Implementation of Practical Algorithms in the Inverse Problem with Unknown Boundary Conditions

Abstract

This work attempts to propose a practical methodology for the one-dimensional heat conduction problems with unknown boundary conditions of the first kind at both ends. The methodology can be effectively applied when the remote boundary condition is inaccessible or internal temperature is required, such as a cutting process with cooling lubricant, and ablative thermal protection system. The proposed methodology utilizes the shifting function method in conjunction with the least squares error method to transform the inverse heat conduction problem into an approximate “well-posed” problem. Consequently, the temperature and the heat flux distributions over the entire time and space domains are determined by using half-range Fourier cosine series solutions. Experimental examples of spray cooling problems are provided to illustrate the advantages of the proposed methodology, including fast convergence of the temperature function, fewer discrete measured times in numerical analysis, and regardless of two interior temperature probes positions.

Acknowledgement

Special thanks are addressed to Prof. Te-Wen Tu (Department of Mechanical Engineering, Air Force Institute of Technology, Taiwan) for his valuable advices for this work, especially on the solution of Fourier series expansion.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Notes on contributors

Jin-Sheng Leu

Jin-Sheng Leu is an Associate Professor in Aircraft Engineering Department at Air Force Institute of Technology, Kaohsiung, Taiwan. He received his Ph.D. in Mechanical Engineering from National Cheng-Kung University, Tainan, Taiwan, in 1994. His main research interests are heat transfer enhancement techniques (including active and passive techniques), liquid film evaporation application, inverse heat conduction problem, and the electrohydrodynamic technique.