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1
Content available Numerical Model of Solidification Including Formation of Multiple Shrinkage Cavities
Skrzypczak T., Sowa L., Węgrzyn-Skrzypczak E.
Archives of Foundry Engineering
|
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.
2
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.
3
Content available Computer Simulation of the Solidification Process Including Air Gap Formation
Skrzypczak T., Węgrzyn-Skrzypczak E., Sowa L.
Archives of Foundry Engineering
|
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.
4
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.
5
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.
6
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
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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.
7
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
|
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.
8
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
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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.
9
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
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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.
10
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
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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.
11
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
|
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.
12
Content available remote Three-dimensional numerical model of solidification with motion of the liquid phase
Skrzypczak T., Węgrzyn-Skrzypczak E.
Archives of Foundry Engineering
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2011
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Vol. 11, iss. 2
127-132
EN
The paper presents numerical modeling of solidification process with convective motion of the liquid phase, generated both in the liquid and mushy zones. The transition region between the areas filled with liquid and solid is treated as a porous medium, which incorporates the suppression of fluid motion caused by the solid phase growth. Mathematical and numerical models of the phenomenon for threedimensional domain are presented. To solve the problem Finite Element Method is used. The results obtained from numerical simulation are presented and discussed.
PL
W pracy przedstawiono model matematyczny i numeryczny procesu krzepnięcia z uwzględnieniem ruchów konwekcyjnych fazy ciekłej, generowanych zarówno w strefie ciekłej jak i stało-ciekłej. Strefa przejściowa pomiędzy obszarem cieczy i ciała stałego traktowana jest jako ośrodek porowaty, w którym uwzględniono tłumienie ruchu cieczy wywołane narastaniem fazy stałej. Przedstawiono model matematyczny i numeryczny rozważanego procesu dla obszaru trójwymiarowego. Do rozwiązania zagadnienia wykorzystano metodę elementów skończonych. Przedstawiono i omówiono uzyskane wyniki.
13
Content available remote Mathematical description of discontinuous Galerkin method in the theory of thermoelasticity
Węgrzyn-Skrzypczak E., Skrzypczak T.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
|
2010
|
Vol. 9, nr 2
235-242
EN
In the paper mathematical description of Discontinuous Galerkin Method (DGM) used in the theory of thermoelasticity is presented. Displacement form of governing equations is introduced as the base of mathematical model. Space discretization methodology for discontinuous finite element method is showed.
14
Content available remote Simulation of shrinkage cavity formation during solidification of binary alloy
Skrzypczak T., Węgrzyn-Skrzypczak E.
Archives of Foundry Engineering
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2010
|
Vol. 10, iss. 1
147-152
EN
Presented paper is focused on numerical modeling of binary alloy solidification process with connection to shrinkage cavity formation phenomenon. Appropriate matching of cooling parameters during solidification process of the cast with raiser is essential to obtain suitable properties of the manufactured part. Localization, structure and depth of the shrinkage cavity is connected to these parameters. The raiser is removed after process, so defect localization in the top part of the manufactured element is of great importance. Mathematical model of solidification process is presented in the paper. The main focus is put on the algorithm of shrinkage cavity creation process. On the basis of mathematical model the numerical approach using finite element method is proposed. On the base of mathematical and numerical model computer program is made. It is able to perform simulation of the shrinkage cavity formation in 2D region. Shape and localization of shrinkage cavity obtained from simulation is compared to defect which was created during experiment.
15
Content available remote Analysis of three-dimensional binary alloy solidification with shrinkage cavity formation
Skrzypczak T., Węgrzyn-Skrzypczak E.
Archives of Foundry Engineering
|
2010
|
Vol. 10, iss. 4
199--204
EN
The paper focuses on modeling of binary alloy solidification process with using Finite Element Method (FEM). The process is characterized by liquidus and solidus temperatures which are constant because solutal segregation is neglected. Analysed region is threedimensional and consists of casting and riser. Mathematical and numerical models of solidification process are presented in the paper. The main focus is put on the algorithm of shrinkage cavity creation process. On the base of mathematical and numerical model computer program has been made. It is capable to simulate shrinkage cavity formation. Two examples show the results of different calculations performed by the program. The first example shows shrinkage cavity created during fast cooling of the top part of the riser while the second one was performed by significantly slower cooling. The shape and localization of shrinkage cavity obtained from simulation is compared to defect which was created during experiment.
16
Content available remote Mathematical and numerical model of 3D natural convection in a cube
Węgrzyn-Skrzypczak E., Skrzypczak T.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2009
|
Vol. 8, nr 1
185-191
EN
In the paper mathematical and numerical model of air convection in 3D region is considered. Governing equations of the model are presented. Main assumptions of numerical approach are discussed. Finite Element Method (FEM) with Characteristic Based Split (CBS) scheme is used to solve the problem. Two examples of numerical calculation are presented and discussed.
17
Content available remote Accuracy of numerical solution of heat diffusion equation
Węgrzyn-Skrzypczak E., Skrzypczak T.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2008
|
Vol. 7, nr 1
193-197
EN
Presented paper is focused on results of numerical solution of transient heat conductivity equation in two-dimensional region. Convection term is neglected in mathematical model of the phenomenon. Solutions based on classical Galerkin finite element formulation obtained for girds of different qualities are compared to discontinuous Galerkin method. Spatial discretization of computational domain and order of basis functions are taken into account.
18
Content available remote Numerical modelling of the binary alloys solidification with solutal undercooling
Skrzypczak T., Węgrzyn-Skrzypczak E.
Archives of Foundry Engineering
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2008
|
Vol. 8, iss. 1 spec.
299--302
EN
In the paper description of mathematical and numerical model of binary alloy solidification is presented. Metal alloy consisting of main component and solute is introduced. Moving, sharp solidification front is assumed. Constitutional undercooling phenomenon is taken into consideration. As a solidification front advances, solute is redistributed at the interface. Commonly, solute is rejected into the liquid, where it accumulates into solute boundary layer. Depending on the temperature gradient, such liquid may be undercooled below its melting point, even though it is hotter than liquid at the front. This phenomenon is often called constitutional or solutal undercooling, to emphasize that it arises from variations in solutal distribution of liquid. An important consequence of this accumulation of solute is that it can cause the front to break down into cells or dendrites. This occurs because there is a liquid ahead of the front with lower solute content, and hence a higher melting temperatures than liquid at the front. In the paper location and shape of undercooled region depending on solidification parameters is discussed. Numerical method basing on Finite Element Method (FEM) allowing prediction of breakdown of moving planar front during solidification of binary alloy is proposed.
PL
W pracy zaprezentowano matematyczny i numeryczny opis krzepnięcia stopu dwuskładnikowego, traktowanego jak mieszanina składnika głównego i domieszki. Wprowadzono ostry front krzepnięcia, którego położenie zmienia się w czasie. W modelu uwzględniono zjawisko przechłodzenia stężeniowego. Podczas krzepnięcia domieszka nie wbudowana w strukturę powstającego ciała stałego zostaje rozprowadzona w cieczy przed czołem frontu, tworząc brzegową warstwę dyfuzyjną. Wzrost stężenia powoduje zmianę gradientu temperatury likwidusu, często powodując przechłodzenie cieczy poniżej punktu krzepnięcia. Zjawisko to zwane jest przechłodzeniem stężeniowym. Jego konsekwencją jest zmiana morfologii frontu krzepnięcia, tzn. przejście od postaci gładkiej do komórkowej i dendrytycznej. W pracy przedstawiono metodę przewidywania położenia oraz śledzenia obszaru utraty stateczności frontu. Do obliczeń zastosowano metodę elementów skończonych.
19
Content available remote Numerical model of the solidification of alloys with natural convection of the liquid
Węgrzyn-Skrzypczak E., Skrzypczak T.
Archives of Foundry Engineering
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2008
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Vol. 8, iss. 1
109-112
EN
The paper deals with comparison of numerical analysis results obtained for binary alloys solidification process in the sand and permanent mould with motion of the fluid in the liquid and mushy zone. The partial differential equations describing mathematical model of the phenomena are presented. Finite Element Method is used for modeling process. Characteristic Based Split (CBS) method is used for solving momentum equation. Such approach allows to uncouple velocities and pressure. Petrov-Galerkin formulation is employed to stabilize heat conductivity equation with convective term. The results of the numerical simulations in the 2D region are discussed. Velocity fields, cooling rates and positions of the liquid, solid-liquid and solid regions are compared.
20
Content available remote Influence of coolant motion on structure of hardened steel element
Kulawik A., Skrzypczak T., Węgrzyn-Skrzypczak E.
Archives of Foundry Engineering
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2008
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Vol. 8, iss. 3s
39--44
EN
Presented paper is focused on volumetric hardening process using liquid low melting point metal as a coolant. Effect of convective motion of the coolant on material structure after hardening is investigated. Comparison with results obtained for model neglecting motion of liquid is executed. Mathematical and numerical model based on Finite Element Metod is described. Characteristic Based Split (CBS) method is used to uncouple velocities and pressure and finally to solve Navier-Stokes equation. Petrov-Galerkin formulation is employed to stabilize convective term in heat transport equation. Phase transformations model is created on the basis of Johnson-Mehl and Avrami laws. Continuous cooling diagram (CTPc) for C45 steel is exploited in presented model of phase transformations. Temporary temperatures, phases participation, thermal and structural strains in hardening element and coolant velocities are shown and discussed.
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