Choosing the Appropriate Level of Coupling in

Choosing the Appropriate Level of Coupling in Multiphysics Modeling of Metallurgical Processes Georgi S. Djambazov and Koulis A. Pericleous School of Computing and Mathematical Sciences University of Greenwich, Park Row, London, SEW 9LS, United Kingdom Abstract. A computational model for the interrelated phenomena in the process of vacuum arc remelting is analyzed and adjusted of optimal accuracy and computation time. The decision steps in this case study are offered as an example how the coupling in models of similar processes can be addressed. Results show dominance of the electromagnetic forces over buoyancy and inertia for the investigated process conditions. Keywords: computational modeling, multiphysics, liquid metals PACS: 41.20.Gz;47.11.Df;47.65.-d

INTRODUCTION Vacuum Arc Remelting (VAR) is a refining metallurgical process where a long (usually cyhndrical) electrode is being melted by the heat of electrical arcs in a vacuum furnace. It is done with special alloys in order to meet the very strict requirements for cleanliness, homogeneity, improved fatigue and fracture toughness of the final product when those cannot be achieved in conventional melting and casting processes. Direct current (DC) power is applied to strike an arc between the electrode and the base plate of a copper mould (or 'crucible') contained in a water jacket. The intense heat generated by the electric arc melts the tip of the electrode and a new ingot is progressively formed in the water-cooled mould. A high vacuum is being maintained throughout the remelting process. The required properties of the final material can only be obtained if the inevitable segregation of the alloy components during solidification is kept to a minimum. This can be achieved if the solidification structure of the newly formed ingot is predominantly directional and dendritic. Such structures are formed when high temperature gradient at the solidification front is maintained during the entire remelting process and the depth of the molten pool at the top of the ingot is kept within optimal limits. The depth of the pool influences its profile which determines the direction of dendrite growth whose angle relative to the ingot axis needs to be within an optimal range in order to achieve the desired structure. The pool shape is also affected by the flow patterns which in turn depend on the balance between electromagnetic and buoyancy forces acting on the liquid metal. CPl 184, Applications of Mathematics in Engineering and Economics edited by G. Venkov, R. Kovacheva, and V. Pasheva © 2009 American Institute of Physics 978-0-7354-0750-3/09/$25.00

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Two distinct stages of the computational modeling of the VAR process can be identified: (a) macro model and (b) micro model. In this paper a macro model of the process is described. Data from the macro model can be supplied to a separate micro model [1] which can predict the likelihood of defects appearing in the cast ingot. If time-averaged variables are considered, VAR would be essentially an axisymmetric process. However, for some conditions, time-varying external magnetic fields have been observed [2]. The analysis [3] of those disturbances shows that offcenter rotation of the 'preferred location' for the electrical centre of the main arc may be occurring. (The actual arc columns are usually more than one at any moment, they move, extinguish themselves and reignite in different places in the narrow space under the electrode - so the preferred spot can only be defined in a time-average sense.) It turns out that the time scale of those 'averaged' motions is comparable to the time scales of the fluid flow in the liquid pool whose changes can influence the whole process. In this work a 3-dimensional (3D), time-dependent computational macro model is used to predict the influence such rotation might have on the process conditions.

PHYSICS IN THE VAR MODEL Multi-physics modehng refers to the simultaneous solution of coupled differential equations belonging to different classes. It is done when it is necessary to correctly represent inter-related physical phenomena in a given industrial process. To what extent the coupling between the various aspects of the physics involved should be taken into account in the model? When deahng with large 3D, time-dependent models one needs to balance the desire for better accuracy with the computational cost involved with the chosen level of couphng. Here a prehminary analysis of the feedback mechanisms in the liquid pool at the top of the VAR ingot is offered as a tool to reach the right decision. buoyancy

Fluid Dynamics uxB

V

Heat Transfer

Lorentz force

temperature

Electromagnetics

Solid Mechanics

FIGURE 1. Relations between physical phenomena at the top of the VAR ingot.

Undoubtedly, the electric arc is the driving force of the process. It provides the heat source that melts the electrode and sustains a liquid pool at the top of the ingot despite the strong cooling of the water on the outer side of the crucible. Another very important influence of the electric current of the arc is the Lorentz (electromagnetic) force which creates recirculating flow in the melt pool. The flow influences the heat transfer to the sohd part of the ingot, and the fluid temperature affects the flow via buoyancy (due to small density variations). At a certain distance below the edge of

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the liquid pool, the newly sohdified ingot is strong enough to resist the hydrostatic pressure of the liquid part and to shrink away from the crucible wall thus forming a narrow gap which breaks the electrical connection between the lower part of the ingot and the crucible. This limited 'contact zone' can change the electric current paths in the ingot and affect the Lorentz force closing the feedback loop shown with thick solid lines in Fig. 1. At first glance it appears that, for the top of the ingot, the VAR process should be modeled as fully-coupled multi-physics. Now it is time to remind ourselves how the 3D Lorentz force will be calculated.

DC Electromagnetics of the VAR Process Using the fact that direct current (DC) produces static magnetic field (without electromagnetic induction) and assuming that the processed alloy is not ferromagnetic (true for at least one type of INCONEL [4] special alloy), a scalar electric potential (p can be defined and solved for with mixed Dirichlet and Neumann boundary conditions. Then it can be used to calculate the electric current density J with the help of the electrical conductivity a: J = -a grad