Transient Convective Heat Transfer in Porous Media
In this study, several methods to analyze convective heat transfer in a porous medium are presented and discussed. First, the method of Fourier was used to obtain solutions for reduced temperatures <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inli...
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MDPI AG
2023-10-01
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| author | Ruben D’Rose Mark Willemsz David Smeulders |
| author_facet | Ruben D’Rose Mark Willemsz David Smeulders |
| author_sort | Ruben D’Rose |
| collection | DOAJ |
| description | In this study, several methods to analyze convective heat transfer in a porous medium are presented and discussed. First, the method of Fourier was used to obtain solutions for reduced temperatures <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>θ</mi><mi>s</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>θ</mi><mi>f</mi></msub></semantics></math></inline-formula>. The results showed an exponentially decaying propagating temperature front. Then, we discuss the method of integration that was presented earlier by Schumann. This method makes use of a transformation of variables. Thirdly, the system of partial differential equations was directly solved with the Finite Difference method, of which the result showed good agreement with the Fourier solutions. For the chosen <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="sans-serif">Δ</mi><mi>τ</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="sans-serif">Δ</mi><mi>ξ</mi></mrow></semantics></math></inline-formula>, the maximum error for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>θ</mi><mi>f</mi></msub><mo>=</mo><mn>3.7</mn><mo>%</mo></mrow></semantics></math></inline-formula>. The maximum error for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>θ</mi><mi>s</mi></msub></semantics></math></inline-formula> for the first <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> and first <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula> is large (36%) but decays rapidly. The problem was extended by adding a linear heat source term to the solid. Again, making use of the change in variables, analytical solutions were derived for the solid and fluid phases, and corrections to the previous literature were suggested. Finally, results obtained from a numerical model were compared to the analytical solutions, which again showed good agreement (maximum error of 6%). |
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| spelling | doaj.art-50996a8f2c794088b9cf03aafaeaa6962023-11-10T15:07:49ZengMDPI AGMathematics2227-73902023-10-011121441510.3390/math11214415Transient Convective Heat Transfer in Porous MediaRuben D’Rose0Mark Willemsz1David Smeulders2Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The NetherlandsDepartment of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The NetherlandsDepartment of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The NetherlandsIn this study, several methods to analyze convective heat transfer in a porous medium are presented and discussed. First, the method of Fourier was used to obtain solutions for reduced temperatures <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>θ</mi><mi>s</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>θ</mi><mi>f</mi></msub></semantics></math></inline-formula>. The results showed an exponentially decaying propagating temperature front. Then, we discuss the method of integration that was presented earlier by Schumann. This method makes use of a transformation of variables. Thirdly, the system of partial differential equations was directly solved with the Finite Difference method, of which the result showed good agreement with the Fourier solutions. For the chosen <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="sans-serif">Δ</mi><mi>τ</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="sans-serif">Δ</mi><mi>ξ</mi></mrow></semantics></math></inline-formula>, the maximum error for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>θ</mi><mi>f</mi></msub><mo>=</mo><mn>3.7</mn><mo>%</mo></mrow></semantics></math></inline-formula>. The maximum error for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>θ</mi><mi>s</mi></msub></semantics></math></inline-formula> for the first <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> and first <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula> is large (36%) but decays rapidly. The problem was extended by adding a linear heat source term to the solid. Again, making use of the change in variables, analytical solutions were derived for the solid and fluid phases, and corrections to the previous literature were suggested. Finally, results obtained from a numerical model were compared to the analytical solutions, which again showed good agreement (maximum error of 6%).https://www.mdpi.com/2227-7390/11/21/4415mathematical modelnumerical methodheat transfer |
| spellingShingle | Ruben D’Rose Mark Willemsz David Smeulders Transient Convective Heat Transfer in Porous Media Mathematics mathematical model numerical method heat transfer |
| title | Transient Convective Heat Transfer in Porous Media |
| title_full | Transient Convective Heat Transfer in Porous Media |
| title_fullStr | Transient Convective Heat Transfer in Porous Media |
| title_full_unstemmed | Transient Convective Heat Transfer in Porous Media |
| title_short | Transient Convective Heat Transfer in Porous Media |
| title_sort | transient convective heat transfer in porous media |
| topic | mathematical model numerical method heat transfer |
| url | https://www.mdpi.com/2227-7390/11/21/4415 |
| work_keys_str_mv | AT rubendrose transientconvectiveheattransferinporousmedia AT markwillemsz transientconvectiveheattransferinporousmedia AT davidsmeulders transientconvectiveheattransferinporousmedia |