## THERMAL SCIENCE

International Scientific Journal

### STOCHASTIC TECHNIQUE FOR SOLUTIONS OF NON-LINEAR FIN EQUATION ARISING IN THERMAL EQUILIBRIUM MODEL

**ABSTRACT**

In this study, a stochastic numerical technique is used to investigate the numerical solution of heat transfer temperature distribution system using feed forward artificial neural networks. Mathematical model of fin equation is formulated with the help of artificial neural networks. 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, interior point technique, pattern search, genetic algorithm and a hybrid approach of pattern search - interior point technique, genetic algorithm - active set technique, genetic algorithm - interior point technique, and genetic algorithm - sequential quadratic programming with different selections of weights. The governing fin equation is transformed into an equivalent non-linear second order ODE. For this transformed ODE model we have performed several simulations to provide the justification of better convergence of results. Moreover, the effectiveness of the designed models 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

**THERMAL SCIENCE** YEAR

**2020**, VOLUME

**24**, ISSUE

**Issue 5**, PAGES [3013 - 3022]

- Long, C., Sayma, N., Heat Transfer, Ventus Publishing, Telluride, Col., USA 2009, p. 156
- Gurrum, S. P., et al., Thermal Issues in Next-Generation Integrated Circuits, IEEE Transactions on Device and Materials Reliability, 4 (2004), 4, pp. 709-714
- Remsburg, R., Advanced Thermal Design of Electronic Equipment, Springer Science and Business Media, New York, USA, 2011
- McGlen, R. J., et al., Thermal Management Techniques for High Power Electronic Devices, Applied Thermal Engineering, 24 (2004), 8, pp. 1143-1156
- Kraus, A. D., et al., Extended Surface Heat Transfer, John Wiley and Sons, New York, USA, 2002
- Chang, M. H., A Decomposition Solution for Fins with Temperature Dependent Surface Heat Flux, International Journal of Heat and Mass Transfer, 48 (2005), 9, pp. 1819-1824
- Saeid, N. H., Natural-Convection in a Square Cavity with Discrete Heating at the Bottom with Different Fin Shapes, Heat Transfer Engineering, 39 (2017), 2, pp. 154-161
- Behbahani, S. W., et al., The 2-D Rectangular Fin with Variable Heat Transfer Coefficient, International Journal of Heat and Mass Transfer, 34 (1991), 1, pp. 79-85
- Ghasemi, S. E., et al., Thermal Analysis of Convective Fin with Temperature-Dependent Thermal Conductivity and Heat Generation, Case Studies in Thermal Engineering, 4 (2014), Nov., pp. 1-8
- Aziz, A., Na, T. Y., Periodic Heat Transfer in Fins with Variable Thermal Parameters, International Journal of Heat and Mass Transfer, 24 (1981), 8, pp. 1397-1404
- Chung, B. T. F., yer, J. R., Optimum Design of Longitudinal Rectangular Fins and Cylindrical Spines with Variable Heat Transfer Coefficient, Heat Transfer Engineering, 4 (1993), 1, pp. 31-42
- Khani, F., Aziz, A., Thermal Analysis of a Longitudinal Trapezoidal Fin with Temperature-Dependent Thermal Conductivity and Heat Transfer Coefficient, Communications in Non-linear Science and Numerical Simulation, 15 (2010), 3, pp. 590-601
- Chiu, C. H., 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 (2002), 10, pp. 2067-2075
- Kim, S., et al., An Approximate Solution of the Non-Linear Fin Problem with Temperature-Dependent Thermal Conductivity and Heat Transfer Coefficient, Journal of Physics D: Applied Physics, 40 (2007), 14, pp. 43-82
- Yang, Y. T., et al., A Double Decomposition Method for Solving the Periodic Base Temperature in Convective Longitudinal Fins, Energy Conversion and Management, 49 (2008), 10, pp. 2910-2916
- Kundu, B., 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 (2010), 1, pp. 254-267
- Aziz, A., Fang, T., Alternative Solutions for Longitudinal Fins of Rectangular, Trapezoidal, and Concave Parabolic Profiles, Energy Conversion and Management, 51 (2010), 11, pp. 2188-2194
- Hosseini, K., et al., Homotopy Analysis Method for a Fin with Temperature Dependent Internal Heat Generation and Thermal Conductivity, International Journal of Non-linear Science, 14 (2012), 2, pp. 201-210
- Singla, R. K., 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), Nov., pp. 496-506
- Duan, J. S., et al., 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), Apr., pp. 137-143
- Parisi, D. R., et al., Solving Differential Equations with Unsupervised Neural Networks, Chem. Eng. Process, 42 (2003), 8-9, pp. 715-721
- Khan, J. A., et al., Stochastic Computational Approach for Complex Non-Linear Ordinary Differential Equations, Chin. Phys. Lett., 28 (2011), 2, pp. 020206-020209
- Hooke, R., Jeeves, T. A., Direct Search Solution of Numerical and Statistical Problems, Journal Assoc. Comput. Mach., 8 (1961), 2, pp. 212-229
- Ganzarolli, M. M., et al., Optimum Fins Spacing and Thickness of a Finned Heat Exchanger Plate, Heat Transfer Engineering, 31 (2010), 1, pp. 25-32
- Arqub, O. A., Abo-Hammour, Z., Numerical Solution of Systems of Second-Order Boundary Value Problems Using Continuous Genetic Algorithm, Information Sciences, 279 (2014), Sept., pp. 396-415
- Arqub, O. A., Approximate Solutions of DAS with Non-Classical Boundary Conditions Using Novel Reproducing Kernel Algorithm, Fundamenta Informaticae, 146 (2016), 3, pp. 231-254
- Arqub, O. A., The Reproducing Kernel Algorithm for Handling Differential Algebraic Systems of Ordinary Differential Equations, Mathematical Methods in the Applied Sciences, 39 (2016), 15, pp. 4549-4562
- Arqub, O. A., Rashaideh, H., The RKHS Method for Numerical Treatment for Integrodifferential Algebraic Systems of Temporal Two-Point BVP, Neural Computing and Applications, 30 (2017), Jan., pp. 2595-2606
- Ahmad, I., Bilal, M., Numerical Solution of Blasius Equation through Neural Networks Algorithm, American Journal of Computational Mathematics, 4 (2014), 3, pp. 223-232
- Ahmad, I., Mukhtar, A., Stochastic Approach for the Solution of Multi-Pantograph Differential Equation Arising in Cell-Growth Model, Appl. Math. Comput., 261 (2015), June, pp. 360-372
- Hooke, R., Jeeves, T. A., Direct Search Solution of Numerical and Statistical Problems, Journal Assoc. Comput. Mach., 8 (1961), 2, pp. 212-229
- Yu, W. C., Positive Basis and a Class of Direct Search Techniques, Sci. Sin. [Zhong-guo Kexue], 9 (1979), S1, pp. 53-68