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1
Content available Numerical Evaluation of the Impact of Riser Geometry on The Shrinkage Defects Formation in the Solidifying Castin
Sowa L., Skrzypczak T., Kwiatoń P.
Archives of Metallurgy and Materials
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2022
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Vol. 67, iss. 1
181--187
EN
The work concerns of modeling the process of manufacturing machine parts by casting method. Making a casting without internal defects is a difficult task and usually requires numerous computer simulations and their experimental verification at the prototyping stage. Numerical simulations are then of priority importance in determining the appropriate parameters of the casting process and in selecting the shape of the riser for the casting fed with it. These actions are aimed at leading shrinkage defects to the riser, so that the casting remains free from this type of defects. Since shrinkage defects usually disqualify the casting from its further use, this type of research is still valid and requires further work. The paper presents the mathematical model and the results of numerical simulations of the casting solidification process obtained by using the Finite Element Method (FEM). A partial differential equation describing the course of thermal phenomena in the process of 3D casting creating was applied. This equation was supplemented with appropriate boundary and initial conditions that define the physical problem under consideration. In numerical simulations, by selecting the appropriate shape riser, an attempt was made to obtain a casting without internal defects, using a simple method of identifying their location. This is the main aim of the research as such defects in the casting disqualify it from use.
2
Content available The Effect of Metal Pouring Conditions on the Formation of Defects in the Casting
Sowa L., Skrzypczak T., Kwiatoń P.
Archives of Foundry Engineering
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2020
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Vol. 20, iss. 2
31--36
EN
The mathematical model and numerical simulations of the solidification of a cylindrical casting, which take into account the process of the mould cavity filling by liquid metal and the feeding of the casting through the conical riser during its solidification, are proposed in the paper. The interdependence of thermal and flow phenomena were taken into account because they have an essential influence on solidification process. The effect of the pouring temperature and pouring velocity of the metal on the solidification kinetics of the casting was determined. In order to obtain the casting without shrinkage defects, an appropriate selection of these parameters was tried, which is important for foundry practice. The velocity fields have been obtained from the solution of Navier-Stokes equations and continuity equation, while temperature fields from solving the equation of heat conductivity containing the convection term. In the solidification modelling the changes in thermo-physical parameters as a function of temperature were considered. The finite element method (FEM) was used to solve the problem.
3
Content available Numerical Model of Solidification Including Formation of Multiple Shrinkage Cavities
Skrzypczak T., Sowa L., Węgrzyn-Skrzypczak E.
Archives of Foundry Engineering
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2020
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Vol. 20, iss. 1
37--42
EN
Presented paper shows the mathematical and numerical approaches for modelling of binary alloy solidification solved by the Finite Element Method (FEM). The phenomenon of shrinkage cavities formation process is included in the numerical model. Multiple macroscopic cavities can be modelled within the single casting volume. Solid, liquid and gaseous phases with different material properties are taken into account during solidification process. Mathematical model uses the differential equation of heat diffusion. Modification of specific heat is used to describe the heat releasing during liquid-solid phase change. Numerical procedure of shrinkage cavities evolution is based on the recognition of non-connected liquid volumes and local shrinkage computation in the each of them. The recognition is done by the selection of sets of interconnected nodes containing liquid phase in the finite element mesh. Original computer program was developed to perform calculation process. Obtained results of temperature and shrinkage cavities distributions are presented and discussed in details.
4
Content available The Influence of Riser Shape on Feeding Effectiveness of Solidifying Casting
Sowa L., Skrzypczak T., Kwiatoń P.
Archives of Foundry Engineering
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2019
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iss. 4
91--94
EN
The mathematical model and numerical simulations of the solidification of a cylindrical shaped casting, which take into account the process of filling the mould cavity by liquid metal and feeding the casting through the riser during its solidification, are presented in the paper. Mutual dependence of thermal and flow phenomena were taken into account because have an essential influence on solidification process. The effect of the riser shape on the effectiveness of feeding of the solidifying casting was determined. In order to obtain the casting without shrinkage defects, an appropriate selection of riser shape was made, which is important for foundry practice. Numerical calculations of the solidification process of system consisting of the casting and the conical or cylindrical riser were carried out. The velocity fields have been obtained from the solution of momentum equations and continuity equation, while temperature fields from solving the equation of heat conductivity containing the convection term. Changes in thermo-physical parameters as a function of temperature were considered. The finite element method (FEM) was used to solve the problem.
5
Content available Analysis of Temperature and Velocity Fields During Filling of Continuous Casting Mould
Sowa L., Skrzypczak T., Kwiatoń P.
Archives of Foundry Engineering
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2018
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Vol. 18, iss. 1
115--118
EN
In this paper, the mathematical model and numerical simulations of the molten steel flow by the submerged entry nozzle and the filling process of the continuous casting mould cavity are presented. In the mathematical model, the temperature fields were obtained by solving the energy equation, while the velocity fields were calculated by solving the momentum equations and the continuity equation. These equations contain the turbulent viscosity which is found by solving two additional transport equations for the turbulent kinetic energy and its rate of dissipation. In the numerical simulations, coupling of the thermal and fluid flow phenomena by changes in the thermophysical parameters of alloy depending on the temperature has been taken into consideration. This problem (2D) was solved by using the finite element method. Numerical simulations of filling the continuous casting mould cavity were performed for two variants of liquid metal pouring. The effect of the cases of pouring the continuous casting mould on the velocity fields and the solid phase growth kinetics in the process of filling the continuous casting mould was evaluated as these magnitudes have an influence on the high quality of the continuous cast steel slab.
6
Content available Analytical and numerical solution of the heat conduction problem in the rod
Węgrzyn-Skrzypczak E., Skrzypczak T.
Journal of Applied Mathematics and Computational Mechanics
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2017
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Vol. 16, nr 4
79--86
EN
In this paper, the results of analytical and numerical solution of the problem of heat transport in the rod of finite length are presented. The analytical solution is obtained with the use of the Fourier series. The numerical model of the problem is based on the Finite Element Method (FEM). In addition, to check the compatibility of both solutions, distributions of the temperature for selected time moments are compared and discussed.
7
Content available Computer Simulation of the Solidification Process Including Air Gap Formation
Skrzypczak T., Węgrzyn-Skrzypczak E., Sowa L.
Archives of Foundry Engineering
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2017
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Vol. 17, iss. 4
147--150
EN
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.
8
Content available remote Local and Global Instability of the Slender Geometrically Nonlinear SystemWith Non-Prismatic Element Subjected to the Euler’s Load
Szmidla J., Skrzypczak T., Jurczyńska A.
Machine Dynamics Research
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2016
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Vol. 40, No. 4
79--88
EN
A theoretical considerations and numerical calculations concerning the issue of the stability of the geometrically nonlinear system with non-prismatic element are presented in this work. The analysed columns were subjected to the Euler’s load. On the basis of the minimum potential energy principle as well as the small parameter method, the differential equations of displacements were formulated and its solutions were obtained. The assumption that the approximation of the non-prismatic rod satisfies the condition of constant total volume resulting from the value of the coefficient of flexural stiffness distribution has been made. The results of the carried out numerical simulations refer to the local and global stability loss. It has been proved that taking into consideration in the geometrically nonlinear system appropriate shaped rod of variable cross-section causes an increase in the value of bifurcation load and in a consequence an „exit” from the area of the local instability (loss of rectilinear form of static equilibrium).
9
Content available Thermomechanical States in Arc Weld Surfaced Steel Elements
Winczek J., Skrzypczak T.
Archives of Metallurgy and Materials
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2016
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Vol. 61, iss. 3
1623--1634
EN
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.
10
Content available The model of temporary temperature field during multi-pass arc weld surfacing. Pt. 1, Analitycal description
Winczek J., Rygał G., Skrzypczak T.
Journal of Applied Mathematics and Computational Mechanics
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2015
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Vol. 14, nr 2
123--130
EN
This work presents a model of a temperature field in a steel element during multi-pass Gas Metal Arc Weld surfacing taking into account heat of the melted electrode material. An analytical solution for a half-infinite body model is obtained by aggregating temperature increments caused by applying liquid metal and heat radiation of a moving electrode. The assumptions are Gaussian distributed heat sources of applied melted electrode material and of an electric arc.
11
Content available The model of a temporary temperature field during multi-pass arc weld surfacing. Pt. 2, Example of computations
Winczek J., Rygał G., Skrzypczak T.
Journal of Applied Mathematics and Computational Mechanics
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2015
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Vol. 14, nr 3
141--148
EN
In this work computations of a temperature field are carried out during multipass Gas Metal Arc Weld surfacing of a cuboidal steel element taking into account heat of the melted electrode material. The results are presented in the form of temporary and maximum temperature distribution in the element’s cross-section and thermal cycles at selected points.
12
Content available Modeling of thermal contact through gap with the use of Finite Element Method
Węgrzyn-Skrzypczak E., Skrzypczak T.
Journal of Applied Mathematics and Computational Mechanics
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2015
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Vol. 14, nr 4
145--152
EN
In the paper the mathematical and numerical descriptions of the general case of thermal contact between two flat bodies are presented. The numerical model of the problem is based on the Finite Element Method (FEM). Variable width of the contact gap between interacting bodies is considered. The model allows the use of independent spatial discretization of the contacting components, which means that the edges of the finite elements lying on the both sides of the contact gap need not be matched. The algorithm of treatment of the fourth kind boundary condition is described in details.
13
Content available Effect Of Natural Convection On Directional Solidification Of Pure Metal
Skrzypczak T., Węgrzyn-Skrzypczak E., Winczek J.
Archives of Metallurgy and Materials
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2015
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Vol. 60, iss. 2A
835--841
EN
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.
PL
Praca porusza problematykę modelowania kierunkowego krzepnięcia czystego metalu. Podczas tego procesu obserwuje się formowanie ostrego frontu krzepnięcia w postaci powierzchni separującej ciecz i ciało stałe w przypadku trójwymiarowym lub krzywej w przypadku płaskim. Położenie oraz kształt interfejsu krzepnięcia zmieniają się w czasie a wartości prędkości lokalnych zależą od różnicy intensywności strumieni ciepła po stronie ciała stałego i cieczy. Krzepnięcie z ostrym frontem należy do grupy procesów z przemianami fazowymi, które warunkowane są zmianami temperatury, ciśnienia, itp. Przejście fazowe z jednego stanu w drugi ma z matematycznego punktu widzenia charakter nieciągły. Procesy tego typu można zidentyfikować podczas zamarzania wody, parowania, topnienia i krzepnięcia metali i stopów, itp. W pracy zbadano wpływ zjawiska konwekcji swobodnej na chwilowy rozkład temperatury oraz ruch granicy narastania fazy stałej podczas krzepnięcia czystej miedzi w obszarze płaskim. Model matematyczny sformułowano na bazie równań różniczkowych transportu ciepła z konwekcją, Naviera-Stokesa i ruchu frontu krzepnięcia. Układ równań uzupełniono odpowiednimi warunkami początkowymi i brzegowymi oraz warunkami ciągłości na froncie. Rozwiązanie obejmuje chwilowe rozkłady temperatury, prędkości oraz położenie granicy międzyfazowej. Sformułowanie matematyczne zagadnienia bazuje na modelu z ruchomymi granicami wewnętrznymi, czyli tzw. modelu Stefana. Równania zostały zdyskretyzowane przestrzennie z wykorzystaniem metody elementów skończonych. W modelu numerycznym wykorzystano siatkę niezmienną w czasie. Do propagacji frontu użyto metody poziomic. Do wyznaczenia prędkości w cieczy wykorzystano metodę rzutowania, która poprzez eliminację ciśnienia z równania pędu pozwala na rozprzężenie prędkości i ciśnień. Równania rozwiązano z wykorzystaniem sformułowania Petrova-Galerkina. Omówiono wyniki analizy numerycznej oraz porównano je z wynikami otrzymanymi z symulacji, w której pominięto ruch cieczy.
14
Content available Mathematical and numerical basis of binary alloy solidification models with substitute thermal capacity. Part 1
Węgrzyn-Skrzypczak E., Skrzypczak T.
Journal of Applied Mathematics and Computational Mechanics
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2014
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Vol. 13, nr 2
135--140
EN
The presented work is focused on the basis of mathematical and numerical descriptions of the binary alloy solidification problem. The mathematical formulation is based on the so-called substitute thermal capacity, which implies a change in the specific heat of solidifying material. In the literature one can find many ways to define this parameter. Five models, differing in the description of the substitute thermal capacity as well as the numerical model using the finite element method (FEM) are considered.
15
Content available Mathematical and numerical basis of binary alloy solidification models with substitute thermal capacity. Part 2
Węgrzyn-Skrzypczak E., Skrzypczak T.
Journal of Applied Mathematics and Computational Mechanics
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2014
|
Vol. 13, nr 2
141--147
EN
In this paper, the results obtained from five models of the solidification with substitute thermal capacity were compared. The calculations were carried out for steel containing 0.35% carbon with using an in-home solver based on the finite element method (FEM). A comparison was made on the base of analysis of the cooling curves at selected nodes.
16
Content available Numerical model with explicit time integration scheme for tracking interfaces
Węgrzyn-Skrzypczak E., Skrzypczak T.
Journal of Applied Mathematics and Computational Mechanics
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2013
|
Vol. 12, nr 2
111--116
EN
In this paper a simple and effective method for tracking interfaces in two-dimensional area is described. The presented approach is very attractive in solving Stefan problems where moving internal boundaries occur. It is based on the level set method (LSM) and uses the so-called distance function. A numerical model based on the finite element method (FEM) is proposed.
17
Content available Investigation of accuracy of the interface tracking method
Węgrzyn-Skrzypczak E., Skrzypczak T.
Journal of Applied Mathematics and Computational Mechanics
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2013
|
Vol. 12, nr 2
105--110
EN
In this paper accuracy of a simple and effective method for tracking interfaces in two-dimensional area is investigated. The method is based on the level set method (LSM) with "brute force" reinitialization algorithm. A comparison of numerical solution with an analytical solution is presented and discussed.
18
Content available Effect of Mesh Quality on the Numerical Solution of the Solidification of Pure Metal
Skrzypczak T., Węgrzyn-Skrzypczak E.
Archives of Foundry Engineering
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2013
|
Vol. 13, iss. 2 spec.
89--92
EN
The paper presents a method of mathematical and numerical modelling of directional solidification process of pure metal in the two-dimensional region. In this case, the thermal conditions associated with the process favours the occurrence of sharp solidification front. The mathematical description of the process is based on the Stefan formulation with appropriate continuity conditions on the solid-liquid interface. The numerical model is based on the finite element method (FEM). The calculations were made on a fixed mesh with diffused solidification front to avoid the difficulties associated with the discontinuity. Temporary position of the interface was calculated with the use of the level set method (LSM). Effect of the quality of the spatial discretization on the accuracy of numerical solution was investigated. Obtained results of the temporary front position were compared with the analytical solution. The correlation between the quality of the spatial discretization and the accuracy of the results was observed. Methods used in the work had significant impact on the computation time and helped avoid the explicit consideration of discontinuity of heat flux on the front.
19
Content available Sharp Interface Numerical Modeling of Solidification Process of Pure Metal
Skrzypczak T.
Archives of Metallurgy and Materials
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2012
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Vol. 57, iss. 4
1189-1199
EN
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.
PL
W pracy skupiono się na badaniu procesu krzepnięcia czystych metali, podczas którego front krzepnięcia pozostaje płaski. W przypadku trójwymiarowym jest on powierzchnia oddzielająca ciecz od ciała stałego, w przypadku dwuwymiarowym ma postać krzywej. Położenie i topologia frontu krzepnięcia zmienia się w czasie, a prędkość przemieszczania zależy od różnicy wartości strumieni cieplnych po stronie ciała stałego i cieczy. Takie sformułowanie klasyfikuje opisywane zjawisko w grupie tzw. zagadnień Stefana. Model matematyczny tego procesu stanowią równania różniczkowe przewodnictwa ciepła oraz ruchu powierzchni międzyfazowej. Układ ten uzupełniają odpowiednie warunki brzegowe, początkowe oraz warunki ciągłości na froncie. Jego rozwiązanie polega na wyznaczeniu chwilowych pól temperatury oraz położenia frontu. Najczęściej nie da się uzyskać rozwiązania tak sformułowanego problemu w sposób dokładny. W pracy zaprezentowano model matematyczny zagadnienia dla przypadku płaskiego. Równanie różniczkowe przewodnictwa ciepła uzupełniono warunkami brzegowymi Dirichleta oraz Neumanna. Ruch interfejsu międzyfazowego opisano tzw. równaniem poziomic (ang. level set equation), którego rozwiązania poszukiwano w postaci chwilowego rozkładu funkcji dystansu. Izolinia zerowa tego rozkładu pokrywa się z położeniem frontu. Otrzymane wartości funkcji dystansu wymagają systematycznej reinicjalizacji. Przedstawiono również model numeryczny procesu bazujący na metodzie elementów skończonych. Opisano schemat postępowania prowadzący do otrzymania dyskretnych równań MES. Wykorzystano jawny schemat całkowania po czasie, co pozwoliło uniknąć konieczności rozwiazywania układu równań zarówno w przypadku równania przewodnictwa ciepła jak i równania poziomic. Szczegółowo opisano metodę reinicjalizowania funkcji dystansu. Zaprezentowano przykłady analizy numerycznej procesu krzepnięcia czystej miedzi w obszarze o złożonej geometrii. Porównano wyniki otrzymane dla stałych własności materiałowych z wynikami uzyskanymi z wykorzystaniem własności zależnych od temperatury.
20
Content available remote Mathematical model and numerical solution of double diffusive natural convection system
Węgrzyn-Skrzypczak E., Skrzypczak T.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2011
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Vol. 10, nr 2
225-229
EN
In this paper a two-dimensional double diffusive natural convection system is considered. A mathematical model of heat and moisture transport driven by combined thermal- and solutal-induced buoyancy forces is described. A numerical model based on the Finite Element Method (FEM) is proposed. The results of numerical analysis are presented and discussed.
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