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ABSTRACT
The present work deals with the application of the golden section search method (GSSM) for predicting the internal rate of heat generation to reconstruct a given temperature distribution within a rectangular fin involving all modes of heat transfer. The thermal conductivity has been assumed to be temperature-dependent. The forward problem is numerically solved using an implicit fourth-order Runge–Kutta method, whereas, an inverse problem has been solved using GSSM. In conjunction with GSSM, for the inverse analysis, the effect of inverse crime has been addressed using a different solver operating on fifth-order accurate Runge–Kutta method than that used for synthesizing the input data. A case study of Hastelloy generally used in gas turbine applications is also presented and the effect of measurement error in the temperature distribution has been reported. For pure temperature data, an exact estimation of the internal heat generation rate is done, whereas, even with noisy data, a satisfactory estimation of the heat generation rate is also achieved which is verified from the reconstructed temperature distributions.
Nomenclature
| Ac | = | area of cross section (m2) |
| ADM | = | Adomian decomposition method |
| BVP | = | boundary value problem |
| bvp4c | = | Matlab solver for boundary value problem implementing fourth-order Runge–Kutta formula |
| bvp5c | = | Matlab solver for boundary value problem implementing fifth-order Runge–Kutta formula |
| e | = | measurement error |
| F | = | objective function |
| Fa, Fb, Fp, Fq | = | respective function values at vertices Va, Vb, Vp, Vq of GSSM |
| GSSM | = | golden section search method |
| h | = | heat transfer coefficient, W/(m2 · K) |
| k | = | thermal conductivity, W/(m · K) |
| ka | = | thermal conductivity at ambient condition, W/(m · K) |
| L | = | length of fin, m |
| N | = | number of terms |
| Nc | = | convection-conduction parameter, see Eq. (Equation6 |
| Nr | = | radiation-conduction parameter, see Eq. (Equation6 |
| P | = | perimeter of the fin, m |
| Q | = | nondimensional rate of internal heat generation (heat generation number), see Eq. (Equation6 |
| q | = | rate of internal heat generation, W/m3 |
| R | = | reduction ratio |
| RK4 | = | fourth-order accurate Runge–Kutta method |
| RK5 | = | fifth-order accurate Runge–Kutta method |
| r | = | inverse golden number |
| SQ | = | sensitivity coefficient of temperature with respect to heat generation number |
| T | = | temperature, K |
| = | exact temperature, K | |
| Ta | = | ambient temperature, K |
| Tb | = | base temperature, K |
| Ts | = | radiation sink temperature, K |
| t | = | thickness of fin, m |
| Va, Vb, Vp, Vq | = | vertices of GSSM |
| W | = | fin width, m |
| x | = | axial coordinate of the fin, m |
| X | = | nondimensional axial coordinate of the fin, x/L |
Greek symbols
| β | = | variable thermal conductivity parameter, see Eq. (Equation6 |
| ϵ | = | emissivity of the fin material |
| λ | = | coefficient of variable thermal conductivity, 1/K |
| θ | = | nondimensional temperature, T/Tb |
| θa | = | nondimensional ambient temperature, Ta/Tb |
| θs | = | nondimensional radiation sink temperature, Ts/Tb |
| = | exact value of nondimensional temperature, | |
| σ | = | Stefan-Boltzmann constant |
Additional information
Notes on contributors
Ranjan Das
Ranjan Das is an Assistant Professor in the Department of Mechanical Engineering at the Indian Institute of Technology Ropar. He has earned Undergraduate degree from Bhavnagar University, Gujarat, in 2003, Masters degree from National Institute of Technology Silchar in 2006 and Doctoral degree from Indian Institute of Technology Guwahati in 2010, all in Mechanical Engineering. From June 2010 to December 2011, he served Tezpur University, Assam (A Central University), as an Assistant Professor in the Department of Mechanical Engineering. Thereafter, as a research fellow from January to November 2012, he has also served the School of Mechanical and Aerospace Engineering at Nanyang Technological University Singapore. He has published many papers in peer-reviewed journals and conferences. His current research area involves inverse problems in heat and mass transfer systems such as extended surfaces, cooling towers radiative transfer in participating media, optimization, along with various applications of renewable energy.
Balaram Kundu
Balaram Kundu is an Associate Professor in the Department of Mechanical Engineering at Jadavpur University, Kolkata. He earned his Undergraduate, Masters, and Doctoral degrees from Regional Engineering College, Durgapur, in 1993, Bengal Engineering College, Shibpur, in 1995, and Indian Institute of Technology, Kharagpur, in 2000, respectively. From 1998–2002, he worked at Jalpaiguri Government Engineering College, India. Thereafter, he held positions of Senior Lecturer, Reader, and Associate Professor from 2003–2007 and 2008–2010, and 2011-present, respectively, at Jadavpur University, Kolkata. He is Recipient of Institutional Medal for 1997–1998 for the best paper published in the Journal of the Institution of Engineers (India) along with Research Assistant Professor, Research Associate Professor, and Research Professor Positions awarded by Brain Korea, South Korea. In 2015, he has been awarded as outstanding scientist award by the Venus International Foundation, India. He has published many papers in prestigious peer-reviewed international journals along with conference proceedings. His research area includes analytical and computational heat transfer, fin-and-tube heat exchanger, flat-plate solar collector, heat transfer in porous materials, and biological heat transfer.