A Stochastic Approach of Thermal Fatigue Crack Growth (LEFM) in Mixing Tees

The assessment of fatigue crack growth due to turbulent mixing of hot and cold coolants presents significant challenges, in particular to determine the thermal loading spectrum. Thermal striping is defined as a random temperature fluctuation produced by incomplete mixing of fluid streams at different temperatures, and it is essentially a random phenomenon in a temporal sense. The objective of this work is to develop a stochastic model to assess thermal fatigue crack growth in mixing tees, based on the power spectral density (PSD) of the temperature fluctuation at the inner pipe surface. Based on the analytical solution for temperature distribution through the wall thickness, obtained by means of Hankel transform, a frequency temperature response function is proposed, in the framework of single-input, single-output (SISO) methodology from random noise/signal theory under sinusoidal input. For the elastic thermal stresses distribution solutions, the magnitude of the frequency response function is first derived and checked against the prediction by FEA. The frequency response of the stress intensity factor (SIF) is obtained by a polynomial fitting of the stress profiles through the wall thickness at various instants of time. The variability in load is given by the statistical properties of thermal spectrum. The temperature spectrum is assumed to be given as a stationary normalized Gaussian narrow-band stochastic process, with constant PSD for a defined range of frequencies. The connection between SIF’s PSD and temperature’s PSD is assured with SIF frequency response function modulus. The frequency of the peaks of each magnitude for K_I, which is supposed to be a stationary narrow-band Gaussian process, is characterized by the Rayleigh distribution, and, consequently, the expected value of crack growth rate in respect to cycles is obtained. The probabilities of failure are estimated by mean of the Monte Carlo methods considering a limit state function, which is based on the developed stochastic model. The results of the stochastic approach of thermal fatigue crack growth in mixing tees is completed with probabilistic input to account for the variability in the material characteristics, and finally an application is given to obtain the probability of mixing tees piping failure as function of time reference period.