536.242We use the finite-element method to study the problem of determining the temperature field and the evolution of the phase-state regions in a hollow cylindrical casting. Using weighted discrepancies we reduce the problem to solving systems of ordinary differential equations. We give a model solution of a specific problem and detailed analysis of the results.The properties and structure of molded parts depends to a significant degree on the thermal conditions under which they are obtained. In this connection there is understandable interest in the study of temperature fields in the structural-phase component of products during casting.It is known that in the manufacture of molded machine parts and products, as a rule, the following transformations occur: crystallization, polymorphic transformations, and recrystallization.In the process of forming a casting the temperature varies over a wide range--from the pouring temperature to the temperature of the surrounding medium. In the process an evolution of the aggregate state occurs in the body. In general during cooling one can distinguish the following characteristic regions of the thermophysical state: the liquid-phase region (1), T > Tt; the transition region (t), Ts < T < Tl, and the solid-phase region (s), T <_ T~, where Tl and T~ are the temperatures of the liquidus and the solidus of the melt [1].In the present paper we study the temperature fields and the evolution of the regions of the phase state in a hollow cylindrical casting.Suppose we have a mold that is a hollow cylinder with internal radius Rc~t and outer radius Rmold. Along the axis of the mold there is a rod of radius Rrod. (1)Here c = c(r, T) is the specific heat capacity per unit volume, ,~ = A(r, T) is the thermal conductivity, h is the specific heat of melting, and ~ is the degree of completeness of crystallization (the relative content of the solid phase at each point of the transition region). Since ~ = 0 for the liquid phase and ~ = 1 for the solid phase, the term ho~ occurs only in the transition region. The initial and boundary conditions of the problem have the form Tm, 0 < r < Rrod,where c~ is the thermal diffusivity. Assuming a linear relation between the degree of completeness of crystallization and the temperature
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