1. Objectives
It has been recognized that fluctuation of fluid temperature in a pipe may induce high cycle thermal fatigue in the pipe, which sometimes results in fracture of the pipe. The information about the temperature and residual stress distributions is important to prevent the high cycle thermal fatigue. It is, however, difficult to measure the inner surface temperature directly. The monitoring of the temperature on accessible outer surface is promising to solve the problem. This paper presents an inverse analysis method for estimating the temperature and residual stress distributions in the pipe from the temperature history measured on the outer surface. A regularization method is proposed for obtaining a reasonable solution.
2. Inverse Method for Estimating Temperature and Thermal Stress Distributions
Based on a mathematical analysis it was found that the relationship between the temperatures on the inner surface and on the outer surface is expressed by a reduction factor R and a phase lag Dp, which are dependent on frequency. An inverse analysis method was proposed, in which the outer surface temperature measured on the outer surface is decomposed of frequency components, and their amplitude is multiplied by a factor of 1/R and the phase is advanced by Dp. The inner surface temperature history is obtained by summing up the frequency components.
3. Regularization Method
The reduction factor R is very small for very high frequency components. Then high frequency fluctuation measured on the outer surface is exaggerated in the estimation of the inner surface temperature history. In the presence of measurement noise, incorrect high frequency components deteriorate the estimation. A regularization method is proposed to exclude the incorrect components. A maximum limit is set to the factor of 1/R. This limit is determined by considering the effect of noise to the coefficient of frequency coefficients.
4. Numerical Simulations
Numerical simulations are made to examine the applicability of the proposed method. A pipe made of stainless steel is considered. The diameter of the pipe assumed to be large compared with the pipe thickness. Is is assumed that the inner surface temperature history is made of three frequency components.
5. Results
It is found that the temperature history on the inner surface is recovered when measurement noise is not included in the measurement o the outer surface. As the noise level is increased high frequency components give rise to big fluctuation in the estimated inner surface temperature history. When the regularization method is applied reasonable estimation is made, although the high frequency components are lost. Using the temperature history the estimation of the thermal stress in the pipe is made.
6. Conclusions
This paper presents an inverse analysis method for estimating the temperature and residual stress distributions in the pipe from the temperature history measured on the outer surface. It is found that the proposed method is useful for obtaining a reasonable estimate of the inner surface temperature and thermal stresses.
