Exact Analytical and Numerical Solutions for Convective Heat Transfer in a Semi-Spherical Extended Surface with Regular Singular Points

Document Type : Research Article


Chemical Engineering Department, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil, I.R. IRAN


In this study, an exact analytical solution for the convective heat transfer equation from a semi-spherical fin was presented. To obtain a mathematical model, the system was assumed to be a lump in the vertical direction and the governing equation in the Cartesian coordinate was transferred to the Mathieu equation. The exact solution was compared with numerical results such as the finite difference method and midpoint method with Richardson extrapolation (Midrich). Not surprisingly, the exact solution prevailed over the numerical solutions in terms of accuracy and ease of use. Furthermore, the effect of Biot number on the heat transfer of the fin and the fine performance was investigated. The relative error of the results obtained from the analytical and numerical solutions at the base, center, and tip of the fin was 0, 7.72, and 40.25 percent, respectively. The results showed that the relative error between the analytical and numerical solutions depends on the Biot number and varies as a function of the fin length. The obtained analytical solution could be encouraging from different mathematical and industrial applications' points of view.


Main Subjects

[1] Kraus, A.D., Aziz A., Welty J., "Extended Surface Heat Transfer", John Wiley & Sons  Inc. (2002).
[2] Yadav, S., Verma K.A., Ray M., Pandey K.M., Thermal Analysis of Semi-Circular Pin Fins for Application in Electronics Cooling, International Journal of Recent Technology and Engineering, 8(2): 2366-2374 (2019).
[3] Subahan K., Siva Reddy E., Meenakshi Reddy R., CFD Analysis of Pin-Fin Heat Sink Used in Electronic Devices, International Journal of Scientific and Technology Research, 8(9): 562-569 (2019).
[4] Saha S.K., Emani M.S., Ranjan H., Bharti A.K.,
Heat Transfer Enhancement in Plate and Fin Extended Surfaces, in SpringerBriefs in Applied Sciences and Technology, 1-145 (2020).
[5] Belinskiy B.P., Hiestand J.W., Weerasena L., Optimal Design of a Fin in Steady-State, Applied Mathematical Modelling, 77: 1188-1200 (2020).
[6] Atouei S., Hosseinzadeh K., Hatami M., Ghasemi S.E., Sahebi S., Ganji D., Heat Transfer Study on Convective–Radiative Semi-Spherical Fins with Temperature-Dependent Properties and Heat Generation Using Efficient Computational Methods, Applied Thermal Engineering, 89: 299-305 (2015).
[7] Hatami M., Ahangar G.R.M., Ganji D., Boubaker K., Refrigeration Efficiency Analysis for Fully Wet Semi-Spherical Porous Fins, Energy Conversion and Management, 84: 533-540 (2014).
[8] Sabbaghi, S., Rezaii A., Shahri G.R., Baktash M., Mathematical Analysis for the Efficiency of a Semi-Spherical Fin with Simultaneous Heat and Mass Transfer, International Journal of Refrigeration, 34(8): 1877-1882 (2011).
[10] Bilal Ashraf, M., Hayat T., Alsaedi A., Shehzad S.A., Soret and Dufour Effects on the Mixed Convection Flow of an Oldroyd-B Fluid with Convective Boundary Conditions, Results in Physics, 6: 917-924 (2016).
[11] Moitsheki R., Steady Heat Transfer Through a Radial Fin with Rectangular and Hyperbolic Profiles, Nonlinear Analysis: Real World Applications, 12(2): 867-874 (2011).
[14] Joneidi A., Ganji D., Babaelahi M., Differential Transformation Method to Determine Fin Efficiency of Convective Straight Fins with Temperature-Dependent Thermal Conductivity, International Communications in Heat and Mass Transfer, 36(7): 757-762 (2009).
[15] Petroudi R.I., Ganji D.D., Shotorban B.A., Nejad K.M., Rahimi E., Rohollahtabar R., Taherinia F., Semi-Analytical Method for Solving Non-Linear Equation Arising Of Natural Convection Porous Fin, Thermal Science, 16(5): 1303-1308, (2012).
[16] Ma J., Sun Y., Li B., Chen H., Spectral Collocation Method for Radiative–Conductive Porous Fin with Temperature-Dependent Properties, Energy Conversion and Management, 111: 279-288 (2016).
[17] Ganji D.D., Ganji Z.Z., Ganji D.H., Determination of Temperature Distribution for Annular Fins with Temperature Dependent Thermal Conductivity by HPM, Thermal Science, 15:  111-115, (2011).
[18] Turkyilmazoglu M., Exact Solutions to Heat Transfer In Straight Fins of Varying Exponential Shape Having Temperature Dependent Properties, International Journal of Thermal Sciences, 55: 69-75 (2012).
[25] Gorla R.S.R., Bakier A.Y., Thermal Analysis of Natural Convection and Radiation in Porous Fins, International Communications in Heat and Mass Transfer, 38(5): 638-645, (2011).
[26] Xu R., Weng A., The Calculation for Characteristic Multiplier of Hill's Equation x″+q(t)x=0 in Case q(t) with Positive Mean, Nonlinear Analysis: Real World Applications, 9(3): 949-962, (2008).
[27] Andrei D. Polyanin , Valentin F. Zaitsev, "Handbook of Exact Solutions for Ordinary Differential Equations", London: CRC Press, Vol. 1. (2002).
[28] Zlatev Z., Dimov Ivan, Faragó, István Havasi, Ágnes, "Richardson Extrapolation: Practical Aspects and Applications", De Gruyter Series in Applied and Numerical Mathematics 2, (2018).
[29] Richardson L.F., The Approximate Arithmetical Solution by Finite Differences of Physical Problems Involving Differential Equations, with an Application to the Stresses in a Masonry Dam, Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, 210: 307-357 (1911).
[30] Richardson L.F., Gaunt J.A., The Deferred Approach to the Limit. Part I. Single Lattice. Part II. Interpenetrating Lattices, Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, 226: 299-361 (1927).
[31] Uri M. Ascher, L.R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, Philadelphia: Society for Industrial and Applied Mathematics, 332: (1998).
[32] Polyanin A.D., Zaitsev V.F., "Handbook of Ordinary Differential Equations: Exact Solutions, Methods, and Problems", CRC Press, (2017).