Mathematics, Vol. 14, Pages 327: Numerical and Experimental Estimation of Heat Source Strengths in Multi-Chip Modules on Printed Circuit Boards

Mathematics doi: 10.3390/math14020327

Authors:
Cheng-Hung Huang
Hao-Wei Su

In this study, a three-dimensional Inverse Conjugate Heat Transfer Problem (ICHTP) is numerically and experimentally investigated to estimate the heat-source strength of multiple chips mounted on a printed circuit board (PCB) using the Conjugate Gradient Method (CGM) and infrared thermography. The interfaces between the PCB and the surrounding air domain are assumed to exhibit perfect thermal contact, establishing a fully coupled conjugate heat transfer framework for the inverse analysis. Unlike the conventional Inverse Heat Conduction Problem (IHCP), which typically only accounts for conduction within solid domains, the present ICHTP formulation requires the simultaneous solution of the governing continuity, momentum, and energy equations in the air domain, along with the heat conduction equation in the chips and PCB. This coupling introduces substantial computational complexity due to the nonlinear interaction between convective and conductive heat transfer mechanisms, as well as the sensitivity of the inverse solution to measurement uncertainties. The numerical simulations are conducted first with error-free measurement data and an inlet velocity of uin = 4 m/s; the recovered heat-sources exhibit excellent agreement with the true values. The computed average errors for the estimated temperatures ERR1 and estimated heat sources ERR2 are as low as 0.0031% and 1.87%, respectively. The accuracy of the estimated heat sources is then experimentally validated under various prescribed inlet air velocities. During experimental verification at an inlet velocity of 4 m/s, the corresponding ERR1 and ERR2 values are obtained as 0.91% and 3.34%, while at 6 m/s, the values are 0.86% and 2.81%, respectively. Compared with the numerical results, the accuracy of the experimental estimations decreases noticeably. This discrepancy arises because the numerical simulations are free from measurement noise, whereas experimental data inherently include uncertainties due to thermal picture resolutions, environmental fluctuations, and other uncontrollable factors. These results highlight the inherent challenges associated with inverse problems and underscore the critical importance of obtaining precise and reliable temperature measurements to ensure accurate heat source estimation.

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