The fast running SFE method draws up an overview of the temperature changes and a qualitative stress estimation for every monitored component. The fatigue handbook and the knowledge of these parameters determine the pertinence of a fatigue analysis for the component. In that way a more detailed and automatic method, FFE, can be used to calculate the CUFs.

The determination of the time-history of loads and the resulting local stresses are the basis of the method. On a measuring section, close to the fatigue relevant location, FAMOS measures the temperature at the outside surface of the pipe. First of all the interpretation of this temperature has to be explained.

In a homogeneous isotropic solid, where the temperature is a function of time and space T = f (x, y, z, t), the equation of heat conduction is as follows:

d2T 82T 82T pc 8T n

—т +—v + —v ———- = 0

8×2 8y2 8z2 X 8t

К… thermal conductivity of the solid [W/(m-K)] p… density of the solid [kg/m3] c… specific heat of the solid [J/(kg-K)]

This equation can be written in a cylindrical coordinate system:

82T 18T 1 82T 82T pc 8T n

82 + + —882 + ~8———— X~8f ~ ° (2)

8r2 r 8r r2 80z 8z2 X 8t

The stratification effects will not be considered here and the application of the method will be restricted to plug flow events. In this case, the temperature evolution is independent of the circumferential direction T = f (r, z, t). Moreover the measuring section on the pipe is located relatively far away from geometrical discontinuities and T (z — Sz) = T (z + dz) holds true. Following this presumption, the temperature in the assessed section can be written as: T = f (r, t). This involves:

82T 1 cT = pc cT 8r2 r 8r X 8t

This equation handles the thermal evolution inside the thickness of the pipe. The solution of the equation depends on the applied boundary conditions. As the varying load is the medium temperature flowing throughout the component, the heat transfer between the fluid and the inner surface is governed by a Newton’s law of cooling:

= h(T — TM )r =ri (4)

r =ri

h… heat transfer coefficient [W/(m2 K)]

For further explanations of the mathematical background of the method see e. g. [12].

All the difficulty to apply this equation during unsteady fluid temperature states is due to the determination of the heat transfer coefficient h. Indeed, this parameter depends on the velocity, the thermo-hydraulics conditions and the geometry of the surface. The time dependent knowledge of all these parameters with sufficient accuracy is hardly compatible with a fast determination of the real loads. To solve this problem, the inverse philosophy was developed to calculate the stresses in the structure. Indeed, to perform a structural analysis, the knowledge of the temperature distribution throughout the wall is sufficient. The FFE method is based on the time history determination of the inner wall temperature by solving the inverse problem of conduction of heat. The according flowchart is shown in Figure 6.

Newton‘s law

Fig. 6. Different way to get inner wall temperature

Solving the inverse conduction equation of heat is done by application of potential functions (unit transients). A unit transient is applied at the inner surface of the pipe (boundary condition), the equation of conduction of heat is solved, the resulting time-history of temperature at the outer wall is observed. The resolution of the equation of heat can be done by means of an analytical method or with the help of a finite element program (ANSYS®). In that case a two-dimensional model of the section of the pipe is generated. The benefit of this last choice is the opportunity to integrate the thermal influences of the thermocouple installation at the outer surface of the pipe in the solution (see Figure 7).

The determined temperature response calculated in the thermocouple (outer wall) will be considered as a reference. Its evolution is characteristic by the applied unit transient at the inside surface of the pipe (characterized by a temperature rate of changes and a thermal amplitude ATref). Thus the FAMOS measured outside temperature will be scanned step by step (typically every second). The temperature difference at the outer wall between two time steps is compared with the simulated outer wall temperature (reference). The factor resulting from this comparison is through linearity properties also available at the inner side of the structure. Thus, step by step the inside temperature of the pipe can be restituted. A computation algorithm of this process was developed. The acquired measured data of FAMOS are read into the FFE program. A preparatory work consists of calculating, for the different observed piping sections, the thermal references. These last ones depend on the material, pipe thickness and measurement thermocouple. After this pre-processing work, the computation of the transient inner wall temperatures is completely automated.

Unit Transient applies at the inside surface of the pipe

Fig. 7. FE calculation of the temperature response at the outside of the pipe

The determined inner wall temperature will be used to calculate the thermal stress at the fatigue relevant locations. An appropriate temperature transfer function can readily be used for correction of the axial dependency of the temperature if the FAMOS section is far away from the stress calculation locations T(z -8z) Ф T(z + 8z). The procedure is shown schematically in Figure 8.

The thermal stress determination is done according to a similar process as previously explained. A two — or three — dimensional finite element model of the monitored component is generated (nozzle, heat exchanger,…). A unit (elementary) transient is used as a reference load of a thermal calculation. Thus, the thermal field in the structure is calculated. Subsequently, the thermal stresses are calculated by a linearly elastic structural analysis.

The resulting thermal stresses are determined for typical fatigue relevant locations. The calculated stress components are the response to a reference load characterized by a temperature rate of change and a thermal amplitude ATref. The exemplary procedure is shown in Figure 9.

Fig. 9. FE calculation of the stress responses at a fatigue relevant location

The inside temperature calculated in the previous step by means of FFE, is scanned step by step. Between two time steps, the temperature difference is interpreted as a unit transient the amplitude of which is compared to the reference unit transient of amplitude ATref. Because of linearity in the thermal stress calculation, the comparison between the measured amplitude and the reference gives a coefficient to be applied to the reference stress matrix in order to obtain the stress contribution resulting from the thermal load at the calculated time. The time-dependent stress components are then obtained by the summation of all these single contributions. The process is also completely automated within the FFE program. The stress matrix references have to be calculated previously in an FE program. The results are then added to the database of FFE: it is the pre-processing work. Subsequently, the calculation at the selected locations can be processed. Within a few minutes, thermal loads and stress components of the entire operating cycles are calculated (see e. g. Figure 10).

If information on the time dependent pressure or piping section forces and moments are available based on operational instrumentation, the resulting mechanical stress components can be calculated equally by means of FFE (scaling of unit loads). Thermal and mechanical
stress components are added and the equivalent stress is calculated. The use of a rain-flow algorithm will classify the stress ranges, a standard conform comparison with the fatigue curves will give the fatigue level of the selected locations.

Finally, if the calculated fatigue usage factor is lower than the allowable limit, the fatigue check will be successfully finished. If not, further analyses according to the detailed code based fatigue check will be performed.

In order to optimize the costs and user flexibility, the FFE program was based on a modular architecture. Thus, only information required by the customer/user is calculated. This architecture also permits an easy upgrade of the program to implement new modules e. g. as a consequence of changes of nuclear standards (new fatigue curves, environmental factor integration,…) or further calculation methods (automated stratification consideration).

Fig. 10. FFE temperature and thermal stresses calculation for shut down event