A Periodically Reversed Flow Driven by a Modulated Traveling Magnetic Field: Part II. Theoretical Model
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A former numerical simulation suggested that the melt flow in a mushy zone is generated by a periodically reversed driving force and might be particularly efficient for reducing the macrosegregation in the solidified material. The companion article[1] presents the results of an experimental investigation. These experimental results suggest that this type of electromagnetically driven flow has specific features that deserve to be understood and interpreted. The details of the experiment and the way the modulated force is generated are qualitatively described in Reference 1 without detailed expressions of either the driving force, namely, its space and time distributions, or the liquid metal velocity. Therefore, a theoretical analysis is necessary in order to interpret the output data[1] and to allow extrapolation to actual solidification devices. A very similar investigation has been performed and was recently published by Eckert et al.[3,4] The principle and the technical approaches are similar. While these authors use a rotating magnetic field, we use a traveling magnetic field (TMF), the electromagnetic force being periodically reversed in both cases. In brief, let us mention just one of the significant differences. Rotating flows always generate a poloidal secondary flow, which is responsible for a significant transfer of angular momentum. In the case of a traveling field, there is no such transfer, and the momentum is delivered to the fluid exactly when the force field acts. [2]
XIAODONG WANG, formerly Post-Doctor, is Research Associate, Metals Processing Center, McGill University, Montreal, PQ, Canada H3A 2B2. Contact e-mail: [email protected] RENE MOREAU, Professor and French Academician, JACQUELINE ETAY, Research Director, and YVES FAUTRELLE, Professor and Head, CNRS/SIMAP/EPM Group, are with the CNRS/SIMAP/ EPM, BP75, 38402 St. Martin d’He`res Cedex, France. Manuscript submitted April 11, 2008. Article published online January 14, 2009. 104—VOLUME 40B, FEBRUARY 2009
Let us briefly summarize the experimental conditions. For the sake of simplicity of measurement, the fluid flow geometry is a parallelepipedic enclosure (Figure 1). However, it is expected that the most important properties of the flow are generic enough to allow for an extrapolation to the more complex geometry of a typical three-dimensional (3-D) or axisymmetric-directional solidification furnace. This fluid enclosure is located above a flat and horizontal wall; the wall mimics the solidifying interface and the liquid metal at room temperature (GaInSn) mimics the melt. This enclosure is located above a linear motor the inductor of which is fed by a three-phase alternating current (AC) at the normal frequency (50 Hz). The electrically conducting fluid is submitted to an electromagnetic force applied in the direction of the main axis (x), which can be periodically reversed. An additional square wave generator allows the order of the three phases to change after a time interval ti 80 ms, making it possible to periodically reverse the sign of the driving force
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