IMPROVED TRANSVERSAL METHOD OF LINES (ITMOL) FOR UNIDIRECTIONAL, UNSTEADY HEAT CONDUCTION IN REGULAR SOLID BODIES WITH HEAT CONVECTION EXCHANGE TO NEARBY FLUIDS

Volumen 12, Edición 2, 2020, pp. 179-189

DOI: 10.1615/ComputThermalScien.2020028420

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Antonio Campo
The University of Texas at San Antonio

Diego Celentano
Departamento de Ingenieria Mecánica y Metalurgica, Pontificia Universidad Católica de Chile, Santiago, Chile

SINOPSIS

The present paper addresses unidirectional, unsteady, heat conduction in regular solid bodies (large plane wall, long cylinder and sphere) with uniform initial temperature, thermophysical properties invariant with temperature and heat convection exchange with a neighboring fluid. A novel analytical/numerical procedure named the improved transversal method of lines (ITMOL) has been implemented to transform the one-dimensional, unsteady heat conduction equations along with the uniform initial temperature and the convection boundary conditions in rectangular, cylindrical and spherical coordinates into equivalent one-dimensional, "quasi-steady" heat conduction equations. The transformed "quasi-steady" heat conduction equations are nonlinear ordinary differential equations of second order with linear boundary conditions, which can be solved with any numerical method. The singular feature of this kind of "quasi-steady" heat conduction equations is that time appears embedded into them. In this work, the temperature profiles in the regular solid bodies are determined by a suitable combination of ITMOL and the finite-difference method. The center, surface and mean temperature profiles, as well as the total heat transfer in the large plane wall, long cylinder and sphere exhibit excellent quality for the full spectrum of mean convective coefficients h (0 < h < ∞) over the entire time domain 0 < t < ∞.

PALABRAS CLAVE: one-dimensional, unsteady heat conduction, large plane wall, long cylinder, sphere, improved transversal method of lines (ITMOL), nonlinear one-dimensional, "quasi-steady" heat conduction equation center, surface and mean temperatures

REFERENCIAS

Al-Gwaiz, M., Sturm-Liouville Theory and Its Applications, London, UK: Springer-Verlag, 2008. Arpaci, V., Conduction Heat Transfer, Reading, MA: Addison Wesley, 1966.
Campo, A., Estimate of Total Heat Transfer in Multi-Dimensional Solid Bodies Applying the Product Solution Method with the Intersection of One-Dimensional Solid Bodies, Heat Mass Transf., vol. 53, no. 12, pp. 3565-3570,2017.
Campo, A., Salazar, A.J., Celentano, D.J., and Raydan, M., Accurate Numerical-Analytical Solution of the Heat Conduction Equation, Int. J. Numer. Meth. Heat Fluid Flow, vol. 24, no. 7, pp. 1519-1536, 2014.
Chapra, S.C. andCanale, R.P., Numerical Methods for Engineers, 7th ed., New York: McGraw-Hill, 2014.
Grigull, U. and Sandner, H., Heat Conduction, Berlin, Germany: Springer-Verlag, 1984.
Grigull, U., Bach, J., and Sandner, H., Naherungslosungen der Nichtstationaren Warmeleitung, Forschungim Ingenieurwesen Part A, vol. 32, no. 1,pp. 11-18, 1966.
Haji-Sheikh, A. and Beck, J.V., An Efficient Method of Computing Eigenvalues in Heat Conduction, Numer. Heat Transf, PartB: Fundamentals, vol. 38, no. 2, pp. 133-156, 2000.
Larsson, S. and Thomee, V., Partial Differential Equations with Numerical Methods, 1st ed., New York: Springer-Verlag, 2003.
Liukov, A.V., Analytical Heat Diffusion Theory, New York: Academic Press, 1968.
Milkhailov, M.D. and Vulchanov, N.L., Computational Procedure for Sturm-Liouville Problems, J. Comput. Phys., vol. 50, no. 3, pp. 323-336, 1983.
Mills, A.F., Heat Transfer, 2nd ed., Upper Saddle River, NJ: Prentice Hall, 1998.
Moffat, R.J., Contributions to the Theory of Single-Sample Uncertainty Analysis, J. Fluids Eng., vol. 104, no. 2, pp. 250-260, 1982.
Ozisik, M.N., Heat Conduction, 2nd ed., Hoboken, NJ: John Wiley & Sons, 1993. Poulikakos, D., Conduction Heat Transfer, Upper Saddle River, NJ: Prentice Hall, 1994.
Rektorys, K., The Method of Discretization in Time and Partial Differential Equations, Dordrecht, Netherlands: D. Reidel Publishing Co., 1982.
Rothe, E.H., Zweidimensionale Parabolische Randwertaufgaben als Grenzfall Eindimensionaler Randwertaufgaben, Math. Ann., vol. 102, pp. 650-670, 1930.