Heat Model for Solidification of

Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer 259

III. Heat Model for Solidification of

Crucible-Molten Pool in the Cast Ingot The heat model used is a quasi steady-state two- dimensional one. Axial-symmetrical geometry is used Fig. 3 and cast ingot boundary surfaces are noted in Fig. 3 by G with respective indexes. Fig. 3. Geometrical conditions: G 1 – top ingot surface, G 2 – interface molten poolcrucible, G 3 – free side ingot surface, G 4 – interface – ingotpuller. The shown dimensions are typical radius and length for the used crucibles In Fig. 4 a typical temperature distribution on a half of the ingot cross-section is shown [2],[3]. In this case T 2 is the melting temperature. The bottom of the molten pool on top of cast ingot is presented by line T 2 if have not a mushy zone due to soft transition between liquid and solid state of melted alloy. Note, that in the top part of the molten-pool-periphery a solid ring could be created [3] see also the discussion bellows with inner diameter of ø2R 1 and outer diameter of ø2R ingot , as a height of h L . Fig. 4. Geometry characteristics of the molten pool there T 2 is melting temperature T m ; h L is height of intensively cooled side wall of the cast ingot by contact with inner cooled crucible wall The computer simulation made is based on a quasi steady-state two-dimensional heat model. There the temperature distribution in limited region of the cast ingot is described by Laplace equation: 2 2 1 T T V T r r r r a z z ∂ ∂ ∂ ∂ ⎛ ⎞ ⋅ ⋅ + + ⋅ = ⎜ ⎟ ∂ ∂ ∂ ∂ ⎝ ⎠ 1 where the last term indicates the casting presented by the heat that is added by the poured molten metal into the crucible and given by the casting velocity of the ingot, moved with a speed of V, coinciding with the z-axis; a is thermal diffusivity a= λC.ρ , where C is the specific heat, λ is the thermal conductivity and ρ is the sample density. It is used an axis-symmetrical thermal geometry. The stirring and mixing processes in the molten pool are presented by a modification of the value of the ingot thermal conductivity. The value of liquid thermal conductivity multiplication factor, used to simulate stirring or mixing processes in the melting is between 1 and 2 [6]. The molecular thermal conductivity of the liquid phase is 1-2 times lower than the thermal conductivity of the solid phase [12]. Taking into account that our goal is to clarify the basic phenomena of the heating, using EB source, the above facts make it possible to assume that the thermal conductivity is temperature independent. A set of appropriate boundary conditions takes into account the radiation losses and the heating beam energy distribution with a correction of the secondary electron energy losses and they are given according the interfaces G 1 , G 2 , G 3 and G 4 Fig. 3 correspondingly: ƒ 4 4 1 z h n st T | P T T z λ α σ = ∂ ⋅ = − + ⋅ ⋅ − ∂ ƒ 1 2 r R r R T T | | r r λ λ = = ∂ ∂ ⋅ = ⋅ ∂ ∂ ƒ 4 4 1 r R st T | T T r λ σ σ = ∂ ⋅ = − ⋅ ⋅ − ∂ ƒ 1 2 z z T T | | z z λ λ = = ∂ ∂ ⋅ = ⋅ ∂ ∂ where α is emissivity, σ is the Stefan-Bolzman constant, λ is the thermal conductivity, T st is the ingot surface temperature. Working with a real Newton heat transfer the values of these coefficients and interface areas’ magnitudes should be estimated more deeply involving an effective coefficient α i presenting the ratio between the transfers in real and ideal cases in order to apply the considered approach. There three types of heat contact area between the ingot and the crucible are assumed: i areas with an ideal heat contact described by the Fourier equation namely a heat conductivity takes place and there is an equality between the heat flows and Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer 260 temperatures of the both sides of the interface as well as of the products of the heat conductivity and the temperature gradients, ii areas with a Newton type heat contact namely the product of the thermal transfer coefficient and the difference between the temperatures of the ingot outer wall and of the inner crucible wall, iii areas in which the radiation losses predominate. The magnitudes of these areas are not known and probably change with the time. Due to this the effective coefficient of the heat transfer α or λ i λ together with the chosen total heat contact area namely the height of the contact ring h define the heat flow to the crucible. In a similar way an effective heat transfer coefficient α λ i λ in the pulleringot interface is used.

IV. Results from Numerical Experiments