Solving the Nonlinear Heat Conduction in a Spherical Coordinate with Electrical Simulation

Authors: A. M. Gheitaghy, H. Saffari, G. Q. Zhang

Abstract:

Numerical approach based on the electrical simulation method is proposed to solve a nonlinear transient heat conduction problem with nonlinear boundary for a spherical body. This problem represents a strong nonlinearity in both the governing equation for temperature dependent thermal property and the boundary condition for combined convective and radiative cooling. By analysing the equivalent electrical model using the electrical circuit simulation program HSPICE, transient temperature and heat flux distributions at sphere can be obtained easily and fast. The solutions clearly illustrate the effect of the radiation-conduction parameter N_rc, the Biot number and the linear coefficient of temperature dependent conductivity and heat capacity. On comparing the results with corresponding numerical solutions, the accuracy and efficiency of this computational method is found to be good.

Keywords: Convective boundary, radiative boundary, electrical simulation method, nonlinear heat conduction, spherical coordinate.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1112310

References:

[1] R. Siegel-J. R. Howell, 2001, Thermal Radiation Heat Transfer, Taylor & Francis, New York, NY 10001, USA
[2] Lall, P. S., and Viskanta, R., 1967, "Transient Energy Transfer in a Gray Radiating Gas During Expansion," Physics of Fluids, 10(1), pp. 98-107.
[3] Viskanta, R., and Merriam, R. L., 1967, "Shielding of Surfaces in Couette Flow against Radiation by Transpiration of an Absorbing-Emitting Gas," International Journal of Heat and Mass Transfer, 10(5), pp. 641-653.
[4] Bayazitoglu, Y., and Suryanarayana, P. V. R., 1989, "Transient Radiative Heat Transfer from a Sphere Surrounded by a Participating Medium," Journal of Heat Transfer, 111(3), pp. 713-718.
[5] Tsai, J. R., and Özişik, M. N., 1987, "Transient, Combined Conduction and Radiation in an Absorbing, Emitting, and Isotropically Scattering Solid Sphere," Journal of Quantitative Spectroscopy and Radiative Transfer, 38(4), pp. 243-251.
[6] Thynell, S. T., 1990, "Interaction of Conduction and Radiation in Anisotropically Scattering, Spherical Media," Journal of Thermophysics and Heat Transfer, 4(3), pp. 299-304.
[7] Trabelsi, H., Sghaier, T., and Sifaoui, M. S., 2005, "A Theoretical Study of Radiation between Two Concentric Spheres Using a Modified Discrete Ordinates Method Associated with Legendre Transform," Journal of Quantitative Spectroscopy and Radiative Transfer, 93(4), pp. 415-428.
[8] A. Haji-Sheikh, E. M. S., 1967, "Solution of Heat Conduction Problems by Probability Methods," Trans. ASME, J. Heat Transfer, 89, pp. 121-131.
[9] Davies, T. W., 1985, "The Cooling of a Plate by Combined Thermal Radiation and Convection," International Communications in Heat and Mass Transfer, 12(4), pp. 405-415.
[10] Parang, M., Crocker, D. S., and Haynes, B. D., 1990, "Perturbation Solution for Spherical and Cylindrical Solidification by Combined Convective and Radiative Cooling," International Journal of Heat and Fluid Flow, 11(2), pp. 142-148.
[11] Sundén, B., 1986, "Transient Heat Conduction in a Composite Slab by a Time-Varying Incident Heat Flux Combined with Convective and Radiative Cooling," International Communications in Heat and Mass Transfer, 13(5), pp. 515-522.
[12] Kessler, G., 2009, "Steady State and Transient Temperature Profiles in a Multishell Spherical System Heated Internally by Reactor-Grade Plutonium," Nuclear Engineering and Design, 239(11), pp. 2430-2443.
[13] Su, J., 2004, "Improved Lumped Models for Transient Radiative Cooling of a Spherical Body," International Communications in Heat and Mass Transfer, 31(1), pp. 85-94.
[14] Liao, S., Su, J., and Chwang, A. T., 2006, "Series Solutions for a Nonlinear Model of Combined Convective and Radiative Cooling of a Spherical Body," International Journal of Heat and Mass Transfer, 49(15-16), pp. 2437-2445.
[15] El-Nahhas, A., 2009, "Analytic Approximations for a Strongly Nonlinear Problem of Combined Convective and Radiative Cooling of a Spherical Body," Journal of Heat Transfer, 131(11), pp. 111703-6.
[16] V. Paschkis, H. D. B., 1942, "A Method for Determining Unsteady-State Heat Transfer by Means of an Electrical Analogy," Transactions ASME, 46, pp. 7-15.
[17] Jordán, J. Z., 2006, "Network Method to Study the Transient Heat Transfer Problem in a Vertical Channel with Viscous Dissipation," International Communications in Heat and Mass Transfer, 33(9), pp. 1079-1087.
[18] González Fernández, C. F., Alhama, F., López Sánchez, J. F., and Horno, J., 1998, "Application of the Network Method to Heat Conduction Processes with Polynomial and Potential-Exponentially Varying Thermal Properties," Numerical Heat Transfer; Part A: Applications, 33(5), pp. 549-559.
[19] Zueco, J., Eguía, P., Granada, E., Míguez, J. L., and Bég, O. A., 2010, "An Electrical Network for the Numerical Solution of Transient Mhd Couette Flow of a Dusty Fluid: Effects of Variable Properties and Hall Current," International Communications in Heat and Mass Transfer, 37(10), pp. 1432-1439.
[20] Nagel, L. W., 1973, Spice, a Computer Program to Simulate Semiconductor Circueqits, University of California.