PSI - Issue 6

Irina Stareva et al. / Procedia Structural Integrity 6 (2017) 48–55 Author name / Structural Integrity Procedia 00 (2017) 000 – 000

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3.2. Choice of integration step

It is reasonable to assume that the optimal time step t  and the spatial step z  should depend on physical data of the problem, such as corrosion rates (related to the tube thickness) and the density of the tube material. Therefore, we investigate this problem for certain data. The corrosion kinetics constants are 0.1   r R a a mm/year, 0.0005   r R m m mm/(year·MPa); the density 7800   kg/m 3 , and 9.8  g m/s 2 . Steel tubes with external radius 0.92 0  R m and internal radius 0.9 0  r m are considered. The lengths of the tubes are 12 m, 25 m, 50 m, 100 m, and 200 m. Time to reach the limit stress *  = 300 MPa and time to reach the residual thickness * h = 5 mm were calculated for various initial data (but not all the results are presented here). The calculation results revealed that t  =1 day gives a relative error less than 1% for different spatial steps z  : 0.24, 0.5, 1, 2, and 4 m. However, since we wanted to receive more accurate numerical solution (to compare it with an approximate analytical formula), for further analysis we selected smaller step. There is good convergence of the method. For t  = 2 hour, relative error is less than 0.001% for all the mentioned spatial steps. To analyze the effect of a spatial step, the time * t to reach the limit stress *  = 300 MPa was calculated for the tubes with different lengths l by the use of different spatial steps z l N /   (see Table 1). The results confirm that the accuracy of calculations depends on the absolute value of z  , but not on the number of the nodal points. Therefore, using the same spatial step for tubes with different lengths (irrespective of the number of nodal points) is justified. The value “er” in Table 1 means the relative difference between the results on the current and previous steps (of course, it is less than the global error). The time step t  = 1 hour was used for Table 1. For our analysis we selected the spatial step z  =2 m. For engineering calculations it can be chosen greater.

Table 1. The time to reach the limit stress (year) z  12 m 25 m

50 m 98.37

100 m 96.706 96.321 96.114 0.22% 95.997 0.12% 95.937 0.06% 95.905 0.03% 0.4%

200 m 93.367 92.552 0.88% 92.104 0.49% 91.846 0.28% 91.714 0.14% 91.642 0.08%

/ 3 l / 6 l

t* t* er t* er t* er t* er t* er

99.614 99.574 0.04% 99.554 0.02% 99.543 0.011% 99.538 0.005% 99.536 0.002%

99.192 99.105 0.09% 99.061 0.04% 99.037 0.03% 99.025 0.012% 99.019 0.006%

98.189 0.18% 98.094

/12 l

0.1%

/ 25 l

98.041 0.05% 98.015 0.03% 98.001 0.014%

/ 50 l

/100 l

3.3. Calculation results and discussion

First of all, it should be emphasized that calculation of the time to reach the limit stress *  = 300 MPa is not justified from the practical point of view (it is just an exercise) because this limit is reached when the residual thickness of the tube is rather small (less than 0.1 mm even for l = 200 m) and the tube ceases to be a tube. Thus, the lifetime * t of the tube should be defined by the criterion of the minimum residual thickness: (0, *) * h t h  . Consider the growth of stresses with time in the steel tube with external radius 0.92 0  R m and internal radius 0.9 0  r m. The corrosion kinetics constants are 0.1   r R a a mm/year, 0.0005   r R m m mm/(year·MPa); the density 7800   kg/m 3 , and 9.8  g m/s 2 . Distribution of stresses in the steel tube with length of 200 m in points z = 0, 25 m, 50 m, 75 m, 100 m, 125 m, 150 m, 175 m, 200 m is represented in Fig. 1. The lower indices at  indicate the z coordinate. These values of