The Levitation Melting Process Using Cold Crucible Technique*

Transcription

1 The Levitation Melting Process Using Cold Crucible Technique* By Annie GAGNOUD,** Jacqueline ETA Y*** and Marcel GARNIER**** Synopsis Classical levitation melting process using conventional conical inductcr can not be used at industrial scale because of the limited size of liquid metal which can be maintained in equilibrium against gravity force. Because of high frequency and axisymmetry of magnetic field distribution, a stagnation point occurs at the bottom of the liquid load, where only surface tension can balance hydrostatic pressure. The use of cold crucibles with particular shape does not suppress the stagnation point but reduces region where magnetic field is deficient. Consequently possible loads of liquid metal which can be levitated are considerably increased. Numerical modeling of cold crucible and of equilibrium shape of levitated liquid metal provides a useful guide for tayloring levitation melting devices. Key words: electromagnetic levitation; cold crucible; numerical simulation. I. Introduction Levitation melting offers many advantages in metallurgy compared with classical techniques : the main advantages are cleanness and superheating. Molten metal never comes into contact with the crucible wall which is in conventional melting technique a source of contamination, particularly by carbon, sulphur, or various oxides. Such contamination is enhanced by strong turbulent stirring motions induced in the melt by rotational part of electromagnetic forces. In levitation melting turbulent stirring is a very efficient way of mixing, useful in the production of pure special alloys. Despite its numerous advantages, levitation melting has never fulfilled its optimistically stated earlier promise, " the full-scale commercial production... of quantities of the world's purest metals and some of its finest alloys ".1) Many theoretical works have been done to provide the metallurgist more than a casual knowledge of electromagnetic theory applied to metal levitation, and to precise assets and limitations of this process. Electromagnetic levitation melting is a typical example of coupled phenomena : a free boundary problem is to be solved to determine the equilibrium shape and the stability conditions of a given molten metal volume with respect to electrical and geometrical parameters of the system. A non linear coupling arises through boundary conditions which have to be expressed along the unknown free surface. This coupling can be defined as follows : a coil generates an alternating magnetic field which weakly diffuses into electroconducting materials to be levitated. Within the skin depth interaction between induced currents and magnetic field results in: (1) Pressure effect related to magnetic field distribution along the free surface. The local balance between magnetic pressure, surface tension, gravity and hydrodynamical pressure governs the shape of the free surface. Since skin depth is small, the geometry of the liquid volume determines magnetic field distribution, and consequently magnetic pressure, introducing a first coupling. (2) Stirring effect : even if skin depth is very thin, electromagnetic forces have a rotational part related to variation of magnetic pressure along the boundary. The resulting source of vorticity generates turbulent recirculating flows in the liquid. The non uniform resulting hydrodynamical pressure is to be taken into account in free surface determination. A second coupling arises then. To avoid interesting but difficult problem of interaction between magnetic field distribution and equilibrium shape of levitated liquid, main of theoretical works only consider solid samples.2'3~ Some recent works take into account fluid dynamical aspects of the process4'5) in some asymptotic cases where the initial spherical sample is only weakly deformed because of strong surface tension. Classical levitation melting device consists in a conical helical inductor with several turns, and one or two counterspires. Till now geometry of the inductor was empirically defined since no theoretical predictive way was existing to solve previously defined non linear coupling. Two methods are presented which make it possible to calculate the equilibrium shape of levitated liquid with respect to both electrical and geometrical parameters of the system,6'7j This provides a very useful tool to optimize classical devices and to define and explain their limits. As a consequence magnetic field distribution necessary to increase possible mass of liquid to be levitated can be predicted and corresponding inductors can be defined. Cold crucible for levitation melting is the result of this analysis. II. Equilibrium Free Surface Shape Computation 1. Local Method Equilibrium shape of the free surface is resulting * ** *** **** Manuscript received on April 6, 1987; accepted in the final form on September 11, Institut National Polytechnique de Grenoble, Grenoble, France. Centre National de la Recherche Scientifique, Paris, France. Laboratoire Madylam, 38042, Saint-Martin-d'Heres Cedex, France C 1988ISIJ ( 36 )

2 2 Transactions ISIJ, Vol. 28, 1988 (37) from the local balance between pressure, irrotational electromagnetic forces, surface tension and gravity. Local balance is expressed by boundary condition: Pi-Pe = WK+aBs/2+z...(1) where, Pe, PZ : external and internal pressure on surface aq for domain Q, respectively (Fig. 1) K: the ratio between typical size of the domain Q, whose volume is V(SO), and local curvature radius BS : local tangential magnetic referred to typical magnetic field Bo generated by the inductor W: Weber number, r/pga2 with r surface a : tension, p density and g gravity non dimensional magnetic pressure: Bo/ppga, with p magnetic permeability of liquid metal z : vertical coordinate referred to typical size a. Condition (1) expressed at stagnation point 0, where magnetic field is zero since electromagnetic skin depth is assumed to be zero, leads to elimination of pressure term : W(K-Ko)+crBs/2+(z-zo) = 0...(2) Equilibrium shape is reached through numerical process : Q is initially spherical. A fictitious irrotational velocity is introduced together with its potential ~b, and free surface is locally displaced during time with respect to normal velocity (aq5/an)s. Allowing for time dependence in the free surface shape, we modify Eq. (2) to introduce non stationary terms (Bernouilli equation) : a~ + -- i v~ I2+W (K-Ko)+(z--zo)+a s = 0 at (3) which may be written: d~ _ 1 dt 2 10O l2+w(k_ko)+(z-zo)+abs 2 = 0...(4) where, d/dt : the time derivative following a fluid Fig. 1. Axisymmetric molten metal. s Since: particle (Stokes derivative). d~5 dt s (t) -- We can find a- an ' Dirichlet problem: V2O = o dt (~s (t)) = dt (~S) (5) knowing s(t) and ~S(t) by solving in Q with 0 _ ~S on asp...(6) Magnetic field B verifies 028=0 with B=0 in Q since a/a=0. Duri ng each timestep in the procedure we have then two similar potential types problems to solve Bs and a~ n s Green's function and integral equation technique are used. Since near the equilibrium position displacement velocity vanishes (Q~5=O), Eq. (4) can be written for each time step At: c'5s(t+4t) _ qi (t)-4t[w(k-ko) +(z-zo)+abs/2]...(7) At each time step magnetic field, curvature and normal direction to aq are computed. A damping coefficient acting on ~is is to be introduced to prevent oscillations. Local position of aq in cylindrical system of coordinate (R, 0, z) is given by: R(t+dt)=R(t)+a~ z(t+at) = z(t)+ a nr dt n S... 8 nz dt a n S 2. Global Method Equilibrium shape of the free surface corresponds for aq to the minimum of total energy F(Q), computed with assumption that intensity of inducting currents remains constant: F(,2) _ - x3(b2/2l) dv + pgz dv + r ds... (9) To find the minimum of this energy functional, derivative with respect to the domain is to be defined.a) Consider the identity application E, with B displacement field keeping volume 2 constant. For given domain Q0, F(8) denotes the expression for F which corresponds to domain Q deduced from Q0 through displacement 0. Q _ (E + 0)Q0 and F(0) = F (Q)... (10) The derivative of F with respect to the domain Qo applied to a given displacement T which keeps volume constant is defined by: af af z...(11) a~ ae B-o

3 (38) Transactions ISIJ, Vol. 28, 1988 Equilibrium shape of did corresponds to stationary zero value for derivative of F(Q) given by: asp 2 \ \ asq2 ~ ~ /...(12) where, n : external normal to domain Q and <X Y> scalar product of vectors X and Y. A new functional G(Q)=~b+AV(Q) is to be defined to ensure volume constancy where A is determined by relation: 9V(Q) asp r= <(z n)>ds = 0...(13) aq Iterative process starting from initial volume S2o, towards equilibrium shape is defined as follows : with aqk+l = a~k+tk+i...(14) zk+l(x) = -s(b2/21~+pgz+yk+a)knk...(15) for any point x belonging to aq and given positive parameter a which is to be fitted to lead to fast decreasing of G(Q). Each iteration needs to compute magnetic field and local curvature of aq. t CM' III. Computed Free Surface Shapes Figures 2 and 3 give typical examples of computed free surfaces with the help of global and local methods. These results are in good agreement with classical observations. The most R=a/w is increasing the most different from a sphere the shape is. When Weber number decreases liquid metal flows down. Parameter a is related to the position of the gravity center which becomes higher when a is increasing. The use of high frequency for levitation melting justifies the very small skin depth hypothesis. Moreover, where o/a or zero, stirring motions do not modify free surface shape and magnetostatic approximation is valid. These stirring motions lead to temperature homogenization and prevent any buoyancy effect from occurring. Global and local methods give similar results. However global method is faster converging and does not need the delicate choice of an artificial damping coefficient. The interest of such a computational method is not only in the possible analysis of the influence of physical parameters but also in the geometry of the inductor. A fundamental difficulty which prevents levitated volumes to be very large clearly appears : with very thin skin depth, which are necessary to promote magnetic pressure, a stagnation point with zero magnetic field arises at the lower part of the liquid. Near this point only surface tension can balance gravity Fig. 2. Levitation in conical inductor. Generatrix of the equilibrium I=1 070 A; 1'=10 Nm-1. shape obtained for Fig. 3. Levitation in conical inductor. Generatrix of the equilibrium shape A; r=0.1 Nm-1. obtained for

4 Transactions ISIJ, Vol. 28, 1988 (39) force. To achieve levitation of large loads of liquid metal it is necessary for a given material to make curvature to increase. This is impossible with conventional inductors : indeed because of the need for water cooling of the inductor the lower turn can not be very small in diameter. As a consequence magnetic field which is zero on the vertical axis increases very slowly towards the inductor. Therefore radius of curvature near the stagnation point can not be small. Moreover, because of the presence of the inductor near the free surface, the equilibrium shape can not be smooth : this is a major obstacle in the stability of the liquid. Computational methods make it possible to define magnetic field distribution suitable for imposing a quasi spherical shape to levitated liquid and to force magnetic field to increase very quickly in the vicinity of the stagnation point. Only one inductor can verify these conditions : cold crucible inductor. Iv. Cold Crucible for Levitation Melting Cold crucible technique was introduced by Battelle Northwest Laboratories9~ for melting reactive materials. A multiturn coil is surrounding a water cooled copper crucible made of a number of vertical sectors with internal water cooling and with gasps between the sectors. Currents induced on the external surface of the sectors have to flow around each sector generating magnetic field inside the crucible. In operation a slag barrier is formed in a very thin layer along the crubile and prevents any contact between crucible and melt : this process is often referred to as inductoslag. Cold crucible for levitation differs from these crucibles in the internal shape of the sectors : the internal diameter of the crucible decreases from the top to the bottom (Fig. 4). Such a crucible offers many advantages : discrete distribution of inducting currents is transformed by the crucible in a continuous distribution of induced currents; magnetic field intensity is increasing from the top to the bottom as diameter decreases; near the bottom cold crucible behaves like a field concentrator : magnetic field strongly increases in radial direction. Such a crucible is very interesting for levitation melting. Computed equilibrium shapes are given on Figs. 4 and 5. These shapes are not very far from spheres due to continuous distribution of induced currents. Moreover strong increasing of magnetic field near stagnation point reduces " magnetic hole " effect. In case of instability, coupling between free surface shape and induced current distribution leads to a local increasing of induced currents when liquid metal comes near the crucible: restoring forces arise then which do not locally exist with conventional technique. Another fundamental advantage of this technique concerns safety conditions of melting : with classical inductors, in case of strong instabilities or in case of energy supply switching off, molten metal falls on the water cooled wires, destroys them and some explosion may occur. With cold crucible in same circumstances liquid metal falls inside the crucible where it solidifies without any damage. If typical mass of levitated load of molten metal is Fig. 4. Fig. 5. Levitation in a copper cold crucible segmented in four sectors. Generatrix of the equilibrium shape obtained with global method for 1=2 500 A; r=0.6 Nm-1; o/a =0.14. Levitation in a copper cold crucible segmented in four sectors. Generatrix of the equilibrium shape obtained with global method for I = A; r =1 Nm-1; o/a = 0.14.

5 (40) Transactions ISIJ, Vol. 28, g with conventional technique, it is g with cold crucible. This is confirmed by experiments. It is to be noticed that internal shape of the crucible is of prime importance on the metal quantity possible to be levitated. Computational code developped in Grenoble makes possible optimization of this shape for a given material. V. Conclusion Till now levitation melting was only used at laboratory scale and was concerning small mass of liquid metals (-.100 g). Empirical considerations were the only guide in tayloring levitation melting device. Thanks to a computational method able to solve coupled problems between fluid mechanics and electromagnetism a new technology has been developed : cold crucible for levitation. This technique offers many advantages and makes possible the use of levitation melting at industrial scale. At least one order of magnitude difference appears between mass possible to be levitated with conventional technique and with cold crucible. Strong stirring motions are induced in the melt which provide excellent uniformity of temperature and composition of the melt. In addition turbulent stirring insures that alloying elements which can be added separately can be mixed effectively. High superheating can be obtained because heat exchanges are only governed by radiation. Casting of liquid metal is possible : central part of the crucible is in this case a cylindrical crucible inserted in the levitation crucible. This part is quickly removed for casting. Because of electromagnetic forces, liquid metal flows without contact with the wall. An interesting field of applications is opened for this technique which concerns amorphous metal manufacturing, small diameter powder manufacturing, pure alloys with high purity or high melting point... REFERENCES 1) H. Motz and H. Wise: J. Chern. Phys., 32 (1980), ) E. C. Okress, D. M. Wrougthon, G. Lomenetz, P. H. Brace and J.C.R. Kelly: J. Appl. Phys., 23 (1952), ) P. Rony: " The electromagnetic levitation of metals; the design, construction and operation ", Ph. D. Dissertation to University of California, Berkeley, (1965). 4) A. D. Sneyd and H. K. Moffatt: J. Fluid Mechanics, 117 (1982), 47. 5) A. J. Mestel: J. Fluid Mechanics, 117 (1982), 27. 6) A. Gagnoud: " Modelisation des installations de fusion en creuset froid pour l'elaboration en continue ou la levitation ", Doctoral Thesis to INPG, Grenoble, (1986). 7) A. Gagnoud, J. Etay and M. Gamier: J. de Mecanique Theorique et Appliquee, 5 (1986), ) 0. Sero-Guillaume: " Sur l'equilibre des liquides magnetiques, Application a la magnetostatique ", Doctoral Thesis to Nancy Univ., Nancy, (1983). 9) G. H. Schippereit, A. F. Leathrman and D. Evers: J. Met., 13 (1961).