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T. Skrzypczak

The paper is focused on the study of the solidification process of pure metals, in which the solidification front is smooth. It has the shape of a surface separating liquid from solid in three dimensional space or a curve in 2D. The location and topology of moving interface change over time and its velocity depends on the values of heat fluxes on the solid and liquid side of it.

Such a formulation belongs to a group called Stefan problems. A mathematical model of the Stefan problem is based on differential equations of heat conduction and interface motion. This system of equations is supplemented by appropriate initial and boundary conditions as well as the continuity conditions at the solidification interface. The solution involves the determination of temporary temperature field and interface position. Typically, it is impossible to obtain the exact solution of such problem.

This paper presents a mathematical model for the two-dimensional problem. The equation of heat conduction is supplemented with Dirichlet and Neumann boundary conditions. Interface motion is described by the level set equation which solution is sought in the form of temporary distribution of the signed distance function. Zero level of the distance field coincides with the position of the front. Values of the signed distance function obtained from the level set equation require systematic reinitialization.

Numerical model of the process based on the finite element method (FEM) is also presented. FEM equations are derived and discussed. The explicit time integration scheme is proposed. It helps to avoid solving the system of equations during each time step. The reinitialization procedure of the signed distance function is described in detail. Examples of numerical analysis of the solidification process of pure copper within the complex geometry are presented. Results obtained from the use of constant material properties are compared with those obtained from the use of temperature dependent properties.

Open access

T. Skrzypczak, E. Węgrzyn-Skrzypczak and L. Sowa

Abstract

The paper presents an approach of numerical modelling of alloy solidification in permanent mold and transient heat transport between the casting and the mold in two-dimensional space. The gap of time-dependent width called "air gap", filled with heat conducting gaseous medium is included in the model. The coefficient of thermal conductivity of the gas filling the space between the casting and the mold is small enough to introduce significant thermal resistance into the heat transport process. The mathematical model of heat transport is based on the partial differential equation of heat conduction written independently for the solidifying region and the mold. Appropriate solidification model based on the latent heat of solidification is also included in the mathematical description. These equations are supplemented by appropriate initial and boundary conditions. The formation process of air gap depends on the thermal deformations of the mold and the casting. The numerical model is based on the finite element method (FEM) with independent spatial discretization of interacting regions. It results in multi-mesh problem because the considered regions are disconnected.

Open access

T. Skrzypczak, E. Węgrzyn-Skrzypczak and J. Winczek

Abstract

The paper is focused on the modeling of the directional solidification process of pure metal. During the process the solidification front is sharp in the shape of the surface separating liquid from solid in three dimensional space or a curve in 2D. The position and shape of the solid-liquid interface change according to time. The local velocity of the interface depends on the values of heat fluxes on the solid and liquid sides. Sharp interface solidification belongs to the phase transition problems which occur due to temperature changes, pressure, etc. Transition from one state to another is discontinuous from the mathematical point of view. Such process can be identified during water freezing, evaporation, melting and solidification of metals and alloys, etc.

The influence of natural convection on the temperature distribution and the solid-liquid interface motion during solidification of pure copper is studied. The mathematical model of the process is based on the differential equations of heat transfer with convection, Navier-Stokes equation and the motion of the interface. This system of equations is supplemented by the appropriate initial and boundary conditions. In addition the continuity conditions at the solidification interface must be properly formulated. The solution involves the determination of the temporary temperature and velocity fields and the position of the interface. Typically, it is impossible to obtain the exact solution of such problem. The numerical model of solidification of pure copper in a closed cavity is presented, the influence of the natural convection on the phase change is investigated. Mathematical formulation of the problem is based on the Stefan problem with moving internal boundaries. The equations are spatially discretized with the use of fixed grid by means of the Finite Element Method (FEM). Front advancing technique uses the Level Set Method (LSM). Chorin’s projection method is used to solve Navier-Stokes equation. Such approach makes possible to uncouple velocities and pressure. The Petrov-Galerkin formulation is employed to stabilize numerical solutions of the equations. The results of numerical simulations in the 2D region are discussed and compared to the results obtained from the simulation where movement of the liquid phase was neglected.

Open access

J. Winczek and T. Skrzypczak

Abstract

The paper presents a model of temperature, phase transformation and stresses fields in a steel element during single-pass Gas Metal Arc Weld (GMAW) surfacing.

Kinetics of phase transformations during heating is limited by temperature values at the beginning and at the end of austenitic transformation, while the progress of phase transformations during cooling is determined on the basis of TTT-welding diagram and Johnson-Mehl-Avrami and Kolomogorov law for diffusive transformations and Koistinen-Marburger for martensitic transformation. Stress state of a bar subjected to thermo-mechanical loads is described assuming the plane cross section hypothesis and using integral equations of stress equilibrium of a bar as well as simple Hook’s law. Stresses in the elastic-plastic state are determined by iteration using solutions with a variable elastic modulus of elasticity, conditioned by tensile curves. Dependence of stresses on strains is assumed on the basis of tensile curves of particular structures, taking into account the influence of temperature. There were performed calculations of the temperature field, phase transformations, strains and stresses for GMAW surfacing of a cuboid element made of S235 steel. Authors’ programs, made in Borland Delphi, were used for calculations.

Open access

T. Banach, M. Bochniarz, P. Łyp, Ł. Adaszek, W. Wawron, B. Furmaga, M. Skrzypczak, J. Ziętek and S. Winiarczyk

Abstract

The aim of this study was to use matrix-assisted laser desorption ionization time-of-flight mass spectrometry (MALDI-TOF MS) for the identification of coagulase-negative staphylococci (CNS) isolated from the milk of cows with subclinical mastitis. The study material consisted of 33 isolates of CNS, identified by the results of API Staph tests, obtained from the milk of cows with subclinical mastitis. Based on the spectra analyses, MALDI-TOF MS tests of 33 bacterial samples allowed identification of the microorganisms in 27 cases (81.8%). The most frequent cause of subclinical mastitis was found to be Staphylococcus sciuri (39%), while S. vitulinus was detected in 15% of the milk samples. The results obtained indicate that MALDI-TOF MS can be used for the identification of CNS isolated from bovine mastitis as a method supplementary to biochemical tests.