Inverse estimation of the inner wall temperature fluctuations in a pipe elbow

Thermal stratification occurs in flows through elbows used for mixing hot and cold fluids. The movement of the interface between the hot and cold fluids over time causes fluctuations of the inner wall temperature, which can lead to the stress variations and structural thermal fatigue. Therefore, accurate inner wall temperature data detailing the fluctuations is essential for analysis and predictions of thermal fatigue of the piping. In the present study, the conjugate gradient method (CGM) is applied to solve the three-dimensional inverse heat conduction problem (IHCP) to estimate the unknown temperature fluctuations on the inner wall of a pipe elbow from simulated outer temperature measurements. First, the direct heat conduction problem is solved using the finite element method (FEM) to produce the outer and inner wall temperatures. Then, the inverse heat conduction problem is solved to estimate the inner wall temperatures based on the outer wall temperature data. The accuracy of the inverse algorithm is then examined by comparing the estimated inner wall temperatures with the exact temperatures from the direct solution. The numerical results show that the inner wall temperature of the elbow can be accurately estimated by using the present algorithm for the test case considered.