THERMAL SCIENCE

International Scientific Journal

Thermal Science - Online First

online first only

Stochastic technique for solutions of nonlinear fin equation arising in thermal equilibrium model

ABSTRACT
In this study, a stochastic numerical technique is used to investigate the numerical solution of the heat transfer temperature dependent system using Feed-forward artificial neural networks (ANN). The Mathematical model of Fin equation is formulated with the help of ANN. The effect of the heat on a rectangular fin with thermal conductivity and temperature dependent internal heat generation is calculated through neural networks optimization with optimizers like active set technique (AST), interior point technique (IPT), pattern search (PS), genetic algorithm (GA) and a hybrid approach of PS-IPT, GA-AST, GA-IPT and GA-SQP with different selections of weights. The governing fin equation is transformed into an equivalent nonlinear second order ordinary differential equation. For this transformed ordinary differential equation model we have performed several simulations to provide the justification of better convergence of results. Moreover, the effectiveness of the designed model is validated through a complete statistical analysis. This study reveals the importance of rectangular fins during the heat transformation through the system.
KEYWORDS
PAPER SUBMITTED: 2018-02-21
PAPER REVISED: 2019-01-27
PAPER ACCEPTED: 2019-02-16
PUBLISHED ONLINE: 2019-03-09
DOI REFERENCE: https://doi.org/10.2298/TSCI180221057A
REFERENCES
  1. Long, C. and. Sayma, N., Heat Transfer, Ventus Publishing, (p. 156), 2009.
  2. Gurrum, S. P., et al., Thermal issues in next-generation integrated circuits." IEEE Transactions on device and materials reliability 4.4 (2004), pp. 709-714.
  3. Remsburg R. Advanced thermal design of electronic equipment, Springer Science & Business Media, 2011.
  4. McGlen R. J., et al., Thermal management techniques for high power electronic devices, Applied Thermal Engineering, 24(8), (2004): 1143-1156.
  5. Kraus, A. D., et al., Extended surface heat transfer. John Wiley & Sons, 2002.
  6. Chang, M. H., A decomposition solution for fins with temperature dependent surface heat flux, International journal of heat and mass transfer, 48(9), (2005), pp. 1819-1824,
  7. Saeid, N.H., Natural convection in a square cavity with discrete heating at the bottom with different fin shapes. Heat Transfer Engineering, (2017). dx.doi.org/10.1080/01457632.2017.1288053.
  8. Behbahani, S. W., et al., Two-dimensional rectangular fin with variable heat transfer coefficient, International journal of heat and mass transfer, 34(1), (1991), pp. 79-85.
  9. Ghasemi, S. E., et al., Thermal analysis of convective fin with temperature-dependent thermal conductivity and heat generation," Case Studies in Thermal Engineering, (2014).
  10. Aziz, A. and Na, T. Y., Periodic heat transfer in fins with variable thermal parameters, International Journal of Heat and Mass Transfer, 24(8), (1981), pp. 1397-1404.
  11. Chung, B. T. F. and Iyer, J. R., Optimum design of longitudinal rectangular fins and cylindrical spines with variable heat transfer coefficient, Heat transfer engineering, 4(1), (1993), pp. 31-42.
  12. Khani, F. and Aziz, A., Thermal analysis of a longitudinal trapezoidal fin with temperaturedependent thermal conductivity and heat transfer coefficient, Communications in Nonlinear Science and Numerical Simulation, 15(3), (2010), pp. 590-601.
  13. Chiu, C. H. and. Chen, C. O. K., A decomposition method for solving the convective longitudinal fins with variable thermal conductivity, International Journal of Heat and Mass Transfer, 45(10), (2002), pp. 2067-2075.
  14. Kim, S., et al., An approximate solution of the nonlinear fin problem with temperaturedependent thermal conductivity and heat transfer coefficient, Journal of Physics D: Applied Physics, 40(14), (2007), pp. 43-82.
  15. Yang, Y. T., et al., A double decomposition method for solving the periodic base temperature in convective longitudinal fins, Energy Conversion and Management, 49(10), (2008), pp. 2910- 2916.
  16. Kundu, B.and Bhanja, D., Performance and optimization analysis of a constructal T-shaped fin subject to variable thermal conductivity and convective heat transfer coefficient, International Journal of heat and mass transfer,53(1), (2010), pp. 254-267.
  17. Aziz, A. and Fang, T., Alternative solutions for longitudinal fins of rectangular, trapezoidal, and concave parabolic profiles, Energy conversion and Management, 51(11), (2010), pp. 2188-2194.
  18. Hosseini, K., et al., Homotopy analysis method for a fin with temperature dependent internal heat generation and thermal conductivity, International Journal of Nonlinear Science, 14(2), (2012), pp. 201-210.
  19. Singla, R. K. and Das, R., Application of Adomian decomposition method and inverse solution for a fin with variable thermal conductivity and heat generation, International Journal of Heat and Mass Transfer, 66, (2013), pp. 496-506.
  20. J. S. Duan, Z. Wang, S. Z. Fu. and T. Chaolu,, Parametrized temperature distribution and efficiency of convective straight fins with temperature-dependent thermal conductivity by a new modified decomposition method," International Journal of Heat and Mass Transfer, 59, (2013), pp.137-143.
  21. D. R. Parisi, M. C. Mariani, and M. A. Laborde, Solving differential equations with unsupervised neural networks," Chem Eng Process, 42(8-9), (2003), pp. 715-721.
  22. Khan, J. A., et al., Stochastic computational approach for complex non-linear ordinary differential equations, Chin Phys Lett, 28(2), (2011), pp. 020206-020209.
  23. Hooke, R. and. Jeeves, T. A., Direct search solution of numerical and statistical problems," J. Assoc. Comput. Mach, 8 (2), (1961), pp. 212-229.
  24. Ganzarolli, M. M., Carlos A.C. Altemani, Optimum fins spacing and thickness of a finned heat exchanger plate. Heat Transfer Engineering 31,1, 25-32 (2010).
  25. Arqub, O.A. and Abo-Hammour, Z. Numerical solution of systems of second-order boundary value problems using continuous genetic algorithm, Information Sciences 279, (2014), pp.396-415.
  26. Arqub, O.A., Approximate solutions of DASs with nonclassical boundary conditions using novel reproducing kernel algorithm. Fundamenta Informaticae, 146(3), (2016), pp.231-254.
  27. Arqub, O.A., The reproducing kernel algorithm for handling differential algebraic systems of ordinary differential equations. Mathematical Methods in the Applied Sciences, 39(15), (2016), pp.4549-4562.
  28. Arqub, O.A. and Rashaideh, H.,The RKHS method for numerical treatment for integrodifferential algebraic systems of temporal two-point BVPs. Neural Computing and Applications, (2017), pp.1-12.
  29. Ahmad, I. and Bilal, M., Numerical Solution of Blasius Equation through Neural Networks Algorithm., American Journal of Computational Mathematics, 4, (2014), pp. 223-232. dx.doi.org/10.4236/ajcm.2014.43019
  30. Ahmad, I. and Mukhtar, A., Stochastic approach for the solution of multi-pantograph differential equation arising in cell-growth model," Appl. Math. Comput. 261, (2015), pp. 36.
  31. Hooke, R., Jeeves, T. A., Direct search solution of numerical and statistical problems, J. Assoc. Comput. Mach, 8 (2), (1961), pp. 212-229.
  32. Yu, W. C., Positive basis and a class of direct search techniques, Sci. Sin.