A Shortcut Method to Solve for a 1D Heat Conduction Model under Complicated Boundary Conditions
The function of boundary temperature variation with time, <i>f</i>(<i>t</i>) is generally defined according to measured data. For <i>f</i>(<i>t</i>), which has a complicated expression, a corresponding one-dimensional heat conduction model was construc...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-10-01
|
| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/11/10/556 |
| _version_ | 1797475309076873216 |
|---|---|
| author | Ting Wei Yuezan Tao Honglei Ren Fei Lin |
| author_facet | Ting Wei Yuezan Tao Honglei Ren Fei Lin |
| author_sort | Ting Wei |
| collection | DOAJ |
| description | The function of boundary temperature variation with time, <i>f</i>(<i>t</i>) is generally defined according to measured data. For <i>f</i>(<i>t</i>), which has a complicated expression, a corresponding one-dimensional heat conduction model was constructed under the first type of boundary conditions (Dirichlet conditions) in a semi-infinite domain. By taking advantage of the Fourier transform properties, a theoretical solution was given for the model, under the condition that <i>f</i>(<i>t</i>) does not directly participate in the transformation process. The solution consists of the product of erfc(<i>t</i>) and <i>f</i>(0) and the convolution of erfc(<i>t</i>) and the derivative of <i>f</i>(<i>t</i>). The piecewise linear interpolation equation of <i>f</i>(<i>t</i>), based on the measured data of temperature, was substituted into the theoretical solution, thus quickly solving the model and deriving a corresponding analytical solution. Based on the analytical solution under the linear decay function boundary condition, the inflection point method and curve fitting method for calculating the thermal diffusivity were introduced and exemplified, and the variation laws of the appearance moment of the inflection point were discussed. The obtained results show that the values of thermal diffusivity calculated by the two methods are basically consistent, and that the inflection point values rise with the increasing values of the initial temperature variation of the boundary, the decrease in boundary temperature velocity, and the distance from the boundary, respectively. |
| first_indexed | 2024-03-09T20:43:16Z |
| format | Article |
| id | doaj.art-f114053d69b74026b67873dcd1e10eed |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-09T20:43:16Z |
| publishDate | 2022-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-f114053d69b74026b67873dcd1e10eed2023-11-23T22:54:16ZengMDPI AGAxioms2075-16802022-10-01111055610.3390/axioms11100556A Shortcut Method to Solve for a 1D Heat Conduction Model under Complicated Boundary ConditionsTing Wei0Yuezan Tao1Honglei Ren2Fei Lin3School of Civil Engineering, Hefei University of Technology, Hefei 230009, ChinaSchool of Civil Engineering, Hefei University of Technology, Hefei 230009, ChinaSchool of Civil Engineering, Hefei University of Technology, Hefei 230009, ChinaSchool of Civil Engineering, Hefei University of Technology, Hefei 230009, ChinaThe function of boundary temperature variation with time, <i>f</i>(<i>t</i>) is generally defined according to measured data. For <i>f</i>(<i>t</i>), which has a complicated expression, a corresponding one-dimensional heat conduction model was constructed under the first type of boundary conditions (Dirichlet conditions) in a semi-infinite domain. By taking advantage of the Fourier transform properties, a theoretical solution was given for the model, under the condition that <i>f</i>(<i>t</i>) does not directly participate in the transformation process. The solution consists of the product of erfc(<i>t</i>) and <i>f</i>(0) and the convolution of erfc(<i>t</i>) and the derivative of <i>f</i>(<i>t</i>). The piecewise linear interpolation equation of <i>f</i>(<i>t</i>), based on the measured data of temperature, was substituted into the theoretical solution, thus quickly solving the model and deriving a corresponding analytical solution. Based on the analytical solution under the linear decay function boundary condition, the inflection point method and curve fitting method for calculating the thermal diffusivity were introduced and exemplified, and the variation laws of the appearance moment of the inflection point were discussed. The obtained results show that the values of thermal diffusivity calculated by the two methods are basically consistent, and that the inflection point values rise with the increasing values of the initial temperature variation of the boundary, the decrease in boundary temperature velocity, and the distance from the boundary, respectively.https://www.mdpi.com/2075-1680/11/10/556Fourier transformconvolutioncurve fitting methodinflection point method |
| spellingShingle | Ting Wei Yuezan Tao Honglei Ren Fei Lin A Shortcut Method to Solve for a 1D Heat Conduction Model under Complicated Boundary Conditions Axioms Fourier transform convolution curve fitting method inflection point method |
| title | A Shortcut Method to Solve for a 1D Heat Conduction Model under Complicated Boundary Conditions |
| title_full | A Shortcut Method to Solve for a 1D Heat Conduction Model under Complicated Boundary Conditions |
| title_fullStr | A Shortcut Method to Solve for a 1D Heat Conduction Model under Complicated Boundary Conditions |
| title_full_unstemmed | A Shortcut Method to Solve for a 1D Heat Conduction Model under Complicated Boundary Conditions |
| title_short | A Shortcut Method to Solve for a 1D Heat Conduction Model under Complicated Boundary Conditions |
| title_sort | shortcut method to solve for a 1d heat conduction model under complicated boundary conditions |
| topic | Fourier transform convolution curve fitting method inflection point method |
| url | https://www.mdpi.com/2075-1680/11/10/556 |
| work_keys_str_mv | AT tingwei ashortcutmethodtosolvefora1dheatconductionmodelundercomplicatedboundaryconditions AT yuezantao ashortcutmethodtosolvefora1dheatconductionmodelundercomplicatedboundaryconditions AT hongleiren ashortcutmethodtosolvefora1dheatconductionmodelundercomplicatedboundaryconditions AT feilin ashortcutmethodtosolvefora1dheatconductionmodelundercomplicatedboundaryconditions AT tingwei shortcutmethodtosolvefora1dheatconductionmodelundercomplicatedboundaryconditions AT yuezantao shortcutmethodtosolvefora1dheatconductionmodelundercomplicatedboundaryconditions AT hongleiren shortcutmethodtosolvefora1dheatconductionmodelundercomplicatedboundaryconditions AT feilin shortcutmethodtosolvefora1dheatconductionmodelundercomplicatedboundaryconditions |