Density-Based Model of Bending Strength for AGR Graphites Irradiated in Oxidising Environments Ernie D. Eason Modeling & Computing Services Boulder, Colorado, USA Graham Hall Barry J. Marsden Nuclear Graphite Research Group, School of Mechanical, Aerospace & Civil Engineering, University of Manchester, UK Presented at INGSM-14 Seattle, Washington, USA September 18, 2013

2 Data Used for Density-Based Strength Models Trepanned data used for model calibration and validation: –All are 3-point bending strength, measured at Windscale Nuclear Laboratories (WNL) –1835 points were used for calibration –203 randomly-selected points were set aside, used to validate the model on data not used for fitting 2013 models and comparisons are based on trepanned bending strength S, not ratio S/S 0

3 Trepanned Bending Strength Irradiated Density Model Model form The coefficients C i and exponents N i vary by reactor The T irr term is a small correction (+3%, -1%) Average T irr =  C over all trepanned data

4 Advantages of an Irradiated Density Strength Model versus a Weight Loss Strength Model Simple power function form Better fit than a model based on weight loss on the same data –Slightly smaller standard error, 3.99 vs MPa –No significant residual error trend in any variable Much smaller “year effect” than a weight loss model (1/3 as large) No need to estimate virgin density or make corrections as with weight loss estimates

5 The “Year Effect”-- Strength Appears to Increase with Trepanning Year 24 MPa MPa MPa 2006 Same reactor, three sets of strength measurements from trepanning campaigns 3 and 6 years apart

6 The “Year Effect” for Weight Loss versus Density-based Strength Models Preliminary Weight Loss ModelPreliminary Density-Based Model Measured Strength Increases Significantly in Newer Data 1/3 as much Increase

7 Method of Fitting the Density-Based Model Preliminary model fitted to Strength, S Final model fitted to log(S) Distribution of residuals is approximately normal AND approximately log-normal, so either fit is statistically reasonable Fitting a power law in the logs is common practice –produces a linear least squares fit –minimizes relative error log(S) fit is practically better – tighter estimates at low S and low  irr (expect lower S at long exposure)

8 Model and Calibration Data Plots

9 Chauvenet Outliers Chauvenet outliers are points so far from the model that they should not be observed in a normal or log-normal distribution of data A few Chauvenet outliers were identified –2 outliers from preliminary models calibrated to S –5 additional outliers from the final model calibrated to log(S) –The outliers represent 0.3% of 2038 points

10 Model and Calibration Data Plots, cont’d

11 Density-Based Strength Model Quality of Fit Standard Error S e = measured as log(S) corresponds to 12.3% relative error Over the range of measured strength in the data set (9 < S ≤ 58 MPa), 12.3% error corresponds to 1.1 to 7.1 MPa Model vs. measured log(S) shows overall agreement of data and model (next slide) All residual plots are flat, with non-significant trends (next several slides)

12 Calibration and Validation Data Sets Calibration DataValidation Data The Validation Data Fit The Model as Well as the Calibration Data

13 Flat Residual Plots

14 Flat Residual Plots, continued

15 The “Year Effect” for Weight Loss versus Density-based Models – Residual Plots Significant Residual TrendNo Significant Residual Trend

16 Flat Residual Plot for Inert-Irradiated Young’s Modulus Ratio Including a function of inert E does not improve the density-based model

17 Significant Residual Trends if Inert- Irradiated Young’s Modulus is Imposed Including (inert E) 0.5 in the density-based model seriously degrades the fit Unconservative (actual S < model S) Conservative (actual S > model S)

18 Newer Data, Received After Calibration Density-Based Model Calibrated to 1998 – 2010 Data Reasonably Predicts 2011 & 2012 Trepanned Data All Reactors 1998 – 2010 Calib. Data 5 Reactors 2011 & 2012 Data Mean log(S) Residual 2.2 x (~0) -9.4 x10 -4 (~0) Residual S d as log(S) Residual S d as S (MPa) Number of Points Differences Not Significant 2011 & 2012 Data Model Prediction, 1:1 Line

19 Conclusions on the Density-Based Strength Model Fitted to log(Strength) The density-based model provides a simple, good fit (12.3% relative error) The density model is better than a weight loss model –Slightly lower standard error on the same data –No significant residual error trends –Much smaller, non-significant “year effect” –No need to estimate virgin density The density-based bending strength model does not need or benefit from including an inert Young’s modulus term Including an (inert E) 0.5 term in either weight loss or density- based bending strength models seriously degrades the fit The density-based bending strength model reasonably predicts data not used for fitting, including –Randomly-selected validation set –Newer trepanned data and several other comparison sets

20 Additional slides follow for answering questions

21 Additional Comparison Data for Density- Based Strength Models Installed samples from HNB R4 and HRA R2, irradiated in HPB R3, measured 3-point bending strength Trepanned data 1996 – 2000, 3-point bending strength measured at Berkeley Technical Centre (BTC) BFB & DIDO test reactor irradiations under oxidising conditions, measured annealed tensile strength S A

22 Density-Based Strength Model Compared with Installed Sample Data Not Used for Fitting (solid black symbols) HNBHRA

23 Density-Based Strength Model and BTC Data Not Used for Fitting The BTC data average 1 MPa below the WNL data and model

24 Density-Based Strength Model & BFB and DIDO Annealed Data Not Used for Fitting (Coefficient recalibrated for annealed tensile strength)