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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

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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

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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

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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

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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

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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

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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)

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8 Model and Calibration Data Plots

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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

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10 Model and Calibration Data Plots, cont’d

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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)

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

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13 Flat Residual Plots

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14 Flat Residual Plots, continued

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

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

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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)

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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

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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

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20 Additional slides follow for answering questions

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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

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

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

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

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