Thermoelastic Stresses in Thick-Walled Vessels Under an Arbitrary Thermal Transient via the Inverse Route Available to Purchase

A common threat to thick-walled vessels and pipes is thermal shock from operational steady-state or transient thermoelastic stresses. As such, boundary conditions must be known or determined in order to reveal the underlying thermal state. For direct problems where all boundary conditions are known, the procedure is relatively straightforward and mathematically tractable as shown by recent studies. Although more practical from a measurement standpoint, the inverse problem where boundary conditions must be determined from remotely determined temperature and/or flux data is ill-posed and inherently sensitive to errors in the data. As a result, the inverse route is rarely used to determine thermal-stresses. Moreover, most analytical solutions to the inverse problem rely on a host of assumptions that usually restrict their utility to timeframes before the thermal wave reaches the natural boundaries of the structure. To help offset these limitations and at the same time solve for the useful case of a thick-walled cylinder exposed to thermal loading on the ID, the inverse problem was solved using a least-squares determination of polynomial coefficients based on a generalized direct-solution to the Heat Equation. Once the inverse problem was solved in this fashion and the unknown boundary-condition on the ID determined, the resulting polynomial was used with the generalized direct solution to estimate the internal temperature and stress distributions as a function of time and radial position. For a thick-walled cylinder under an internal transient with external convection, excellent agreement was seen with various measured temperature histories. Given the versatility of the polynomial solutions advocated, the method appears well suited for many thermal scenarios provided the analysis is restricted to the time interval used to determine the polynomial and the thermophysical properties that do not vary with temperature.