Study of conduction heat transfer in semi-elliptic cross sections using analytical and bivariate Chebyshev pseudospectral methods

• Published in: Journal of thermal analysis and calorimetry Vol. 149; no. 1; pp. 243 - 264
• Main Authors: Sarkar, U. K., Kundu, K.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 2024; Springer Nature B.V
• Subjects: Analytical Chemistry; Bivariate analysis; Boundary conditions; Chebyshev approximation; Chemistry; Chemistry and Materials Science; Collocation methods; Conduction heating; Conductive heat transfer; Cross-sections; Cylindrical coordinates; Exact solutions; Heat; Heat flux; Inorganic Chemistry; Measurement Science and Instrumentation; Physical Chemistry; Polymer Sciences; Spectral methods; Temperature distribution; Wall temperature; Heat conduction; Bivariate Chebyshev pseudospectral method; Analytical solution; Semi-elliptic cross section; Elliptic-cylindrical coordinates
• ISSN: 1388-6150; 1588-2926
• DOI: 10.1007/s10973-023-12718-9

Exact analytical solutions have been obtained for conduction heat transfer in a long rod or duct, having cross section of a semi-ellipse, using the “elliptic-cylindrical coordinate system”. Results are presented for two possible cross-sectional configurations of the rod: in one case, the cross section is bounded by a semi-ellipse with the straight edge aligned with the major axis, whereas, in other case, it is bounded by a semi-ellipse with the straight edge coincident with the minor axis. Expressions of temperature distribution, heat flux and heat line are determined for constant wall temperature as well as constant heat flux boundary conditions. The analytical results are illustrated graphically to highlight the salient physics associated with the problem. Apart from the analytical results, the “bivariate Chebyshev collocation spectral method” has been used to determine the numerical solution of the problem of heat conduction in the semi-elliptical geometries; numerical results are found to be consistent with the analytical expressions. The study opens up avenues for obtaining exponentially accurate numerical solution of energy equation in complex elliptic geometries using Chebyshev spectral method.