Mathematical Model Of Solidification Research Articles

Mathematical Modeling of Technological Modes of Operation of Liquid Stamping of Aluminum Alloy AK7

In this paper we examine a mathematical model of solidification of the melt in a two-phase zone during the formation of forgings through liquid stamping. From the simultaneous solution of the Fourier equations of thermal conductivity for the melt, the two-phase zone, and the solid crust, the solidification kinetics has been established and the effect of external pressure on the temperature fields and components of the solidification time of hollow thin-walled forgings has been thoroughly assessed.

Application of the Swarm Intelligence Algorithm for Reconstructing the Cooling Conditions of Steel Ingot Continuous Casting

This paper presents a proposal to apply one of the swarm intelligence algorithms, the artificial bee colony (ABC) algorithm, to solve the inverse problem of steel ingot continuous casting. The discussed task consists of retrieving the cooling conditions of the process on the basis of temperature measurements and by taking into account the macrosegregation phenomenon. The examined process was modeled by using the mathematical model of solidification within the temperature interval. The solution method was based on the implicit scheme of the finite difference method supplemented by the procedure of correcting the field of temperature in the vicinity of liquidus and solidus curves, which was then used for solving the appropriate direct problem. The computational example, illustrating the stability and accuracy of the proposed method, is also presented in the paper.

Mathematical modeling of macrosegregation during solidification of binary alloy by Control Volume Finite Element Method

• A mathematical model for solidification with CVFEM is implemented in a computer code. • Coupled heat and mass transfer in solidification of lead-tin alloy is described. • Thermo-solutal convection and freckle formation in phase-change are reported. • Effects of cavity aspect ratio and cooling direction in solidification are studied. • Macrosegregation maps for Pb-48%Sn alloy in rectangular cavities is investigated. Unsteady coupled natural convective heat and mass transfer with liquid to solid phase change is described to analyze the macrosegregation of a Pb–Sn alloy in a rectangular two-dimensional cavity. The effect of cooling direction is investigated, by comparing the numerical simulation of solidification in rectangular molds with one vertical side cooled and with bottom wall cooling, both with the cold wall temperature changing with time. Fluid mechanics, heat and mass transfer coupled equations with unsteady, convective, diffusion and sources terms are solved with the Control Volume Finite Element Method (CVFEM) in cavities with aspect ratios A = H / B = 1 / 3 , 0.6 and 3. Results for the evolution of the distribution of streamlines, solidification front, isotherms and iso-concentrations are obtained with a code developed to characterize coupled fluid, heat and mass transport phenomena. Complex physical processes such as thermo-solutal convection during solidification, freckle formation and the final pattern of macrosegregation are described in terms of the cavity aspect ratio by the mathematical model. A good description of freckle formation and its relation to direction of cooling and aspect ratio has been obtained by the transient-convective–diffusive mathematical model solved by CVFEM.

A novel multi-dimensional model for solidification process with supercooling

It is well known that many materials do not solidify at their nominal phase-change temperature. Rather, nucleation occurs in them at a lower temperature. This phenomenon is usually termed “supercooling” or “subcooling” in the literature. Understanding, prediction and, if possible, prevention, or at least reduction, of supercooling are very important specifically to latent heat thermal energy storage (LHTES) systems, because the temperature differences in them must be small in order to achieve higher efficiency.In the present study, a novel mathematical model of solidification with supercooling and heat transfer is developed. For the first time, it is multidimensional in space. The model encompasses all possible stages of the process, namely, single-phase liquid cooling from the initial state to the nucleation temperature, kinetic nucleation accompanied by a rapid temperature rise to the nominal phase-change temperature, regular solidification and finally cooling of the solid phase. The kinetic solidification speed, based on the activation energy, is temperature- and, as a result, time-dependent. The model ensures a smooth, physically meaningful transition from the kinetic to regular solidification. Local and overall energy balance preservation at all stages of the process is ensured.The model is based on the enthalpy formulation, resolved using an in-house numerical code based on finite volumes. For the single phase cooling, it is validated using the well-known solutions from the literature. The model is then compared to experimental results of solidification of supercooled gallium in a vertical cylindrical mold. Accordingly, heat transfer in the mold is also included. It is shown that the model reflects the experimental results fairly well, in particular when predicting temperatures at various locations inside the material. Also, physically sound solidification patterns are obtained.

Optimization of strand and final electromagnetic stirrers of round bloom casters with multiple sections

Strand electromagnetic stirring (S-EMS) and final electromagnetic stirring (F-EMS) are the main methods used to improve the center porosity and segregation for round blooms. To optimize the stirring conditions, nail shooting tests were conducted for three sections of large round blooms with diameters of ϕ380 mm, ϕ450 mm, and ϕ600 mm. Acid leaching and sulfur print tests were used to investigate the shell thickness. Based on the results of nail shooting tests, a mathematical model of solidification was established, and the variation of shell thickness and the central solid fraction were exactly calculated by the model. By taking all sections into account, the locations of S-EMS and F-EMS were optimized for each section. In the results, the macro-segregation of various sections is improved after the locations of S-EMS and F-EMS systems are changed.

Numerical Model Of Binary Alloys Solidification Basing On The One Domain Approach And The Simple Macrosegregation Models

AbstractIn the paper the thermal processes proceeding in the domain of solidifying binary alloy are considered. The mathematical model of solidification and cooling processes bases on the one domain method (or fixed domain method). In such a model the parameter called a substitute thermal capacity (STC) appears. At the stage of STC construction the macrosegregation process described by the lever arm rule or the Scheil model is taken into account. In this way one obtains the formulas determining the course of STC resulting from the certain physical considerations and this approach seems to be closer to the real course of thermal processes proceeding in domain of solidifying alloy. In the final part the examples of numerical solutions basing on the finite difference method are presented.

Study of phase transformations in CMSX-6 and CMSX-8 superalloys

Nickel-based superalloys are extensively used mainly in the aircraft and aeronautic industry, particularly in the hottest parts of engines or turbo-reactors. The phase reactions occurring in these heat-resistant materials play a crucial role in many aspects of the processing and service of the highly alloyed materials. Cast Ni-based superalloys are obtained in a complex way and their structure is complicated. Differential scanning calorimetry (DSC) technique was applied for determination of temperature ranges of the phase transformations occurring in the CMSX-6 and CMSX-8 superalloys during heating/cooling processes. Thermophysical properties, including temperatures of the phase transformation, are the critical input parameters in mathematical models of solidification and casting of metallic materials. The literature data concerning phase transformations and performance of the heat treatment for CMSX-6 and CMSX-8 are incomplete and ambiguous. DSC results accompanied by scanning electron microscopy characterization of microstructure of CMSX-6 and CMSX-8 superalloy was applied. The present study will improve the understanding of the fundamental mechanisms of phase transformations of single-crystal nickel-based superalloys.

Phase transformations in CMSX-4 nickel-base superalloy

Thermophysical properties (including transformation temperatures) are the critical input parameters in mathematical models of solidification and casting of metallic materials. The technique of Differential Scanning Calorimetry (DSC) was selected for the study of precipitation and phase changes in CMSX-4 alloy system. The results of the calorimetric measurements together with phase transformation temperatures determined from DSC heating and cooling curves was described. In the present paper, DSC results accompanied by scanning electron microscope characterization of CMSX-4 superalloy is applied to understand significance of precipitation or phase changes in alloy system for design heat treatment for the nickel based superalloy.

Forced crystallization of drops in front of a body moving in a supercooled cloud

A physical and mathematical model for solidification of a metastable drop in an incompressible gas flow around a revolving cylinder that simulates the leading edge of a wing of an aircraft moving in a supercooled cloud is proposed. The formulas of classical hydrodynamics are used to find the velocity field of the carrying medium (the air). Against this background, the dynamics of a drop that determine the intensity of its blasting and extraction of the heat of crystallization (at the stage of solidification), as well as the rate of cooling of the solidified particle, are studied. The results of the numerical treatment for a typical set of parameters are given. One possible application is the forced crystallization of drops in a supercooled cloud prior to their collision with the surface of an aircraft (for instance, with the aid of advanced exposure to ultrasound) as a means of icing control.

About Kolmogorov's statistical theory of phase transformation

Many mathematical models of solidification use the following equation for the estimation of solid volume f S = 1 − exp ( − n 1 τ n 2 ) , where τ is time, n 1 and n 2 are constants. The fundamentals of the statistical theory of metal solidification were developed in classic papers written by Kolmogorov [A.N. Kolmogorov, Bull. Acad. Sci. USSR. Ser. Math., 3 (1937) 355–359 (in Russian)], Johnson and Mehl [W.A. Johnson, R.F. Mehl, Trans. Metall. Soc. AIME 135 (1939) 416–442], Avrami [M. Avrami, J. Chem. Phys. 7 (1939) 1103–1112; M. Avrami, J. Chem. Phys. 8 (1940) 212–224; M. Avrami, J. Chem. Phys. 9 (1941) 177–184]. The above equation in contemporary literature is named after these authors (K-J-M-A), but this equation is only a particular solution of Kolmogorov's general solution f S = 1−exp(− Ω), where the function Ω is the so-called total extended volume (per unit volume) of the growing grains, if their overlapping is neglected. Kolmogorov's general solution for the Ω-function is an accurate development only in the case of geometric similarity of the growing grains, their uniform distribution and equal orientation in space. Moreover, for an arbitrary moment and direction all grains must have the same absolute value of the growth velocity vector. If the conditions stated above are neglected, the above equations can give overestimated results. The source of this overestimation is the, so-called, screening effect. The presented solution takes into account the screening effect for the calculation of Ω-function. This solution expands the scope of the Kolmogorov's statistical theory of transformation. The proposed solution takes into consideration the growth of the grains of any shape, assuming their uniform distribution in space.

Application of high temperature DSC technique to nickel based superalloys

Heat capacity is a critical input parameter in mathematical models of solidification and casting. It appears in its own right, and is required for the extraction of thermal conductivity from thermal diffusivity data. It also provides a measure of the latent heat associated with changes of phase, and the precipitation or dissolution of precipitates. Calorimetry is a well-established technique and is used to solve a wide range of materials problems such as studying precipitation or phase changes in alloy systems, and the kinetics of these phase transformations. The quality and range of data that can be obtained will be demonstrated by applying it to nickel base superalloys. This paper discusses important aspects of the measurement including control of the atmosphere, suitable reference materials for temperature and enthalpy calibration, size of the sample and the rate of heating and cooling. The optimised DSC technique was used to investigate the properties of two nickel based superalloys (CMSX4 and IN738LC), comparing the results with previous values and determining sources of error in the method.

NUMERICAL SIMULATION OF POLYMER CRYSTALLIZATION

We consider a mathematical model for solidification of semicrystalline polymers, describing the evolution of temperature, crystalline volume fraction, number and average size of crystals. In turn, the model couples a suitable kinetics of nonisothermal crystallization, taking into account both formation and growth of nuclei, with the thermal energy balance equation. We also present a model of secondary crystallization. The numerical approximation is performed by semiexplicit finite differences in time and finite elements in space. The fully discrete scheme amounts to solve, at any time step, a symmetric positive definite linear system preceded by an elementwise explicit computation. The computed numerical crystal structures match qualitatively the experimental ones.

Simulation of freckles during vertical solidification of binary alloys

A mathematical model of solidification that simulates the formation of channel segregates or freckles is presented. The model simulates the entire solidification process starting with the initial melt to the solidified cast, and the resulting segregation is predicted. Emphasis is given to the initial transient, when the dendritic zone begins to develop and the conditions for the possible nucleation of channels are established. The mechanisms that lead to the creation and eventual growth or termination of channels are explained in detail and illustrated by several numerical examples. Predictions of the pattern and location of channels in different cooling situations are in good agreement with experimental observations.

Crystal Anisotropy Directs Solidification

Crystal Anisotropy Directs Solidification