Thermoelastic stability analysis of solidification of pure metals on a coated planar mold of finite thickness: Effects of the coating layer

Abstract

During the solidification process, initially liquid material is started to solidify when its temperature reduce below its melting temperature by heat transfer from its surfaces which are in contact with a surrounding mold. Therefore the solidification process starts with the formation of the solid at the surfaces of the mold and spreads all of the liquid mass as the time progressed. The heat transfer from liquid to mold continues until all liquid mass solidify. If the process is interrupted and remaining liquid is drawn away from the process, some periodic variations are observed on the thickness of the solidified shell. This undulatory structure is seen in the solidification of many metals as inevitable result of cooling. Such uneven undulations are not desirable in the manufacturing because the associated thermal distortions in the solidified shell cause severe cracks in the solidified ingot. Undesirable cracks have significant effects on the strength and microscopic structure of the ingot and according to this the final product has poor quality. If the solidification is not interrupt, at the end of the process the periodic variations at the freezing front tends to die out since the conditions at the liquid/solid interface becomes less dependent upon the conditions at the mold/shell interface due to the increasing of the solidified shell thickness. These non-uniform undulations are occurred as a consequence of the non-uniform heat extraction along the mold/shell interface. There is some thermal resistance at this interface due to the surface roughness and contaminants films. Thermal contact resistance is changed as a function of contact pressure. There is an inverse proportion between the contact pressure and thermal resistance. The non-uniform temperature and stress fields in the solidified material and mold associated with non-uniform heat extraction cause thermal distortions in the casting and mold. Thermal distortions have a great influence on the contact pressure at the shell/mold interface. As it seen the thermal and mechanical problems xvi must be investigated together and the system behaves like a positive feedback system, therefore it may have potential to being unstable. The purposes of this thesis are that investigating the instability mechanism in the solidification process during casting of pure metals and improving our knowledge about this thermos-elastic instability mechanism, determining the conditions which eliminate or minimize the profile defects such as cracks in the surface and internal structure of casting product associated with the coupled thermo-mechanic events, analyzing the effects of coating layer and other system parameters on the solidification process and specify the conditions which improve the quality of product. The fluid flow in the melt metal, adjusting the freezing range of the metal, solidification rate of the continuous casting process and superheat of the melted metal affect the non-uniform structure at the moving interface. But researches about the solidification process show that undulations depend considerably the geometry of the mold/shell interface. Mold coating at the mold-shell interface is one of the most important factors controlling the heat transfer and, hence, it has very important role on the solidification rate and development of microstructure. Other advantages of coating layer protecting the mold during the casting process because of expensiveness of the mold manufacturing technology and the additional manufacturing requirements are prevented before the using of final product by controlling the unstable growth of the solidified shell. At the end coating layer affects the quality of casting and provides energy, time and workforce savings. In this study thermal and mechanical problems are coupled through the contact pressure dependent thermal contact resistance at the coating/shell interface. The thermal diffusivities of the mold, coating and solidified shell materials are assumed infinitely large. The physical meaning of this assumption is that the thermal capacities of the materials of the mold, coating and shell are zero. This assumption is correct at the beginning of the solidification since the materials properties have no significant effects on the process and it allows us to solve the heat conduction problem analytically because the temperature distributions in the each layer changes linearly in space coordinate. The model is restricted to the solidification of pure metals and the solidification occurs at a distinct temperature. Therefore, there is a sharp interface between solid and liquid phases, no mushy zone occurs. For modeling the changes in the heat flux drawn from the lower surface of the mold, small spatial perturbations are added and amplitude of this perturbation are much small than its wavelength. The liquid initially at its melting temperature and the liquid phase is assumed to be at constant hydrostatic pressure. The determination of the stress distribution in the solidified layer is very complex, because the material is continually being solidified while the solid is in a deformed state, and therefore the final cast product exhibits residual stress, even after the temperature has been reduced to zero. Linear constitutive model has been employed which assumes the behavior of the solidified material to be elastic during solidification. Thermo-elastic displacement potential and Airy stress function are used to obtain particular solution and homogeneous solutions of the stress fields in each layer, respectively. Linear perturbation method is used to obtain the governing equations of solidification problem on coated planar mold of finite thickness. Two-dimensional problem is modelled with two one dimensional problem called as zeroth and first order problem. Zeroth order solution is the nonlinear unperturbed solution of the problem. As long as the perturbation is small compared with the zeroth order values, the perturbation will be linear. The zeroth and first order problems involve fields varying in only one spatial dimension and time, they are considerably easier to implement and more efficient computationally than a full two-dimensional numerical solution of the solidification problem. After the modeling of thermal and mechanical problems with respect to thermal and mechanical boundary conditions, two second-order differential equations with variable coefficients which involving amplitude of perturbation at the moving interface, its time derivative, residual stress and its derivative with respect to mean shell thickness. The problem solve in dimensionless form for generality of the solution. These equations are written in state-space form and solving simultaneously with the given initial conditions. A variable step variable-order predictor-corrector algorithm is used to solve this stiff problem. The effects of the system parameters such as thickness of the coating layer and mold, thermal conductivity ratio between the solidified shell material and coating material, thermal conductivity ratio between the coating material and mold material, coupling between thermal and mechanical problems at the interfaces between solidified shell, coating and mold on the growth of the solidified shell thickness are investigated. The results of this study use in the development of the numerical solution of general problem with adding thermal diffusivities, as well as in providing an initial conditions for the general case and use to control validity of the numerical solution.