Applied Scientific Research Volume 44 Issue 1-2 1987 [Doi 10.1007_bf00412016] P. a. Davidson_ F. Boysan -- The Importance of Secondary Flow in the Rot

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Applied Scientific Research 44:241-259 (1987) © Martinus Nijhoff Publishers, Dordrecht - Printed in the Netherlands 241 The importance of secondary flow in the rotary electromagnetic stirring of steel during continuous casting P.A. DAVIDSON 1 & F. B O Y S A N 2 1 Department of Engineering, Cambridge University, Cambridge, UK; 2 Department of Chemical Engineering and Fuel Technology, Sheffield University, Sheffield, UK Abstract. This paper considers some aspects of the flow generated in a c
  Applied Scientific Research 44:241-259 (1987) 24 © Martinus Nijhoff Publishers, Dordrecht - Printed in the Netherlands he importance of secondary flow in the rotary electromagnetic stirring of steel during continuous casting P.A. DAVIDSON 1 F. BOYSAN 2 1 Department of Engineering, Cambridge University, Cambridge, UK; 2 Department of Chemical Engineering and Fuel Technology, Sheffield University, Sheffield, UK Abstract. This paper considers some aspects of the flow generated in a circular strand by a rotary electromagnetic stirrer. A review is given of one-dimensional models of stirring in which the axial variation in the stirring force is ignored. In these models the magnetic body force is balanced by shear, all the inertial forces being zero (except for the centripetal acceleration). In practice, the magnetic torque occurs only over a relatively short length of the strand. The effect of this axial dependence in driving force is an axial variation in swirl, which in turn drives a secondary poloidal flow. Dimensional analysis shows that the poloidal motion is as strong as the primary swirl flow. The principle force balance in the forced region is now between the magnetic body force and inertial. The secondary flow sweeps the angular momentum out of the forced region, so that the forced vortex penetrates some distance from the magnetic stirrer. The length of the recirculating eddy is controlled by wall shear. This acts, predominantly in the unforced region, to diffuse and dissipate the angular momentum and energy created by the body force. otation B magnetic field strength F angular momentum uor E electric field 8 boundary layer thickness unit vector ~ viscous dissipation rate F 8 force 0 angular coordinate f z/R) dimensionless variation of force v viscosity with depth v eddy viscosity J current density p density k turbulence kinetic energy o conductivity k' wall roughness ~j shear stress L axial length scale in ~2 angular velocity unforced region w vorticity p pressure w frequency R radius Re Reynolds number Subscripts r radial coordinate R wall T torque r radial t time 0 azimuthal u_ velocity z axial V characteristic velocity B~oR~ 0 core V. shear velocity p poloidal v fluctuating velocity z axial coordinate  242 P A Davidson and F Boysan 1 Introduction Continuous casting has become an increasingly common means of producing steel ingots. The process is shown schematically in Fig. 1. There is a metal- lurgical requirement to stir the melt as it solidifies, and this has led to the use of electromagnetic stirring [3]. Stirrers have been placed around the mould, below the mould and near the point of final solidification. Typically these stirrers resemble the stator of an induction motor, producing a travelling or rotating magnetic field. The field induces motion in the melt with peripheral velocities of the order of 20 cm/s. We shaU be concerned with rotary stirrers, whose primary purpose is to induce swirl in the melt. The cost of implementation of magnetic stirring is considerable; yet the optimum configuration for stirring is often assessed on an empirical basis [4]. The question of how many stirrers are required, and where they should be placed, frequently arise in the literature [3,4]. In order to give general answers to these, it is necessary to understand not only the metallurgical processes at work, but also the nature of the velocity field induced by stirring. In particu- lar; the following hydrodynamic questions arise. i) How does the magnitude of the induced swirl scale on magnetic field strength, mould size and melt properties? i_i) How far beyond the stirrer does the induced vortex extend? iii) Do secondary flows develop and are they important? Incomin 9 steel 1, Copper mould Melt Solid steel ~ii~, m Fig 1 Diagrammatic representation of the continuous casting process.   he importance of secondary low 243 ould stirring surface ~ - _ _ orced .... 7-- [J .... .~ Magnetic force Z Sub-mould stirring Line ~ ~ of symmetly_ lregion i Magnetic Fig 2 Typical magnetic orce distributions force In order to simplify the problem attention is restricted to flow in a circular strand and entrance effects of the melt in the mould are ignored. Typical magnetic force distributions are shown in Fig. 2. Often rotation rates are sufficiently low that the surface of the melt remains flat. In this situation the surface may be treated as a plane of symmetry and the problems of mould and sub-mould stirring become hydrodynamically identical. We shall consider two models of stirring. Firstly a one-dimensional model is reviewed. In this analysis the axial variation in the magnetic force and hence swirl is ignored. In such a situation the secondary flows are by definition zero. This flow has been analysed several times [1 5 7 9] and is a popular model of stirring. We shall show however that it is misleading in the context of continuous casting. A more realistic two-dimensional axisymmetric model is then considered in which all three velocity components are non-zero. The coordinate system used is shown in Fig. 2 and notation is given at the start of this paper. The radius R shown in Fig. 2 refers to the outer radius of the melt. It is assumed to be constant the taper resulting from the increasing shell thickness being ignored.  244 P.A. Davidson and F. Boysan II. The magnetic body force The magnetic field within the melt is governed by the advection diffusion equation, 3B 7=v The relative size of the advection to diffusion terms is given by the magnetic Reynolds number, R = uRl~. We shall assume that Re m is small. This is generally true in both laboratory and industrial situations, and allows us to ignore advection of the magnetic field. In this approximation the melt is treated as a solid conductor, and the magnetic field determined by the standard eddy-current equation, 3B 1 = ~7 2B. 3t /~rr It is worth noting that, in this approximation, there is no charge distribu- tion within the melt, and that charges will not be deposited on the free surface of the melt by eddy currents, since there is no component of current normal to the surface [2]. The ratio of the time derivative term to the diffusion term in the equation for B is given by the skin depth parameter, A = R2~oI ~ = 2(R/3) 2 where = 2///~o¢o) 1/2, Our second assumption is that A is also small, although larger than Re m. Re m << A << 1. This requires the skin depth to be large relative to R, and is referred to as a low frequency approximation. It is shown in [9] that, in this context, a low frequency analysis is a good approximation for values of A as high as unity. This covers most 50 Hz laboratory experiments and some industrial applica- tions. Since we have assumed that Re m << A, a condition almost invariably met in practice, we may deduce, u << ~0R This implies that the field advection term in the advection-diffusion equation for B is much smaller than the time derivative term. This expression is sometimes taken as an alternative condition for field advection to be negligi- ble.
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