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1
Reconstruction of the heat transfer coefficient in the inverse Stefan problem
Matlak J., Słota D., Zielonka A.
Hutnik, Wiadomości Hutnicze
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2018
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Vol. 85, nr 1
6--9
EN
In the paper we will present the method of finding the heat transfer coefficient in the inverse problem of pure metal solidification. In the considered model the shrinkage of metal and the air-gap between material and mold will be taken into account. The method is based on the algorithm for solution of the direct problem and on the Artificial Bee Colony algorithm. In the algorithm for solving the direct problem we use the finite element method supplemented by the procedure allowing to define the position of the moving interface and the change of material size associated with the shrinkage. To solve the inverse problem, a functional defining the error of approximate solution must be minimized. To minimize this functional we use the Artificial Bee Colony algorithm. Then we present the computational example illustrating precision and stability of the presented method.
PL
W pracy zaprezentowana zostanie metoda wyznaczania współczynnika wnikania ciepła w zagadnieniu odwrotnym krzepnięciem czystego metalu. W rozważanym modelu uwzględniony będzie skurcz metalu oraz szczelina powietrzna pomiędzy odlewem i wlewkiem. Prezentowana metoda wykorzystuje algorytm rozwiązania zagadnienia bezpośredniego oraz algorytm pszczeli. W algorytmie rozwiązania zagadnienia bezpośredniego wykorzystano metodę elementów skończonych uzupełnioną o procedurę pozwalającą określić położenie granicy rozdziału faz oraz zmianę wymiarów wlewka spowodowaną skurczem metalu. W rozwiązaniu zagadnienia odwrotnego należy zminimalizować funkcjonał określający błąd rozwiązania przybliżonego. W tym celu wykorzystano algorytm pszczeli. Przedstawiono także przykład obliczeniowy ilustrujący dokładność i stabilność prezentowanej metody.
2
Content available The Taylor transformation hybrid method applied for solving the Stefan problem
Grzymkowski R., Hetmaniok E., Pleszczyński M.
Silesian Journal of Pure and Applied Mathematics
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2016
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Vol. 6, iss. 1
111--123
EN
The paper presents the analytic-numerical hybrid method using, among others, the Taylor transformation, thanks to which the solution of the Stefan problem is replaced by the solution of a nonlinear system of equations.
3
Content available Numerical scheme for one phase 1D fractional Stefan problem using the similarity variable technique
Błasik M.
Journal of Applied Mathematics and Computational Mechanics
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2014
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Vol. 13, nr 1
13--21
EN
In this paper we present a numerical method to solve a one-dimensional, one-phase extended Stefan problem with fractional time derivative described in the Caputo sense. The proposed method is based on applying a similarity variable for the anomalous-diffusion equation and the finite difference method. In the final part, examples of numerical results are discussed.
4
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.
5
Zagadnienie Stefana dla żebra pracującego przy wrzeniu
Orzechowski T.
Inżynieria i Aparatura Chemiczna
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2006
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Nr 6s
179-180
EN
The Stefan problem dealing with the determination of a boundary shape has been presented. A numerical analysis with the use of heat polynomials has been conducted. On the example of a triangular fin a methodology and its effectiveness for boiling heat transfer were shown.
6
Content available remote Application of generalized finite difference method in numerical modelling of moving boundary problems
Mochnacki B., Lara S.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2003
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Vol. 2, nr 1
129--143
EN
The application of generalized finite difference method for numerical modelling of thermal processes proceeding in the solidifying casting domain is presented. The solidification of pure metals and eutectic alloys is considered. In such case the solidification process takes place at the constant temperature (the Stefan problem). From the numerical point of view the solution of this task is very complex; in particular, in the case of 2D or 3D domains; and in literature one can find the procedures enabling to avoid the difficulties with the direct modelling of the problem discussed. The part of them consist in the substitution of the solidification point T* by the certain interval [T* - ΔT, T* + ΔT]. In this way the subdomain of artificial mushy zone is introduced and the fixed domain approach [1] can be used. On the stage of numerical algorithm construction and numerical simulation the generalized finite difference method is used. In the final part of the paper the examples of computations are shown.
7
Content available remote The influence of artificial mushy zone parameters on the numerical solution of the Stefan problem
Mochnacki B., Lara S.
Archiwum Odlewnictwa
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2003
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R. 3, nr 10
31--36
EN
The solidification of pure metals and eutectic alloys proceed at the constant temperature (the Stefan problem). From the numerical point of view the solution of this task is very complex and in literature one can find the procedures enabling to avoid the difficulties with the direct modelling of the problem discussed. The part of them consist in the substitution of the solidification point T* on the certain interval [T*-ΔT, T*+ΔT]. In this way the subdomain of artificial mushy zone (AMZ) is introduced and the fixed domain approach [1, 2, 3] can be used. In the paper the analysis of AMZ assumed parameters on the results of solidification process simulation is presented. On the stage of numerical computations the generalized FDM has been applied.
PL
Krzepnięcie czystych metali i stopów eutektycznych zachodzi w stałej temperaturze (problem Stefana). Z matematycznego punktu widzenia rozwiązanie takiego zadania jest bardzo skomplikowane i w literaturze można znaleźć procedury umożliwiające ominięcie trudności związanych z bezpośrednim modelowaniem rozważanego problemu. Część z nich bazuje na zastąpieniu temperatury krzepnięcia pewnym przedziałem o szerokości 2ΔT. W ten sposób w modelu pojawia się podobszar nazywany sztuczną strefą dwufazową. W pracy analizowano wpływ parametrów tej strefy na numeryczne rozwiązanie procesu krzepnięcia. Na etapie obliczeń numerycznych wykorzystano uogólnioną metodę różnic skończonych.
8
Content available remote Application of the enthalpy approach in numerical modelling of solidification process
Mochnacki B., Lara S., Pawlak E.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2002
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Vol. 1, nr 1
163--176
EN
The numerical modelling of the Stefan problem using directly the typical description of the process is rather difficult. In this connection one can find in literature the models basing on the enthalpy convention and the artificial mushy zone introduction. This paper summarizes the state of the art of the enthalpy models of phase change problems. The part of the approaches discussed has been modified by the authors of the paper, we also present the new interpretations of some models.
9
Wybrane metody modelowania procesów wymiany ciepła
Białecki R., Fic A., Rudnicki Z., Składzeń J., Skorek J.
Gospodarka Paliwami i Energią
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1999
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nr 1
22-28
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