Closed-Form Solutions for Locating Heat-Concentrated Sources Using Temperature Difference

The closed-form solution, one of the effective and sufficient optimization methods, is usually less computationally burdensome than iterative and nonlinear minimization in optimization problems of heat source localization. This work presents two-dimensional, closed-form solutions for locating heat-concentrated sources using temperature differences for known and unknown temperature gradient systems. The nonlinear location equations for heat-concentrated source location are simplified to linear equations, and they are solved directly to obtain the analytical solution. To validate the accuracy of the proposed analytical solutions, three numerical examples of heat source localization were conducted. Results show that the proposed analytical solutions have a higher accuracy than iterative results by Levenberg–Marquardt. The locating accuracy for the three sources using AS-KTG improved by 94.82%, 90.40%, and 92.77%, while the locating accuracy for the three sources using AS-UTG improved by 68.94%, 16.72%, and 46.86%, respectively. It is concluded that the proposed method can locate the heat sources using temperatures and coordinates of sensors without the need for a heat transfer coefficient, a heat transfer rate, and thermal conductivity. These proposed analytical solutions can provide a new approach to locating heat sources for more complicated conditions using temperature differences, such as the localization of geothermal sources and nuclear waste leak points.