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Analysis of Free Surface Flows with Solidification
by
Andrey Kuznetsov
Department of Mechanical and Aerospace Engineering, North Carolina State University
The objective of this paper is to present a theory of free surface flows with continuous solidification. This type of open-channel flows is relevant to a number of important technological processes, such as the horizontal continuous casting of carbon steel. Since carbon steel is a binary alloy, in formulating a mathematical model in addition to accounting for fluid flow and heat transfer it is also necessary to account for the solute transport and for the two-phase region (the mushy zone) effects. Extensive numerical simulations provide valuable insight into this process.
Classical free surface flows received considerable attention in the literature. A good overview on open-channel flows is given in Chow [1]. Both analytical and numerical investigations of open-channel flows is presented in Garcia-Navarro et al. [2], Thomas et al. [3], Rahman et al. [4]. However, open channel flows with solidification have not received so far sufficient attention. This gap needs to be filled, because these flows are relevant to a number of important industrial applications, such as the strip casting of carbon steel.
In recent years, there has been a number of papers devoted to modeling of the heat transfer and fluid flow for different schemes of both vertical [5-9] and horizontal [10-14] strip casting processes. These papers present extensive investigations of both fluid flow and heat transfer in the solidifying strip. However, further insight into this process is needed, such as investigation of coupling flow and heat transfer with solute transport and accounting for the free surface behavior.
In recent publications a number of models for describing fluid flow of binary alloys during solidification have been proposed, mainly for the case when the flow is caused by natural convection. Unlike pure substances, binary alloys solidify over extended temperature ranges and solid formation usually occurs within a two-phase region (the mushy zone), where solid and liquid phases coexist. Sound theories for transport processes in the mushy zone have been developed only recently.
Derivation of the set of governing equations for the mushy zone based on the mixture theory approach was originally reported in [15-16] and recently extended to account for microscopic phenomena in [17-18]. The derivation of the set of governing equations based on a volume-averaging procedure is presented in [19-21]. An excellent review of different models with basic features of each model summarized is given in [22]. Very recently, a three-phase model (solid, liquid and gas phases) of the mushy zone has been proposed [23, 24] and comparisons against the two-phase model have been carried out. Since the appearance of these models, the solidification of alloys has been extensively investigated. Numerical results of these studies along with the main features of the numerical procedures are reported in [25-37]. Reference [38] is one of the first research works on modeling flow, heat and solute transport in a multicomponent steel. Different from the investigations reported in [25-38], where fluid flow is mainly caused by a relatively weak natural convection, in this research we consider the case of a strong forced convection, caused by the change of the height of the free surface.
In this paper, we consider free surface flow of a binary alloy, for example, carbon steel, on a horizontal surface (casting table) which moves with a constant velocity, U. We assume that the binary alloy enters the moving surface with the fully developed relative velocity profile, where relative means relative to the surface. In the beginning of the computational domain the binary alloy is completely in the liquid state, that is its temperature is above the liquidus temperature. A constant heat flux is withdrawn from the surface, and this causes solidification of the alloy as it flows downstream. At the end of the computational domain the alloy is completely solidified, and the solid strip leaves the moving surface with the same constant velocity, U. This process is an example of a free surface flow with solidification. It also should be noted that in the beginning of the casting table, alloy at the free surface is in the liquid state while farther downstream, alloy at the free surface is in the mushy state. In establishing the mathematical model for this process, the following assumptions and simplifications are utilized:
- The properties of the solid and liquid phases are homogeneous and isotropic, the solid phase is stationary and rigid, no microporosity forms in the strip;
- The solid and liquid in the mushy zone are in local thermal and phase equilibrium, the thermophysical properties are constant, but may be different for liquid and solid phases;
- No species diffusion in the solid phase between the averaging volumes and complete diffusion in the solid phase within the averaging volume (lever rule) is assumed;
- Heat transfer by radiation and convection from the free surface is negligible;
- The thin layer approximation can be invoked;
- The surface tension effects are negligible;
- The flow resistance due to the growing dendrites is accounted for only in the direction perpendicular to the primary dendrite arms (in the x-direction), resistance in the y-direction is neglected because of the small thickness of the strip;
- The density difference between the fluid and solid phases is accounted for only in the continuity and the species transport equations, but it is neglected in the energy equation. It other words, the term accounting for the density change is incorporated into the latent heat term and the temperature dependence of the ëffective latent heat" is then neglected. Thus the energy equation then takes the form suggested in Beckermann and Viskanta [27].
Thus this paper suggests a model of free surface flow with solidification. This model is applied to numerically investigate coupled fluid flow, heat transfer and solute transport in horizontal continuous casting process. It is shown that the solute diffusion in the liquid phase causes a formation near the casting table a thin diffusion boundary layer. This boundary layer is essentially depleted of the solute. The formation of this boundary layer can be explained by considering the effect of diffusion near an intensively cooled impermeable wall. It is established that increasing of heat transfer rate from the casting table results in considerable decrease of macrosegregation level in the strip. This is because larger heat transfer rate results in smaller width of the mushy zone, which in turn results in smaller macrosegregation. It is also established that increasing of casting velocity results in slight increase of macrosegregation level in the strip. This is because larger casting velocity results in larger width of the mushy zone, which in turn results in larger macrosegregation.
REFERENCES
1. Chow, V.T. 1959. Open-channel hydraulics. McGraw-Hill, NY.
2. Garcia-Navarro, P., Alcrudo, F.
3. Thomas, S., Hankey, W.L., Faghri, A.
4. Rahman, M.M., Hankey, W.L.
5. Shiomi, M., Mori, K.
6. Raihle, C.-M., Fredriksson, H.
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9. Hwang, J.D., Lin, H.J., Hwang, W.S.
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15. Bennon, W.D.
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22. Viskanta, R. 1990. JSME Int. J., Series II 33, 409-423.
23. Kuznetsov, A.V.
24. Kuznetsov, A.V.
25. Bennon, W.D.
26. Voller, V.R.
27. Beckermann, C.
28. Bennon, W.D.
29. Engel, A.H.H.
30. Voller, V.R., Brent, A.D.
31. Prakash, C.
32. Neilson, D.G., Incropera, F.P.
33. Felicelli, S.D., Heinrich, J.C.
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35. Felicelli, S.D., Heinrich, J.C.
36. Lee, S.L.
37. Schneider, M.C.
38. Böhmer, W.F.A., Schneider, M.C., Beckermann, C.
Date received: February 12, 1999
Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caco-04.