7.34: Predicted velocity profile in the bath at Q = 371 Ncnrs 3o-l Chapter 7. Model Predictions and Discussion 72 Q=876Ncm''s\" hw=*00mm 3? '2&L*^> \\ V i i i • f J i t 1 f *-i a j i i » t / / 4 i f / / / i f * • / / i < S f i iO o d b a in o d ° (3 a) in \"5 d a) > o cm t I d o d >n o o o • Radial position, m Figure 7.35: Predicted velocity profile in the bath at Q = 876 Ncm's - 1 . Chapter 7. Model Predictions and Discussion 73 © Q=1257NcmV hb=400mm dg—6.35mm 1 V ** ' i ^ ^ • J in © o in — o Radial position, m in ' 6 o _ •* ' d 9) N N O c > o &0 0) X o ' d m o o o S t I ' I o © d Figure 7.36: Predicted velocity profile in the bath at Q = 1257 Ncm 3 s - 1 . Chapter 7. Model Predictions and Discussion 74 0.5 0.4 -.IT 0 3 'o o IS > ^ 0.2 -S3 v Z 0.1 Q=371Ncm3s~1 hb=400mm d0=6.35mm • Measured — Flat surface model Present work s 0.4 0.6 0.8 Non-dimensionalized Height, z/ hb Figure 7.37: Comparison between measurements and model predictions of near-wall bath velocity at Q = 371 Ncm's - 1. Chapter 7. Model Predictions and Discussion 75 Figure 7.38: Comparison between measurements and model predictions of near-wall bath velocity at Q = 876 Ncm 3s _ 1 . Chapter 7. Model Predictions and Discussion 76 Figure 7.39: Comparison between measurements and model predictions of near-wall bath velocity at Q = 1257 Ncm 3s _ 1 . Chapter 7. Model Predictions and Discussion 77 o 13 > < 2.6 2.4 2.2 2 1.8 1.6 H 1.4 1.2 -1 -0.8 -0.6 0.4 0.2 H 0 Q=371NcmV hb=400mm d0=6.35mm Measured Rat surface model Present work 0 n 1 1 1 1 r~ 0.2 0.4 0.6 — i 1 1 1 0.8 1 1.2 Non-dimensionalized Height, zl hb Figure 7.40: Comparison between measurements and model predictions of axial jet velocity at Q = 371 Ncm 3 s - 1 . Chapter 7. Model Predictions and Discussion 78 Figure 7.41: Comparison between measurements and model predictions of axial jet velocity at Q = 876 Ncm 3s _ 1 . Chapter 7. Model Predictions and Discussion 79 1! 'o o 13 > Is < 2.6 2.4 -2.2 -2 -1.8 1.6 -1.4 -1.2 -1 0.8 0.6 0.4 H 0.2 0 0 0.2 Q=1257NcmV 1 h b=400mm dQ=6.35mm Measured Flat surface model Present work -] 1 1 T -0.4 0.6 i r 0.8 1.2 Non-dimensionalized Height, z/ / i 6 Figure 7.42: Comparison between measurements and model predictions of axial jet velocity at Q = 1257 Ncm 3s _ 1 . Chapter 7. Model Predictions and Discussion *ft O O Ift o C\\J W - H o o o o 6 ° d d Radial position, m Figure 7.43: Contours of predicted turbulent kinetic energy (Joules/kg) distribution the plume, at Q = 371 Ncm 3s _ 1 . Chapter 7. Model Predictions and Discussion Figure 7.44: Contours of predicted turbulent kinetic energy (Joules/kg) distribution the plume, at Q = 876 Ncm 3s _ 1 . Chapter 7. Model Predictions and Discussion Radial position. Figure 7.45: Contours of predicted turbulent kinetic energy (Joules/kg) distribution the plume, at Q = 1257 Ncm 3s _ 1 . Chapter 7. Model Predictions and Discussion Figure 7.46: Contours of predicted turbulent kinetic energy (Joules/kg) distribution the spout, at Q = 371 Ncm 3s _ 1 . Chapter 7. Model Predictions and Discussion Figure 7.47: Contours of predicted turbulent kinetic energy (Joules/kg) distribution the spout, at Q = 876 Ncm 3s _ 1 . Chapter 7. Model Predictions and Discussion Figure 7.48: Contours of predicted turbulent kinetic energy (Joules/kg) distribution the spout, at Q = 1257 Ncm 3s _ 1 . Chapter 7. Model Predictions and Discussion 86 10 o *n O lT3 o OJ c\\j -> -. o o O O O O o o Radial position, m Figure 7.49: Contours of predicted effective viscosity non-dimensionalized with molec-ular viscosity in the plume, at Q = 371 Ncm 3s _ 1 . Chapter 7. Model Predictions and Discussion 87 Radial position, m Figure 7.50: Contours of predicted effective viscosity non-dimensionalized with molec-ular viscosity in the plume, at Q = 876 Ncm 3 s - 1 . Chapter 7. Model Predictions and Discussion 88 in o in o uo o __