The Stationary Thermal Field in a Multilayer Elliptic Cylinder
An analytical–numerical method for determining the two-dimensional (2D) thermal field in a layer-inhomogeneous elliptic cylinder (elliptical roller) was developed in the article. A mathematical model was formulated in the form of a boundary problem for Poisson equations with an external boundary con...
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MDPI AG
2023-02-01
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| Online Access: | https://www.mdpi.com/2076-3417/13/4/2354 |
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| author | Jerzy Gołębiowski Marek Zaręba |
| author_facet | Jerzy Gołębiowski Marek Zaręba |
| author_sort | Jerzy Gołębiowski |
| collection | DOAJ |
| description | An analytical–numerical method for determining the two-dimensional (2D) thermal field in a layer-inhomogeneous elliptic cylinder (elliptical roller) was developed in the article. A mathematical model was formulated in the form of a boundary problem for Poisson equations with an external boundary condition of the third kind (Hankel’s). The conditions of continuity of temperature and heat flux increment were assumed at the inner boundaries of material layers. The eigenfunctions of the boundary problem were determined analytically. Hankel’s condition was subjected to appropriate mathematical transformations. As a result, a system of algebraic equations with respect to the unknown coefficients of the eigenfunctions was obtained. The above-mentioned system of equations was solved numerically (iteratively). As an example of an application of the aforementioned method, an analysis of the thermal field in an elliptical electric wire was presented. The system consists of an aluminum core and two layers of insulation (PVC and rubber). In addition to the field distribution, the steady-state current rating was also determined. The thermal conductivities of PVC and rubber are very similar to each other. For this reason, apart from the real model, a test system was also considered. Significantly different values of thermal conductivity were assumed in individual layers of the test model. The temperature distributions were presented graphically. The graphs showed that the temperature drop is almost linear in the insulation of an electrical conductor. On the other hand, in the analogous area of the test model, a broken line was observed. It was also found that the elliptical layer boundaries are not isothermal. The results obtained by the method presented in this paper were verified numerically. |
| first_indexed | 2024-03-11T09:12:10Z |
| format | Article |
| id | doaj.art-925ae24b603e4c388f6d7086c5e2849b |
| institution | Directory Open Access Journal |
| issn | 2076-3417 |
| language | English |
| last_indexed | 2024-03-11T09:12:10Z |
| publishDate | 2023-02-01 |
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| spelling | doaj.art-925ae24b603e4c388f6d7086c5e2849b2023-11-16T18:54:32ZengMDPI AGApplied Sciences2076-34172023-02-01134235410.3390/app13042354The Stationary Thermal Field in a Multilayer Elliptic CylinderJerzy Gołębiowski0Marek Zaręba1Faculty of Electrical Engineering, Technical University of Bialystok, 15-351 Białystok, PolandFaculty of Electrical Engineering, Technical University of Bialystok, 15-351 Białystok, PolandAn analytical–numerical method for determining the two-dimensional (2D) thermal field in a layer-inhomogeneous elliptic cylinder (elliptical roller) was developed in the article. A mathematical model was formulated in the form of a boundary problem for Poisson equations with an external boundary condition of the third kind (Hankel’s). The conditions of continuity of temperature and heat flux increment were assumed at the inner boundaries of material layers. The eigenfunctions of the boundary problem were determined analytically. Hankel’s condition was subjected to appropriate mathematical transformations. As a result, a system of algebraic equations with respect to the unknown coefficients of the eigenfunctions was obtained. The above-mentioned system of equations was solved numerically (iteratively). As an example of an application of the aforementioned method, an analysis of the thermal field in an elliptical electric wire was presented. The system consists of an aluminum core and two layers of insulation (PVC and rubber). In addition to the field distribution, the steady-state current rating was also determined. The thermal conductivities of PVC and rubber are very similar to each other. For this reason, apart from the real model, a test system was also considered. Significantly different values of thermal conductivity were assumed in individual layers of the test model. The temperature distributions were presented graphically. The graphs showed that the temperature drop is almost linear in the insulation of an electrical conductor. On the other hand, in the analogous area of the test model, a broken line was observed. It was also found that the elliptical layer boundaries are not isothermal. The results obtained by the method presented in this paper were verified numerically.https://www.mdpi.com/2076-3417/13/4/2354field theory methodssteady-state temperature fieldmultilayer elliptic cylindercurrent-carrying capacity |
| spellingShingle | Jerzy Gołębiowski Marek Zaręba The Stationary Thermal Field in a Multilayer Elliptic Cylinder Applied Sciences field theory methods steady-state temperature field multilayer elliptic cylinder current-carrying capacity |
| title | The Stationary Thermal Field in a Multilayer Elliptic Cylinder |
| title_full | The Stationary Thermal Field in a Multilayer Elliptic Cylinder |
| title_fullStr | The Stationary Thermal Field in a Multilayer Elliptic Cylinder |
| title_full_unstemmed | The Stationary Thermal Field in a Multilayer Elliptic Cylinder |
| title_short | The Stationary Thermal Field in a Multilayer Elliptic Cylinder |
| title_sort | stationary thermal field in a multilayer elliptic cylinder |
| topic | field theory methods steady-state temperature field multilayer elliptic cylinder current-carrying capacity |
| url | https://www.mdpi.com/2076-3417/13/4/2354 |
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