A simplex search method for a conductive–convective fin with variable conductivity

Abstract An inverse problem is solved for simultaneously estimating the convection–conduction parameter and the variable thermal conductivity parameter in a conductive–convective fin with temperature dependent thermal conductivity. Initially, the temperature field is obtained from a direct method using an analytical approach based on decomposition scheme and then using a simplex search minimization algorithm an inverse problem is solved for estimating the unknowns. The objective function to be minimized is represented by the sum of square of the error between the measured temperature field and an initially guessed value which is updated in an iterative manner. The estimation accuracy is studied for the effect of measurement errors, initial guess and number of measurement points. It is observed that although very good estimation accuracy is possible with more number of measurement points, reasonably well estimation is obtained even with fewer number of measurement points without measurement error. Subject to selection of a proper initial guess, it is seen that the number of iterations could be significantly reduced. The relative sensitiveness of the estimated parameters is studied and is observed from the present work that the estimated convection–conduction parameter contributes more to the temperature distribution than the variable conductivity parameter.

[1]  D. P. Sekulic,et al.  Extended surface heat transfer , 1972 .

[2]  G. Adomian.  Solving frontier problems modelled by nonlinear partial differential equations , 1991 .

[3]  A. Muzzio,et al.  Approximate Solution for Convective Fins With Variable Thermal Conductivity , 1976 .

[4]  J. Weiner,et al.  Fundamentals and applications , 2003 .

[5]  Cha'o-Kuang Chen,et al.  A decomposition method for solving the convective longitudinal fins with variable thermal conductivity , 2002 .

[6]  E. Jaynes.  The well-posed problem , 1973 .

[7]  H. S. Kang,et al.  Optimization of a Pin Fin With Variable Base Thickness , 2010 .

[8]  A. Aziz,et al.  Perturbation Solution for Convecting Fin With Variable Thermal Conductivity , 1975 .

[9]  Cha'o-Kuang Chen,et al.  Estimation of Time-Varying Inlet Temperature and Heat Flux in Turbulent Circular Pipe Flow , 2006 .

[10]  P. J. Heggs,et al.  Design charts for radial rectangular fins in terms of performance ratio and maximum effectiveness , 2004 .

[11]  K. I. M. McKinnon,et al.  Convergence of the Nelder-Mead Simplex Method to a Nonstationary Point , 1998, SIAM J. Optim..

[12]  A. Al-Garni,et al.  The optimal dimensions of circular fins with variable profile and temperature-dependent thermal conductivity , 1996 .

[13]  G. Adomian.  A review of the decomposition method in applied mathematics , 1988 .

[14]  A. Aziz,et al.  Application of perturbation techniques to heat-transfer problems with variable thermal properties , 1976 .

[15]  J. Grace,et al.  A new method for solving the inverse conduction problem in steady heat flux measurement , 1997 .

[16]  Jeffrey C. Lagarias,et al.  Convergence Properties of the Nelder-Mead Simplex Method in Low Dimensions , 1998, SIAM J. Optim..

[17]  Richard Pasquetti,et al.  Inverse heat conduction applied to the measurement of heat transfer coefficient on a cylinder: Comparison between an analytical and a boundary element technique , 1991 .

[18]  Two-dimensional inverse problem in estimating heat flux of pin fins , 2001 .

[19]  K. Cole,et al.  Analysis of flux-base fins for estimation of heat transfer coefficient , 2009 .

[20]  John A. Nelder,et al.  A Simplex Method for Function Minimization , 1965, Comput. J..

[21]  Min‐Hsing Chang,et al.  The Inverse Estimation of Local Heat Transfer Coefficient in a Vertical Plate Fin with Its Base Subjected to Periodically Oscillated Temperature , 2001 .

[22]  Tzer-Ming Chen.  Numerical solution of hyperbolic heat conduction in thin surface layers , 2007 .

[23]  Abdul Aziz,et al.  Alternative Solutions for Longitudinal Fins of Rectangular, Trapezoidal, and Concave Parabolic Profiles , 2010 .

[24]  Frank P. Incropera,et al.  Fundamentals of Heat and Mass Transfer , 1981 .

[25]  O. Nelles.  Nonlinear System Identification , 2001 .

[26]  Yeh Rong-Hua.  An analytical study of the optimum dimensions of rectangular fins and cylindrical pin fins , 1997 .

[27]  J. V. Beck,et al.  Parameter Estimation Method for Flash Thermal Diffusivity with Two Different Heat Transfer Coefficients , 1995 .