Analytic Solutions to Determine Critical Magnetic Fields for Thermoelectric Magnetohydrodynamics in Alloy Solidification
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bility to control microstructural evolution of solidifying alloys is of fundamental importance for modifying and tailoring material properties. Thermal gradients and stirring are examples of controls currently used in industry for manufacturing alloys. The introduction of magnetic fields into the solidifying process offers another control for alloy production. To increase the knowledge of these controlling mechanisms, numerical models have been constructed (for example, Kao et al.[1–4]) to solve the equations governing the complex physical processes of solidification under the influence of magnetic fields. These equations describe mass, energy, and momentum transport in the vicinity of the solid– liquid interface coupled with the thermoelectrically induced Lorentz forces. In this paper, the author presents another approach for solving these equations with the objective of deriving analytic solutions that can be readily evaluated to provide useful previews of results prior to initiating the time-consuming but necessary numerical simulations for parametric studies on solidification under the influence of magnetic fields. Experiments[5,6] and numerical models[7,8] have shown that forced convection can have a significant impact on ANDREW KAO, Senior Lecturer in Applied Mathematics, is with the Centre of Numerical Modelling & Process Analysis, University of Greenwich, Old Royal Naval College, Park Row, London SE10 9LS, U.K. Contact e-mail: [email protected] Manuscript submitted September 30, 2014. Article published online July 1, 2015 METALLURGICAL AND MATERIALS TRANSACTIONS A
the microstructure evolution with effects such as preferential growth, grain refinement, and macrosegregation all being observed and predicted. Typically, fluid flow is introduced through traditional electromagnetic stirring using an AC field. Any DC field present will act as a damping mechanism. Under certain thermal conditions, natural and inherent thermoelectric currents can be generated as a result of the Seebeck effect. When these currents interact with an external DC magnetic field, a Lorentz force is formed which becomes the driver of fluid motion. This effect, known as Thermoelectric Magnetohydrodynamics (TEMHD), was first detailed by Shercliff who demonstrated that processes with large thermal gradients and a significant thermoelectric power could attain relatively high fluid velocities.[9] Shercliff applied the TEMHD theory to several phenomena related to nuclear fusion reactors[10,11] and showed, as an example, that under a moderate magnetic field strength, velocities of O(10 cm/s) in liquid Lithium could be achieved. Indeed recent experiments[12] support Shercliff’s theoretical work on TEMHD. TEMHD has gained recognition as a potential costeffective, low-energy, stirring mechanism. For example, experiments with a moderate thermal gradient of 2.8 K/ cm and a thermoelectric power of 20 lK/V have demonstrated that a rare earth Neodymium magnet alone would be sufficient to achieve velocities of O(33 mm/s) in a conducting fluid without the need for electromagnets.[13,14] F
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