Material data

The material X8CrNiMoNb-16-16 was selected for the pressure vessel design since there were existing experimental creep data for the material from which a fitting parameter could be obtained. The material is an austenitic high-temperature and high strength steel used for elevated temperature applications. The chemical composition of the material is given in Table 12.

Table 12. Chemical composition of X8CrNiMoNb 16-16 in Wt% [62]

C Ni Cr Mn P S Si Mo Nb Ta

Min 0.04 15.5 15.5 0.3 1.6 10xC

Max 0.1 17.5 17.5 1.5 0.035 0.015 0.6 2 1.2 1.2

The mechanical properties of the material X8CrNiMoNb-16-16 at room temperature (RT) and at 700

°C are given in Table 1. The properties at 700 °C were obtained by linear extrapolation of the available material data which was obtained from a material database [62]. The density change due to material expansion at elevated temperature was neglected and the material density at 700 °C was assumed to be equal to the density at RT.

Table 13. Properties of X8CrNiMoNb-16-16 at RT and at 700°C [62]

Property Symbol Unit Temperature

RT 700 °C

1) Mean coefficient of thermal expansion between RT (20°C) and 700°C

Available material creep data [62] was used for making a Larson-Miller parameter plot for creep resistance, 𝜎_𝑅𝑝1 and creep rupture strength, 𝜎_𝐶𝑅𝑆 so both time-to-rupture and time-to-1% strain could be obtained for different stress levels. Data for 𝜎_𝑅𝑝1was included since many designs are based on a maximum allowable amount of creep strain, for example 0.1 or 1% during the expected lifetime of a component [64]. As were also previously mentioned the 1% strain limit is also typically used to avoid damage due to lack of ductility. All material data that was used are noted in Appendix A.

3.2.1.1 Accuracy of the Larson-Miller parameter plot

The Larson-Miller parameter, 𝑃_𝐿𝑀 which is given in Eq. 2.2 was used to correlate the temperature and time with stress so the creep data could be described in one fit from which allowable time corresponding to a certain stress and temperature could be obtained. The fit can be optimized by changing the variable 𝐶 to optimize the correlation coefficient, 𝑅². However, predictions should only be made within the stress range of the data [65]. Although there are other more complex creep data correlation methods which might be more accurate, the Larson-Miller parameter was chosen because of its relatively easy data correlation.

Fits, both with and without logarithmic scale on the stress axis, were tested. By changing the 𝐶-value, four fits each were selected for the creep resistance, 𝜎_𝑅𝑝1 and the creep rupture strength, 𝜎_𝐶𝑅𝑆 based on the 𝑅²-value. The 𝑅² value can range between 0 and 1 where a value closer to 1 indicates a better

55 correlation between the data and the fit. The best fits were obtained with a second order polynomial fit and a Gaussian fit. Graphical representations of the obtained fits are presented in Figure 41.

Figure 41. Larson-Miller parameter plots with Gaussian and second order polynomial fits for σRp1 and σCRS

The 𝑅² value alone does not necessarily guarantee if the fit is good and useful for predictions and there are other statistics which can be used for evaluating the appropriateness of the fit such as the sum of squares due to errors (SSE) and the root mean squared error (RMSE) given below [66] [67]:

𝑆𝑆𝐸 = ∑ 𝑤_𝑖(𝑦_𝑖− 𝑦̂_𝑖)²

𝑛

𝑖=1

Eq. 3.6

𝑅𝑀𝑆𝐸 = √ 𝑆𝑆𝐸

𝑛 − 𝑚 ^{Eq. 3.7}

Where 𝑤_𝑖 are the weights which regulates how much each response value impact the final parameter, 𝑦_𝑖 is the data points, 𝑦̂_𝑖 is the value from the fit and (𝑦_𝑖− 𝑦̂_𝑖) represent the residuals. In the expression

Gauss1 R²: 0.9989 Poly 2 R²: 0.9983

Gauss1 R²: 0.9976 Poly 2 R²: 0.9981

Gauss1 R²: 0.9976 Poly 2 R²: 0.9964

Gauss1 R²: 0.9939 Poly 2 R²: 0.9964

56 for RMSE, 𝑛 repsesent the number of data points and 𝑚 is the number of parameters in the function.

For SSE and RMSE a value closer to zero is an indication that the fit is useful for prediction [67]. MATLAB curve fitting application in was used for obtaining the statistical curve fitting data. The statistics results from each evaluated fit for 𝜎_𝑅𝑝1 and 𝜎_𝐶𝑅𝑆 are given in Table 14 and Table 15.

Table 14. Statistics of CRS-fits from MATLAB

Name 𝑪 Fit type SSE 𝑹^𝟐 𝒏 − 𝒎 RMSE

Log CRS 13.9 Poly 2 0.0197 0.9964 51 0.01965 3

Log CRS 13.9 Gauss 1 0.03325 0.9939 51 0.02553 3

CRS 15.4 Poly 2 742 0.9964 51 3.814 3

CRS 15.4 Gauss 1 498.5 0.9976 51 3.126 3

Table 15. Statistics of Rp1-fits from MATLAB

Name 𝑪 Fit type SSE 𝑹^𝟐 𝒏 − 𝒎 RMSE 𝒎

Log Rp1 13.4 Poly 2 0.005157 0.9981 33 0.0125 3

Log Rp1 13.4 Gauss 1 0.006412 0.9976 33 0.01394 3

Rp1 12.7 Poly 2 115.5 0.9983 33 1.871 3

Rp1 12.7 Gauss 1 73.03 0.9989 33 1.488 3

A bad fit can severely over and underestimate the allowable time duration under a certain stress and temperature. As seen from the tables, the best fits for both creep resistance and creep rupture strength was obtained with the second order polynomial fits when stresses on the ordinate were plotted with a logarithmic scale. These are highlighted in yellow and the equations for the selected fits are given by:

𝑙𝑜𝑔𝜎_𝑅𝑝1= −2.721 ∙ 10⁻⁸ 𝑃_𝐿𝑀² + 6.564 ∙ 10⁻⁴𝑃_𝐿𝑀− 1.495 Eq. 3.8

𝑙𝑜𝑔𝜎_𝐶𝑅𝑆 = −2.901 ∙ 10⁻⁸ 𝑃_𝐿𝑀² + 7.302 ∙ 10⁻⁴𝑃_𝐿𝑀− 1.98 ^{Eq. 3.9} 𝑃_𝐿𝑀(𝜎_𝑅𝑝1) = 𝑇 (𝑙𝑜𝑔𝑡_{𝑓𝑅𝑝1}+ 𝐶) ^{Eq. 3.10}

𝑃_𝐿𝑀(𝜎_𝐶𝑅𝑆) = 𝑇(𝑙𝑜𝑔𝑡_{𝑓𝐶𝑅𝑆}+ 𝐶) ^{Eq. 3.11}

The above equations do not display all decimal places. The equations were solved with respect to the allowable time duration for the corresponding stress and temperature condition using MATLAB where all decimal places in the fits were used. Solving without all decimal places the obtained results can differ from the ones calculated by the software.

The error between the allowable time obtained from the fits and from the data points was calculated using Eq. 3.12 to get an indication of how much the rupture-time and 1%strain-time deviated from the actual values. The time calculated from the fits, the actual time from the creep data and the error between them for some of the extremes are given in Table 16 and Table 17 respectively.

57

%_{𝑒𝑟𝑟𝑜𝑟} = |𝑡_{𝑓𝑓𝑖𝑡}− 𝑡_{𝑓𝑑𝑎𝑡𝑎}

𝑡_{𝑓𝑑𝑎𝑡𝑎} | ∙ 100 Eq. 3.12

Table 16. Sample points for testing the accuracy of the fitted CRS curves Temperature

Table 17. Sample points for testing the accuracy of the fitted Rp1 curves Temperature

As seen from the tables the chosen fits overestimated and underestimated the time duration for some stress and temperature combinations. However, the average error between all the data points and the corresponding values from the fits were 6.19% for the creep resistance and 13.08% for the creep rupture strength. The residual plots for the fits given in Figure 42 shows the values of the Larson-Miller parameter that produced most error.

Figure 42. Residual plots for σRp1 and σCRS

58