Transfer-Matrix Method for Calculus of Long Cylinder Tube with Industrial Applications
The Transfer-Matrix Method (TMM) is an original and relatively simple mathematical approach for the calculus of thin-walled cylindrical tubes presented in this work. Calculation with TMM is much less used than calculation with the Finite Elements Method (FEM), even though it is much easier to apply...
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MDPI AG
2023-08-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/17/3756 |
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| author | Luminita Codrea Mihai-Sorin Tripa Daniel Opruţa Robert Gyorbiro Mihaela Suciu |
| author_facet | Luminita Codrea Mihai-Sorin Tripa Daniel Opruţa Robert Gyorbiro Mihaela Suciu |
| author_sort | Luminita Codrea |
| collection | DOAJ |
| description | The Transfer-Matrix Method (TMM) is an original and relatively simple mathematical approach for the calculus of thin-walled cylindrical tubes presented in this work. Calculation with TMM is much less used than calculation with the Finite Elements Method (FEM), even though it is much easier to apply in different fields. That is why it was considered imperative to present this original study. The calculus is based on Dirac’s and Heaviside’s functions and operators and on matrix calculation. The state vectors, the transfer-matrix, and the vector corresponding to the external efforts were defined, which were then used in the calculations. A matrix relation can be written, which gives the state vector of the last section depending on the state vector of the first section, a relation in which the conditions of the two end supports can be set. As an application, a heat exchanger was studied, with a large cylinder subjected to a uniformly distributed internal load, and from the inner cylinder bundle, a cylinder subjected to both uniform internal and external loads was considered. For the second cylinder, two possibilities of action for the external forces were considered, a successive action and a simultaneous action, achieving the same results in both situations. The TMM is intended to be used for iterative calculus in optimization problems where rapid successive results are required. In the future, we want to expand this method to other applications, and we want to develop related programs. This is an original theoretical study and is a complement to the research in the field on thin-walled cylinder tubes and their applications in heat exchangers. |
| first_indexed | 2024-03-10T23:17:22Z |
| format | Article |
| id | doaj.art-5e091b3456514fb08b33f337274d9ca4 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T23:17:22Z |
| publishDate | 2023-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-5e091b3456514fb08b33f337274d9ca42023-11-19T08:31:43ZengMDPI AGMathematics2227-73902023-08-011117375610.3390/math11173756Transfer-Matrix Method for Calculus of Long Cylinder Tube with Industrial ApplicationsLuminita Codrea0Mihai-Sorin Tripa1Daniel Opruţa2Robert Gyorbiro3Mihaela Suciu4Department of Mechanical Engineering, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, RomaniaDepartment of Design Engineering and Robotics, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, RomaniaDepartment of Mechanical Engineering, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, RomaniaDepartment of Mechanical Engineering, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, RomaniaDepartment of Mechanical Engineering, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, RomaniaThe Transfer-Matrix Method (TMM) is an original and relatively simple mathematical approach for the calculus of thin-walled cylindrical tubes presented in this work. Calculation with TMM is much less used than calculation with the Finite Elements Method (FEM), even though it is much easier to apply in different fields. That is why it was considered imperative to present this original study. The calculus is based on Dirac’s and Heaviside’s functions and operators and on matrix calculation. The state vectors, the transfer-matrix, and the vector corresponding to the external efforts were defined, which were then used in the calculations. A matrix relation can be written, which gives the state vector of the last section depending on the state vector of the first section, a relation in which the conditions of the two end supports can be set. As an application, a heat exchanger was studied, with a large cylinder subjected to a uniformly distributed internal load, and from the inner cylinder bundle, a cylinder subjected to both uniform internal and external loads was considered. For the second cylinder, two possibilities of action for the external forces were considered, a successive action and a simultaneous action, achieving the same results in both situations. The TMM is intended to be used for iterative calculus in optimization problems where rapid successive results are required. In the future, we want to expand this method to other applications, and we want to develop related programs. This is an original theoretical study and is a complement to the research in the field on thin-walled cylinder tubes and their applications in heat exchangers.https://www.mdpi.com/2227-7390/11/17/3756mathematical model of thin-walled cylindrical tubeload densitystate vectorTransfer-Matrix MethodDirac’s functionHeaviside’s function |
| spellingShingle | Luminita Codrea Mihai-Sorin Tripa Daniel Opruţa Robert Gyorbiro Mihaela Suciu Transfer-Matrix Method for Calculus of Long Cylinder Tube with Industrial Applications Mathematics mathematical model of thin-walled cylindrical tube load density state vector Transfer-Matrix Method Dirac’s function Heaviside’s function |
| title | Transfer-Matrix Method for Calculus of Long Cylinder Tube with Industrial Applications |
| title_full | Transfer-Matrix Method for Calculus of Long Cylinder Tube with Industrial Applications |
| title_fullStr | Transfer-Matrix Method for Calculus of Long Cylinder Tube with Industrial Applications |
| title_full_unstemmed | Transfer-Matrix Method for Calculus of Long Cylinder Tube with Industrial Applications |
| title_short | Transfer-Matrix Method for Calculus of Long Cylinder Tube with Industrial Applications |
| title_sort | transfer matrix method for calculus of long cylinder tube with industrial applications |
| topic | mathematical model of thin-walled cylindrical tube load density state vector Transfer-Matrix Method Dirac’s function Heaviside’s function |
| url | https://www.mdpi.com/2227-7390/11/17/3756 |
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