Mathematical and Physical Modeling of Magnetic Flow Control in a Vertical Slot Nozzle Metal Delivery System for a Horizontal Single Belt Caster

Transcription

1 Mathematical and Physical Modeling of Magnetic Flow Control in a Vertical Slot Nozzle Metal Delivery System for a Horizontal Single Belt Caster by Mohammad Mahdi Aboutalebi Mining and Materials Engineering McGill University, Montreal August 2014 A thesis submitted to McGill University in partial fulfilment of the requirements of the degree of Master of engineering Mohammad Mahdi Aboutalebi 2014

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3 ABSTRACT One of the major aspects that quantify the quality of the strips in near net shape casting processes, including the Horizontal Single Belt Caster (HSBC), is the metal feeding system. The metal delivery system in HSBC controls liquid metal feeding onto the water-cooled belt, can be considered as the heart of the caster. The feeding system must promote an even distribution of melt on the water cooled moving substrate so as to create an iso-kinetic flow. In this exploratory study, an Electromagnetic Braking (EMBr) system was used as a flow modifier, so as to create an iso-kinetic velocity profile along the thickness and across the width, of the strips in a HSBC, using a vertical slot nozzle to improve the surface quality of strips. In this regard, a three dimensional mathematical model was developed to study the effects of EMBr on the flow field. In this model, time-averaged momentum equations with the standard k-ε turbulence model, along with the magneto hydrodynamics equations, were solved in order to analyze, and visualize, the effect of EMBr on the flow field. Beside the mathematical model, a physical cold model was also developed to visualize the effect of EMBr on the fluid flow. The effect of EMBr was simulated using a full scale salt water model using a vertical slot nozzle. To impose a magnetic field on the liquid, a pair of permanent Neodymium (NdFeB) magnets was used. Due to the low electrical conductivity of salt water compared to liquid steel, an additional electrical field was used to satisfy the dynamic similarity requirements. The results obtained from the mathematical model, confirmed the effect of magnetic field in modifying the flow field. Based on the computed results, it was shown that the imposition of a magnetic field caused the flow to become iso-kinetic. This can result in an improvement in the quality of the final strip products. I

4 RÉSUMÉ L'un des principaux aspects qui permettent de quantifier la qualité des bandes dans les processus de coulée de forme nets proches, y compris La Coulée Continue à Bande Horizontale (HSBC), est son système d'alimentation du métal liquide (ou le system de distribution du métal liquide). Le système de distribution de métal dans HSBC qui contrôle la déposition du métal liquide sur une bande à couler refroidie à l'eau, peut être considéré comme le cœur du system. Le système d'alimentation doit favoriser une distribution uniforme du métal liquide sur le substrat en mouvement de manière à créer un écoulement iso cinétique. Dans cette étude de recherche, un système électromagnétique de freinage (EMBr) a été utilisé comme modificateur d'écoulement. Le system EMBr va créer un profil de vitesse iso-cinétique des bandes coulée par le system HSBC. On utilise une «nozzle» à fente verticale pour améliorer la qualité de la surface des bandes. À cet égard, un modèle mathématique tridimensionnel a été développé pour étudier les effets de l EMBr sur le champ d'écoulement. Dans ce modèle, les équations moyennées en temps du «momentum», le modèle de turbulence k-ε standard, ainsi que les équations magnéto hydrodynamique, ont été résolus afin d'analyser et de visualiser l'effet de l EMBr sur le champ d'écoulement. A côté du modèle mathématique, un modèle physique à froid a aussi été développé pour visualiser l'effet de l EMBr sur le flux de fluide. L'effet de l EMBr a été simulé en utilisant un modèle de l'eau salée à grande échelle à l'aide d'une «nozzle» à fente verticale. Pour imposer un champ magnétique sur le liquide, une paire d aimants permanents au néodyme (NdFeB) a été utilisée. En raison de la faible conductivité électrique de l'eau salée par rapport à l'acier liquide, un champ électrique supplémentaire a été utilisé pour satisfaire aux exigences de similarité dynamique. II

5 Les résultats obtenus à partir du modèle mathématique, ont confirmé l'effet du champ magnétique dans la modification du champ d'écoulement. Sur la base des résultats calculés, on a montré que l'application d'un champ magnétique a déterminé que l'écoulement devienne iso-cinétique. Il améliore aussi de manière significative la qualité des bandes métalliques fabriquées en utilisant le system HSBC. III

6 ACKNOWLEDGMENT I would like to express the deepest appreciation to my supervisor Professor. R.I.L. Guthrie and my co-supervisor Dr. Isac for their advice and guidance throughout this project. Thanks to Dr. Luis Calzado for his help about the experimental set-up and his technical knowledge for the water-model experiments. I would also like to thank my other colleagues and friends for their discussions and constructive criticisms. My sincere thanks to Don Pavlasek and Jozef Boka for fabricating the components of the experimental set-up, and for their suggestions which made the design better. I would like to extend my sincere gratitude to the McGill Metals Processing Center (MMPC) for the experimental facility and financial support. I owe a very important debt to Professor M. Reza Aboutalebi, for his help and fruitful discussions about the mathematical modeling, and about the experimental set-up. Finally, a special thanks to my family and friends. Words cannot express how grateful I am to my mother Tahereh and my Father Reza for their unconditional help and patience and to my beloved sister Sahba for her encouragement. IV

7 Table of Contents 1.0. Introduction Metal Delivery Systems for the Horizontal Single Belt Casting Process Objectives of the present work Literature Review Evolution of Continuous Casting in the Steel Industry Direct Belt Casting (DBC) Horizontal Single Belt Casting at McGill Metals Processing Center (MMPC) Applications of MagnetoHydroDynamics (MHD) in Metallurgy Applications of Magnetohydrodynamics (MHD) in Continuous Casting Mathematical Model Problem Statement Mathematical Formulation Numerical Solution Procedure Physical Model Principles of MHD Water Model Development Results and Discussion Results of Mathematical Model Fluid flow of liquid steel on to the moving belt without EMBr Evaluation of mesh dependency Typical Computed Results Verification of MHD Mathematical Model Parametric study Typical Results of the Physical Model Flow Visualization Studies Conclusions References V

8 NOMENCLATURE ρ v Density, Velocity, kg m 3 m s P τ Static Pressure, Stress tensor, N m 2 N m 2 F External body force(lorentz Force), N m 3 B Applied magnetic field vector, Tesla b Induced magnetic field vector, Tesla E Electric fields, V m H Induction magnetic field, A m D q Induction electric filed, Electric charge density, C m 2 C m 3 j Electric charge density, A m 2 μ m Magnetic permeability, N A 2 μ Dynamic Viscosity, Pa. s θ Kinematic Viscosity, m 2 s l Charactristic length, m VI

9 g Gravitational acceleration vector, m s 2 ε m Electric permittivity, A 2. s 4 kg. m 3 φ Scalar elecrical potential, Volts A Vector electrical potential, Volts σ Electrical conductivity, 1 Ω. m ω Vorticity, 1 s k Kinetic energy of turbulence per unit mass, m 2 s 2 ε Rate of dissipation of "k", m 2 s 3 Re = ρvl μ, Reynolds Number Ha = Bl σ μ, Hartmann number Fr = v2 gl, Froude Number N = Ha2 Re, Stuart number Re m = μσvl, Magnetic Reynolds number VII

10 Chapter Introduction Steel companies are always looking for processes to reduce operating costs and to pioneer energy and capital cost savings. One solution that has been proposed to meet this demand is to develop novel casting processes to eliminate the number of intermediate processing steps using conventional slab casters and to directly fabricate the products with dimensions near to the final products. This is generally known as near net shape casting. In this regard, the technologies of thin slab casting and strip casting have been recently introduced to the steel industry and this has started a new era in the field of castings. Compared to ingot or conventional slab casting, these newer processes possess the following advantages: Elimination of stands for roughing passes, which leads to energy savings Production of smaller tonnages of various steel grades and quality Reduction of operating costs and capital investment using less and smaller equipment High potential for production of strips with good mechanical properties due to the high rate of solidification Ideal for both integrated steel mills and mini-mills with high rate of production Environmentally-friendly approach to the production of strips Thin slab casting and strip casting are defined by their thicknesses. The thickness of thin slab is in the range of mm. To take advantage of this process, various thin slab casting facilities using oscillating molds, have been attempted in several countries such as Germany, Japan, and USA. Thin slab casting (TSC) became industrialized in 1990, when NUCOR Corporation started operating the Compact Strip Production (CSP) caster at Crawfordsville in the USA. However, one of the drawbacks of thin slab casting is the friction problem of moving solidified shells against the mold s walls. [1] The ideal near net shape casting process is to cast a thin strip directly from the molten metal. The design of the first strip caster which was invented by Henry Bessemer in 1857, is shown in Figure 1. Much more recently, twin roll strip casting (TRC) has been upgraded by changing the 1

11 design of the caster such as changing the diameters or the angle of the rolls. The significant characteristic of these twin roll casters is the moving mold technology, which overcomes the friction problem between the static mold, and solidified shell that occurs in thin slab casting. Nevertheless, it has been reported that one of the major problems in twin roll casters is the production limitation for low carbon steels. In order to catch up with the productivity and the quality of a conventional slab casting operation, an increase in the roll diameters up to 2.5m plus another hot rolling machine is needed, respectively. [1, 2] Fig Henry Bessemer s original drawing of the twin-roll casting process An alternative approach to the twin roll process, that has been proposed by research groups at Clausthal and McGill universities twenty four years ago, is the horizontal single belt casting process. This consists of a metal delivery system, a horizontally moving belt, and a number of finishing mills. The process has been studied intensively by MEFOS in Sweden, Clausthal Technical University in Germany, and the McGill s Metals Processing Center (MMPC). This technology is reported to have no limitations in terms of any types of steel production. A commercial HSBC caster can be reached an annual productivity of up to 2.5 Mt/m-width, which comparable with the integrated mills. [1, 3] 1.1. Metal Delivery Systems for the Horizontal Single Belt Casting Process In the proposed horizontal single belt casting process, the metal delivery system plays a significant role. This system determines how liquid metal is fed onto the moving belt and is responsible for an even distribution of liquid metal across the width and thickness of the cast 2

12 strip. The metal delivery system for single belt casting must satisfy two requirements in order to increase the quality of the finished products. The metal should be laid down on the moving substrate in an iso-kinetic manner and also be fed evenly onto the belt with the desired geometry in terms of width and thickness. In the case of having a stable thickness of strip, the exit velocity of molten metal through the nozzle should be the same as the belt speed in order to avoid a hydraulic jump. Therefore, for having stable formation of the initial solidified shell, the impinging velocity of liquid steel onto the moving belt should be compatible with that of belt. Considering the importance of the metal delivery system to provide stable conditions during the feeding process, several methods have been tested for this purpose. The inclined multi-hole arrangement at MEFOS, the argon rake with its integrated siphon at Clausthal, and the multichannel metal delivery system as well as the slot nozzle at the MMPC, have all been tested to overcome problems associated with metal feeding. However, the multi-hole nozzle leads to hydraulic jumps and causes unstable menisci whilst the argon rake is complicated to set up over the belt and is associated with operating costs and maintenance difficulties.[4] The multi-channel nozzle avoids the strongly impinging jet and non-uniform velocities. However, this design is a complicated system to use and has disadvantages such as early solidification and nozzle clogging. Thus the fluid flow control in the feeding system has a vital role in casting the strip products, but is not yet fully optimized. However, the vertical slot nozzle is the proposed method and is used at the MMPC for the horizontal single belt caster feeding system Objectives of the present work A pilot scale Hazelett Horizontal Single Belt Caster (HSBC) has been installed since 1997 in the MMPC s foundry at McGill University, in order to study fluid flow for a liquid metal delivery system, and subsequent solidification phenomena on the belt. The most important part of this caster is the metal delivery system which needs to be properly designed. For the design of metal delivery and feeding systems, a comprehensive survey of possible design concepts is essential. Following this, water modeling is necessary for the precise design and validation of the equipment s components for the flow delivery system. Mathematical modeling 3

13 is also an essential tool in order to study and predict flow fields in the metal delivery system proposed. Because of the importance of fluid flow control in the melt feeding system of the HSBC system, the present study has been focused on controlling the fluid flow in a metal delivery system with a vertical slot nozzle, using Electromagnetic Brake (EMBr). A mathematical model and a physical model were both developed in order to study the flow of liquid metal through a vertical slot nozzle delivery system, in the presence of a DC magnetic field, and an applied current. 4

14 Chapter Literature Review 2.1. Evolution of Continuous Casting in the Steel Industry The steel industry has discovered new approaches to sheet production by asking the question: Instead of casting large steel ingots, letting them cool, and then expending large amounts of energy for reheating the ingots and then rolling them into sheets, why not continuously cast sheet metal in the first place? In the 1970 s and 1980 s, continuous casting and later near net shape casting was introduced to the steel industry. In the early 70 s, continuous casting provided only 10% of the total steel worldwide [5]. By the mid-80 s, this number had increased to 50% and by the year 2011, this ratio had climbed to represent over than 90% of the total 1518 million tons of steel produced in the world, as shown in Figure 2.1 for total worldwide steel production. Following extensive work in the area of continuous casting during the last 20 years, alternative processes including the Near Net Shape Casting (NNSC) processes for steel, have also been researched. The main objective of these new processes is to cast steel in a dimensional form close to that of the final products. By using these processes, several hot and cold deformation steps can be eliminated for producing the finished products. This could significantly decrease capital and production costs, by scaling down equipment. Fig World s crude steel production [5] 5

15 2.2. Direct Belt Casting (DBC) The single belt strip casting is a promising technology for strip production. This process was independently conceived by Herbertson and Guthrie [6], and by Reichelt, Scheulen, Schwerdtfeger, Voss-Spilker and Feuerstacke [7], in The process was commercialized in September 2013, in Germany. Two R&D centers have been working on Horizontal Single Belt Casting (HSBC) for the past twenty odd years: Clausthal University in Germany was later supported by MEFOS located in Sweden, under the European Steel Research Funding Program, while the second is at the McGill Metals Processing Center (MMPC) in Canada. These two R&D centers have been working on commercialization and solving the problems of Direct Steel Casting (DSC) or Horizontal Single Belt Casting (HSBC) An HSBC caster comprises the following parts : A metal delivery system A horizontally moving belt A run-out table and finishing stands In the Horizontal Single Belt Casting process, the liquid metal in the tundish is fed into the metal delivery system. This plays a significant role in the process. From there the melt is deposited onto a moving, water cooled, horizontal belt. For more reduction in the strip thickness and for better surface quality, hot rolling may be used directly after the casting process. One of the main problems in the HSBC process is controlling turbulent flows in the metal delivery system. In the HSBC process, molten metal is converted directly into 5-10 mm thick strips. As such, the metal feeding system plays a remarkable role in this process. In order to increase the quality of finished products, the metal should be laid down on the moving substrate in an iso-kinetic manner, and also fed evenly onto the belt of the desired geometry in terms of width and thickness of the metal stream. Based on the important role of the metal delivery system and to provide for stable conditions during the feeding process, several methods have been tested for this purpose. [8-10] 6

16 Low Pressure System (LPS) developed at Clausthal. Tube Feeding System (TFS) developed at MEFOS. Extended nozzle system developed at McGill. Multi-channel type of nozzle developed at McGill Zig-zag nozzle system developed at Clausthal. Argon rake nozzle developed at Clausthal. Three-chamber tundish developed at McGill. Vertical slot nozzle proposed at McGill Table 2.1 shows a few of the main problems involved in direct steel casting processes, considering the reasons and the corresponding solutions.[3, 4] Table Main defects, Main reasons and solution for the problems in direct steel casting processes Process Main Defect Main Reason Countermeasures HSBC Surface Porosity Residual solidification and Use of Ar+CO 2 gas for oxidation of steel decarburization of upper surface 60-80% in-line hot rolling Nozzle clogging Freezing of liquid at nozzle Slot type nozzle with argon rake holes Thickness nonuniformity Bulging of belt Pressure reduction in the water cooling chamber Transverse flow marks Non-uniform feeding of liquid Zig-zag nozzle Unstable meniscus High speed feeding (Strong hydraulic jump) Reduction of jump intensity 7

17 As demonstrated in the Table 2.1, for any defect a possible solution is proposed, but for the reduction of hydraulic jumps and for obtaining a stable meniscus profile, very few studies have been done and more research work needs to be carried out. Figure 2.2 illustrates the Direct Belt Casting (DBC) pilot caster developed and tested at Clausthal.[2] Fig DBC at Clausthal pilot. [2] Figure 2.3 depicts the MMPC Horizontal Single Belt Caster, now operational, off-campus, at the new Stinson laboratories. [3] Fig HSBC pilot caster at the MMPC. [11] 8

18 Figure 2.4 illustrates the Direct Belt Casting (DBC) machine installed at MEFOS. [12] Fig DBC at MEFOS pilot plant. [12] The main advantages of this process are: Casting speed is about m/min Direct production of hot strip of usual quality and dimensions by help of In-Line-Rolling. Suitable for medium and large production rates up to 2 Mt/year. Low investment costs, low production costs. In Table 2.2, the process features and characteristics of strips produced by the above single belt casters are given. [1] 9

19 Table HSBC features in different R&D centers. [1] Process Clausthal pilot MEFOS MMPC Variables Belt length (mm) Belt width (mm) Belt material Low C steel Cu, Low C steel Cu, Low C Steel Cast thickness (mm) (For Al) Cast width (mm) , Casting speed m/min (For Al) Furnace capacity (Kg) In line rolling Yes No Yes Horizontal Single Belt Casting at McGill Metals Processing Center (MMPC) There have been a number of studies carried out at the MMPC aiming at the improvement of strip quality produced by the Horizontal Single Belt Caster (HSBC). C. Jefferies et. al, developed a mathematical model along with a water model to study the effects of ceramic filters in the metal delivery system with an extended- nozzle. They reported the flow modifier can play an effective role in controlling the turbulent flow. The predicted velocity profiles for the metal delivery system are shown in Figure 2.5 in which the positive effect of a filter is clearly seen. [13] Fig Predicted velocity profiles in Metal delivery system of HSBC: a) without filter, b) with filter 10

20 K. Moon et al, developed a new metal delivery system for the single belt caster and evaluated this using a full scale water model of the Horizontal Single Belt Caster at the MMPC. They also developed a mathematical model to study the flow field and temperature distribution in the system. They proposed a slot nozzle combined with a multi-channel type flow modifier instead of using an extended nozzle with a ceramic filter as a flow modifier. In this regard, they claimed that the new metal delivery system has the potential to reduce the strongly impinging jet on the belt and create a uniform, lateral flow of liquid onto the cooling belt. Two types of metal delivery systems considered in their study are given in Figure 2.6. [4] Fig Predicted velocity profiles for the multi-channel type of nozzles: a) Type FC, b) Type FD. Product quality and caster productivity is directly dependent on the interfacial heat transfer between the melt being cast and the moving, water-cooled, substrate. Thus, besides studying the fluid flow, extensive studies at MMPC have been devoted to heat transfer analysis of the strips in different metal delivery systems. In this regard, important variables which affect the rate of interfacial heat transfer between the solidifying strip and the water cooled moving substrates, 11

21 namely; melt superheat, thermophysical properties of the melt and the substrate, texture of the substrate, and casting speed, were investigated. J. Kim studied the interfacial heat transfer and solidification of magnesium alloys (AZ91 and AM50), and aluminum alloy (AA6111) on Horizontal Single Belt Caster. In this study different types of substrates, including copper, low carbon steel, alumina coated steel, nickel coated steel, and zirconia coated steel, were used to study the interfacial heat fluxes between the solidifying strip and the water cooled moving substrate. They reported that the major thermal resistance to heat extraction is the generation of a thin, interfacial layer of gas, which separates the strip from the moving substrate. This interfacial layer can be formed by the shrinkage of solidified skin and possibly the entrainment of a thin air film. They showed that by changing the wettability of the substrate with oil spraying, the quality of cast surface of strips could be dramatically improved. The improvement of surface strip quality is shown in Figure 2.7. [14] Fig Improved surface quality by oil spraying the copper surface D. Li et al. continued J. Kim s work and studied the effect of the surface roughness of the water cooled moving substrate on the quality of aluminum alloys strip. [15] Significant research works have been carried out on the effect of various substrate textures and various interfacial agents on heat transfer, solidification and strip quality. Another field of research pursued by the MMPC is the study of microstructure and mechanical properties of strips produced by the Horizontal Single Belt Casting technique. D. Li et.al. 12

22 extensively studied the microstructure and mechanical properties of aluminum alloy (AA6111) strips produced by Horizontal Single Belt Caster (HSBC) and compared them with the strips produced by Twin Belt Casting (TBC) technique. They have reported, based on the tensile properties and microstructure of the AA6111 strips, that HSBC can be an alternative for TBC process. [16, 17] 2.3. Applications of MagnetoHydroDynamics (MHD) in Metallurgy MHD is the acronym for magnetohydrodynamics. The term metallurgical MHD was first used in the International Union of Theoretical and Applied Mechanics (IUTAM) conference held in Cambridge in In the years following this conference, applications of magnetic fields in industrial metallurgy have become significant and vivid. The use of this phrase covers four groups of research on MHD and consequently leads to the applications in the metallurgical industry. These applications can be categorized as: [18] i. Magnetic stirring induced by a rotating magnetic field. Magnetic stirring is nothing but an induction motor, where the liquid metal takes the place of the rotor. ii. The magnetic braking of jets, vortices, and natural convection. The magnetic braking principle takes advantage of the motion of an electrically conductive fluid in the presence of a magnetic field, in which the Lorentz force opposes the relative motion. iii. Interfacial instabilities which arise when a current is passed between two conductive fluids. This application causes the bar to pinch in on itself, and there would be a similar result if current passes through a liquid-metal pool. iv. Magnetic levitation and heating induced by high-frequency magnetic fields. This application relies on the fact that an induction coil carrying a high frequency current will tend to oppose currents in any adjacent conductor. 13

23 Applications of Magnetohydrodynamics (MHD) in Continuous Casting In continuous casting, many of the casting defects such as mould powder entrainment (MPE) are associated with the flow pattern in the mold region. There, by controlling the flow near the meniscus, the product s quality is considerably improved. For instance, an excessive surface velocity can cause severe slag entrapment, whilst insufficient surface flow allows meniscus freezing and related surface defects. In the design of submerged Entry Nozzles (SEN), deep penetration encourages the capture of the meniscus inclusions. There have been a number of solutions presented for this problem; some of them deal with the designs and operating conditions and some of them take advantage of the first and second categories of metallurgical MHD applications. In terms of mold design and operating conditions, there are a lot of parameters involved. These include the submerged entry nozzle (SEN) and port design shape, the SEN submergence depth (distance from top of nozzle ports to mold top surface), mold size, casting speed, position of the flow-control mechanism (slide gate or stopper rod), the rate of inert gas injection, and the application of electromagnetic forces. Each of these factors must be adjusted with regard to the other factors, in order to produce a good flow pattern [19]. In the case of MHD applications in continuous casting, much research has been carried out, which can be divided into two main categories: i. Electromagnetic stirring (EMS) ii. Electromagnetic braking (EMBr) Electromagnetic Stirring in Continuous Casting Electromagnetic stirrers employ an alternating current to generate a continuously- varying magnetic field to control the flow in the mold cavity. In slab- mold, the EMS is located on each wide side of the mold, near the meniscus. The magnetic field generated by this system forces the flow to circulate around the mold edge, which decreases the temperature gradient in the meniscus region, and improves the final quality of the slab. Okazawa et al. developed a mathematical model and simulated the turbulent flow using the Large Eddy Simulation (LES) model and studied the influence of EMS magnet location on flow circulation. For this study, they 14

24 used a low temperature (mercury) physical model to verify the mathematical model. Figure 2.8 shows the experimental set-up for this mercury model. Based on the numerical and experimental results, it was concluded that the location of EMS plays a significant part in influencing the circulating flow. The particular, the number of poles in the motor has a significant effect on the circulation flow. [20] Fig Experimental set-up [20] The multimode EMS(MM-EMS) uses two stirrers on each of the wide sides located near the SEN outlet ports [21] This multimode EMS has three different modes of operation, which namely: i. Electromagnetic Level Stabilizer (EMLS): The electromagnetic level stabilizer (EMLS) mode directs the forces so as to oppose high speed flows exiting the SEN, reducing surface velocity, and flattening the meniscus profile ii. Electromagnetic Level Accelerator (EMLA): The electromagnetic level accelerator (EMLA) mode accelerates liquid metal flows exiting the SEN, so as to increase both surface velocity and heat transfer at the meniscus. 15

25 iii. Electromagnetic Rotary Stirrer (EMRS): the electromagnetic rotary stirrer (EMRS) mode sequences the flow-to-stir and encourages mixing. Ishii et al, [21] developed a numerical model using the Reynolds-averaged Navier-Stokes (RANS) model to simulate the effects of EMLS on the meniscus velocity field. Based on the results they have reported, EMLS can effectively suppress meniscus velocity, particularly for thin slab casters. Kubota et al. [22] model, have reported that using EMLS and EMLA simultaneously, can successfully control the meniscus flow and reduce mold-slag entrapment, depending on casting conditions. In research by Kunstreich et al.[23], they have shown that MM- EMS can effectively maintain a favorable double-roll flow pattern in the mold cavity, and reduce the number of inclusions, cracks, and slivers, present in the final product Electromagnetic Braking (EMBr) in Continuous Casting Electromagnetic Braking (EMBr) has been used to diffuse and suppress the strong flows exiting from submerged entry nozzle (SEN), in order to decrease the surface fluctuation within the molds [21]. In terms the applications of EMBr in continuous casting, there have been multiple types of EMBr systems designed to achieve various objectives, as follows: i. Local EMBr : In this type, two permanent magnets were applied so as to create a rectangular static magnetic field near the SEN ports. This system has been applied to decrease and suppress the flows exiting from the SEN and also decrease the height fluctuations at the surface. [24] ii. Ruler EMBr : This method uses two rectangular magnets set across the width of a structure just located below the SEN ports on opposite sides of the mold. Like local EMBr, this is used to stabilize the meniscus velocity and profile.[25] iii. Flow-control-mold (FC-mold) : This system uses two pairs of permanent magnets, in order to control the flow both exiting the nozzle and also the meniscus region, and to control meniscus velocity. In terms of the locations of these two magnetic fields, one is located at the meniscus while the other one is located beneath the SEN ports. [25] 16

26 Ha et al. [24] developed a three dimensional mathematical model using the Reynolds-averaging Navier-Stokes (RANS) turbulence model. In this model, they simulated the effects of EMBr on the flow, heat transfer, and solidification of steel within the continuous casting mold. The influences of the magnetic field on the transport phenomena were simulated by solving threedimensional conservation equations for mass, momentum, energy, together with Maxwell equations for the electromagnetic field coupled with a flow field. Figure 2.9 shows the predicted velocity field and effects of EMBr on two different vertical planes. B max = 0 T B max = 0.15 T B max = 0.3 T Fig Velocity fields at the x-y planes at z=0m and z=0.027m for casting speed of 1.8m/min and B max = 0, 0.15, and 0.3 T It has been reported, the EMBr, effectively slows down the flow at the outer ports of SEN. It was also found by applying the EMBr to the continuous casting system; it reduces the impingement force of the jet on the mould face, and shortens the penetration depth of the lower recirculation zone. They reported significant improvements in flows and temperatures by imposing the magnetic field. They also mentioned that due to implementing of EMBr, temperature gradients near the recirculation cells decreased. This reduction in temperature gradient has beneficial effects on uniform quality of slab products. Surface temperature increases due to the application of EMBr and this effect leads to a reduction in the need to superheat the molten steel as a lubrication effect. 17

27 Takatani et al. [25] used a similar method and developed a three-dimensional model to analyze heat transfer and fluid flow in a continuous casting mold, in the presence of magnetic field. They verified the model by comparison of the flow field between the experimentally measured and computed results, under isothermal conditions. The governing equations have been solved in this model are; Continuity, momentum and thermal energy including solidification phenomena. The Lorentz force was considered in the momentum equations as source term. In this model, for calculating the Lorentz force, the authors took advantage of Maxwell equation and Ohm s law and neglect the induced magnetic field. Consequently, the electrical potential of the flow has been solved by the Poisson equation of electric potential, from which the induced current density was calculated and the resulting Lorentz force was also calculated. (σ φ) =. σ (v t B) 2.1 Figure shows the flow field results with the EMBr. B max = 0 T B max = 0.17 B max = 0.35 Fig Effect of the intensity of the magnetic flux density on the flow field. (at 0.25 thickness of the slab) Based on the results, they concluded that EMBr can be used effectively for controlling the flow field in the mold and for imposing a strong magnetic field. EMBr can cause the jet to bend and dissipate before impinging against the narrow face [25]. 18

28 Kim et al. [26] studied the influences of EMBr by numerical simulation of the coupled turbulent flow equations for the continuous casting system. In this model, a revised low-reynolds number k-ε turbulence model was used to consider the turbulent effects on the fluid flow. They reported that applying a local EMBr cause a major decrease in jet momentum and velocity. It was also found, that increasing the magnetic flux density mainly effects on the surface temperature of the solidified shell. Figure 2.11 demonstrates the effects of EMBr on the surface and vertical central symmetrical plane [26]. (1) (2) Fig Temperature fields at; (1) the free surface level, and (2) the vertical centrally symmetric plane parallel to the wide faces for various magnetic flux densities; (a) B=0 T, (b)= B= 0.13 T, and (c) B= 0.25 T Harada et al. [27] have studied the effects of different types of in-mold electromagnetic braking (EMBr) technique, which forms a local magnetic field in the width direction of a mold, on the fluid flow phenomena in the strand pool was examined. They developed a numerical model and verified it with a mercury model experiment. Based on the experimental results, it has been revealed that the level magnetic field causes a lower meniscus velocity in comparison with the local magnetic field. In case of local magnetic field they have reported, the surface velocity has been greatly affected by the nozzle condition. Figure 2.12 shows a measured velocity in the nozzle condition for two different cases. 19

29 (a) (b) (c) Fig Measured velocity distribution at half thickness in the mercury pool for; (a) Without EMBr, (b) With local EMBr of B= 0.3 T, and (c) with level EMBr of B= 0.3 T Takeuchi et al. [28] developed physical and numerical models to investigate and compare the effects of the local EMBr and the ruler EMBr on the liquid metal flow field. It was found from experimental results that a large recirculating flow below the ruler EMBr was suppressed by imposing magnetic field and causes plug like (nonrecirculating) flow at the meniscus region. Figure 2.13 shows a schematic image of the electromagnetic brakes locations. (a) (b) Fig Schematic representation of the in-mold electromagnetic brake (EMBr) technique applied to continuous casting process ; (a) Level magnetic field, and (b) Local magnetic field. 20

30 Idogawa et al. [29] developed a numerical model to investigate the effects of a newly designed EMBr on the fluid flow in the continuous casting mold cavity. The numerical model has been verified by mercury model experiments. They observed that the EMBr suppresses the flow velocity at the meniscus and decreases the penetration depth of inclusions. Figure 2.14 demonstrates the calculated results of the velocity field at the mid-face of the mold. Fig Calculated results of the flow patterns at the center face of the mold thickness In 1996 Idogawa et al. [30] carried on their studies on the effects of EMBr in a continuous casting mold. The study was carried out in order to optimize the process. As their last study, the first magnetic field imposed at the meniscus region for controlling the surface flow of the molten steel and the second one has been applied below the Submerged Entry Nozzle (SEN) to suppress the flow in the lower part of the continuous casting mold. The flow velocity was measured in mercury model experiment in order to investigate the macroscopic flow. Figure 2.15 indicates the results of the flow velocity measurement at 3 different depths. 21

31 Fig Profiles of downward flow velocity in the mercury a) without magnetic field, b) with 1 pair of magnetic poles and c) with 2 pairs of magnetic poles As shown in Figure 2.15 (a), there is a sharp velocity peak at Z=50mm which represents the discharge flow from the nozzle. This peak becomes weaker as magnetic fields are applied. Based on these results, a two-field, full-width design was adapted and used in plant experiments at Kawasaki Steels Chiba Works. More recently, Haiqi et al. [31] studied the flow field in a continuous casting mold under the influence of a magnetic field, using a three-dimensional mathematical model. Figure 2.16 shows the position of EMBr in this model. 22

32 (a) (b) (c) Fig Geometry structure of the model. a) Structure of nozzle, b) the position of EMBr, and c) model grid division In this model, the standard k-ε two equation model was used to simulate the turbulent flows in the mold region. The continuity, momentum, k, and ε equations represent the governing equations in this model and have been discretized using commercial software, FLUENT ANSYS, which adopts the Control Volume (CV) method. Based on the results, the braking effect becomes more significant as the external magnetic field increases and as the re-circulation zones gradually becomes shallower. Due to the significant influence of casting speed on the quality of the products, they reported that at high casting speeds, EMBr may effectively suppress the flow velocity at the mold region and increase the product quality. Li et al. [32] developed a mathematical model to study the effects of EMBr on vortex flows in thin slab continuous casting. A finite difference model was used to study the fluid flow in the mold region. For considering the effect of turbulence, the low Reynolds number k-ε model was used to simulate turbulent flow. The governing equations used in this model are continuity, momentum, k, and ε equations. A FORTRAN computer code was used for discretization of the governing equations. In order to investigate the behavior of vortices in the flow field, the vortices defined as follows, 23

33 ω = v In Cartesian coordinates 2.2 ω = 2 ψ In two dimensional coordinates 2.3 where ψ represents the stream function. Figure 2.17 shows simulated flow field with and without EMBr, on the surface and of the half thickness of the continuous casting strand. (a) (b) Fig Simulated flow field on surface and center plane: a) without EMBr, b) with EMBr By comparing these two flow fields, they reported that EMBr had successfully suppressed the flow, and the vortices in the surface are pushed into the back of the SEN in the mold. In this work, they also investigated the effects of Magnetic field strength on the surface fluctuations and reported that the meniscus in the mold becomes remarkably steady. [32] 24

34 The efforts of B. Thomas and co-workers aimed at understanding the behavior of fluid flow at the continuous casting mold in presence of EMBr are undeniable. In 2007, they investigated the influences of varying SEN submerged length and EMBr field strength by development of a three-dimensional, steady k-ε model, considering the magnetic induction method, on fluid flow in the continuous casting mold cavity. Fluent software was used to discretize the governing equations. Their study has been verified by comparing the results with an analytical solution and with experimental results. For experimental validation, nail board and oscillation mark measurements were collected at the plant. Based on the numerical and experimental results, they reported that by increasing the EMBr strength with a constant SEN depth, the upper recirculation zone and meniscus velocity decreases, as well as leading a smaller meniscus wave. Similar results were reported by just increasing the SEN depth without imposing a magnetic field. But in the case of increasing the SEN depth at a constant EMBr, this brought unexpected results. That condition caused higher meniscus velocities, greater meniscus fluctuations, and deeper penetration depths [21]. In terms of the effects of EMBr on transport phenomena, Kang et al. [33] developed a 2- dimensional coupled model incorporating fluid flow, heat transfer, and EMBr effects to analyze solidification in a continuous casting mold. In this model, a modified k-ε turbulent model was used to investigate the effects of turbulence on transport phenomena in the continuous casting system. The governing equations solved in this model were continuity, momentum, energy, k, and ε. Solidification was modeled on the basis of a continuum model. The Navier-Stokes equation was modified by adding Darcy terms to account for the mushy zone and the Lorentz force term. Based on the results obtained, they reported that EMBr successfully suppresses the melt flow and results in velocity reduction in the meniscus region. The suppression of melt flow in the mold region caused a large temperature gradient near the solidified shell. Figure 2.18, 2.19, and 2.20 demonstrate velocity vectors, temperature contours, and turbulent energy contours due to the three different cases of magnetic imposition respectively. [33] 25

35 (a) (b) (c) Fig Velocity vectors with the variation of magnetic flux density. a) B max = 0.0 T, b) B max = 0.17 T, and c) 0.35 T (a) (b) (c) Fig Temperature contours with the variation of magnetic flux density. a) B max = 0.0 T,b) B max = 0.17 T, and c) 0.35 T 26

36 (a) (b) (c) Fig Turbulent energy contours with the variation of magnetic flux density. a) B max = 0.0 T,b) B max = 0.17 T, and c) 0.35 T Tian et al. [34] developed a numerical model to analyze the effects of casting speed on fluid flow in a funnel shaped mould with a new type EMBr. Fluent software was used for discretization of governing equations, based on the control volume method. A schematic view of this new type of magnet is shown in Figure Fig Schematics of a new type of magnet 27

37 They reported that this new type of coiled magnet can produce larger magnetic flux density under the same electrical parameters compared to the conventional type of magnet. In their study fluid flows were considered to be turbulent and were modeled by solving the k-ε turbulence model. Based on their results, they reported the new type of magnet successfully increased the magnetic flux density owing to the irregular and inhomogeneous distribution in the mould region. They also stated that turbulent flows are significantly controlled with the help of this new type of magnetic. The nail dipping method has been a simple and inexpensive method to determine the height of the steel top surface as well as the slag layer thickness. This method has also been used to evaluate the fluid flow in the meniscus region of a continuous casting mold.. Figure 2.22 shows how fluid flows is evaluated in this technique.[35] Fig Demonstration of fluid flow evaluation. Ji et al. [35] studied the effects of EMBr on flow in a slab continuous casting mold using the nail dipping measurement. Figure 2.23 shows a schematic picture of a nail board measurement. 28

38 Fig Schematic of nail board measurement Based on the results they reported, the electromagnetic force suppresses the high-speed flow and decreases the meniscus flow velocity. They also reported that by increasing the SEN depth and increasing the casting speed without EMBr, the meniscus velocity was decreased and caused a smaller wave height at the surface. Hernandez et al. [36] modeled a fluid flow and investigated the effects of EMBr position, mould curvature and slide gate of steel in the mold of slab casting. This model was verified by classical one-dimensional MHD Couette flow, which simply allows a direct comparison with the numerical calculations. The positions for the magnetic field at the same level as the discharging ports of the submerged entry nozzle (SEN) and the second one is below the SEN tip. By locating the magnetic field at the same level as the discharging port and increasing the magnetic flux leads to rise of the discharging jets and the elimination of the two upper roll flows with lower roll flows with smaller velocities. In the case of the second position for locating the EMBr, this eliminated the effects of the biased flow in the meniscus velocity profile and induced a uniform downward flow. Fluid flow in the mold considered as a turbulent flow. In this work, Launder and Spalding k-ε models was used for simulating turbulence flow. The governing equations were discretized in Fluent software using an implicit, first order up-wind scheme and the Simple C algorithm for pressure-velocity coupling. They also considered the effect of induced magnetic field by solving the equation given below: B t + (U. )B = 1 μσ 2 B + (B. )U

39 Based on the results they found that EMBr does not eliminate the effect of the slide gate inside the SEN in either of the positions studied. They also found that the mold curvature created a flow towards the outer mold walls. In this case EMBr in the second position had better effects in controlling this flow. Extensive modeling and experiments by Timmel et al. [37] on fluid flow in continuous casting mold in the presence of a DC magnetic field, revealed that with the magnetic field, the issuing jet of liquid steel became more horizontal and the impingment point on the opposite side wall was shifted upward. In this model, they built a physical model using a low melting point liquid metal to investigate the flow structure. The alloy chosen to simulate the continuous casting process was GaInSn, as it is a liquid at room temperature. The experiments were performed in a discontinuous mode. Laser and an ultrasonic distance sensor were used during the process to monitor the liquid level of both the tundish and the mold. For measuring the fluid velocity in the mold, Ultrasound Doppler Velocimeter (UDV) was used. Based on the results they reported the significance change of the flow under the EMBr. Figure 2.24 presents the results based on UDV at the nozzle outlet. (a) (b) Fig Ultrasound Doppler Velocimeter (UDV) measurement of the horizontal flow at the nozzle outlet. a) without magnetic field, b) B max = 0.31 T 30

40 WU et al. [38] came up with a computational fluid flow model to simulate the effects of EMBr and nozzle outlet angle on fluid flow in the continuous casting process, heat transfer, and inclusion removal in continuous casting mold. The continuity, momentum, energy, k, and ε equations represent the governing equations in this model and have been discretized by commercial software based on the Control Volume method. Magnetic induction model in Fluent was adopted to model the localized-type of static magnetic field, to incorporate effects of EMBr on the steel flow at the mold region. They observed that, by increasing the magnetic field strength, the surface velocity of the fluid increases. They also reported that by increasing the angle of the nozzle or imposing the magnetic field, the temperature at the meniscus increases, which is significantly helpful to decrease hooks and oscillation marks, also facilitate the melting of the top slag. According to the free surface level profile results, increasing the magnetic field had slight effects on the surface fluctuations. However, by increasing the magnetic field and raising the nozzle angle up to 15 o degree, the surface fluctuations were considerably decreased. In terms of inclusion removal they stated that increasing the magnetic field can increase the inclusion removal fraction.[38] Application of EMBr in Channel Flows In terms of the effects of magnetic fields in channel flows, there have only been a few studies of fluid flows in ducts and free surface flows in ducts where the effects of magnetic fields have been studied, in comparison with the continuous casting process. In this section, a number of relevant papers are reviewed. Thomas et al. [39] developed a numerical model to analyze the effects of a magnetic field on mean velocities and turbulence parameters in molten metal flows through a square duct. The physical and computational domains considered in their study are shown in Figure

41 Fig Physical and computational domain In this model, Direct Numerical Simulation (DNS) was used to characterize the three dimensional transient flow. It was found that in the case of turbulent flows, the magnetic field tends to significantly re-laminarize the flow and restructure the turbulence. They observed that the magnetic field suppresses the turbulent flow and flattens the velocity profile across the crosssection. In terms of Hartmann flow, Depuy et al. [40] developed a three dimensional model to investigate the effects of magnetic field on fluid flow and heat transfer for a channel flow. In this study, they used the commercial software COMSOL for discretization of the governing equations based on the finite element model. They have also coupled Navier-Stokes with the MHD equations to study the effects of EMBr on the velocity profile. The Lorentz force, velocity profile, and the Nusselt number results were compared to the exact solution for the Hartmann problem. The analytical and computed velocity profiles for various magnetic field strength are shown in Figure

42 Fig Analytical and computed velocity profile for various Hartmann Numbers As shown in the Figure 2.26, the velocity magnitude in the channel flow decreases with increasing EMBr strength. Lofgren et al. [41] have made a major effort at understanding liquid metals flowing in an open channel in the presence of EMBr. In 1997, they investigated the effects of EMBr on liquid metal flowing over a moving substrate. They used an analytical solution to investigate the influences of EMBr on the flow field. For simplification, they divided the flow into two regions, the impinging region and the film region. It was assumed that in the film region the boundary layer equations are valid. The schematic central symmetric plane is shown in Figure Fig Schematic two dimensional slot jet impingement 33

43 They proved in their research study that the flow was dependent on the Reynolds number, Re, the Froude number, Fr, the Hartman number, M, and the dimensionless β number. β has been defined as the ratio of the belt velocity to jet velocity. Based on the results obtained from this work, it has been shown for β <Fr -2/3, a hydraulic jump appears and the thickness of the fluid grows suddenly on the moving substrate. They also reported that with the help of EMBr, this hydraulic jump vanishes and this could help the stability of the flow field and the solidification process in the Horizontal Single Belt Casting (HSBC) process. [41] In 1999, they published another paper and investigated the stabilizing effects of a transverse magnetic field on the film flow. They used the same two dimensional flow, considering the upper wall as a free surface. A schematic of the HSBC caster is shown in Figure Fig Horizontal belt strip casting feeding trough a slit It has been found that the transverse magnetic field has two damping mechanisms. For a small interaction number N 1, the main mechanism is the braking of the flow, reducing the shear stress. The other damping mechanism is due to dissipation by the Joule effect. [42] In 2003, Lofgren et al. used a set of three-dimensional shallow water equations, to simulate the metal flow introduced on to a moving substrate. They investigated the effects of a transverse magnetic field on the fluid flow. It was shown that the effects of the magnetic field are to brake the flow and damp the standing hydraulic jump patterns, and that these could also be modified using different feeding methods. [43] Extensive experiments by Andreev and Thess. [44] have revealed the effects of non-uniform magnetic fields on liquid metal flow in a plane channel with insulating walls. In their experiments, they used eutectic alloy GaInSn as the working fluid, which is conveniently a liquid 34

44 at room temperature. In terms of inhomogeneous magnetic field, magnets with finite dimensions were used. The conceptual structure of this case is shown in Figure Fig Test section used for the experimental investigation As can be seen in Figure 34, based on velocity measurements, they divided the channel in the three different regions; namely: i. a turbulence suppression region ii. a vortical region iii. a wall jet region. They observed the effects of EMBr in the first region brakes the incoming flow and turns the velocity profile to the M-shaped form, and this decreases the velocity fluctuations significantly. In terms of section two, which has the strongest magnetic field, this transformation became more significant. In the third region because the magnetic field is negligible and the increase in the intensity of the flow increase depends on the fluid Reynolds number. 35

45 Ji et al. [45] developed a numerical model to study the effects of a transverse magnetic field on damping the turbulent pipe flow field. For this model they used the standard k-ε turbulence model to consider the turbulence effects. The model was verified by data available from the literature. The numerical results vividly demonstrated the effects of turbulence suppression by the magnetic field and correctly predicted the opposing Lorentz force. Studies of fluid flow and solidification in the presence of a magnetic field in the Horizontal Single Belt Caster were carried out at the MMPC by R. Aboutalebi et al. [46]. In that research, they developed a numerical model to simulate turbulent fluid flow and solidification in the presence of EMBr in an extended nozzle metal delivery system. The simulation was carried out for plain carbon steel. The low-reynolds k-ε turbulence model was used for simulating the turbulence flow and the Darcy-porosity approach was adopted to simulate fluid flow within the mushy solidification region. The electrical potential approach was taken into account in terms of the low magnetic Reynolds number for simulating the effects of the DC magnetic field. The MHD model generated two source terms in the Navier-Stokes and Energy Equations, namely; the Lorentz Force and the Joule heating term, respectively. In this study, three different heterogeneous DC magnetic designs were tested numerically and the best one is shown here based on the numerical results. The three types of designs are shown in Figure Fig Typical configurations of magnetic devices considered for the model (a) type A, (b) type B, (c) type C, (d) magnetic flux density profile for type B. 36

46 Fig The effect of magnetic flux density on the flow pattern in the symmetry plane (type B) (a) B = 0.2 T, (b) B = 0.5 T, (c) B = 0.7 T and (d) B = 1.0 T. 37

47 Chapter Mathematical Model Chapter 3.0 considers the mathematical model needed to solve for the velocity fields with and without EMBr for a typical slot metal delivery system being studied at McGill Problem Statement A typical HSBC with a vertical slot nozzle metal delivery system has been considered in this study. A schematic of the system is shown in Figure 3.1. As can be seen, the liquid metal is continuously fed into a tundish from which the melt is deposited onto the moving belt through a slot nozzle. The geometrical and operational parameters associated with the caster are given in Tables 3.1 and 3.2, respectively. Fig Schematic of the Horizontal Single Belt Caster with a vertical slot nozzle metal delivery system Table 3.1. The geometrical parameters of the caster. Parts Measures(mm) Belt length 300 Belt width 50 Nozzle width 50 Nozzle height 12 Nozzle thickness(size) 4 Reservoir width 50 Reservoir height 30 Reservoir length 50 38

48 Table 3.2. The operational parameters of the pilot HSBC caster. Parameter Unit Value Belt speed (m/s) 1 Strip thickness (mm) 10 Strip width (mm) 50 Nozzle dimension (mm mm) 50 4 Magnetic flux density (T) Mathematical Formulation In the mathematical formulation of the Magnetohydrodynamic (MHD) system, the following assumptions have been made: The fluid is incompressible and Newtonian. The two-equation k-ε model was used to simulate the turbulent flow. A steady state flow was assumed. Constant physical properties are considered for fluid MHD turbulence is not considered. A DC magnetic field is applied The interaction of fluid flow and the magnetic field can be described by a complete set of MHD equations, including the turbulent Navier-Stokes equations, the continuity equation, the k-ε turbulence model, Ampere s law, Faraday s law, Ohm s law, Gauss s law, and the current conservation equation [18, 47-52]. These equations can be generally expressed as: Gauss s law for magnetism. B = where B is magnetic field. This equation represent the net magnetic flux out of any closed surface is zero. 39

49 Faraday s law E = B t 3.2 where E and B represent electrical field and magnetic field, respectively. Faraday s law shows a quantitative relation between a changing magnetic field and the electrical field. Ampere s law B μ m = J 3.3 where B, J, and μ m are the magnetic field, the current density, and the magnetic permeability, respectively. Ampere s equation reveals the magnetic field generated by a given distribution of current. Current conservation. J = This equation describes the conservation of current density and can be derived by taking the divergence of Ampere s law. Ohm s law J σ e = E + v B 3.5 where J, E, B, v, and σ e are current density, electrical field, magnetic field, velocity, and electrical conductivity, respectively. This equation represents that current density is in direct correlation with the electrical field and the cross product of the velocity and the magnetic field. Continuity equation ρ t +. (ρv ) =

50 where ρ and v are fluid density and fluid velocity respectively. This equation shows in any steady state process, the rate at which mass enters a system is equal to the rate at which mass leaves the system. The Navier-Stokes(N-S) equation (ρv ) t +. (ρv v ) = P +. (τ ) + J B 3.7 where ρ, v, P, τ, J, and B are fluid density, fluid velocity, pressure, shear stress, current density, and magnetic field, respectively. The solution of the N-S equations is a velocity field, which is essential to study the fluid flow. The first five equations 3.1 to 3.5, govern the induced effect of an applied external magnetic field on the liquid metal flow. In numerical calculations, rather than using these five equations independently, the magnetic induction equation is often used as a fundamental governing equation, because it links B directly to the velocity field. The magnetic induction equation can be obtained by eliminating current density and electrical field from the equations. Taking the curl of both sides of Ohm s law and substituting Ampere s law and Faraday s law into it, the magnetic induction equation can be obtained: B t + (v. )B = 1 μ m σ e 2 B + (B. ) v 3.8 B is defined as: B = B 0 + b 3.9 where b and B 0 respectively represent the induced magnetic field and the imposed magnetic field. After the magnetic field is solved from the induction equation, the current density can next be calculated, using Ampere s law. Then, having the current density, the Lorentz force, which represents the source term in the N-S equations, is calculated. The Lorentz force equation is expressed as below: F = J B

51 In summary, the governing equations of MHD flow for incompressible liquid metal with constant properties can be expressed as: ρ +. (ρv ) = 0 t (ρv ) t +. (ρv v ) = P +. (μ v ) + J B B t + (v. )B = 1 μ m σ e 2 B + (B. ) v 3.11 B μ m = J F = J B Another way to calculate the induced current density and Lorentz force is by the electrical potential method, when the magnetic Reynolds number (Re m ) has a very small value. In this method, by ignoring the induced magnetic field, the induced current density and the Lorentz force can be determined by using the following set of equations. F = J B J = σ ( φ + v B 0 ) 3.13 where φ denotes the electrical potential.. J =

52 By inserting current density from equation 3.13 into the conservation of current density equation 3.14, a Poisson equation for electric potential can be derived as, 2 φ =. (v B 0 ) = B 0. v = B 0. ω 3.15 where ω vorticity and external magnetic field is given as B 0. The Standard k-ε Model of Launder and Spalding was used for simulating the turbulent flow. In the k-ε model, k represents the kinetic energy of turbulence per unit mass and ε represents its rate of energy dissipation. The kinetic energy of turbulence is expressed as follows : k = 1 2 (u2 + v 2 + w 2 ) 3.16 where u, v, and w are the fluctuating components of the velocities in X, Y, and Z, directions, respectively. So, in addition to continuity and momentum equations, two extra equations for k and ε must be solved to simulate the effects of turbulence. (ρk) t + div(ρkv ) = div [ μ t σ k grad k] + 2μ t E ij. E ij ρε 3.17 (ρε) t + div(ρεv ) = div [ μ t ε grad ε] + C σ 1ε 2μ ε k te ij. E ij C 2ε ρ ε2 k 3.18 where μ t and E ij,is a turbulence viscosity and the rate of deformation, respectively, and can be calculated from these equations: E ij = 1 2 [ u i x j + u j x i ] 3.19 μ t = ρ C μ k2 ε

53 where C μ is a dimensionless constant. Equations 3.13 and 3.14 require five adjustable constants C μ,σ k, σ ε, C 1ε, and C 2ε. The recommended constants, used in this model are given in Table [53] Table 3.3. Recommended constant for k-ε turbulent model Constant C μ σ k σ ε C 1ε C 2ε value These values for the constants were arrived by comprehensive data fitting by Launder and Spalding for a wide range of turbulent flows Numerical Solution Procedure The governing equations associated with appropriate boundary conditions were solved numerically using the following steps: The computational domain was defined. Mesh generation was made. The mesh was exported to an appropriate CFD package. Scale the geometry and quality test of the mesh. The various models needed to be adopted in the modeling were set up. (This step represents the equations that need to be solved in the model) The material properties were set. The boundary and operational conditions were set. The solution parameters were set up. Iterative solution of the discretized equations was made until convergence was achieved. Post processing and graphical representation was carried out based on computed results. 44

54 Commercial software called Gambit was used for creating the domain and generating the mesh for the CFD solver. The present calculations were carried out using the CFD FLUENT 14.5 package. The geometry and the mesh are shown in Figure 3.2 and Figure 3.3, respectively. (a) (b) (c) (d) Fig The geometry of computational domain using GAMBIT a) Plan view, b) Isometric view, c) Side view, and d) Width view (a) (b) (c) (d) Fig The mesh generation of the computational domain using GAMBIT a) Plan view, b) Isometric view, c) Side view, and d) Width view 45

55 Because of symmetry, calculations were carried out in only one half of the metal delivery system of the caster. This was done to save on computational time and costs. All velocity components were set to zero at the walls using the non- slip boundary condition, except for the moving belt. The boundary condition for the moving belt was a non-slip shearing condition and for wall motion it set a moving wall. At the free surface of the liquid, the shear stresses were set to zero. For the MHD boundary condition, all walls were considered as being electrically insulating, expressed by: J n = 0 where n, and J represent the normal vector to the boundaries and the current density, respectively. In terms of number and type of cells, a number of non-uniform hexahedron cells were used. This has a better compatibility with the selected geometry and the meshing was made finer near the walls. In numerical modeling, the steady state flow field solution was first obtained without considering the magnetic effects. After obtaining the steady state velocity field, the MHD calculation was introduced using the previously converged flow field. For better convergence in MHD calculations, the strength of the externally imposed magnetic field was under-relaxed properly. Regarding the solution parameters, the SIMPLE algorithm was used for pressure-velocity coupling. In terms of discretization of the convective terms in the equations for the first 1000 iterations, we used the first order upwind scheme. For the second 1000 iterations, we used the second order upwind scheme, and from 2000 iterations until convergence, the QUICK scheme was performed for momentum, k, and ε. In the case of under relaxation factors for pressure, momentum, k, and ε, we started from 0.4, 0.5, 0.6, and 0.6 respectively, and slowly increased them to 0.6, 0.7, 0.8, and 0.8. For the second part, when the external magnetic field was imposed, the same algorithm for pressure-velocity coupling was used but in terms of the discretizing equations for momentum, k, ε, and the MHD equations, a first order scheme was used. The discretization scheme for the flow equations evolved using the same approach, as explained before, but for the MHD equations, the scheme for discretization did not change. For the under 46

56 relaxation factor for the pressure, momentum, k, ε, and MHD, we started with 0.4, 0.5, 0.6, 0.6, and 0.8, respectively, and gradually increased them to 0.6, 0.7, 0.8,0,8, and 0.9. After all the simulations were completed, the results were post-processed to generate velocity fields with, and without EMBr. For all the simulations, a 288 core High Performance Computer Cluster from SGI with 64 GB RAM was used. 47

57 Chapter Physical Model Chapter 4 provides the details of the physical model developed to study the magnetic flow control in a typical metal delivery system adopted for a pilot caster. A water model of the pilot Horizontal Single Belt Caster (HSBC) was constructed and used to physically study the effects of EMBr on the fluid flow. Liquid steel was replaced by aqueous sodium chloride solution, due to its dynamic similarities and ease to be use at low temperature. A schematic representation of the water model is shown in Figure 4.1. Fig Schematic of an HSBC Metal Delivery System using a salty water model This physical model consists of four parts, namely, a reservoir, a vertical slot nozzle, a moving belt, and an electromagnetic braking system (EMBr). The reservoir and the vertical slot nozzle 48

58 were used for control the inflow rate of water onto the moving belt and the EMBr was tested to control the flow of NaCl solution deposited onto the moving belt. It is believed that the damping of the flow as a result of the imposed magnetic field regulated the flow and affected the free surface level. A picture of the water model constructed at the MMPC is shown in Figure 4.2. Figure 4.2. An image of the physical model at MMPC 4.1. Principles of MHD Water Model Development The damping magnetic force is induced in the liquid as a consequence of interactions between the imposed magnetic flux density and an induced current in the moving conductive fluid. This effect mainly depends on the electrical conductivity of fluid (σ), the fluid s velocity (v), and the imposed magnetic field (B). The MHD flow of electrically conducting fluids exposed to a DC magnetic field B 0 is basically characterized by the following dimensionless numbers: 49

59 1. The Hartmann Number is defined as the ratio of the Lorentz force to the viscous force. The Hartmann number is expressed as below: Ha = BL c ( σ ) 4.1 μ where B, L c, and μ are the magnetic field, the characteristic length, and the dynamic viscosity, respectively. 2. The Interaction Number, or Stuart Number, is defined as the ratio of the magnetic force to the inertial force. The Stuart Number is presented as : N = Ha2 Re = B2 L c σ e ρu 4.2 where u, L c, σ e, B, ρ, and Re respectively represent the fluid velocity, the characteristic length, the fluid electrical conductivity, the magnetic field, the fluid density, and the Reynolds number. 3. The Magnetic Reynolds Number, which is defined as the ratio of magnetic advection to magnetic diffusion. Re m = μ m σ e ul c 4.3 where μ m and σ e are magnetic permeability and the electrical conductivity of the liquid respectively. For this modeling the governing equations can be non-dimensionalized as follows: x = x D, y = y D, z = z D, u = u u 0, v = v u 0, w = w u 0, P = P ρu 0 2 k = k u 2, ε = εd 0 u 3, B = B, J = 0 B 0 J, t = tu 0 σ 0 u 0 B 0 D where, D is the hydraulic diameter, u 0 is the speed of the moving belt, B 0 is the imposed magnetic field, μ is the dynamic viscosity of liquid steel, σ 0 is the electrical conductivity of liquid steel, and μ m is the magnetic permeability in vacuum. 50

60 The non-dimensionalized equations can be written as: u. u = t +. (u u ) =. P + 1 Re. u + Ha2 Re ( J B ) 4.6 B t + (u. )B = 1 Re m 2 B + (B. )u 4.7 k + t. (k u ) = 1 Re. [ k 1 Re m B = J 4.8 σ k ] + 2 Re E ij. E ij ε 4.9 ε + t. (ε u ) = 1 Re. [ ε ] + 2 (C σ ε Re 1ε)E ij. E ij (C 2ε ) ε k where the value of four dimensionless constants σ k, σ ε, C 1ε, and C 2ε are equal to, 1.00, 1.30, 1.44, and 1.92, respectively. Based on the non-dimensionalized equations, there are essentially three independent dimensionless parameters (Reynolds, Stuart, and Magnetic Reynolds numbers), appearing. These govern the flow field in presence of the imposed magnetic field. In magnetic damping of moving conductive fluids, the damping impact is mainly dictated by the Interaction Number (or Stuart Number), which is a combination of the Ha and Re numbers. Thus, the Stuart number is considered as the main similarity criterion in the development of the present MHD physical model. In this model, a 15% NaCl solution with an electrical conductivity of 25 1 Ω.m was used as the conductive fluid. Since the electrical conductivity of sodium chloride solution is much lower than the electrical conductivity of liquid metals (e.g. liquid steel), an extra current can be imposed by applying a constant voltage in the same direction as the induced current, in order to enhance the Lorentz force. In this case, the Interaction number is rewritten as: [54] N = (σ e EBL c ) ρu where E is the electrical potential gradient ( V ) applied in the direction of induced current. Table m 4.1 compares the physical properties of liquid steel and 15% NaCl solution. As can be seen, the 51

61 electrical conductivity of NaCl solution is about 4 orders of magnitude lower than that of liquid steel. Table 4.1. The properties of salt water and liquid metal Symbols Definition Unit Liquid steel 15%NaCl solution ρ σ density conductivity ( kg m 3) ( S m ) μ Dynamic viscosity (Pa. s) ν Kinematic viscosity ( m2 s ) It is well known that the magnetic permeability μ m is almost constant for non-ferromagnetic materials, and is equal to the magnetic permeability value in a vacuum: 4π 10 7 N A2. We limit the liquid metals under consideration of nonferromagnetic materials, thus, μ m is taken as a constant. [55] According to the solution proposed for producing an iso-kinetic flow, a magnetic region was created so as to impose the magnetic field. In terms of the imposition of the magnetic field, two NdFeB permanent magnets have been used. They were located on the top and the bottom of the moving belt, along the width of the strip. Figure 4.3 shows a two dimensional schematic of the water model and the proposed location of the magnets. The South Pole of the magnet is located above, and the North Pole below, the strip. Fig Schematic of cross section of the water model with proposed magnets locations 52

62 The dimensions of the permanent NdFeB magnets are given in Table 4.2. Table 4.2. Dimensions of the permanent magnets Length(cm) Width(cm) Height(cm) A frame made from extruded aluminum tubing was designed and used for imposition of the magnetic field and to fix the location of the magnets with respect to the flowing stream of salt water. The design of magnets holder is shown in Figure 4.4. Fig Design of magnet holder system 53

63 Based on the operating parameters and assuming a magnetic flux density (B) of 0.3 T, a characteristic length of 28 mm, the Interaction number for magnetic flow control on the pilot caster can be calculated as: N = σ eb 0 2 L C ρu = = 0.25 For the same Interaction number in the physical model and using equation 4.11 one can obtain: EB u 2 ~400 Assuming a magnetic field of 0.7 T and applying a voltage of 12 volts in the width (Z direction) of the strip on the physical model, one will obtain a belt speed of about 0.65 m/s. Based on the above discussion the characteristics of the proposed physical model in comparison with the prototype are given in Table 4.3. Table 4.3. The characteristics of the Prototype and the Physical Model Parameter Unit Prototype Model Working fluid - Liquid steel 15% NaCl solution Electrical conductivity of liquid (S/m) Belt speed (m/s) (m/s) Magnetic field strength (Tesla) Imposed electrical potential (V/m) Characteristic length (m) Hartmann Number Interaction Number Reynolds Number - 28,823 14,530 Magnetic Reynolds - < 1 < 1 There is a relation between the electrical conductivity of salt water and the temperature of the solution as [56]: σ T = σ T0 (1 + α (T T 0 ))

64 where α has been measured as 0.02 or 0.03 S/m; this means that the electrical conductivity of salt water would change by +2% to +3% by increasing the solution temperature by 1 Kelvin. So, in this case, by increasing the solution temperature we could raise the electrical conductivity of the sodium chloride solution up to 70 (S/m) which is almost four times greater than that at the room temperature. So, in this new approach the Imposed electrical potential will be about 100 V/m which requires the imposition of the electrical potential of 5 volts in the z-direction. An image of the physical model is shown in Figure 4.5. Figure 4.5. An image of the physical model at MMPC 55

65 In terms of running the experiments on the physical model, at first 15% NaCl solution was made in a cylindrical plexiglass tank with the capacity of 20 liters. A pump with the power of 190 watts was used for pumping the salt water from the reservoir into the tundish. A flow rate of 0.15 liter sec was maintained at the inflow to the tundish to achieve a steady state level in the tundish. The solution was drained into a square tank underneath the belt from which the NaCl solution was discharged into the reservoir. In order to maintain a steady state flow in the system it was required that the level of water in the square tank was maintained at the height of 5cm. A Tachometer was used to control and fix the speed of the moving substrate. Regarding the imposition of magnetic field, two Neodymium permanent magnets providing with the magnetic flux density of 0.7 Tesla perpendicular to the moving belt has been used. Two graphite electrodes with the electrical conductivity of S m and a 12 Volts DC power supply were used for imposing the electrical filed through the width of the channel within the imposed magnetic region. It is noted that different types of electrodes such as Titanium, stainless steel, copper, lead and graphite were tested, and which graphite showed a better performance in NaCl solution. A photograph of the experimental set-up is shown in Figure 4.5 representing the permanent magnets at the top and bottom of the moving belt as well as the graphite electrodes for imposing the electric field. Flow visualization studies were carried out using suspended threads in the proper locations in the system. 56

66 Chapter Results and Discussion This chapter provides readers with the results and a discussion of the mathematical and physical modeling used to analyze of the effect of DC magnetic fields on fluid flow in a metal delivery system using a vertical slot nozzle for the HSBC process. It is important for the convenience of readers, to state that the X, Y, and Z directions are respectively in the length, thickness, and width of the strips and plane close to the surface and plane close to the belt are planes 1mm below the surface and 1mm above the moving belt respectively Results of Mathematical Model This section on the results of the mathematical model is divided into four subsections. It starts with studying the fluid flow without an imposed magnetic field followed by the mesh dependency test. The next section provides some typical results in the presence of EMBr and demonstrates the effects of an opposing Lorentz force on the fluid flow. Subsequently, the validation of our numerical model was assigned with a similar case. In the last part, a parametric study is done with respect to the effect of EMBr location and magnetic flux density on the fluid flow Fluid flow of liquid steel on to the moving belt without EMBr In the pilot-scale Horizontal Single Belt Caster (HSBC) with a vertical slot nozzle designed at the MMPC, the liquid metal is discharged vertically onto the moving belt via a slot jet flow with a cross sectional area of 4mm x 50mm wide and with a nozzle-to-plate spacing of three times the width of the slot nozzle. The flow behavior in this system can be presented in various forms such as a velocity field, or streamline contours on a certain plane of the system, velocity profiles along the different lines of desired planes, and pressure distribution within the area of interest. In this metal delivery system, the melt impinges onto the moving surface, where a stagnation point is formed. This is followed by developing a circulating flow within the impingement zone. The Impingement zone is extended up to the point where the effect of impingement vanishes and 57

67 further downstream flow becomes parallel. Within the impingement zone, a reverse flow is formed due to pressure changes along the spreading jet flow. Velocity and pressure fields vary from stagnation region with high pressure to lower pressure within the impingement zone. Additionally, by moving the impingement surface, the stagnation point shifts slightly downstream.[57, 58] The velocity pattern in the longitudinal symmetry plane of the system is shown in Figure 5.1 wherein the ratio of belt speed to the average jet velocity is equal to 0.4. Fig The velocity contour overlaid with the velocity vectors in the longitudinal symmetry plane. As can be seen from Figure 5.1, the impinging jet deviated slightly downstream so that the impingement point moves to about 1/3 of the slot width, downstream, while a small dead zone is formed at the back wall. A downstream recirculation flow pattern is clearly observed in the upper part of wall jet at free surface. Impingement of fluid on a stationary surface makes a rather large circulation flow in the impingement region while the moving surface causes the circulation flow to shrink both upwards and backwards [57, 59]. Therefore this recirculation flow presumably shrinks by increase of belt speed. The velocity profiles in the z-direction on the horizontal sections (z-x plane) are drawn to demonstrate the development of the fluid flow along the length of the strip. These velocity profiles picked at two different horizontal planes, the first plane is the horizontal plane located 1mm above the moving belt and the second one is the plane positioned 1mm underneath of the surface of the strip. The predicted velocity profiles are shown at different positions along the 58

68 length of the belt namely; 4.5, 5.4, 7.5, 10.4, and 24.5 cm ahead of the back-wall which demonstrated in Figures 5.2 and 5.3. Fig Predicted velocity profile of liquid steel in the width (Z direction) on the horizontal plane 1mm below the surface with distance from the back wall: (a) 4.5, (b) 5.4, (c) 7.5, (d) 10.4, and (e) 24.5 cm. Fig Predicted velocity profile of liquid steel in the width (Z direction) on the horizontal plane 1mm above the moving belt with distance from the back wall: (a) 4.5, (b) 5.4, (c) 7.5, (d) 10.4, and (e) 24.5 cm. The velocity profiles at different distances starting from the impingement zone along the moving belt clearly represent the flow development downstream. It can be readily understood from Figures 5.2 and 5.3 that fully developed flow is obtained at the distance of 24.5 cm from the back-wall. The jet spreads towards the side dams from symmetry plane results in a higher flow rate close to the side dam while the flow is slightly damped in the mid area of the horizontal 59

69 plane due to the stagnation point. This effect is more appreciable in the plane just above the moving belt than in the plane just underneath the free surface, as it is closer to the stagnation point. Figure 5.4 represents the predicted pressure field together with velocity pattern in the longitudinal symmetry plane. The higher pressure is experienced in the stagnation region planes and the pressure decreases in an anisotropic way within the impingement region as seen in the Figure 5.4. This makes the jet flow non-uniformly spread over the impingement zone. A similar pattern can be seen in longitudinal section close to side dam as shown in Figure 5.5. However, the pressure which developed by the impingement jet, is lower at the side-dam plane compared to the symmetry plane and this leads to spread out the flow and increase the velocity near the side dams. Fig The pressure contour overlaid with the velocity vectors at the symmetry plane Fig The pressure contour overlaid with the velocity vectors at the plane near the side dam 60

70 Z (m) The flow development within the impingement region is presented in Figure 5.6 in which the y- direction velocity profiles in the longitudinal symmetry plane at different locations along the impingement region are compared. Fig Velocity profiles in the y-direction in the symmetry plane with distance from the back wall: (a) 4.5, (b) 5.4, (c) 7.5, (d) 10.4, and (e) 24.5 cm Evaluation of mesh dependency Different mesh densities were examined to evaluate the effect of grid size on computed results. Four different cell numbers of 67860, , , and were tested. The computed velocity profile along the width of the strip in the horizontal plane using different mesh sizes revealed that the velocity profile for the last two mesh sizes ( , and cells) are almost identical as depicted in Figure 5.7. The maximum difference between the results obtained by using the last two mesh sizes is less than 3%. Thus to meet reasonable accuracy and computational costs, a non-uniform cell number of elements were selected for all of the numerical simulations reported here Velocity (m/s) cells cells cells cells Fig Grid dependency of computed velocity profile along the width of the strip at X= 5.4mm ahead of back wall and 1mm below the surface of strips. 61

71 Typical Computed Results The velocity contours with demonstration of velocity bands in X direction without the EMBr at two different planes along the thickness of the strip which are 1mm below the top surface of the strip and 1mm above the moving belt are shown in Figure 5.8. Fig Plan view velocity contours without EMBr at a) near the moving belt, and b) near the top surface As shown in this figure, the predicted velocity bands near the moving belt gradually decreased in velocity. Unlike the bottom plane, at near the top surface, because there are no shear stress effects due to the free surface boundary condition, velocity bands steadily increased. To achieve iso-kinetic flows, the flow along the thickness and width of the strip needed to approach the belt velocity. It can be concluded based on these velocity bands that the velocity field is maximized near the side dams at the beginning of the feeding with a vertical slot nozzle. This subsequently changes along the length of the forming strips (in the X direction) and the flow becomes a fully developed turbulent channel flow. This flow behavior can be explained by the creation of 62

72 B 0 (Tesla) stagnation point due to the vertical impingement of the liquid steel onto the horizontal moving substrate which damps the flow over the moving belt and pushes the fluid towards the sidewalls. This phenomenon has also been observed in the physical model developed in this study. In these circumstances, the dynamic pressure is dominant near the stationary sidewalls as compared to the symmetry plane. This type of feeding causes a non-uniform flow field across the width of the forming strips. Thus, the EMBr concept could be the solution for creating iso-kinetic flows both across the width, as well as through the thickness, of the forming strip. The imposed magnetic field is shown in Figure 5.9. The applied magnetic field shown in this figure is assumed to be constant in the region of the magnet location with a value of B 0 =0.5 T, while the magnetic fluxes decays away exponentially on either side, representing the fringing effect at the magnetic pole edges. The exponential function assumed here was similar to those adopted by other researchers. [46, 60] X (m)

73 Fig Local magnetic field and magnetic intensity distribution along the length of the system. Based on Ohm s law, the induced current density results from the interaction of the flow velocity with the imposed magnetic field and it is conserved in this domain. The conservation of current density is shown in Figure 5.10 for two different planes. It is noted that the induced current density causes a magnetic field to be induced within the system. The induced magnetic field is shown in Figure As can be appreciated from this figure, the level of induced magnetic field can be neglected compared to the imposed magnetic field. As mentioned in Table 4.3, the magnetic Reynolds number (Re m ) for liquid steel is less than unity. Thus, the induced magnetic field due to the induced electric current can be neglected. 64

74 The interaction between the induced current density and the applied magnetic field creates the Lorentz force, which leads to homogenous flow field across the width and thickness of the belt. Fig Induced current density at a) near the moving belt, and b) near the top surface Fig The induced magnetic field along the length of the system 65

75 The Lorentz forces on horizontal planes 1mm below the top surface and 1mm above the moving substrate are shown in Figure (a) (b) Fig Lorentz force a) near the moving belt, and b) near the top surface As demonstrated in this Figure, the maximum Lorentz force, which is equal to N/m 3, is created on the plane near the belt since the velocity magnitude near the belt is larger than that in the plane near the free surface. This occurred before X= 5.4cm ahead of the back-wall and it is opposed to the fluid flow with the inward direction. The magnitude of Lorentz force is proportional to the velocity of liquid steel. 66

76 Figure 5.13 shows a plane view velocity contours with EMBr for two different horizontal planes. Fig Plan view velocity contours with EMBr a) near the moving belt, and b) near the top surface As shown in Figure 5.13, the velocity bands have been affected by imposing the magnetic field in this region. The Lorentz forces demonstrated in Figure 5.12, assisted the flow development in positions close to the impingement region, and have a positive effect in suppressing the flow, and flattening the velocity profile across the width of the strip. Correspondingly, the magnetic flow controllers increased the velocity near the top surface and decreased it near the moving belt, which leads to produce iso-kinetic flow condition in the early stages of the casting process. The Lorentz force in the location of the magnets creates two large vortices. The decrease in the velocity and the appearance of two vortices inside the magnet gap in the range of < x < can be physically explained by an increase in the pressure gradient in the central part of the flow due to the braking effect of the Lorentz force. In terms of quantitative study of the effect of EMBr, velocity profiles in the z-direction on horizontal planes were drawn at different positions along X axis which is the flow direction. Figures 5.14 and 5.15 compare the velocity profiles across the width of strip at different positions in z-direction. 67

77 Fig Calculated velocity distribution on the horizontal plane 1mm above the moving belt with distance from the back wall: (a) 4.5, (b) 5.4, (c) 7.5, (d) 10.4, and (e) 24.5 cm. Fig Calculated velocity distribution a on the horizontal plane 1mm below the free surface with distance from the back wall: (a) 4.5, (b) 5.4, (c) 7.5, (d) 10.4, and (e) 24.5 cm. As can be seen from the Figures 5.14 and 5.15, the EMBr has an effect in controlling the flow on the moving belt and increased the velocity at the plane near the top surface and making the flow flat across the width of the strip and keep the velocity close to 1 m/s. The effect of magnetic field creates a more stable surface of the strip. 68

78 The velocity profiles in the longitudinal symmetry plane are shown in Figure As depicted in this Figure, the velocity magnitude across the thickness of the strip becomes nearly 1m/s after the fluid exit the constant magnetic region. The significant point is that, the same flow field remains along the flow direction, downstream of the magnetic braking. Fig Velocity profiles in the y-direction in the symmetry plane with distance from the back wall: (a) 4.5, (b) 5.4, (c) 7.5, (d) 10.4, and (e) 24.5 cm Verification of MHD Mathematical Model Introduction An experiment has been done by Thess et al. which demonstrates the effects of an electromagnetic brake on liquid metal flows in a closed channel. This study is similar to the problem for the metal delivery system proposed for the HSBC process. It is therefore the correct case to validate our numerical model with its results. In that study, they used an eutectic alloy, Ga 68% In 20% Sn 12%, which has a melting temperature of o C. The characteristics of the alloy are given in Table 5.1.[2] Table 5.1. The properties of the liquid alloy (Ga 68% In 20% Sn 12% ) Properties Value Density (ρ) 6360 kg/m 3 Kinematic viscosity (ν) m 2 /s Electrical Conductivity (σ) Ω -1 m -1 69

79 Operating Conditions The experiments were carried out in a Plexiglass rectangular channel 0.5 meter long with a vertical cross section 2W x H = 100mm x 20 mm. A schematic demonstration of the system is given in Figure Fig Test section used for the experimental investigation As illustrated in Figure 5.17, in terms of imposition of a heterogynous magnetic field, two NdFeB permanent magnets were applied to the top and bottom walls of the channel Numerical Model In the present work, we developed a numerical model to investigate the fluid flow under the effects of magnetic field for the same case, to validate our model. We took advantage of ANSYS Fluent 14.5 software for discretization of the conservations equations. In this model, we use a standard k-ε turbulent model to simulate the fluid flow. The inlet mean velocity value of the closed channel, as reported in the paper, was U 0 = 8cm/s, which corresponds to a Reynolds number Re =

80 The schematic of the calculated domain is shown in Figure 5.18 and the characteristics of the closed channel case are given in Table 5.2. Inlet Magnetic Region Outlet Fig The schematic of calculated domain Table 5.2. The dimensions and operating conditions Parameter Unit Value Inlet (m/s) 0.08 Magnetic Flux Density (Tesla) Channel Width (cm) 10 Channel Thickness (cm) 2 The magnetic system in the experiments consists of two rectangular NdFeB permanent magnets with the dimensions of 40mm height, 30 mm width, and 100mm length, so as to cover the entire width of the channel. As we are dealing with permanent magnets, the magnetic region will be heterogeneous, so the Y component of the magnetic flux density decreases in both directions along the X axis from the maximum of at the origin to a very small value at X=60mm and X= -60mm. Figure 5.19 shows the magnetic region in the cross section plane normal to Y axis. The location of this plate is Y=0, X= Entire length, and Z= entire width. 71

81 Fig The magnetic region in the cross section plane normal to Y axis Verification of results Figure 5.20 shows the velocity contour in the Y cross section plane before and after imposing the magnetic field. X Z Fig Velocity contours in the cross section plane normal to the Y axis, (a)before applying EMBr, and (b) after applying EMB 72

82 As shown in the velocity contours in the magnetic region, the velocity near the wall increases whilst in the middle, we observe a reduction in velocity, causing the M shaped velocity profile. We now present the velocity profile in the region of the applied magnetic field. In terms of illustration of the velocity profiles, the following dimensionless coordinates are used: x/h and y/l where H=20mm is the height and L=50mm is the half-width of the channel and also the velocity has been dimensionless according to the inlet mean velocity U 0. In Figure 5.21 the velocity profile are plotted as a function of spanwise coordinates y/l, for x/h=-5.3, -1.5, -0.75, 0.75 and 3. Fig Velocity profile in middle plane of the channel at Re=4000 As shown in Figure 5.21.a, the opposing force has no effects on the flow at x/h -5.3 and it shows the maximum velocity value is in the center of the flow. At x/h= -1.5 where x=60mm, one enters the magnetic field, as observed in Figure 5.21.b. The velocity magnitude starts to increas at the sidewalls and starts to create an M-shaped velocity profile. In Figure 5.21.c, a modest further 73

83 deformation of the streamwise velocity is observed. Figure 5.21.d continues to illustrate the velocity transformation in the downstream direction. Figure 5.21.e, also demonstrates the effect of EMBr for the flow exiting the magnetic region. It shows the velocity profile has been transformed completely and changed from the undesirable jet-like structure to the M-shape velocity profile for the liquid metal. Comparing our numerical results with the results reported in Thess et al. s paper [44], which is also based on experiments, there is a good agreement. This in term, validates our model Parametric study The effects of magnetic intensity and the magnetic brake location on the flow patterns in the metal delivery system were studied using the present model to evaluate the performance of the proposed magnetic flow control device. It has been assumed that the magnetic device used in this work provides a constant DC magnetic field within the region of the magnetic brake, while this decays exponentially on either side of the magnet, along the strip. The magnetic device was installed at two different positions in the system and three different magnetic intensities were tested in this model. Velocity profiles were evaluated in the z-direction (width of the strips) and y-direction (thickness of the strips) at different positions along the x-axis. For the profiles in the z-direction (width of the strip) on a horizontal plane, two different positions namely, 1mm above the moving belt and 1mm below the free surface, were considered. In terms of velocity profiles in the y-direction (across the thickness), the longitudinal symmetry plane was selected Magnetic Field Location Two different EMBr locations were adopted to control the flow field in the system. In the first case, the magnetic field of 50 mm long was applied along the strip ahead of the nozzle after the tundish (position (a)). The other one was 50 mm long which was installed just after the impinging slot jet (position b). The top view plan of the EMBr intensity distribution along the system for imposed magnetic intensity of 0.5 T at two different locations is shown in Figure

84 Fig The imposed magnetic field at two different positions. a) after the tundish, and b) just after the impinging slot jet Figure 5.23 compares the velocity field in the horizontal plane near the free surface and moving belt in which the EMBr is located at two different positions (a and b) with applied magnetics intensity of 0.5 T. The velocity contours in the corresponding planes without magnetic field are also given in this Figure. As shown in Figure 5.23, the velocity bands have been affected by imposing the magnetic field in both locations. The Lorentz forces demonstrated in Figure 5.24, assisted the flow development in position (a) close to the impingement region, and have a positive effect in suppressing the flow, and flattening the velocity profile across the width of the strip. Correspondingly, the magnetic flow controllers increased the velocity near the top surface and decreased velocities near the moving belt, which leads to having an iso-kinetic flow in the early stage of the casting process. It is noted that the magnetic flow control device installed at position (b) works much more properly considering that solidification is negligible in this region and the effect of any solidified layer can be safely ignored. 75

85 Fig Velocity contours in the longitudinal horizontal sections of the system after the slot nozzle 1mm above the moving belt: (a) without EMBr, (b) with EMBr in location (a), and (c) with EMBr at location (b) 76

86 Fig Velocity contours in the longitudinal horizontal sections of the system near the free surface: (a) without EMBr, (b) with EMBr in location (a), and (c) with EMBr at location (b) The electromagnetic force fields for EMBr located at two different regions (position (a) and (b)) are shown in Figure As demonstrated in the Figure 5.25, in both EMBr locations the direction of the Lorentz force is opposite to the liquid steel flow and its magnitude is proportional to the velocity of liquid steel. As seen in this figure the intensity of electromagnetic 77

87 force near the moving belt is higher than the Lorentz force at the free surface because the velocity magnitude at the belt is larger than at the plane near the free surface. The Lorentz force induced in the EMBr region at location (b) results in increasing the velocity in the surface layers of the flow and decrease the velocity near the bottom of the belt, so as to make all the layers of fluid close to 1m/s, equal to the belt speed. The direction of Lorentz force at the top surface in the magnetic field located right after the nozzle is in the same direction as the moving substrate and this is due to the creation of circulation flow in this region. Fig Lorentz force fields within the imposed magnetic region for the EMBr located in position (a); (a): near the surface, and (b): near the moving belt 78

88 Fig Lorentz force fields within the imposed magnetic region for the EMBr located in position (b); (a): near the surface, and (b): near the moving belt In terms of visualizing the effect of EMBr locations, velocity profiles were drawn at different positions from the back wall of the tundish. Figure 5.27 and 5.28 compare the velocity profiles across the width of strip at different positions in X-direction for two EMBr locations adopted in this model and Figure 5.29 indicates Velocity profiles in the y-direction in the symmetry plane. 79

89 Fig Velocity Profiles in the width direction near the free surface for; (1) EMBr located after the tundish, and (2) EMBr located after the slot jet nozzle with distance from the back wall: (a) 4.5, (b) 5.4, (c) 7.5, (d) 10.4, and (e) Fig : Velocity Profiles in the width direction near the moving belt; (1) EMBr located after the tundish, and (2) EMBr located after the slot jet nozzle with distance from the back wall: (a) 4.5, (b) 5.4, (c) 7.5, (d) 10.4, and (e)

90 Fig. 5.29: Velocity profiles in the y-direction in the symmetry plane (1) EMBr located after the tundish, and (2) EMBr located after the slot jet nozzle with distance from the back wall: (a) 4.5, (b) 5.4, (c) 7.5, (d) 10.4, and (e) As can be seen from the above Figures (Figures 5.27, 5.28, and 5.29), the EMBr at position (b) has a better effect on controlling the flow in the impinging zone. As shown in the velocity profiles demonstrated in Figure 5.27 and 5.28, the EMBr imposed at the second location( close to the slot jet) causes the velocity at the surface of the flow to increase and the flow near the moving belt to decrease, which leads to an iso-kinetic fluid flow in cross section. The magnetic field imposed at the second location close to the slot jet also makes the free surface more stable and diminish the surface circulation flow at the surface. On the other hand, the magnetic flow control in the region close to the impinging nozzle is presumably more efficient in real cases because of null solidification in this region. On the whole, it is proposed to install the magnetic device at the location near the impinging jet, which makes the flow more iso-kinetic in cross section. Figure 5.30 indicates the velocity contours overlays with the velocity vectors at the symmetry plane without an EMBr and with an EMBr at location (b). This figure clearly 81

91 demonstrated the effect of EMBr in the second location which caused suppression in the circulation flow at the top surface. Fig. 5.30: The velocity contours overlays with velocity vectors at the symmetry plane (1) without an EMBr (2) with EMBr located after the slot jet nozzle wall Magnetic Intensity The effect of magnetic intensity on fluid flow, for EMBr located at position (a) is shown in Figures 5.31 and 5.32 at two different horizontal planes near the top surface and near the moving belt, respectively. As demonstrated in this figure, by increasing the magnetic intensity, the effect of the braking force increases, resulting in increasingly iso-kinetic flow condition. 82

92 Fig Velocity contours in the longitudinal horizontal sections of the system near the moving substrate for different EMBr intensities at location (a): (a) B= 0 T, (b) B= 0.3 T, (c) B= 0.5 T, and (d) B=0.7 T 83

93 Fig Velocity contours in the longitudinal horizontal sections of the system near the free surface for different EMBr intensities the EMBr at location (a): (a) B= 0 T, (b) B= 0.3 T, (c) B= 0.5 T, and (d) B=0.7 T 84

94 The modification of velocity contours, for EMBr location (b) with three different magnetic intensities is presented in Figures 5.33 and As depicted in these figures, the magnetic intensity direct correlates with the development of iso-kinetic flow, along the fluid flow. Fig Velocity contours in the longitudinal horizontal sections of the system near the moving substrate for different EMBr intensities at location (b): (a) B= 0 T, (b) B= 0.3 T, (c) B= 0.5 T, and (d) B=0.7 T 85

95 Fig Velocity contours in the longitudinal horizontal sections of the system near the free surface for the EMBr at location (b): (a) B= 0 T, (b) B= 0.3 T, (c) B= 0.5 T, and (d) B=0.7 T 86

96 The effect of magnetic intensity on velocity profile on a line across the width of the strip in a horizontal cross section below the free surface and above the moving substrate for EMBr in location (b) are presented in Figures 5.35 and Fig Velocity Profiles in the width direction near the free surface for various EMBr intensities at location (b): (1) B= 0.3T, (2) B= 0.5T, and (3) B= 0.7T; with distance from the back wall: (a) 4.5, (b) 5.4, (c) 7.5, (d) 10.4, and (e)

97 Fig Velocity Profiles in the width direction near the moving belt for surface for various EMBr intensities at location (b): (1) B= 0.3T, (2) B= 0.5T, and (3) B= 0.7T; with distance from the back wall: (a) 4.5, (b) 5.4, (c) 7.5, (d) 10.4, and (e) Similarly Figure 5.37 represents the influence of magnetic flux density on the velocity profile in the y-direction (thickness direction) for different position along the x direction. It can be concluded from Figure 5.37 that the increase in the magnetic flux density makes for a more uniform axial velocity in the y-direction. For having a better strip quality during solidification, iso-kinetic flow plays a significant role in the HSBC process. Increasing the magnetic flux density, leads to a quicker development of the flow and results in iso-kinetic velocity profiles at the early stage of this process. On the whole, an increase in the magnetic strength results in more uniform flow pattern in strip cross section. 88

98 Fig Velocity Profiles in the y-direction in the symmetry plane for various EMBr intensities at location (b): (1) B= 0.3T, (2) B= 0.5T, and (3) B= 0.7T; with distance from the back wall: (a) 4.5, (b) 5.4, (c) 7.5, (d) 10.4, and (e)

99 5.2. Typical Results of the Physical Model The physical model developed in this work was run to evaluate the effect of the DC magnetic brake on the flow field in the slot-nozzle delivery system. The photograph of the aqueous solution model used to study the flow behavior in the system without, and with an imposed electromagnetic field, is given in Figure Y Z X Fig The Physical model Set-Up a) Without EMBr system, and b) With an EMBr installed on the system. 90

100 This section provides information on the flow visualization studies carried out on the physical model Flow Visualization Studies One of the most common, simple, and inexpensive techniques to visualize flow fields in water modeling studies, is to suspend silk or cotton tufts or threads at different locations over the water model system. This approach has been adopted by many other researchers to qualitatively visualize the flow patterns developed in water models of various metallurgical systems [61].Therefore, the same approach was adopted in our physical model for initial study of the flow field in the system. In this regard, we tested the flow field at different locations along the length of the channel (X direction), in the absence of an electromagnetic field. Some of the observed flow patterns are illustrated in Figures Fig Flow patterns at two different locations along the channel a) close to the nozzle, and b) far from the nozzle 91

101 As shown in these Figures, in locations closer to the slot nozzle, the fluid (water) is directed towards the side dams indicating higher flow intensity near the side dams. This can be seen by the deviation of the suspended threads towards the outer stationary walls. On the other hand, in the position far ahead of the tundish front wall, the flow is almost uniform in the middle of the channel, due to the fully developed flow field and therefore the suspended tufts are equally pushed along the stream direction. In order to evaluate the effect of electromagnetic brake on the flow field, flow visualization was made within the same region, in order to compare the flow pattern without, and with, an electromagnetic field. As mentioned in Chapter 4, in addition to an imposed DC magnetic field, an imposed electric field was also used to compensate for the low conductivity of NaCl solution, as compared to liquid metals. As pointed out earlier, MHD physical model has been developed based on the equality of Interaction number between prototype and physical model. A pilotscale single belt caster has been considered as the prototype in which liquid steel introduced on a moving belt from a tundish via a slot nozzle flowing on the belt surrounded by two side dams and a DC magnetic field of 0.3 T is imposed in the y-direction within a specific region close to the tundish along the belt. The Interaction number, defined as the ratio of electromagnetic force to inertial force, in prototype with belt speed of 1m/s is obtained as In order to obtain the same interaction number in aqueous model, one has to apply an extra electric field in the same direction of induced current (i.e. in the z-direction). This requires a voltage gradient of 240 V/m in z-direction corresponding to applying 12V between two electrodes and keeping the belt speed to the value of 0.65 m/s. The current passing the solution between two electrodes is about 2 Amps in the present case. Figure 5.40 shows the EMBr system installed on the physical model representing the locations of the top permanent Neodymium magnets and two graphite electrodes installed in the physical model. The bottom Neodymium magnet sits below the belt, vertically below the upper one, and is hidden from view by the belt. 92

102 Fig The EMBr system used in physical model. By applying the EMBr to the system, the induced Lorentz force pushes the flow towards the side dams, as was confirmed by the mathematical model developed for this study. Figure 5.41 compares the flow pattern in a certain location within the channel without, and with, EMBr. As can be seen from this Figure, by imposing the EMBr on the flow system, the induced electromagnetic force directs the flow towards the side walls and this causes the two floated side tufts to be pushed towards the side dams and align closer with the side walls. This effect may regulate the flow in the cross section and make the flow more uniform. It is noted that the intensity of EMBr depends on the Interaction number in such a way that by increase of Interaction number the Lorentz force increases and consequently the flow is more suppressed and EMBr will be more efficient. In prototype with a constant casting speed the Interaction number can be exponentially enhanced by increasing the magnetic intensity as the Interaction Number is a function of B 2 as shown in Equation 5.2. N = B2 L c σ e ρu

103 In the physical model the Stuart number is a function of B and voltage gradient between two electrodes (E) and it can be controlled by changing the voltage between the electrodes as B is constant in this case which presented as: N = (σ e EBL c ) ρu In this physical model, applying an electric field of 240 V/m, the induced Lorentz force in EMBr region which affects the flow field, however, more effective brake can be achieved by increasing the gradient of voltage between two electrodes by increase of the voltage input. Fig Flow patterns at a) without an EMBr, and b) With an EMBr From the above discussions, it can be deduced that the EMBr system can positively influence the flow field in the metal delivery system of a Horizontal Single Belt Caster. This can be achieved by generating an opposing Lorentz force, which can make the fluid flow, iso-kinetic with the moving belt. As mentioned above, for better achievement of the effect of EMBr in flow control 94