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Hybrid Analytical/Numerical Solution of the Unsteady Heat Conduction Equation Subject to Unequal Robin Boundary Conditions

Antonio Campo

School of Mechanical Engineering (Escuela de Ingeniería Mecánica), Pontifical Catholic University of Valparaíso (Pontificia  Universidad Católica de Valparaíso), Viña del Mar, Valparaíso, Chile.

*Corresponding author: Antonio Campo

Date: November 23,2021 Hits: 1072


The primary objective of the present study is to utilize the Method of Lines (MOL) for the analysis of the unsteady, heat conduction in a slab with different Robin boundary conditions. A hot fluid is in contact with the left side of the slab and a cold fluid is in contact with the right side. In the heat conduction equation, MOL discretizes the second spatial derivative while leaving the time derivative continuous. This operation leads to a system of adjoint ordinary differential equations of first order for each line in the special computational domain that is solved analytically (not numerically) with the potent eigenvalue method. The computational procedure uses a symbolic algebra code that produces the collection of eigenvalues and eigenvectors. Based on this, a sequence of piecewise temperature-time variations at each line are expressed in terms of linear combinations of exponential functions of time. A limiting test case involves a slab with asymmetric Dirichlet boundary conditions, a particular case of asymmetric Robin boundary conditions. The combination of MOL, the eigenvalue method and a symbolic algebra code delivers a set of analytical/numerical temperature-time variations exhibiting excellent quality for the entire time domain.


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[16] A. Campo and M. Arıcı. (2019). Semi-analytical, piecewise temperature–time distributions in solid bodies of regular shape affected by uniform surface heat flux: Combination of the Method Of Lines (MOL) and the eigenvalue method, International Communications in Heat and Mass Transfer, Vol. 108, Article #104276, 2019. 

[17] A. Campo. The Numerical Method Of Lines (NMOL) facilitates the instruction of unsteady heat conduction in simple solid bodies with convective surfaces, International Journal of Mechanical Engineering Education. First Published 25 Mar 2020. https://doi.org/10.1177/0306419020910423.

[18] A. Campo and J. Garza. (2014). Transversal Method Of Lines (TMOL) for unsteady heat conduction with uniform surface heat flux, ASME Journal of Heat Transfer, Vol. 136, Paper No. 111302, 2014.

[19] A. Campo and Y. Masip-Macía. (2019). Semi-analytical solution of unsteady heat conduction in a large plane wall with convective boundary conditions for the “small-time” sub-domain using the Transversal Method Of Lines (TMOL), International Journal of Numerical Methods in Heat and Fluid Flow, Vol. 29, Issue 2, pp. 536-552, 2019. 

[20] A. Campo and J. Sieres. (2020). Semi-analytical treatment of the unsteady heat conduction equation with prescribed surface temperature: The Transversal Method Of Lines (TMOL) delimited to the “small time” sub-domain, International Communications in Heat and Mass Transfer, Vol. 116, July 2020. Article number 1046872020. 

[21] A. Campo and D. J. Celentano. (2020). Improved Transversal Method Of Lines (ITMOL) for unidirectional, un-steady heat conduction in regular solid bodies with heat convection exchange to nearby fluids, Computational Thermal Sciences: An International Journal, Vol. 12, Issue 2, pp. 179-189, 2020.

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Hybrid Analytical/Numerical Solution of the Unsteady Heat Conduction Equation Subject to Unequal Robin Boundary Conditions

How to cite this paper: Antonio Campo. (2021) Hybrid Analytical/Numerical Solution of the Unsteady Heat Conduction Equation Subject to Unequal Robin Boundary ConditionsJournal of Applied Mathematics and Computation5(4), 303-314.

DOI: http://dx.doi.org/10.26855/jamc.2021.12.008